Homotopy lattice gauge fields 1: The fields and their properties

Homotop y lattice gauge ﬁelds 1: The ﬁelds and their prop erties Juan Orendain ∗ 1 , Iv an Sanc hez † 2 , and Jos ´ e A. Zapata ‡ 2 1 Case W estern Reserv e Universit y , USA. 2 Cen tro de Ciencias Matem´ aticas, Univ ersidad Nacional Aut´ onoma de M ´ exico, C.P . 58089, Morelia, Michoac´ an, M ´ exico. Abstract W e in tro duce homotopy lattice gauge ﬁelds (HLGFs), a v ersion of gauge ﬁelds o ver a discretized base, based on a notion of higher parallel transp ort that enriches the usual parallel transp ort along paths on a lattice to also consider higher dimensional paths. Higher dimensional data k eeps information ab out the parallel trans- p ort along homotopies of curves. With this data, a HLGF on a base space of dimension t wo or three determines a principal bundle ov er the base manifold. This data is also responsible for our form ulas for the topo- logical c harge on tw o-dimensional bases. Our framew ork is an application of a nonab elian algebraic topology framew ork dev elop ed to solve the lo cal to global problem in higher di- mensional homotop y . No previous kno wledge of higher category theory is assumed. The second part will be devoted to the space of ﬁelds as an arena for doing Quan tum Field Theory , and to giv e the ﬁrst examples of ho w our framew ork reﬁnes standard lattice gauge theory . Con ten ts 1 In tro duction: The lattice homotop y cutoﬀ, a better cutoﬀ for gauge ﬁelds 2 2 Preparing gauge ﬁelds in the con tin uum for the homotop y lat- tice cutoﬀ 4 2.1 P arallel transp ort . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ∗ e-mail: orendain@proton.me † e-mail: isanchez@matmor.unam.mx ‡ e-mail: zapata@matmor.unam.mx 1 2.2 Higher homotop y parallel transp ort in the con tinuum . . . . . . . 8 2.3 The homotop y lattice cutoﬀ: Essential ingredients . . . . . . . . 11 2.4 The homotop y lattice cutoﬀ: A preliminary example . . . . . . . 13 3 HLGFs as higher parallel transp ort maps on the homotopy lat- tice 17 3.1 Prelude to nonab elian algebraic top ology . . . . . . . . . . . . . 17 3.2 Abstract HLGFs and the higher homotopy Atiy ah group oid . . . 27 3.3 HLGFs after a trivialization ov er v ertices . . . . . . . . . . . . . 31 3.4 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 The homotop y lattice cutoﬀ . . . . . . . . . . . . . . . . . . . . . 34 3.6 A set of generators for HLGFs: ELGFs . . . . . . . . . . . . . . 34 3.7 General higher gauge ﬁelds on the homotop y lattice . . . . . . . 37 4 The topological charge and other top ological considerations 39 4.1 The topological c harge of gauge ﬁelds o ver S 2 . . . . . . . . . . . 39 4.2 The bundle induced b y a HLGF o ver 2 or 3 dimensional base spaces 41 5 Summary and discussion 42 6 App endix 43 1 In tro duction: The lattice homotop y cutoﬀ, a b etter cutoﬀ for gauge ﬁelds In this series of tw o papers, we in troduce a new discretization of gauge ﬁelds. In this ﬁrst part we deﬁne our gauge ﬁelds ov er a discrete base space, and describe their properties. The second part (w ork in progress) is devoted to the space of ﬁelds, and to sho w that it is a go o d arena for doing Quantum Field Theory . Gauge ﬁelds are ubiquitous in fundamen tal physics. They are used to de- scrib e the so-called Standard Mo del of high energy ph ysics, some systems in condensed matter physics and some approaches to quan tum gravit y . Quan tum ﬁeld theory requires a cutoﬀ. Cutoﬀs are not considered to pro- vide simpliﬁed framew orks; they are a necessary ingredien t in the Wilsonian construction of interacting quantum ﬁeld theories. As far as quantum physics is concerned, w e could say that frameworks in volving a cutoﬀ are more funda- men tal than framew orks based on the smo oth category , which is well motiv ated b y classical considerations. Lattice gauge theory (LGT) is a framew ork for w orking with quan tum gauge ﬁelds based on a lattice discretization of the base space. The usual lattice cutoﬀ of gauge theories a voids the diﬃculties in pro ducing a gauge independent proce- dure faced by p erturbative approaches. The focus is on parallel transp ort, and the cutoﬀ consists of selecting P ( L ) ⊂ P ( M ) a ﬁnitely generated subgroupoid 2 of the path group oid selected b y an em b edded lattice L ⊂ M . LGT is a tremen- dously successful framework: Its predictions about the masses of elemen tary particles hav e b een veriﬁed with accuracies higher than 99%. It is, how ever, known that lattice gauge ﬁelds do not keep track of the top o- logical information present in a G -gauge ﬁeld in the contin uum. F or example, the gauge ﬁeld in the contin uum determines a principal G -bundle ov er the base space, and this prop ert y do es not hold for lattice gauge ﬁelds (LGFs); see Sec- tion 2. A symptom is that a form ula for the top ological c harge that is free of am biguities cannot be giv en b efore a con tinuum limit is reached. No w we brieﬂy describe the contributions of this w ork. W e introduce a v ersion of gauge ﬁelds on a discretized base space endow ed with a sp ecial struc- ture designed to k eep track of homotopies of curves, they are called homotop y lattice gauge ﬁelds (HLGFs). They can be seen as a particular type of “higher gauge ﬁelds” on a discrete base space describing a “higher parallel transport op- eration”. W e can sa y that the diﬀerence b et ween HLGFs and standard lattice gauge ﬁelds is that the focus is on higher parallel transport, which includes ordi- nary parallel transp ort and keeps more information ab out the gauge ﬁeld in the con tinuum associated to parallel transp ort along homotopies of paths. HLGFs giv e a (higher) algebraic structure to previous v ersion of homotop y a ware lat- tice gauge ﬁelds prop osed by one of the authors in collab oration with Claudio Meneses [1, 2]. This algebraic structure provides the geometrical clarit y that lets uncov er the parallel transp ort op eration natural to HLGFs, and pro vide a coarse graining map (that we will in tro duce in the second article of this se ries). On a separate note, the restriction of HLGFs to dimension 2 repro duces a pre- vious notion of higher gauge ﬁeld on a discretized base prop osed by Pfeiﬀer [3]. W e will see that, for base spaces of dimensions 2 or 3, a HLGF determines a G -bundle o ver the base space. W e also giv e a simple form ula to calculate the top ological charge in dimension 2. Reading this pap er do es not require previous kno wledge of category theory . W e develop the necessary algebraic tools providing geometrical motiv ation. The reader will learn the essence and basic tools of a “non ab elian algebraic top ology” framew ork developed by Bro wn, Higgins and collaborators [4]. This language is also used to introduce a higher v ersion of the gauge group oid (or Atiy ah group oid) that giv es a ric h algebraic con text to our higher parallel transp ort op eration. There is a profound reason b ehind the “coincidence” of ﬁnding that our mathematical needs demanded b y ph ysical requiremen ts had already b een de- v elop ed. Researc hers in algebraic top ology had big hop es for using higher ho- motop y groups to study spaces. They then realized that for any k > 1 the groups π k X are ab elian. Another problem they faced w as the lo cal to global problem: understanding the homotopy type of a space that is decomposed in to a collection of pieces in terms of homotopy prop erties of its pieces, is a diﬃcult a wkward problem. Whitehead [5] was the ﬁrst to realize that extending the study of homotopy to the study of relative homotop y leads to nonab elian struc- tures. Later Brown, Higgins and collab orators after working for decades solv ed the lo cal to global problem for relativ e homotop y on ﬁltered spaces [6]; see [4] 3 for a p edagogical account of nonab elian algebraic top ology . Filtered spaces add structure to top ological spaces (or to based top ological spaces) that make it natural to replace groups with groupoids. This change was crucial for solving the lo cal to global problem. Additionally , ﬁltered spaces facilitate the goal of talking about relative homotopy . In ph ysics, we are interested in lo cal descrip- tions of systems, as w ell as in considering relative observ ables, and w e need a cutoﬀ (at least as an intermediate step). In this w ork and in [7], w e imp ort the setting and to ols of nonabelian algebraic topology to lattice gauge theory and harv est the ﬁrst fruits. In the second part of the series, we deﬁne and giv e structure to the space of ﬁelds. W e show that it is a go od arena for QFT at a giv en cutoﬀ. W e pro vide examples, and show that our framework reﬁnes standard LGT in the sense that its predictions regarding observ ables living in the standard lattice is the same, but we ha ve access to higher dimensional observ ables that let us k eep track of homotop y data dismissed by LGT. Among the new observ ables that we ha ve at our disp osal is the top ological charge for tw o-dimensional bases. W e also in tro duce coarse-graining maps linking diﬀerent cutoﬀs in such a w a y that an in verse limit lets us “get rid of the cutoﬀ ”. The organization of this paper is as follows: In Section 2 w e geometrically motiv ate the main ideas and sho w ho w the homotop y lattice cutoﬀ brings a ﬁeld in the con tinuum to a HLGF. Section 3 introduces higher algebraic structures in a p edagogical wa y relying on the motiv ation given in the previous section. Section 4 con tains all the top ological asp ects of this work. W e sho w that in dimensions 2 or 3 a HLGF determines a G -bundle ov er the base manifold, and giv e a formula for the top ological charge in dimension tw o. The ﬁnal section con tains a summary and discussion. 2 Preparing gauge ﬁelds in the con tin uum for the homotop y lattice cutoﬀ W e will describ e ho w an ordinary gauge ﬁeld in the contin uum induces a higher homotop y parallel transp ort map. This is no more than an observ ation, but it is the cornerstone of the homotopy lattice cutoﬀ and its ability to locally store information regarding the bundle structure induced b y a gauge ﬁeld in the con tinuum for base spaces of dimension 3 or 2. 2.1 P arallel transp ort W e assume that the reader is familiar with the diﬀerential geometric notion of principal bundles, connections and the induced parallel transport map. W e describ e it brieﬂy to set up our notation in a familiar context. Then w e talk ab out parallel transport from an alternative p oin t of view. Consider M a smo oth n -dimensional manifold with a smo oth triangulation X and G a Lie group. The term curv e will b e used for elemen ts of a groupoid. F rom 4 no w on a piecewise smo oth map 1 c : [ − 1 , 1] → M will b e called a singular curve on M 2 . Singular curv es ha ve a source and a target s ( c ) = c ( − 1), t ( c ) = c (1). A princip al bund le with a c onne ction . There ma y be inequiv alen t G -principal bundles π : P → M o ver M . F or the momen t w e will c ho ose one of them. A connection ω on π determines a lift for any singular curve c in M once an initial condition is speciﬁed in the ﬁber o ver s ( c ). Consider a singular curv e c in M ; the lift from an y initial condition induces a map P T ω ( c ) : π − 1 s ( c ) → π − 1 t ( c ) that comm utes with the right G action on π . W e say that a connection ω determines a parallel transp ort map P T ω [9]. A visual summary is given in Figure 1. Figure 1 Par al lel tr ansp ort fr om a c onne ction on a bund le . ˜ P ( M ), the space of sin- gular curves in M , has tw o important structures: (i) It has a smo oth structure that will let us talk about smo oth maps with that space as domain. The idea is to consider ˜ P ( M ) as a parametrized space in whic h a sp ecial set of parametrizations are deﬁned to b e smo oth with the ob jectiv e of deﬁning a map f : ˜ P ( M ) → R to be smo oth if and only if precomp osition with an y of the smo oth parametrizations results in a smo oth map. One realization of such a smooth structure is the concept of a diﬀeological space [10], which we will use when we deal with the space of ﬁelds in the second part of this series. This notion of a smo oth space extends the notion of a smo oth manifold: All smo oth manifolds are diﬀeological spaces. Imp ortan tly , that structure descends to quotient spaces and certain limits that we will use in the second part of this series. 1 In Section 3 we introduce formally cubical and globular shapes that let us compose in higher dimensions. T o relate the tw o shap es it is easier to parametrize curves with domain [ − 1 , 1] instead of the traditional domain [0 , 1]. 2 In the smo oth category , it is more common to consider lazy paths [8] as singular curves instead of piecewise smo oth curves. These paths are smo oth everywhere, but they are allow ed to hav e points where their velocity vector v anishes. Our treatment of the smo oth category is heuristic, and at this level we can describ e things easier by following our conv entions. 5 (ii) ˜ P ( M ) becomes a group oid after taking a quotien t b y the equiv alence relation of thin homotopy relative to vertices. It has a set of op erations (source, tar- get, comp osition, inv erse and iden tities) that descend to the quotien t P ( M ) = ˜ P ( M ) / ∼ thin and give it the structure of a group oid. The source and target maps were in tro duced earlier. Composition corresponds to singular curv e con- catenation; when it makes sense, c 2 ◦ c 1 is another singular curve corresp onding to following ﬁrst c 1 and later c 2 . The inv erse corresp onds to in verting the direc- tion in which the singular curv e is trav ersed c − 1 ( t ) = c ( − t ). A constan t curve on x ∈ M is called an identit y and it is denoted by id x . Thin equiv alence is an equiv alence relation among singular curv es such that c − 1 ◦ c ∼ thin id s ( c ) . F or gauge ﬁelds the deﬁnition has a clear motiv ation: If w e can go from one singular curv e to another one without sweeping an y area, an appropriately deﬁned in- tegrated curv ature would v anish. It is a very nice equiv alence relation b ecause in some sense it is the w eakest equiv alence relation that makes the quotient a group oid and b ecause it has natural generalizations for the higher dimensional analogs of singular curv es 3 . The elements of P ( M ) will b e called curves or paths. A parallel transp ort map P T ω is deﬁned on ˜ P ( M ) but descends to P ( M ). Giv en c ∈ ˜ P ( M ), the induced transp ort map transp orts the ﬁb er ov er the source to the ﬁb er o ver the target P T ω ( c ) : π − 1 s ( c ) → π − 1 t ( c ) in a G -equiv ariant w ay . Additionally , given comp osable singular curv es, P T ω ( c 2 ◦ c 1 ) = P T ω ( c 2 ) ◦ P T ω ( c 1 ), and P T ω ( c − 1 ◦ c ) = id π − 1 s ( c ) . These are the algebraic prop erties of parallel transp ort maps. W e w ould lik e to describe P T ω as a ﬁeld with the smo oth groupoid P ( M ) as domain. P T ω w ould b e a groupoid homomorphism that after ev aluation k eeps tract of the ev aluation p oin t as it happ ens for any ﬁeld mo deled as a section. This w ould pro vide an elegan t summary of what w e describ ed ab o ve. The question is what is the appropriate group oid that is the target of P T ω . No w w e heuristically describe this idea. F or any curve c , P T ω ( c )’s outcome is a G -equiv ariant transport map π − 1 s ( c ) → π − 1 t ( c ), and we also know that the map dep ends on [ c ]. Th us, the target group oid should hav e ob jects that can b e in terpreted as ﬁb ers o v er points of the base π − 1 x (for x ∈ M ), and its morphisms should ha ve the in terpretation of G -equiv ariant maps π − 1 x → π − 1 y , while remembering that they are asso ciated to curv es c (with x = s ( c ) and y = t ( c )). It turns out that these clues are suﬃcient to deﬁne the Atiy ah group oid At ( π ). Let us describ e At ( π ) with sligh tly more detail; a more formal introduction will be given in the follo wing section. Ob jects are where initial conditions for parallel transp ort can liv e. There is one ob ject for eac h point x ∈ M ; it will b e con venien t to denote ob jects as pairs ( x, F x ). Morphisms will dictate the parallel transp ort of initial conditions along paths on the base. It will b e conv enient to denote morphisms by pairs of the form ( c ∈ P M , T c : F x → F y ) where x is the source of c and y is the target of c and T c comm utes with the global right 3 In the next section, we will deﬁne thin equiv alence precisely within a sp ecial context in which the concept is simply stated. 6 G action on the ﬁb ers of π . The source and target maps of these morphisms are ob vious. Identit y morphisms are of the form ( id x ∈ P M , id : F x → F x ) for all x ∈ M . Composition of morphisms, when deﬁned, is also clear. In version rev erses the direction of the curve and inv erts the map. F orgetting the second en try pro vides a pro jection At ( π ) → P ( M ). F orget- ting the ﬁrst entry pro vides a pro jection to the following groupoid: Its set of ob jects is the set of ﬁb ers of π ; that is, ob jects are of the form F x for some x ∈ M . Its morphisms are maps T ( x,y ) : F x → F y that commute with the global right G -action on π . The op erations on this group oid are obvious. In this context, a parallel transp ort map is a smo oth group oid homomor- phism P ( M ) P T ω − → At ( π ) , whic h is a section of the mentioned pro jection At ( π ) → P ( M ). The presentation giv en ab o ve starts with a connection ω on a bundle and constructs P T ω . A trivial, but interesting, observ ation is that P T ω determines a distribution of horizon tal spaces on π ; that is, P T ω determines ω . Par al lel tr ansp ort on ( M , X 0 ). The following observ ation is the origin of imp ortan t decisions that shap ed the w ork presented in this article. It suggests w orking on a more economical space of singular curves in M . Parallel transp ort on the restricted space of paths still determines the connection ω , while leading to simpler gauge transformations and a conceptually simpler framework. Let ˜ P ( M , X 0 ) ⊂ ˜ P ( M ) b e the space of piecewise smo oth curv es c : [0 , 1] → M with source and target p oin ts, s ( c ) = c (0), t ( c ) = c (1), restricted to lie in X 0 the set of v ertices of the triangulation X . P ( M , X 0 ) = ˜ P ( M , X 0 ) / ∼ thin is a subgroup oid of P ( M ). See Figure 2. Figure 2: The blue singular curv es belong to ˜ P ( M , X 0 ), and the red do es not. The restricted parallel transport map P T ω | ˜ P ( M ,X 0 ) also descends to the quo- tien t P ( M , X 0 ). This group oid has a discrete set of ob jects X 0 ⊂ X . On the other hand, the set of paths (morphisms) has not acquired extra restrictions apart from having restricted endpoints. A question arises: Do es P T ω | ˜ P ( M ,X 0 ) determine ω ? Geometric in tuition tells us that it should b e enough to distinguish betw een any 7 t wo diﬀerent connections: If the tw o connections hav e diﬀeren t curv ature, there m ust be a p oin t on the base space and a neighborho o d of that p oint where parallel transp ort along small lo ops would detect the diﬀerence in curv ature. The mentioned small lo op can b e extended to hav e a base p oints in X 0 ; this path would b e able to distinguish the tw o connections under consideration. Before pro ceeding or giving any details notice that since X 0 is a discrete set, information ab out the ﬁb ers ov er X 0 cannot enco de the structure of a bun- dle ov er M . Th us, parallel transport along paths in P ( M , X 0 ) is an op eration that do es not rely on a preexisting bundle structure. Then parallel transp ort on P ( M , X 0 ) lets us distinguish betw een an y t w o connections on a given bun- dle and, at the same time, p ermits a framework in whic h connections on any p ossible G bundle ov er M are treated at once. It is natural to conjecture that P T ω | ˜ P ( M ,X 0 ) can also distinguish betw een diﬀeren t bundles. The heuristic argument of the previous paragraph was transformed into a rigorous construction by Barrett [11] follo wing a comment from Koba yashi [9]. An explicit construction of the bundle and the connection from the parallel transp ort map is in [11] (considering a base with a single selected base p oin t ( M , ∗ )). A “group oid version” that is more directly related to our exp osition can b e found in [2] 4 . In summary , the Barrett-Kobay ashi construction tells us that the map ( π , ω ) ↔ P T | ˜ P ( M ,X 0 ) is inv ertible. 2.2 Higher homotop y parallel transp ort in the con tinuum Here w e heuristically describe ho w an ordinary gauge ﬁeld acts on homotopies of curv es on a base space. W e call the resulting notion “higher homotopy parallel transp ort”. A single curv e lets the gauge ﬁeld parallel transp ort initial condi- tions on the ﬁb er ov er its source point, to ﬁnal conditions on the ﬁber ov er its target p oin t. Homotopies of curves let the gauge ﬁeld transport homotopies of initial conditions, and yield homotopies of ﬁnal conditions. If we assign globular shap es to the homotopies of initial conditions and of curv es and of ﬁnal condi- tions, the resulting higher homotopy parallel transp ort op eration has p o werful gluing and comp osition properties. The transp ort of globular homotopies of ini- tial conditions of dimensions 1 captures non trivial information ab out the gauge ﬁeld. In Section 4 we will see that there is no interesting information to capture from the transport of 2 dimensional homotopies of initial conditions, and that the transp ort of k dimensional homotopies of initial conditions ma y con tain in- teresting information ab out the gauge ﬁeld, but the globular machinery that we use do es not capture it for k ≥ 3. 4 The construction giv en in [2] uses a triangulation arising from baricen tric sub division, but it is easily adapted to any triangulation with a triangulation X associated with a simplicial set. The main diﬀerence with a simplicial complex is that k-simplices are identiﬁed with a totally ordered set of (k+1) elements, and the induced partial order of vertices p ermeates the whole structure. 8 Consider P T a smo oth parallel transport map on ( M , X 0 ). A homotopy of curv es { c t ∈ ˜ P ( M , X 0 ) } [0 , 1] sharing their source and target p oin ts in X 0 will b e called a singular 2-glob e. Now consider an initial condition u ∈ F s ( c t ) in the ﬁb er ov er s ( c t ) ∈ X 0 . The parallel transp ort along the singular 2-glob e { c t } [0 , 1] leads to { P T ( c t )[ u ] } [0 , 1] a homotop y in F t ( c t ) in terp olating b et w een P T ( c 0 )[ u ] ∈ F t ( c t ) and P T ( c 1 )[ u ] ∈ F t ( c t ) . Notice that this transp ort op eration started with an ob ject of one type (a single initial condition) and ended up with an ob ject of another type. On the other hand, a homotopy of p oin ts in the initial ﬁb er { u t ∈ F s ( c t ) } [0 , 1] is transp orted along the singular 2-glob e to a homotop y of points in the ﬁnal ﬁb er P T ( { c t } [0 , 1] )[ { u t } [0 , 1] ] = { P T ( c t )[ u t ] ∈ F t ( c t ) } [0 , 1] . In summary , P T (( { c t ) } [0 , 1] ) . = { P T ( c t ) } [0 , 1] is a homotopy interpolating b e- t ween P T ( c 0 ) and P T ( c 1 ) taking homotopies of initial conditions on F s ( c t ) and transp orting them to homotopies of ﬁnal conditions on F t ( c t ) . Our ob jective is to in terpret P T (( { c t ) } [0 , 1] ) as a transport map in itself acting on the appropriate t yp e of initial conditions. See Figure 3. Figure 3: A homotopy of singular curv es is a singular 2-globe sharing endp oin ts. The gauge ﬁeld transports homotopies of initial conditions o ver the source along a singular 2-glob e to a homotop y of ﬁnal conditions o ver the target. No w that we kno w what kind of ob jects can b e transp orted by a singular 2-glob e to ob jects of the same t yp e, let us see if the p ossible compositions of singular 2-glob es get along with parallel transp ort. Singular 2-globes can b e composed in t wo directions. See Figure 16 in the next section for a self explanatory picture. Notice that singular 2-glob es ha ve 0-dimensional source and target points, and they also ha ve 1-dimensional source and target singular curv es. First, singular 2-glob es { c 1 t } [0 , 1] , { c 2 t } [0 , 1] suc h that the 0-dimensional target of c 1 coincides with the 0-dimensional source of c 2 can b e comp osed { ( c 2 ◦ 0 c 1 ) t } [0 , 1] = { c 2 t ◦ c 1 t } . Parallel transport along the resulting 2-glob e is, clearly , the comp osition of the parallel transp ort op erations along the original 2-glob es. Second, comp osition of c 1 and c 2 is p ossible when the 1-dimensional target of c 1 coincides with the 1-dimensional source of c 2 (i.e. if 9 c 1 1 = c 2 0 ); we write { ( c 2 ◦ 1 c 1 ) t } [0 , 1] } for the concatenation of homotopies relativ e to ﬁxed endp oin ts b y lifting the condition of b eing ﬁxed of the middle p oin t (and redeﬁning the parameter t ). Parallel transport along { ( c 2 ◦ 1 c 1 ) t } [0 , 1] } acting on homotopies in the initial ﬁb er of the t yp e { ( u 2 ◦ u 1 ) t } [0 , 1] (where ( u 2 ◦ u 1 ) t denotes the clas s of a curve in F s ( c t ) resulting from comp osing representativ es of u 1 t , u 2 t ) is a distributive op eration as stated below. In summary , these t wo comp osition op erations of singular 2-glob es are compatible with parallel transp ort. Since the parallel transp ort map P T in the con tinuum is assumed to be smo oth, in particular it is contin uous leading to compatibilit y as stated by the simple algebraic rules P T ( { ( c 2 ◦ 0 c 1 ) t } [0 , 1] ) = P T ( { c 2 t } [0 , 1] ) ◦ P T ( { c 1 t } [0 , 1] ) , P T ( { ( c 2 ◦ 1 c 1 ) t } [0 , 1] )[ { ( u 2 ◦ u 1 ) t } [0 , 1] ] = { ( v 2 ◦ v 1 ) t } [0 , 1] , with v i t = P T ( c i t )[ u i t ] . Analogously , P T acts on higher homotopies of curves leading to maps par- allel transp orting higher homotopies of initial conditions. T o hav e conv enien t algebraic gluing rules, w e must choose a shap e; we will contin ue to use glob es. 5 Higher homotopies of curves in the base will be higher dimensional globes, and initial conditions on the ﬁb er o ver the source point will b e glob es of the corre- sp onding dimension. T o sharp en the presented ideas, we in tro duce a higher homotopy extension of the Atiy ah group oid. Here we will not be rigorous y et; what we present here is a prelude to what we will in tro duce formally in the following section. W e wan t a higher dimensional extension of P ( M , X 0 ) describing homotopies of paths in P ( M , X 0 ). In the globular structure: The ob jects w ould b e points in X 0 , the 1-morphisms would be paths in P ( M , X 0 ), the 2-morphisms would be 2- glob es interpolating betw een 1-morphisms, etc. After taking a quotien t by thin homotop y , singular k-glob es b ecome elements of a k-group oid, and are called k-glob es. The collection of k-glob es for all k ≥ 0 get organized in to the structure of an strict inﬁnit y groupoid [12]. A singular (k+1)-glob e can b e regarded as a homotop y of k-glob es sharing their source and target (k-1)-glob es; in the next section, w e give a formal deﬁnition in a restricted context. Clearly , the task of making this construction rigorous in the smo oth category is nontrivial; see [8]. W e will deﬁne the appropriate notion of thin homotopy in our con text in the next section. F or our heuristic description, let us use the notation ρ  ( M , X 0 ) and ρ  F x for the inﬁnity groupoid of glob es on M (to b e constructed in the next section) and for the fundamen tal homotop y inﬁnit y groupoid of the ﬁb er F x [12] resp ectiv ely . W e would lik e to enric h ρ  ( M , X 0 ) for describing the higher homotop y parallel transport describ ed ab o ve, where the initial conditions that ma y b e transported are higher homotopies in the ﬁbers ov er X 0 . Th us, there should be one ob ject p er vertex x ∈ X 0 , but they should be thought of as ρ  F x , and the morphisms should hav e the in terpretation of describing globular higher homotopies of parallel transport maps, which means comm uting with the 5 A cubical structure is another choice; the resulting formalism is transparent and p ow erful [7]. The globular choice is more economical, and the notion of gauge transformations is simpler. 10 righ t ρ  G action on the set of ob jects { ρ  F x } x ∈ X 0 . Let us call the resulting structure At 1 ,  ( M , X 0 ; ρ  G ). In this language, w e hav e heuristically describ ed that a smo oth parallel trans- p ort map P T in the contin uum induces a higher homotopy parallel transport map ρ 1 ,  ( M , X 0 ) P T − → At 1 ,  ( M , X 0 ; ρ  G ) , whic h is a section of the pro jection At ( M , X 0 ; ρ  G ) → ρ 1 ,  ( M , X 0 ). This picture will b e describ ed with some detail in the follo wing section. Ph ysical motiv ations call for a cutoﬀ in quan tum ﬁeld theories. The homo- top y lattice cutoﬀ sk etched b elo w leads to a con text of parallel transport along restricted path homotopies. That reduced context will let us give precise and simple deﬁnitions of the notions heuristically described ab o ve. 2.3 The homotop y lattice cutoﬀ: Essen tial ingredien ts The cutoﬀ . Cutoﬀs are not considered to pro vide simpliﬁed framew orks; they are a necessary ingredient in the Wilsonian construction of interacting quan tum ﬁeld theories. W e could say that as far as quantum physics is concerned, cut- oﬀ framew orks for ﬁeld theory are more fundamental than a formalism in the smo oth category , which is w ell motiv ated by classical considerations. The usual lattice cutoﬀ of gauge theories a voids the diﬃculties in pro ducing a gauge inde- p enden t procedure faced b y perturbative approaches. The focus is on parallel transp ort, and the cutoﬀ consists of selecting P ( L ) ⊂ P ( M ) a ﬁnitely generated subgroup oid of the path groupoid selected b y an embedded lattice L ⊂ M . This determines a space of gauge ﬁelds in the lattice and a cutoﬀ map c L : Gauge ﬁelds in the contin uum → Gauge ﬁelds in the lattice . Our goal in this section is to describ e the homotop y lattice cutoﬀ of gauge theories, which reﬁnes the standard lattice cutoﬀ as w e explain later on. The fo cus is on higher parallel transp ort along path homotopies selected b y a cellular decomp osition X of M . This determines a space of gauge ﬁelds in the homotop y lattice and a cutoﬀ map c X : Gauge ﬁelds in the contin uum → Gauge ﬁelds in the homotop y lattice . The fundamental higher homotopy gr oup oid of a ﬁlter e d sp ac e . It is an ap- propriate ﬁnitely generated subinﬁnit y-group oid of ρ  ( M , X 0 ) that will lets us extend the cutoﬀ P ( L ) = P ( X 1 ) ⊂ P ( M ) leading to standard lattice gauge ﬁelds. A truncation should lead to a rigorous, pow erful and simple formal- ism. It ma y look lik e a lot to ask, but it turns out that w e are lo oking for a structure that already exists: It is ρ  ( M , X ∗ ) the inﬁnity fundamental group oid of the ﬁltered space ( M , X ∗ ). A ﬁltered space is a topological space together with a collection of nested subsets such that their union equals the space X 0 ⊆ X 1 ⊆ . . . ⊆ X n = M . W e are interested in skeletal ﬁltrations; that is, the space M is assumed to ha ve cellular decomposition X , and the subset X k 11 of the ﬁltration is the k-th skeleton of X . A cubical v ersion was developed b y Bro wn and Higgins starting in the early 80s [13] (see also the b ook [4]), and [12] giv es a globular version that will be essential in this paper. W e will describ e a minimal v ersion of it in the next section. The truncation consists of not allow- ing singular glob es to roam everywhere in M : Singular k-globes are restricted to hav e image in X k the k-skeleton. The enormous simpliﬁcation, as compared with the construction of ρ  ( M , X 0 ) in the smo oth category , comes from using the skeletal structure to deﬁne thin equiv alence: Two singular k-glob es are thin equiv alent if there is a homotopy interpolating betw een them with image con- tained in X k . Figure 4 shows t wo singular curves sharing endp oin ts that are thin equiv alent. Figure 4 The higher homotopy Atiyah gr oup oid of a ﬁlter e d sp ac e . Recall the heuristic description of At ( π ) given in Section 2.1. Now we presen t a v ariant of At ( π ) that pro vides a framework for describing our higher homotopy parallel transp ort with in ternal group G . Let us call it At 1 ,  ( X ∗ , ρ  G ). There are three imp ortan t c hanges: The ﬁrst is that w e reduced the set of allo wed base p oints to b e X 0 ⊂ M . As discussed earlier, this implies that the set of ﬁbers o ver the base p oin ts do es not know the bundle structure. The second is the cutoﬀ induced by X ∗ , whic h carries a simple deﬁnition of thin equiv alence. The third is that the goal of the structure is to parallel transp ort homotopies of elemen ts of the ﬁb ers F x along appropriate homotopies of paths. The resulting structure has ob jects and morphisms denoted by pairs. The second en try of the ob jects ( x, ρ  k F x ), where the initial conditions to b e parallel transp orted live, is appropriate for describing k -dimensional globular homotopies in F x (for some k ). The ﬁrst en try of the morphisms ( c ∈ ρ  k +1 X ∗ , T c : ρ  k F x → ρ  k F y ) is a k + 1 glob e with 0- dimensional source x ∈ X 0 and 0-dimensional target y ∈ X 0 ; the second entry is a map comm uting with the right ρ  k G action. The ﬁrst en try of those morphisms mo dels a homotopy of curves, and the second one models the corresp onding homotop y of transp ort maps appropriate for acting on homotopies of initial conditions. Imp ortan tly , the homotopy structures of the ob jects and that of the mor- phisms are co ordinated precisely as needed for us to describ e homotop y parallel 12 transp ort. The conditions are stated at a tec hnical level, but the idea is simple. No w we describe an instance of these conditions through an example depicted in the Figure 5: Consider a situation in which we ha ve initial conditions at x , and they are a homotopy (relativ e to ﬁxed endp oin ts) written as the comp osition u ◦ 0 v ∈ ρ  1 S 1 x . The initial condition is transported along a 2-glob e that is also expressed as a comp osition c ◦ 1 d ∈ ρ  2 X ∗ . The morphisms of the group oid that w e are constructing are associated with pairs; in the described situation we ha ve three diﬀerent pairs ( c, M ), ( d, N ) and ( c ◦ 1 d, P ). The compatibility condition sa ys that we can transport the glued initial condition u ◦ 0 v using P , or w e can transp ort the unglued pieces u, v using M , N resp ectiv ely and then glue them, obtaining the same result P ( u ◦ 0 v ) = ( M u ) ◦ 0 ( N v ) . Similar conditions ensure that if w e ha ve initial conditions at x corresp onding Figure 5 to a homotopy u ∈ ρ  1 S 1 x that are transp orted along a 2-glob e c ∈ ρ  2 X ∗ . Then w e can calculate the initial p oint (or source) of the transp orted homotopy in t wo equiv alent w a ys: W e can calculate the source of u and transp ort it using the 1-dimensional source of c , or we can ﬁrst transp ort u using c and then calculate its source. Etc. 2.4 The homotop y lattice cutoﬀ: A preliminary example As a preliminary example, consider the case where we consider M = S 2 and G = S O (2). See Figure 6. The truncation is induced by a triangulation X . The triangulation has a set of vertices denoted by X 0 , a 1-dimensional skele- ton consisting of edges and v ertices and denoted by X 1 and a t wo-dimensional sk eleton consisting of faces, edges and v ertices denoted b y X 2 . The sk ele- tal ﬁltration induced b y the triangulation is X ∗ = ( X 0 ⊆ X 1 ⊆ X 2 = M ). As shown in Figure 6, our triangulation X has ﬁve vertices; tw o of them lo- cated on the North and South P oles and other three vertices on the Equator lo cated at equidistant p oin ts. The Greenwic h meridian lies on X 1 . As men- tioned earlier, w e need a triangulation asso ciated with an underlying simplicial set. T o provide this structure we n umber the vertices: The vertex in the in- tersection of the Greenwic h meridian and the equator will b e v 3 . V ertices v 1 13 and v 2 lie on the equator with the line v 1 → v 2 → v 3 running from w est to east. Finally , v 4 = v N and v 5 = v S corresp ond to the north and south p oles, resp ectiv ely . This completes the description of X 0 . The set of edges is { [ v 1 , v 2 ] , [ v 2 , v 3 ] , [ v 1 , v 3 ] , [ v 1 , v 4 ] , [ v 2 , v 4 ] , [ v 3 , v 4 ], [ v 1 , v 5 ] , [ v 2 , v 5 ] , [ v 3 , v 5 ] } . Notice that w e hav e written 1-simplices, edges, as a pair of vertices resp ecting the order of vertices. Figure 6: The triangulation, the 1-globe Γ 12 and the 2-glob e Γ 135 . A set of generators of the path group oid in X 1 is { Γ ij } with i < j , where Γ ij represen ts a singular curv e with image [ v i , v j ] and running in the opp osite direction. That is, s (Γ ij ) = v j and t (Γ ij ) = v i . This con ven tion is compatible with the conv entions follo wed in [2], as explained in Section 3. If w e wan t to refer to a singular curv e running in the opposite direction w e write − Γ ij . The set of faces is { [ v 1 , v 2 , v 4 ] , [ v 2 , v 3 , v 4 ] , [ v 1 , v 3 , v 4 ] , [ v 1 , v 2 , v 5 ] , [ v 2 , v 3 , v 5 ] , [ v 1 , v 3 , v 5 ] } (where triplets of v ertices are written resp ecting their order). A set of generators of the 2-group oid of 2-globes in X 2 is { Γ ij k } with i < j < k , where Γ ij k represen ts a singular 2-glob e going from the singular c urv e Γ j k to the singular curv e ( − Γ ij ) ◦ Γ ik . The speciﬁc homotop y of singular curv es c t is shown in Figure 13 (in Section 3.1). Additionally , a concrete form ula is giv en in [7]; again, we adopt this conv ention to b e compatible with [2]. If we wan t to refer to a singular 2-glob e with source and target singular curves running in the opposite direction, or what is the same with 0-dimensional source and target in terchanged, we write − 0 Γ ij k , and − 1 Γ ij k denotes the singular 2-glob e where 1-dimensional source and target ha ve exchanged places. This completes the description of the triangulation X , the induced sk eletal ﬁltration X ∗ and our conv entions for asso ciating generating singular globes to simplices. The inﬁnity group oid ρ  ( S 2 , X ∗ ) has three nontrivial levels ρ  0 ( S 2 , X ∗ ) = X 0 the ob jects (or v ertices), ρ  1 ( S 2 , X ∗ ) = P ( X 1 ) the path groupoid and ρ  2 ( S 2 , X ∗ ) the 2-group oid of 2-glob es. No w we will undertak e the task of describing one given parallel transp ort map using the example of P T round the parallel transp ort of unit vectors o ver p oin ts in X 0 resulting from the Levi-Civita connection asso ciated with the met- ric of round unit sphere S 2 . The situation is geometrically clear, but to describ e the action of P T round w e need the reference pro vided by a trivialization on the ﬁb ers o ver X 0 . Parallel 14 (a) (b) Figure 7: (a) The triangulation, the 1-glob e Γ 12 and the 2-globe Γ 135 . (b) P arallel transp ort on the round sphere. transp ort along a curve will corresp ond to an elemen t of S O (2), and parallel transp ort along a singular 2-globe will corresp ond to a homotopy of parallel transp ort maps (considered up to relative homotop y) describ ed by a curv e in S O (2) considered up to homotopy relative to having ﬁxed endpoints. W e will c ho ose an arbitrary trivialization o ver the north p ole v 4 = v N . In order to help us give a simple description, the trivialization on the ﬁb ers o ver the other v ertices will b e chosen using P T round . The trivializations o ver v 1 , v 2 and v 3 are c hosen transp orting the trivialization from v 4 using P T round along the link that joins them. The lo cal trivialization ov er v 5 = v s is chosen transp orting the trivialization on v 4 = v N with P T round along the Greenwic h meridian. According to the conv ention describ ed ab ov e, parallel transp ort on paths corresp onding to edges is P T round (Γ ij ) = R ( n ij θ f ) ∈ S O (2) with θ f = 4 π 6 , (1) and the num b ers n ij ∈ Z sho wn in Figure 7. When the arrow in the ﬁgure is opp osite to Γ ij , we inv ert the sign of the integer shown in the ﬁgure to compute n ij . Homotop y parallel transp ort along singular 2-glob es, as mentioned ab o ve, is described b y curv es in S O (2) considered up to homotop y relativ e to ha ving ﬁxed endp oin ts. In our notation, suc h classes of curves in S O (2) are denoted b y a triple ( R ∈ S O (2) , R ′ ∈ S O (2); x ∈ R ) satisfying R ′ = exp( ix ) R . The triple determines the singular curv e c ( t ) = exp( itx ) R , and ( R, R ′ ; x ) denotes 15 the homotopy class of the curve relative to ha ving ﬁxed endpoints. In our example, homotopy parallel transp ort along singular 2-glob es asso ci- ated with faces is P T round (Γ ij k ) = ( R ( n j k θ f ) , R (( − n ij + n ik ) θ f ); x ij k ) , (2) with x 124 = θ f , x 234 = θ f , x 134 = − θ f , x 125 = − θ f , x 235 = − θ f , x 135 = θ f . A visual represen tation of homotopy parallel transp ort on 2-glob es is shown in Figure 8. Figure 8: The ﬁrst panel shows a 2-glob e as a homotopy of curv es sharing endp oin ts. In the second panel, the transp ort map acts on a 2-globe, shown as in terp olating b et ween 1-glob es. The outcome is shown as the blue curve up to homotop y rel. to endpoints. P arallel transport along any singular curv e con tained in X 1 is calculated b y composing parallel transp ort along the edges. Similarly , as w e will sho w in Section 3, homotop y parallel transp ort along an y singular 2-glob e ﬁtting in X 2 can be calculated comp osing homotop y parallel transp ort along singular 2- glob es asso ciated with faces and along degenerate singular 2-glob es with image con tained on the edges. As an example, consider singular 2-glob es interpolating b et w een the Green wich meridian and meridians at longitude 2 π 3 , 2 2 π 3 and 2 π ; see Figure 7. Let us call the describ ed singular 2-globes c λ with λ ∈ { 2 π 3 , 2 2 π 3 , 2 π } . Because the metric on the unit round sphere has constan t curv ature, we know that P T round ( c 2 π 3 ) = ( id, R (2 θ ); 2 θ f ) , P T round ( c 2 2 π 3 ) = ( id, R (4 θ ) = R ( θ ); 4 θ f ) , P T round ( c 3 2 π 3 ) = ( id, R (6 θ ) = id ; 6 θ f = 4 π ) . Notice that for any p ar al lel tr ansp ort map P T the last entry of the ab o ve ev aluations m ust hav e the structure P T ( c 2 π ) = ( P T (Green wich) ∈ S O (2) , P T (Greenwic h); x c 2 π ), and this requires that Q ( P T ) = 1 2 π x c 2 π is an integer. This in teger is called the top ological charge of P T ; in the case of the connection induced by the round metric on S 2 , w e obtain Q ( P T round ) = 2. In Section 4, we giv e more details ab out the calculation of the topological charge 16 and the c haracterization of the bundle class in tw o or three dimensions. W e also explain why a generalization to four dimensions requires extra ingredien ts. The homotop y cutoﬀ takes the gauge ﬁeld induced b y restricting the parallel transp ort of the Levi-Civita connection on the round sphere to act on unit v ectors and forgets all the information in it except for (1) and (2). This is an example of the ev aluation of the homotopy cutoﬀ map c X : A ∞ S 2 ,S O (2) → A H L X ∗ ,S O (2) from the space of smooth S O (2) connections o v er S 2 to the space of gauge ﬁelds on the homotop y lattice. X ∗ , deﬁnes a homotop y lattice for S 2 , that considers paths on X 1 and homotopies among those paths restricted to X 2 (considered only up to homotopy), and in general homotopies among k-homotopies of paths restricted to X k +1 . 3 HLGFs as higher parallel transp ort maps on the homotop y lattice Firstly in this section, we deﬁne Homotopy Lattice Gauge Fields (HLGFs) at an abstract lev el. Secondly , we use a trivialization on the ﬁb ers ov er p oints in the discrete set X 0 to pro vide a useful description of HLGFs. Finally , we give a set of generators of k-glob es that provide a set of elementary building blo c ks for HLGFs. Since generating 1-glob es are links in X 1 , w e will see that elementary data asso ciated with 1-dimensional ob jects by an HLGF is exactly standard lattice gauge ﬁeld data. 3.1 Prelude to nonab elian algebraic top ology Remark 1. It turns out that the c onventions use d in diﬀer ential ge ometry and in physics fol low the c onvention of c omp osing curves fr om right to left. This for c es us to adher e to this c onvention in our short intr o duction to the to ols of nonab elian algebr aic top olo gy. Standar d c onventions in that ﬁeld use c ate gory the ory c onventions c omp osing fr om left to right. In [7] we use d standar d c ate gory the ory c onventions but arrive d at awkwar d c onventions in which G -bund les had a glob al left action and (after lo c al trivialization) p ar al lel tr ansp ort along c omp osite curves c ould b e describ e d by right multiplic ation in the gr oup. Filter e d sp ac es and ﬁlter e d maps . Consider X a space. Here our fo cus is on ﬁnite dimensional manifolds possibly with boundary and corners, which are equipp ed with a nested collection of subsets X 0 ⊆ X 1 ⊆ . . . suc h that X is the union of the subsets. Such a space is called a ﬁltered space and is denoted as X ∗ ; the collection of nested subsets is called a ﬁltration. A concrete example w as given in Section 2.4. Filtered spaces together with maps f ∗ : X ∗ → Y ∗ among them, whic h preserv e eac h lev el of their ﬁltrations ( f ( X k ) ⊆ Y k for all k ), are organized into the category FTOP . **** Here I stopped adding corrections from Mary (un til the end) **** 17 W e will use t wo kinds of ﬁltrations: Skeletal ﬁltrations, where X k is the k-dimensional skeleton of a cellular decomp osition, and trivial ﬁltrations, where X k = X for all k . Filtered homotopy relativ e to vertices in the case of a skeletal ﬁltration b ecomes thin homotopy rel. vertices, whic h we need to use in the base space, and in the case of a trivial ﬁltration it becomes standard homotopy rel. v ertices, which w e use for the gauge group. Skeletal ﬁltr ations . First we brieﬂy describ e the homotop y theory resulting from skeletal ﬁltrations. Our space X will hav e a triangulation or a cellular decomp osition (by a CW complex, as would b e the case of a cubiculation of the n-torus by a “Cartesian cubiculation” with p erio dic b oundary conditions), and X k will denote its k-sk eleton. Then dim X k = k . An example of a space with a sk eletal ﬁltration is I n the standard n-cub e [ − 1 , 1] n . The ﬁltration comes from the cellular decomp osition in whic h the k- cells are the closed k-dimensional faces of I n . See Figure 9 for standard 1 and 2 cub es with our conv ention for ho w to picture their standard parametrization. Figure 9 Another family of examples that we will use is the family of standard n- glob es. Our conv entions are based on those of [12]; the diﬀerence is motiv ated b y agreeing with the conv en tions of diﬀerential geometry as stated in Remark 1. In this paragraph, the diﬀerence will show up only in the picture. The standard n-glob e G n is the n-dimensional disc G n = { x ∈ R n | || x || ≤ 1 } with the sk eletal ﬁltration inherited from the cellular decomp osition G n = e 0 − ∪ e 0 + ∪ e 1 − ∪ e 1 + ∪ . . . e n − 1 − ∪ e n − 1 + ∪ e n with one n-cell e n = I nt ( G n ) and for 0 ≤ k < n tw o k- dimensional cells e k + , e k − con tained in the in tersection of ∂ G n = S n − 1 and the k-dimensional subspace determined by x j = 0 for every j < n − k . Cell e k ± satisﬁes ± x n − k ≥ 0. See Figure 10. Figure 10 No w w e giv e examples of ﬁltered maps. They will b e relev ant in the algebraic 18 structures deﬁned b elo w (see Figure 11) D α i : I k → I k +1 , ( x 1 , . . . , x k ) D α i − → ( x 1 , . . . , x i − 1 , α 1 , x i , . . . , x k ) , E i : I k +1 → I k , ( x 1 , . . . , x k +1 ) E i − → ( x 1 , . . . , x i − 1 , x i +1 , . . . , x k +1 ) , G i : I k +1 → I k , ( x 1 , . . . , x k +1 ) G i − → ( x 1 , . . . , max( x i , x i +1 ) , . . . , x k +1 ) . Figure 11 In [12] a collection of ﬁltered maps ϕ n : I n → G n , for all n ≥ 1 are deﬁned inductively . They satisfy: (1) ϕ 1 ( x ) = x . (2) If we write x = ( t, y ) ∈ I × I n − 1 then ϕ n ( t, y ) = ( t p 1 − || ϕ n − 1 ( y ) 2 || , ϕ n − 1 ( y )). (3) ϕ n ( ∂ I n ) = ∂ G n . (4) ϕ n sends the ( n − 1) dimensional face of I n transv ersal to direction k ≥ 1 in R n and with ± x k > 0 to the ( n − k ) dimensional cell e n − k ± . See Figure 12 for n = 2. Figure 12 A singular 1-glob e (whic h is the same as a singular 1-cube) in a ﬁltered space X ∗ is a parametrized curv e in X ∗ ; at a more precise lev el, it is a ﬁltered map c : [ − 1 , 1] → X ∗ . Thus, it sends [ − 1 , 1] 0 to X 0 and [ − 1 , 1] 1 = [ − 1 , 1] to X 1 . Notice that this is not the same as sending vertices to v ertices and edges to edges b ecause the image of [ − 1 , 1] needs to lie in X 1 , but is not restricted to b e a single edge. In Section 2.4 eac h edge Γ ij is a singular 1-globe, and also the Green wich meridian running from north to south and the equator running from w est to east and starting at longitude λ = 0 are singular 1-glob es (see Figure 6). The space of 1-globes in X ∗ is denoted by R  1 ( X ∗ ); previously w e had called it ˜ P ( X 1 ). Similarly , a k-glob e in X ∗ is a ﬁltered map c : G k → X ∗ , and a k-cub e in X ∗ is a ﬁltered map c : I k → X ∗ . The spaces of k-glob es and k-cub es in X ∗ are denoted by R  k ( X ∗ ) and R □ k ( X ∗ ) resp ectiv ely . Each face Γ ij k in the example of Section 2.4 is a singular 2-glob e. In [7] we giv e an explicit map from 19 [0 , 1] 2 to a triangulation with the images of { 0 } × [0 , 1] and { 1 } × [0 , 1] equal resp ectiv ely to the highest and second highest vertices of a face, as shown in Figure 6. (That map must b e adapted to the conv entions that w e are no w using with I = [ − 1 , 1].) It is easy to see that this t yp e of degenerate singular 2-cubes can also b e seen as singular 2-glob es. k-Glob es and k-cub es are parametrized k-dimensional subspaces of X ∗ (p os- sibly with singular image) ﬁtting into the sk eletal ﬁltration. Their sp ecial shape (glob es or cub es) will let us talk ab out homotopies among them, and precisely describ e the gluing of those homotopies. Cubic al sets with “c onne ctions” . Now we will in tro duce ∂ α i , ϵ i , Γ i algebraic op erations in R □ ( X ∗ ) that will let us talk about homotopies among cub es. F ace maps: Giv en c ∈ R □ k ( X ∗ ), we deﬁne ∂ α i c ∈ R □ k − 1 ( X ∗ ) (for any 1 ≤ i ≤ k and α ∈ {− , + } ) as ∂ α i c ( x 1 , . . . x k − 1 ) = c ◦ D α i ( x 1 , . . . x k − 1 ). As its name suggests, ∂ α i c is the α -face in direction i of c . In our preliminary example of Section 2.4, we see that ∂ − 1 Γ ij = v j , ∂ 2 1 + Γ ij = v i . Also, at the pictorial level 6 , w e can see that ∂ − 1 Γ ij k = Γ j k . In order to describ e ∂ α 2 Γ ij k w e need notation for degenerate cub es that we will give in a moment, and in order to describ e ∂ + 1 Γ ij k w e need notation for gluing that we will see in a few paragraphs. Degeneracy maps: Giv en c ∈ R □ k ( X ∗ ), we deﬁne ϵ i c ∈ R □ k +1 ( X ∗ ) by ϵ i c ( x 1 , . . . x k +1 ) = c ◦ E i ( x 1 , . . . x k +1 ). As its name suggests, ϵ i c is a ( k + 1) dimensional cub e with the same image as c whic h is degenerate in direction i . Notice that w e can comp ose degeneracy maps to construct cub es with a higher degree of degeneracy . W e see from Figure 13 that in our preliminary example of Section 2.4 ∂ − 2 Γ ij k = ϵ 1 v k , ∂ + 2 Γ ij k = ϵ 1 v j . Connection maps: Giv en c ∈ R □ k ( X ∗ ), we deﬁne Γ i c ∈ R □ k +1 ( X ∗ ) by Γ i c ( x 1 , . . . x k +1 ) = c ◦ G i ( x 1 , . . . x k +1 ). This map pro duces degenerate cub es, but the degeneracy is associated with the pair of directions ( i, i +1). It is b ecause w e hav e these maps that cubical gluing rules do not hav e a cartesian ﬂav or; see Figure 15 b elo w. It is not diﬃcult to verify that the deﬁned op erations ( ∂ α i , ϵ i , Γ i ) deﬁned on R □ ( X ∗ ) satisfy the relations of a “cubical set with connections” listed in the appendix 6; here are some of mentioned relations: ∂ α i ϵ j = id if i = j , ∂ α i ϵ j = ϵ j − 1 ∂ α i if i < j , ∂ α i ϵ j = ϵ j ∂ α i − 1 if i > j . Globular sets . The set of glob es R  ( X ∗ ) admits an algebraic structure for talking ab out homotopies among glob es. Below we will introduce op erations d α i , s i,n on R  ( X ∗ ) satisfying the appropriate set of relations, giving it the structure of a globular set. The collection of maps ϕ send globes to cubes by pullback ϕ ∗ : R  ( X ∗ ) → R □ ( X ∗ ) . Clearly , this map trivially sends 0-glob es in to 0-cub es and 1-globes into 1-cubes. Notice that for n ≥ 2 the map is not surjective: its image ϕ ∗ n ( R  n ( X ∗ )) is the set of cub es c ∈ R □ n ( X ∗ ) such that ∂ α i c ∈ Im ϵ i − 1 1 for i ∈ { 2 , . . . , n } . 6 The example can be completely precise follo wing the deﬁnition of Γ ij k as a singular 2-cube given in the App endix 6. 20 It is simple to verify that the op erations ∂ α 1 , ϵ 1 preserv e ϕ ∗ n ( R  n ( X ∗ )). Using those op erations w e will deﬁne globular operations d α i , s i,n in R  ( X ∗ ) that mak e it into a globular set. Recall that G n has tw o k -dimensional cells for k < n ; one is called the source cell of that dimension, and the other one is called the target cell of that dimension. Our preliminary example of Section 2.4 can b e made precise in a simple w ay using glob es. The 0-globes are the vertices in X 0 . The 1-glob es Γ ij sho wn in Figure 6 are con tinuous maps Γ ij : [ − 1 , 1] → X 1 with Γ ij ( − 1) = j and Γ ij (1) = i . The 2-globes Γ ij k are the con tinuous maps Γ ij k : G 2 → X 2 sho wn in Figure 13. Figure 13: A homeomorphism G 2 → 2-Rhombus dividing the standard 2-glob e in to t wo 2-simplices lets us prescrib e Γ ij k as the gluing of tw o maps that are aﬃne in eac h simplex. The blue curve shows ho w Γ ij k mo dels a homotopy of curv es. F rom the picture we can read the face maps given b elo w. F ace maps: Given c ∈ R  k ( X ∗ ), for i < k we deﬁne d α i c ∈ R  i ( X ∗ ) as c | e α i , or equiv alently as d α i c = ( ϕ ∗ ) − 1 ( ∂ α 1 ) k − i ( ϕ ∗ c ). They are the i -dimensional source and target faces of the glob e. W e will use again the preliminary example of Section 2.4, but this time we will use the notation for glob es. d − 1 Γ ij k = Γ j k , d − 0 Γ ij k = v k , d + 0 Γ ij k = v j (see Figure 13). d + 1 Γ ij k will b e giv en b elo w. Degeneracy maps: Giv en c ∈ R  i ( X ∗ ), for k > i w e deﬁne s i,k c ∈ R  k ( X ∗ ) as s i,k c = ( ϕ ∗ ) − 1 ( ϵ 1 ) k − i ( ϕ ∗ c ). The idea is simple, we need to extend c from b eing deﬁned in dimension i to a higher dimension, and we do it one step at a time: Recall that G n ⊂ R n . Consider the natural embedding of R i in to R i +1 at the x 1 = 0 subspace; using that embedding we can extend maps from G i ⊂ R i ⊂ R i +1 to G i +1 as maps that are indep endent of the x 1 co ordinate. In order to bring c from R  i ( X ∗ ) to R  k ( X ∗ ), the describ ed extension has to b e p erformed k − i times. These maps follow the “globular relations”: d α i d β j = d α i for i < j, s j,k s i,j = s i,k for i < j, d α j s i,k = id for i = j, d α j s i,k = d α j for j < i, d α j s i,k = s i,j for j > i. Higher dimensional gr oup oid structur es. Now w e are ready to glue homotopy maps. Here is where our con ven tion diﬀers from the usual con v ention follo wed in category theory treatments. The reason for our c hoice is given in Remark 1. 21 W e describ e the structure ﬁrst for cubical homotopies. The operations + i , − i on R □ ( X ∗ ) are deﬁned extending the usual deﬁnitions of concatenation of curv es and direction rev ersal of curv es to higher dimensions. Consider a, b ∈ R □ k ( X ∗ ) suc h that ∂ + i a = ∂ − i b , then w e deﬁne b + i a ∈ R □ k ( X ∗ ) b y a simple adjustment that makes the i -th co ordinate of the parameter space run t wice as fast and use a follow ed by b . Figure 14 illustrates gluing in tw o diﬀerent directions for k = 2 and the all important “interc hange rule”. Figure 14: P anel (i) gives the pictorial image of cubical sum. Panel (ii) Is the picture corresp onding to the interc hange rule: ( b + 1 a ) + 2 ( c + 1 d ) = ( c + 2 b ) + 1 ( d + 2 a ). The operation − i a ∈ R □ k ( X ∗ ) is deﬁned by following the the i th co ordinate of parameter space in reversed direction. These op erations (+ i , − i ) together with the cubical op erations and the con- nection, let us comp ose (or glue) homotopies among cub es in a ﬂexible manner. As an example, Figure 15 shows the diagram corresp onding to the building in- structions for a cellular decomp osition of S 2 as the b oundary of a 3-dimensional cub e. A t this point, we do not y et hav e a higher dimensional group oid structure. This structure will emerge after taking a quotient by thin homotop y . Relation ϕ ∗ : R  ( X ∗ ) → R □ ( X ∗ ) shows us that w e can consider singular glob es as particular cases of singular cub es. The gluing and direction c hanging op erations + i , − i preserv e the subset ϕ ∗ ( R  ( X ∗ )) ⊂ R □ ( X ∗ ), inducing corre- sp onding gluing op erations for glob es. The picture for gluing glob es is sho wn in Figure 16. It could appear that gluing singular glob es along higher co dimensional cells w ould necessarily lead to new singular glob es in whic h the gluing region is sp ecial. W e will show, with an example, that after taking thin homotopy equiv- alence into account, this is not the case. Thin homotopy . In gauge theories, talking ab out equiv alence of curves b y thin homotop y is essential. Parallel transp ort as described in the previous sec- tion acting on t wo curv es c 1 , c 2 sharing their source and target p oin ts may diﬀer ev en if c 1 and c 2 are homotopic rel. endp oints. Only in the case of ﬂat connec- tions parallel transport is a function of homotopy classes of curves relativ e to ha ving ﬁxed endp oints. On the other hand, consider the case of the constan t curv e id x with x ∈ X 0 , and the curve c − 1 ◦ c for any curv e c with c (0) = x ; 22 Figure 15: The diagram shows 2-cub es (or squares) glued together using op er- ations + 1 and + 2 . The in terchange rule implies that w e do not hav e to sp ecify the order of the gluing op erations. There are 6 non-degenerate squares, four degenerate squares and tw o doubly degenerate squares. The doubly degener- ate squares are lab eled by a small square. The degenerate square lab eled by an L corresp onds to a connection acting on a 1-dimensional face of a square, L = Γ 1 ( ∂ + 1 f r ont ). The result is that the square L identiﬁes the edges ∂ + 2 lef t and ∂ + 1 f r ont . The reﬂected v ersions of L are obtained using Γ 1 together with − 1 and − 2. this pair of curves are related by a homotopy retracting the curve c − 1 ◦ c to x within the image of curv e. This is an example of a pair of curves related b y thin homotopy . It turns out that for any G -connection ω in M ∗ , we ha ve P T ω ( c − 1 ◦ c ) = P T ω ( id x ) = id F x . In general, for an y connection and an y t wo curv es c, c ′ with the same endp oin ts and related by thin homotopy relative to their endp oin ts w e hav e P T ω ( c ) = P T ω ( c ′ ). A clearly related phenomenon is that the space of singular curves modulo thin homotopy relative to endp oints b ecomes a groupoid. In the con text of a ﬁltered space with a sk eletal ﬁltration, tw o singular k- cub es c, d ∈ R □ k ( X ∗ ) such that c | I k 0 = d | I k 0 are declared to b e thin homotopic relativ e to v ertices if and only if there is a contin uous map h : I k × [0 , 1] → X ∗ with h ( t, v ) = c ( v ) = d ( v ) for all v ∈ I k 0 suc h that h ( I k × [0 , 1]) ⊂ X k and h ( x, 0) = c ( x ), h ( x, 1) = d ( x ). The set of classes of singular cubes by thin homotop y ρ □ X ∗ = R □ X ∗ / ∼ thin inherits the set of op erations ( ∂ α i , ϵ i , Γ i , + i , − i ), following the set of relations listed in The App endix 6, that make it in to an ω -group oid. Its elemen ts, singular cub es up to thin homotopy , are called cubes. Apart from the adv antage of using a shorter name, considering cubes is motiv ated by parallel transport. Additionally , the gluing of cub es has a better b eha vior than gluing b efore considering equiv alence classes b y thin homotopy rel. vertices. Recall the preliminary example of Section 2.4; it is not diﬃcult to 23 Figure 16: The + 2 op eration for cub es induces the ◦ 0 op eration for 2-glob es (gluing along 0-dimensional faces), and + 1 induces ◦ 1 for 2-globes. F or k-globes, the cubical + i induces the globular ◦ k − i , gluing along ( k − i )-dimensional cells. see that, using cubical conv entions, c 2 π 3 ∼ thin Γ 134 + 2 ( − 2 Γ 135 ), where all the elemen ts of the equation are deﬁned in Section 2.4. The resulting category of ω -groupoids is complete and co complete. This prop ert y is inherited by the category of inﬁnity group oids, that w e will presen t b elo w, and this will be crucial when w e deal with the space of ﬁelds in the second part of this series. W e return to glob es. Again we can consider the preliminary example of Section 2.4. Recall that w e show ed explicitly how the 1-simplices of paths Γ ij k are singular 2-globes; that now w e consider up to thin homotopy and call them just 2-glob es. W e can use globular notation to see that c 2 π 3 ∼ thin Γ 134 ◦ 0 ( − 0 Γ 135 ). This shows that gluing glob es along higher co dimensional cells (e.g. using ◦ 0 ) does not alw ays lead to glued regions sho wing their seams; in other words, ha ving distinguishable p oints asso ciated with the place where the gluing to ok place as it could app ear from Figure 16. As another example of how glob es can be glued, consider the b oundary of the 3-cube can be built using glob es. W e ﬁrst assign 2-glob es to the squares in the b oundary as shown in Figure 17. Degenerate 2-glob es are also necessary . A glob e G cov ering S 2 once can b e constructed gluing three globular pieces, named R F, B B , T L for Right-F ront, Bac k-Bottom and Left-T op respectively , G = T L ◦ 1 B B ◦ 1 RF , RF ≃ thin F ront ◦ 0 A ◦ 0 Rig ht, etc . The set of classes of singular globes by thin homotop y ρ  X ∗ = R  X ∗ / ∼ thin inherits the set of op erations ( d α i , s i,k , ◦ i , − i ), following the globular relations stated abov e and relations stating compatibilit y with the gluing op erations (stated below), making it in to a strict ∞ -groupoid. Its elements, singular globes up to thin homotop y , are called glob es. The relations stating compatibilit y of 24 Figure 17: A cellular decomp osition of S 2 with square cells. Eac h square is asso ciated a glob e with the structure and names shown in the picture. Also one degenerate 2-glob e, named A , is sho wn. the gluing op erations with the globular op erations and internal compatibility of the gluing op erations are d α i ( a ◦ j b ) = d α i a ◦ j d α i b, s i,k ( a ◦ j b ) = s i,k a ◦ j s i,k b, ( a ◦ j b ) ◦ i ( c ◦ j d ) = ( a ◦ i c ) ◦ j ( b ◦ i d ) whenev er the expressions are deﬁned. T rivial ﬁltr ations . Our ph ysical motiv ation needs a cutoﬀ for quantum ﬁeld theory , making the space of histories (also called the space of ﬁeld conﬁgurations) ﬁnite dimensional. W e achiev e this goal introducing a ﬁltration in spacetime’s path groupoid and considering paths up to ﬁltered homotop y rel. vertices. On the other hand, the gauge group playing the central role in the target space of our HLGFs cannot hav e a ﬁltration where G 0  = G . The reason is that our mathematical structure only works when G 0 is a group, and in general there is only a ﬁnite collection of nonab elian subgroups of a Lie group G . W e could w ork with a ﬁltration using a discrete subgroup G 0 , but we w ould ha ve to liv e with that structure without the p ossibility of getting rid of that cutoﬀ b y using “ﬁner and ﬁner discrete subgroups”. In the 80s discrete subgroups were used in sim ulations within lattice gauge theory , but as so on as the computer p o w er allow ed them to abandon that drastic simpliﬁcation the use of discrete subgroups b ecame a historical curiosity . A ﬁltration in G with G 0 a discrete subgroup and G i = G for i > 0 could b e in teresting for the calculation of top ological in v ariants; we do not pursue this 25 route in this w ork. Belo w we brieﬂy describ e the fundamental higher homotop y group oid ρ  G in the case where the ﬁltration in the space G is the trivial ﬁltration; i.e., G i = G for all i . W e omit the subscript ∗ to show in the notation that the trivial ﬁltration is used. It is unfortunate that the standard notation uses G for the gauge group and the standard globular shap e for k-glob es is also denoted using the same letter. W e will k ee p the standard notation. The con text should help the reader a void confusion; also the standard k-dimensional globular shap e G k has a sup erscript and the gauge group do esn’t. k-glob es in ρ  G are equiv alence classes of ﬁltered maps from G k to G relative to v ertices. The set of vertices in the domain G k 0 has t wo elemen ts e − 0 and e + 0 . Their images c ( e − 0 ) , c ( e + 0 ) ∈ G are arbitrary p oin ts in G 0 = G . Thus, k-glob es are ﬁltered homotop y clases of maps c : G k → G relative to having the points c ( e − 0 ) , c ( e + 0 ) ∈ G ﬁxed during the homotopy . Since we are using the trivial ﬁltration, ﬁltered homotopy rel. vertices b ecomes usual homotop y rel. vertices. In the case of 1-glob es with c ( e − 0 ) = c ( e + 0 ), homotopy rel. v ertices b ecomes the standard notion of homotopy of closed based curves rel. their base p oin t. In other words, ρ  1 G has subgroups of the t yp e π 1 ( G, ∗ ) for ∗ being any p oin t in G . F or the momen t forget that our G is a Lie group. Could we capture the in- formation ab out the second homotop y group of a space π 2 ( G, ∗ ) from ρ  G trun- cated at lev el t wo? The candidate singular glob es containing that information are 2-singular globes with ˜ c ( e − 0 ) = ˜ c ( e + 0 ) = ∗ and ˜ c ( e − 1 ) = ˜ c ( e + 1 ) = s 1 ∗ . W e add a tilde to remind us that they are not 2-globes b ecause we ha ve not considered their equiv alence class up to ﬁltered homotop y rel. vertices. ˜ c could b e consid- ered as a based map from ( S 2 , ∗ ) to ( G, ∗ ), and we could consider << ˜ c >> its homotop y class among suc h maps. W e remark that << ˜ c >> ∈ π 2 ( G, ∗ ) is not the same kind of equiv alence class as c = [ ˜ c ] ∈ ρ  2 G . The big diﬀerence is that during the homotopies deﬁning c the image of ˜ c ( e − 1 ) and ˜ c ( e + 1 ) can b e diﬀeren t from the constan t curv e s 1 ∗ . On the other hand, the deﬁnition of << ˜ c >> as the equiv alence class of maps from G 2 requires that during the homotopy the image of the complete b oundary of G 2 remains to b e the base point. The same phenomenon happ ens for higher dimensional glob es: c = [ ˜ c ] is the same type of equiv alence class as << ˜ c >> only for 1-glob es. Since π 2 G is trivial for all Lie groups, w orking with ρ  G fails to give us information ab out the homotop y type of G starting at the third fundamental group. In Section 4 w e will see that this is the reason that a HLGF determines a G bundle for base spaces of dimension tw o and three, but in general not for higher dimensional base spaces. The category of ∞ -group oids turns out to be equiv alent to the category of ω -groupoids. 26 3.2 Abstract HLGFs and the higher homotop y A tiyah group oid In section 2 we describ ed gauge ﬁelds in the contin uum as smo oth parallel transp ort maps. In order to state the properties of parallel transport maps in con venien t algebraic terms, w e presented the A tiyah groupoid: P arallel trans- p ort maps are sections of the canonical pro jection from the Atiy ah group oid to the path groupoid. Later, w e motiv ated that HLGFs should be parallel trans- p ort maps along glob es transp orting relativ e homotopies (of initial conditions) in the ﬁb er ov er the source p oin t to relative homotopies (of ﬁnal conditions) in the ﬁb er o ver the target p oint. Here w e will formalize these ideas. W e will start describing in detail what At ( X 1 ; G ) the A tiyah group oid o ver X 1 is. This is the appropriate groupoid to talk ab out parallel transp ort on X 1 . Suc h parallel transp ort maps corresp ond to standard lattice gauge ﬁelds. In section 3.6 we will describ e these ﬁelds in terms of generators; the resulting picture will b e recognizable as a standard gauge ﬁeld, where a gauge ﬁeld is giv en in terms of data asso ciated with the links of X 1 . Consider F a torsor for the Lie group G ; that is, it is a smooth manifold with a transitiv e and free action of G . Left and right actions will b oth pay diﬀerent roles b elo w. F will b e the typical ﬁb er of the G -bundles that we will construct. The pair groupoid P air ( X 0 ) has X 0 as set of ob jects and its set of morphisms is the set of ordered pairs of ob jects. Source and target maps are obvious, and comp osition of composable morphisms is also clear. At ( X 0 ; G ) the Atiy ah group oid ov er P air ( X 0 ) is a mo diﬁcation of P air ( X 0 ) with set of ob jects and set of morphisms corresp onding to ﬁb ers ov er p oin ts in X 0 and G equiv ariant maps among them. In detail, it is con venien t to de- ﬁne ob jects to b e pairs ( x, F x ) for some x ∈ X 0 , and morphisms to b e pairs (( x, y ) , M ( x,y ) ) where the ﬁrst entry is an ordered pair of p oin ts in X 0 and the second one is a G -equiv ariant map M ( x,y ) : F x → F y . The source map is s ((( x, y ) , M ( x,y ) )) = ( x, F x ) and the target map is t ((( x, y ) , M ( x,y ) )) = ( y , F y ). Comp osition of G equiv ariant maps is another G equiv ariant map; the com- p osition of morphisms in At ( X 0 ; G ) is a new pair where the ﬁrst entry is a comp osition of pairs and the second one a comp osition of G equiv ariant maps. Finally , notice that the set of G equiv ariant maps is con venien tly described by its 1 to 1 corresp ondence with the set ( F x × F y ) /G , where the quotien t is b y the diagonal right G action. W e will summarize the construction by writing At ( X 0 ; G ) = ( P air ( X 0 ) × F ) /G mor . At ( X 1 ; G ) the A tiyah groupoid o v er X 1 is the enrichmen t of ρ  1 ( X ∗ ) = P ( X 1 ), the path group oid of X 1 , needed for considering its morphisms as parallel transp ort maps. Thus, it is deﬁned b y the follo wing pullbac k diagram (Figure 18) The meaning of the previous deﬁnition “by pullback” is the follo wing: W e kno w all the corners of the diagram except for the top left corner. W e also know all the arrows except for the ones starting in the top left corner. Consider the 27 Figure 18 problem of ﬁnding a group oid in the top left corner together with its arrows whic h make the diagram into a commuting square. W e would like to ﬁnd a solution to the problem whic h is minimal and suﬃcien t. A solution of this problem is considered minimal and suﬃcient if and only if any other solution S (together with the corresponding arro ws) can be obtained from it (by means of a map lab eled by s ) as sho wn below (Figure 19) Figure 19 This pro cedure is considered a deﬁnition b ecause, in the con text of a com- plete and cocomplete category lik e the one w e are using, it is kno wn that the problem has an answ er, and it is unique. An explicit description of At ( π ) at a heuristic lev el was given in section 2.1. That description is easily adapted to describ e At ( X 1 ; G ). The reader is in vited to verify that the resulting description do es satisfy the deﬁnition given ab o ve. Before pro ceeding, let us clarify a p oin t where our language and the previ- ous technical deﬁnitions do not exactly match. W e rep eatedly mentioned the parallel transp ort of initial conditions on the ﬁb er ov er the source of a curve in the base space; w e referred to the initial conditions as “the ob jects” being par- allel transp orted. Due to the required G equiv ariance of the parallel transport map and the prop erty of the right G action b eing transitive, the information con tained in a map transp orting one initial condition u ∈ F s ( c ) along a curv e c is the same as that contained in a map transp orting another initial condition 28 v ∈ F s ( c ) along the same curve c . Because of this, we may consider that the ob jects b eing transp orted along a curve c are en tire ﬁb ers F s ( c ) . The technical terminology used ab o ve calls entire ﬁbers the ob jects. The parallel transp ort map asso ciated with a curve is a map from one ﬁber to another one; using that map w e can transport individual initial conditions. This is the picture p ortray ed when we deﬁne parallel transport maps as sections P ( X 1 ) → At ( X 1 ; G ) of the pro jection At ( X 1 ; G ) → P ( X 1 ). This is the picture that will b e extended to construct homotopy parallel transp ort. Parallel transp ort maps asso ciated with paths are maps from the source ﬁber to the target ﬁb er. T ransp ort along 2-glob es should b e thou ght of as homotopies of parallel transp ort maps relativ e to having their endp oin ts ﬁxed; as suc h, they are maps from a source ﬁber with the interpretation of b eing possible homotopies of initial conditions to a target ﬁber with the in terpretation of being the resulting homotopies of ﬁnal conditions. Using these parallel transport maps asso ciated with 2-glob es, from the source ﬁber (hosting relative homotopies of initial conditions), we can transport a particular homotopy initial conditions. No w we pro ceed to construct At 1 ,  ( X ∗ ; ρ  G ) the appropriate structure to describ e homotopy parallel transport in the homotopy lattice. The construction requires a group oid cov ering P air ( X 0 ) such that its ob- jects ha ve elements corresp onding to higher homotopies in F x (of diﬀeren t di- mensions) of initial conditions, and morphisms correspond to homotopies of G equiv ariant maps F x → F y . Thus, w e will consider pairs ( x, ρ  k F x ) with x ∈ X 0 and k ≥ 0 as ob jects. Morphisms among these ob jects are pairs (( x, y ) , T k, ( x,y ) ) where the ﬁrst entry is a pair of points in X 0 and the second entry is a ρ  k G - equiv ariant map T k, ( x,y ) : ρ  k F x → ρ  k F y for SOME ?????? k ≥ 0. The construction mirrors precisely that of At ( X 0 ; G ), and it is summarized as At 1 ,  ( X 0 ; ρ  G ) = ( P air ( X 0 ) × ρ  F ) /ρ  G mor . A t a closer level, w e notice an important diﬀerence: The ob jects ov er a giv en p oin t x ∈ X 0 are of the form ( x, ρ  k F x ) for any giv en k ; this set of ob jects may b e denoted by ( x, ρ  F x ) and has the structure of an inﬁnity groupoid. Simi- larly , the set of morphisms o ver a giv en pair of p oints x, y ∈ X 0 organizes into an inﬁnity group oid that may b e denoted by (( x, y ) , ( ρ  F x × ρ  F y ) /ρ  G ). Notice that, as is clear from the deﬁnition, the source and target maps inter- t wine (( x, y ) , ( ρ  F x × ρ  F y ) /ρ  G ) with ( x, ρ  F x ) at eac h level. This internal structure is resp onsible for the superscript in At 1 ,  ( X 0 ; ρ  G ). – JUAN: Check if this structure is the same as group oids internal to inﬁnit y group oids. – Since we ga ve At 1 ,  ( X 0 ; ρ  G ) the structure of an inﬁnity group oid in ternal to group oids, and w e need it to somehow co ver P air ( X 0 ), we will need to trivially promote P air ( X 0 ) to an inﬁnity group oid in ternal to group oids. By 29 construction, there is a surjective morphism At 1 ,  ( X 0 ; ρ  G ) → P air 1 ,  ( X 0 ) . No w we need to tell the structure that parallel transp ort happens along glob es of arbitrary dimension mo deling homotopies of paths in the base. ** W e need to reinterpret (sligh tly mo dify) the structure o ver the underlying set of ρ  X ∗ to make it an inﬁnit y groupoid internal to a group oid. ** The set of ob jects is just X 0 but seen as a trivial inﬁnit y group oid. k-Morphisms in ρ  X ∗ for an y k ≥ 1 pro ject onto morphisms of the group oid P air ( X 0 ); the pro jection given b y the 0-dimensional source and target maps ( d − 0 , d + 0 ). W e will use this pro jection to make ρ  X ∗ in to a group oid. Morphisms ov er a given pair ( x, y ) are asso ciated with ( k + 1)-glob es c ∈ ρ  k +1 X ∗ (for SOME ???? k ≥ 1) such that d − 0 c = x and d + 0 c = y . This set of morphisms ov er ( x, y ) is itself an inﬁnity group oid, but notice that its zeroth level starts with 1-glob es. This inﬁnit y group oid describ es higher homotopies of paths from x to y in the cutoﬀ provided by the skeletal ﬁltration. A morphism c will b e called a k-glob e of paths if c ∈ ρ  k +1 X ∗ . Source and target maps are s ( c ) = d − 0 c , t ( c ) = d + 0 c . The identit y elements are degenerate k-glob es with image x for some k ≥ 1 and some x ∈ X 0 . The inv erse map is − c = − 0 c . Composition in the group oid is d ◦ c = d ◦ 0 c . The shift in the level by one in the morphisms plays an imp ortan t role in gauge ﬁelds, as we will see b elo w. Let us call this structure ρ 1 ,  X ∗ , where the 1 sup erscript helps us distinguish it from the inﬁnity group oid ρ  X ∗ and it reminds us of the lev el shift in the inﬁnity groupoid structure describ ed ab o v e. Clearly , there is a pro jection map (surjective morphism) ρ 1 ,  X ∗ → P air 1 ,  ( X 0 ) . At 1 ,  ( X ∗ ; ρ  G ) is deﬁned by the follo wing pullbac k diagram of inﬁnity group oids internal to groupoids Figure 20 T o gain a more concrete understanding, let us explicitly describe the struc- ture of At 1 ,  ( X ∗ ; ρ  G ). Its ob jects are of the form ( x, ρ  k F x ) for some x ∈ X 0 30 and any given k ≥ 0. Notice that the set of such ob jects for a given x is organized in to the inﬁnit y groupoid ρ  F x . The set of morphisms is in correspondence with morphisms in At 1 ,  ( X 0 ; ρ  G ) and in ρ 1 ,  X ∗ . Given a pair ( x, y ), morphisms are of the form ( c, T k c ) where c ∈ ρ 1 ,  k +1 X ∗ is a k-glob e of paths with ( s ( c ) , t ( c )) = ( x, y ) and T k c : ρ  k F x → ρ  k F y (for any giv en k ≥ 0) is a ρ  k G -equiv ariant map. The set of morphisms co vering ( x, y ) in P air 1 ,  ( X 0 ) organizes into an inﬁnit y groupoid. Source and target maps in At 1 ,  ( X ∗ ; ρ  G ) are obvious; for example, s ( c, T k c ) = ( s ( c ) , ρ  k F s ( c ) ). Our ﬁnal observ ation regarding the structure of At 1 ,  ( X ∗ ; ρ  G ) is that as a co ver of P air 1 ,  ( X 0 ), the preimages of the pro jection hav e the struc- ture of an inﬁnity group oid, and at the low est lev el, w e ﬁnd At ( X 1 ; G ); all the diﬀeren t levels will be denoted by At 1 ,  ( X k +1 ; ρ  k G ). P arallel transp ort maps in the homotopy lattice, abbreviated as HLGFs, corresp ond to sections of the pro jection in the top row of the diagram P T : ρ 1 ,  X ∗ → At 1 ,  ( X ∗ ; ρ  G ) . (3) Let us see ho w, giv en a k-glob e of paths c ∈ ρ 1 ,  k +1 X ∗ with ( s ( c ) , t ( c )) = ( x, y ), suc h section P T induces a parallel transp ort map taking any initial condition u ∈ ρ  k F x to a resulting ﬁnal condition v ∈ ρ  k F y . The section yields P T ( c ), a morphism in At 1 ,  ( X ∗ ; ρ  G ) at level k . According to the preceding description of At 1 ,  ( X ∗ ; ρ  G ), this is a pair ( c, T k c ) consisting of the glob e of paths itself and a ρ  k G -equiv ariant map P T ( c ) = ( c, T k c : ρ  k F x → ρ  k F y ) . Clearly , the second entry is a map that can transport a k-dimensional relative homotop y in F x mo deled as a k-glob e u ∈ ρ  k F x to a resulting k-globe in v ∈ ρ  k F y . This is the parallel transp ort map asso ciated with c b y P T . W e invite the reader to work out in detail ho w is that for paths P T induces the parallel transp ort map asso ciated with standard lattice gauge ﬁelds. Lik e with ordinary ﬁelds, our gauge ﬁelds are now seen as sections. Since the non trivial information is all contained in the second entry of the image, a short hand notation, similar to the one used for ordinary ﬁelds writing ϕ ( x ) instead of ( x, ϕ ( x )), is c P T 7− → P T ( c ) : ρ  k F x → ρ  k F y . 3.3 HLGFs after a trivialization ov er v ertices In the diﬀerential geometric treatment of G bundles and connections, a lo cal trivialization is used to describe an y particular connection with resp ect to the trivialization. The result is a concrete description. Concreteness comes at the price of ha ving to tak e care of relating descriptions that diﬀer only b y the use of diﬀerent lo cal trivializations. In this subsection w e will present a concrete 31 description of HLGFs relying on the choice of a trivialization on the ﬁb ers ov er X 0 . It is very imp ortan t to realize that b ecause X 0 is a discrete set, we can c ho ose a trivialization on the ﬁbers ov er X 0 without the need to w ork lo cally and leading with compatibilit y at chart in tersection that is unav oidable when allo wing a con tinuum set of ﬁb ers 7 . A trivialization ϕ is a set of identiﬁcations { ϕ ( x ) : F x → G } x ∈ X 0 compatible with the right and left actions of G . Consequently , a G equiv ariant map F x → F y translates into a group homomorphism from G to G commuting with the righ t multiplication. All suc h maps can b e obtained b y left multiplication L g : G → G . Recall that the comp osition of tw o G equiv ariant maps is describ ed b y the corresponding group elemen ts m ultiplied in a left-wise order. Similarly , a ρ  G equiv ariant map ρ  F x → ρ  F y translates into an automorphism of ρ  G commuting with the righ t m ultiplication, and all such automorphisms can b e obtained by left m ultiplication. Thus, relative to the trivialization, a HLGF P T acting on a k-globe of paths P T ( c ) : ρ  k F s ( c ) → ρ  k F t ( c ) is enco ded by an elemen t P T ϕ ( c ) ∈ ρ  k G acting by left m ultiplication on ρ  k G . W e hav e a collection of these assignments for every level of glob es of paths. They are summarized by an assignmen t P T ϕ from ρ 1 ,  X ∗ to ρ  G , where the assignment sends k-globes of paths (whic h is the same as k + 1 globes in ρ  X ∗ ) to k-glob es in ρ  G . That P T : ρ 1 ,  X ∗ → At 1 ,  ( X ∗ ; ρ  G ) is a morphism must translate into some property of P T ϕ . The clue to understanding this property is the mismatc h b et w een the structures in the expression that we wan t to understand: On one side we ha ve ρ 1 ,  X ∗ that has the structure of an inﬁnit y groupoid in ternal to a group oid as describ ed in the previous subsection. On the other side, w e ha ve ρ  G that we are using for t wo reasons, its group structure (inherited from the group structure of G ) and an inﬁnit y groupoid structure whose stratiﬁcation in to lev els lets us enco de ρ  G equiv ariant map ρ  F x → ρ  F y b y left m ultiplication on ρ  G . Since we may consider groups G as groupoids B G with a single ob ject, w e see that ρ  G can b e giv en the structure of an inﬁnity group oid in ternal to a groupoid (the groupoid structure inherited from that of B G , and the inﬁnit y groupoid structure follo wing from higher homotop y). W e will denote te corresp onding structure by ρ 1 ,  B G . By construction, we see that a trivialization ϕ transforms P T into a mor- phism of inﬁnity groupoids internal to group oids where the comp osition ◦ and in verse − 1 of the groupoid is in correspondence with the m ultiplication and m ul- tiplicativ e inv erse in B G , and the internal inﬁnit y group oid structure in ρ 1 ,  X ∗ is in corresp ondence with the higher fundamental group oid structure of ρ  B G . 7 In [7] w e presen ted t wo equiv alent descriptions of HLGFs working with trivializations, one uses glob es as we hav e done here. The other one is based on cubes instead of glob es, and it was very useful for proving key properties, but it has the inconv enience that it needs lo cal trivializations with non trivial intersections as in the usual approach in the con tinuum. In that context, a HLGF is a consisten t collection of lo cal HLGFs relativ e to lo cal trivializations. 32 In summary , a trivialization ϕ transforms P T in to a morphism P T ϕ : ρ 1 ,  X ∗ → ρ 1 ,  B G. (4) Con versely , one such morphism of inﬁnity group oids internal to groupoids in- duces a HLGF. In [7] we had deﬁned HLGFs as morphisms of inﬁnit y group oids A ϕ : ρ  X ∗ → ρ  G [ − 1] . (5) W e will not give the deﬁnition of ρ  G [ − 1] here, but a simple comparison mak es it obvious that the deﬁnitions agree. A change to a diﬀerent trivialization ϕ 2 of the ﬁb ers ov er X 0 results in a morphism P T ϕ 2 related to P T ϕ 1 b y conjugation. First, changing from trivi- alization { ϕ 1 ( x ) : F x → G } x ∈ X 0 to trivialization { ϕ 2 ( x ) : F x → G } x ∈ X 0 is accomplished by left multiplication by the left action of the transition functions { ψ 12 ( x ) ∈ G } x ∈ X 0 on G as follows: { ϕ 2 x = L ψ 12 ( x ) ◦ ϕ 1 ( x ) = ψ 12 ( x ) · ϕ 1 ( x ) } . Then for any k-glob e of paths w e ha ve P T ϕ 2 ( c ) = s 0 ,k ( ψ 12 t ( c ) ) · ( P T ϕ 1 ( c )) · s 0 ,k ( ψ 12 s ( c ) ) − 1 . 3.4 Gauge transformations In the diﬀerential geometric picture, gauge transformations are induced b y bun- dle automorphisms. A bundle automorphism transforms a given connection to another one. A t a lo cal level with resp ect to a given lo cal trivialization, these transformations act by conjugation as in the last formula of the previous subsection (but within the given lo cal trivialization). F or each given bundle, these transformations form a group deﬁning gauge equiv alence classes in the space of connections on that bundle. If all G -bundles o ver a bases space M are treated at once in the standard contin uum treatment, gauge transformations form a group oid that is a disjoin union of the automorphism groups of each separate bundle (tw o bundle automorphisms can be comp osed only if they act on the same bundle). F rom the algebraic p oin t of view in the contin uum picture heuristically describ ed in Subsection 2.2, considering a groupoid of paths in M with a discrete collection of base p oints, we describ ed gauge ﬁelds on all possi- ble G -bundles o ver a giv en base M as a group oid homomorphisms (the heuristic coun terpart in the contin uum of deﬁnition (3) in our discrete setting). Notice ho wev er, that HLGFs are only sensitive to the restriction of bundle automor- phisms to the restriction of the bundle o v er the discrete set X 0 , and that the resulting automorphisms ov er the G bundle ov er X 0 do form a group. An automorphism of the bundle o ver X 0 is characterized b y a map g : X 0 → G . The induced gauge transformation acting on HLGFs is P T ′ = g ▷ P T describ ed using a trivialization ϕ o ver X 0 yields P T ′ ϕ ( c ) = s 0 ,k ( g t ( c ) ) · ( P T ϕ ( c )) · s 0 ,k ( g s ( c ) ) − 1 . Considering the abstract deﬁnition of a HLGF of equation (3), P T : ρ 1 ,  X ∗ → At 1 ,  ( X ∗ ; ρ  G ), the gauge transformation is a natural transformation g : 33 P T ⇒ P T ′ . Let us recall that b oth P T and P T ′ are sections of the pro- jection At 1 ,  ( X ∗ ; ρ  G ) → ρ 1 ,  X ∗ , and as such they send ob jects x in ρ 1 ,  X ∗ to ob jects ( x, ρ  k F x ) (for any given k ) in At 1 ,  ( X ∗ ; ρ  G ) and morphisms c in ρ 1 ,  X ∗ to morphisms ( c, T k c ) in At 1 ,  ( X ∗ ; ρ  G ). At a given lev el k , the comp onen t of the natural transformation ov er x is g x : ( x, ρ  k F x ) → ( x, ρ  k F x ) the morphism acting on ρ  k F x ) by left multiplication b y s 0 ,k ( g x ). Given an y morphism c in ρ 1 ,  X ∗ at lev el k with s ( c ) = x , t ( c ) = y the following diagram in At 1 ,  ( X ∗ ; ρ  G ) commutes ( X, ρ  k F X ) ( Y , ρ  k F X ) ( X, ρ  k F Y ) ( Y , ρ  k F Y ) P T ( c ) s 0 ,k ( g X ) s 0 ,k ( g X ) P T ( c ) Notice that the deﬁnition of homotop y parallel transp ort maps as comm uting with the right action of ρ  G on ﬁb ers ov er ob jects forces all natural transfor- mations connecting HLGFs to b e determined by left multiplication by some g : X 0 → G . Also notice that this type of natural transformations form a group acting on HLGFs. 3.5 The homotop y lattice cutoﬀ No w that HLGFs hav e b een formally introduced, it is clear that X a cellular decomp osition of the base space induces a cutoﬀ map sending any gauge ﬁeld in the con tinuum A ∈ A ∞ M ,G to its corresp onding HLGF A ∈ A H L X ∗ ,G constructed ev aluating the higher parallel transp ort of A on higher dimensional globes ﬁtting in X ∗ the skeleton of the appropriate dimension c X : A ∞ M ,G → A H L X ∗ ,G . W e will describ e the space of HLGFs in detail in the second part of this series. 3.6 A set of generators for HLGFs: ELGFs In this section, except where we explicitly say otherwise, we assume that the cellular decomp osition X is a triangulation by a simplicial set. Extended Lattice Gauge Fields (ELGFs) were introduced b y Claudio Mene- ses in collab oration with one of the authors of this pap er [1, 2]. The motiv ation and the main result was to reﬁne the lattice cutoﬀ for gauge ﬁelds in suc h a w ay that the resulting extended lattice gauge ﬁelds retain the ability of c haracteriz- ing a G -bundle (up to bundle equiv alence) after the cutoﬀ 8 . In this subsection 8 Article [1] prop oses tw o notions of extended lattice gauge ﬁelds. The ﬁelds in the ﬁrst of them tautologically c haracterize a bundle, and the second turns out to characterize the ﬁrst for tw o and three dimensional base spaces, but not in the general case . The second notion is the one further developed in [2], and it is what we call ELGFs in this pap er. 34 w e describ e how the v ersion of ELGFs given in [2] can b e understo od as HLGFs ev aluated on a set of generators of ρ 1 ,  X ∗ ; thus pro viding a 1-1 relationship b et w een HLGFs and ELGFs. The original w ork on ELGFs did not pro vide a notion of higher homotop y parallel transport, nor the complete algebraic to ol- b o x for gluing simplices of paths. The mentioned relationship will let us imp ort the top ological results of ELGFs to algebraically sup erior structure of HLGFs. Simplic es of p aths Simplices of paths w ere describ ed in Section 2.4. The 0-simplex of paths Γ ij is a path (i.e. 1-glob e) whose image is contained in the 1-simplex of X with ordered v ertex set [ ij ]. It has j as source and i as target. The 1-simplex of paths Γ ij k is a 2-glob e whose image is contained in the 2-simplex of X with ordered vertex set [ ij k ]. It represen ts a 1-homotopy of paths, all of them with k as source and j as target. The homotopy uses the geometric 1-simplex | [ ij ] | as parameter space: Ev ery x ∈ | [ ij ] | determines a path passing by x in its wa y from k to j . The globular structure of Γ ij k w as explicitly given in Section 3.1 (see Figure 13). Similarly , the 2-simplex of paths Γ ij kl has image in the 3-simplex with ordered set of vertices [ ij k l ], and it represen ts a homotopy of paths using the geometric 2-simplex | [ ij k ] | as parameter space (which has the structure of a 2-glob e). The general construction directly extends the one shown in Figure 13, making it the case k = 2. W rite Γ k = N k ◦ M k where M k : G k +1 → Y k +1 is a homeomorphism from the standard (k+1)-glob e to a closed (k+1)-ball presen ted as the union of t wo (k+1)-simplices with v ertices [1 , . . . , k + 1 , a ] and [1 , . . . , k + 1 , b ]. The c hoice of M k m ust b e compatible with the choice of M k − 1 . Our choice satisﬁes M k | e − k = M k − 1 (where we hav e identiﬁed Y k with the union of the tw o k- simplices in the b oundary of Y k +1 as suggested by the v ertex num b ering) and v k ∈ M k ( e + k ). On the other hand, N k | [1 ,...,k +1 ,a ] is an injective simplicial map, and N k | [1 ,...,k +1 ,b ] is the simplicial map determined by N k ( b ) = N k (1) and N k | [1 ,...,k +1 ,a ] . The original construction presen ted in [2] uses a triangulation b y a simplicial set constructed as the baricen tric subdivision of a simplicial complex. In that reference a k-simplex of paths contained in simplex ν of X is denoted b y Γ τ ν where τ is the k-simplex serving as parameter space for the homotopy; the 2-simplex of paths Γ ij kl w ould hav e b een written as Γ τ ν with ν = [ ij k l ] and τ = [ ij k ]. In [7] we use a similar notation (reversing the roles of the Greek letters), Γ ν τ with ν ⊂ τ and the parameter space for the homotopy b eing ν . Simplic es of p aths gener ate ρ 1 ,  X ∗ In [7] w e prov ed that simplices of paths are generators for ρ 1 ,  X ∗ when X is a triangulation by a simplicial set. A sketc h of the pro of is the following: (i) The HHSvK theorem [12] (solving the lo cal to global problem in higher homo- top y) tells us that a solution of the problem at a local level yields a solution of the general problem. More precisely , an op en co ver U with elemen ts that are inﬂated top dimensional simplices of X satisﬁes the condition that let us con- struct ρ 1 ,  X ∗ from the set of its restrictions to the open sets { ρ 1 ,  ( X | U ) ∗ } U ∈U . Th us, if simplices of paths generate ρ 1 ,  ( X | U ) ∗ } for eac h element of the cov er they also generate the whole ρ 1 ,  X ∗ . (ii) It is prov en that every glob e of paths 35 with image con tained in a single k-simplex of X is generated using r-simplices of paths with 1 ≤ r ≤ k − 1. This proof relies in the simplicial appro ximation theorem allowing us to construct any desired ﬁltered map up to homotopy as a comp osition of simplicial maps (after reﬁning the domain of the map). The evaluation of HLGFs on simplic es of p aths Let us deﬁne A E L i 0 i 1 ...i k +1 = A H L (Γ i 0 i 1 ...i k +1 ) for ev ery 0 ≤ k ≤ n − 1 = dim X − 1. This is our notation for the ev aluation of A H L on all the simplices of paths in X . The set of ev aluations is an ELGF. Because the set of simplices of paths is a generating set for glob es in X we kno w that an ELGF determines a HLGF. The set of ev aluations is not indep enden t, how ever. Algebraic relations that connect all the diﬀeren t simplices of paths as elements of ρ 1 ,  X ∗ imply relations among the ev aluations. Ev aluation on 0-globes of paths are indep enden t of each other. Ev aluations on k-globes of paths need to be compatible with the ev aluations on lo wer dimen- sional glob es of paths on their b oundary . Additionally , the set of ev aluations of a HLGF on simplices of paths that bind a higher dimensional simplex of paths are sub ject to an extendibilit y condition. It states that there is an HLGF com- patible with them that can also b e ev aluated on the b ounded higher dimensional simplex of paths. The ELGF as a set of c onsistent data ELGFs were introduced b efore HLGFs. When they were deﬁned they could ha ve b een though of as consistent ev aluations for gauge ﬁelds in the contin uum. W e sa w that a gauge ﬁeld in the contin uum can b e ev aluated on homotopies of paths. Thus the simplices of paths in X determined a set of ev aluations resulting from the parallel transport along the homotopies of paths corresp onding to path simplices. The set of such ev aluations was not an indep endent set for the same reasons as the ones giv en ab o ve. An ELGF could hav e been deﬁned as a set of consisten t data for the ev aluation of some gauge ﬁeld in the contin uum. Th us, in this paragraph we think of A E L i 0 i 1 ...i k +1 as the ev aluation of a homotopy of parallel transp ort maps on the simplex of paths Γ i 0 i 1 ...i k +1 calculated using a gauge ﬁeld in the contin uum. An ELGF is a set of consisten t ev aluations { A E L i 0 i 1 ...i k +1 } sp eciﬁed follo wing a particular order: W e ﬁrst specify the ev aluations on 0-simplices of paths (i.e. giv e standard lattice gauge ﬁeld data by ev aluation on lattice links). This is free data. After prescribing consistent ev aluations on k-simplices of paths for 0 ≤ k ≤ m , c haracterizing the ev aluation on any k-globe of paths with k ≤ m , the next step is prescribing compatible ev aluations on (m+1)-simplices of paths. A set of ev aluations { A E L i 0 i 1 ...i m +2 } on (m+1)-simplices of paths is consistent if it satisﬁes tw o t yp es of conditions: 1. d − m +1 A E L i 0 i 1 ...i m +2 and d + m +1 A E L i 0 i 1 ...i m +2 are compatible with the ev aluations on m -globes of paths (kno wn from the ev aluations on lo wer dimensional 36 simplices of paths. d − m +1 A E L i 0 i 1 ...i m +2 = A E L ( d − m +2 Γ i 0 i 1 ...i m +2 ) , d + m +1 A E L i 0 i 1 ...i m +2 = A E L ( d + m +2 Γ i 0 i 1 ...i m +2 ) . (6) 2. F or each µ , (m+3)-simplex of X , there is a consistency condition implying that a gauge ﬁeld in the contin uum with the prescrib ed ev aluation can b e extended from ∂ µ to the interior of µ = [ i 0 i 1 . . . i m +3 ]. Consider the (m+1)-glob e of paths c = − 0 s 1 ,m +2 ( d − 1 Γ i 0 i 1 ...i m +3 ) + 0 ( d − m +2 Γ i 0 i 1 ...i m +3 + m +1 d + m +2 Γ i 0 i 1 ...i m +3 ); it can b e seen as the image of S m +2 with base p oin t v m +3 winding around ∂ µ once. Then A E L ( c ) = ( s 0 ,m ( id ) ∈ ρ  m G, s 0 ,m ( id ) ∈ ρ  m G ; a ∈ π m +1 G ⊂ ρ  m +1 G ) F or µ = [ i 0 i 1 . . . i m +3 ] the condition ma y b e written as a = s 0 ,m +1 ( id ) . (7) The deﬁnition of ELGFs given abov e rewrites the conditions originally writ- ten in [2] using the algebraic language describ ed in this article. In [1] it was pro ven that for ev ery ELGF there are gauge ﬁelds in the contin uum whic h w ould induce that ELGF after homotopy lattice cutoﬀ. In terestingly , this set of conditions arose in [1, 2] as conditions guaranteeing the v alidity of ˇ Cec h co cycle conditions for transition functions calculated from an extension of the ELGF to a gauge ﬁeld in the contin uum deﬁne d in the interior of the domain b ound by the set simplic es of p aths . 3.7 General higher gauge ﬁelds on the homotop y lattice In this article, our m ain ob jectiv e is to describ e ordinary gauge ﬁelds within a cutoﬀ as explained previously . In this subsection we deviate from this goal to consider “higher gauge ﬁelds” not necessarily asso ciated with relative higher homotop y classes of ordinary parallel transp ort. Considering our deﬁnition (3) of a HLGF as a section of a group oid with higher homotopy internal structure, P T : ρ 1 ,  X ∗ → At 1 ,  ( X ∗ ; ρ  G ) , w e see that a route for generalizing our higher gauge ﬁelds is to ﬁnd inﬁnity group oids F , K replacing ρ  F and ρ  G respectively , and with the additional in ternal group structure in K and the transitiv e and free right action of K on F . F rom the construction given here, w e see that the resulting higher gauge ﬁelds would come with a notion of higher parallel transport along globes of paths: The ◦ 0 gluing map of ρ  X ∗ w ould be in corresp ondence with the internal m ultiplication in the replacement of ρ  G . 37 This generalization exercise w as studied, using deﬁnition (5), in [7]; we found agreemen t with an existing notion of higher gauge ﬁelds (using 2-groups) in the lattice [3]. Additionally , we found in [7] that any such generalization ( F , K ) can b e realized b y an appropriate ﬁltered group G . That is, giv en ( F , K ) with the sp eciﬁed structure, there is a ﬁltered group G such that K = ρ  G and F = ρ  T G where T G is a torsor for G . 38 4 The top ological c harge and other top ological considerations The top ological charge is a functional of the gauge ﬁeld that is a top ological in v ariant of the G -bundle. Since it happ ens that the ev aluations of the func- tional can only achiev e v alues in a discrete set, the functional may carry non trivial information only if the space of gauge ﬁelds has more than one connected comp onen t; in other words, the top ological charge is relev ant only if there is more than one G -bundle class o ver the giv en base space. A physical situation of interest tak es place in the base space R n , which represen ts spacetime, and there are “fall-oﬀ conditions” for the gauge ﬁeld ensuring at at inﬁnity the ﬁeld is trivial up to gauge. In Euclidian quan tum ﬁeld theory the fall-oﬀ conditions are naturally implemented considering a compactiﬁcation of the base space to S n . In the con tinuum, the topological c harge can b e ev aluated in several wa ys. The most common one is as an in tegral of a polynomial function of the curv a- ture; we could try to regularize the curv ature and the mentioned p olynomial function in terms of HLGFs. The alternative wa y , which we review b elo w, is to calculate the topological charge in S 2 n as the winding n um b er of a map from S 2 n − 1 to G representing the transition function betw een the lo cal trivializations (constructed using the gauge ﬁeld) ov er the tw o hemispheres. The equiv alence of these tw o wa ys to calculate the top ological c harge is prov en in [9]. At the end of Section 3.1 we sa w that the homotop y lattice cutoﬀ leading to HLGFs k eeps homotopical information of the gauge ﬁeld for base spaces of dimension 2 and 3. This will let us giv e a simple form ula for the top ological charge calculated as a winding num b er for bases of dimension 2. 4.1 The top ological c harge of gauge ﬁelds ov er S 2 W e start with the example of U (1) ≃ S O (2) HLGFs on S 2 with the triangulation X describ ed in Section 2.4. The main idea of the calculation of the topological charge as a winding n umber in the contin uum [9] has the following ingredients: (i) Separate the base S 2 in to tw o charts H S , H N with the top ology of the disc in tersecting in a neighborho o d with the top ology of S 1 Equator × ( − ϵ, ϵ ). (ii) Use the gauge ﬁeld and a path system to setup a trivialization o ver H S . The path system is the assignmen t of a path m x S to ev ery p oin t x ∈ H S from the base p oin t S (the South P ole) to x . Then the gauge ﬁeld A parallel transp orts any u S ∈ F S ≃ U (1) to A ( γ x S ) u S ∈ F x . Moreo ver, since the parallel transp ort map A ( m x S ) : F S → F x is 1 to 1, any p oin t in F x can b e parametrized using F S . This pro cedure determines a trivialization ov er H S ; the analogous pro cedure yields a trivialization o ver H N . (iii) The transition function ψ S N : H S ∩ H N → U (1) ≃ S 1 restricted to Eq = S 1 Equator ×{ 0 } is a con tinuous map determined by the gauge ﬁeld A characterizing the bundle ψ S N | Eq ( A )( x ) = ( A ( m x N )) − 1 · A ( m x S ) , 39 where in this formula we see eac h factor in the righthand side as v alued in G . A crucial prop erty is that a homotopy b et ween tw o such transition functions ψ S N | Eq determines a bundle map b etw een the corresponding bundles, making them equiv alent. Homotopy classes are classiﬁed b y their winding num b er, and the top ological c harge is deﬁned to be winding n umber Q ( A ) = W ( ψ S N | Eq ( A )) . (8) F or the HLGF the imp ortan t principle is that the homotopy lattice cutoﬀ k eeps the exact homotopical information when it is casted in a format that is compatible with their algebraic structure. Thus, the regularization of (8) for HLGFs follows three steps: 1. Cho ose a triangulation X with one v ertex in S , another vertex in N and suc h that the Equator is contained in X 1 . 2. Realize ψ S N | Eq as the ev aluation of A the gauge ﬁeld on m the singular 1-glob e of paths (the family of meridians) with source in S , target in N and paramertized by a singular 1-globe winding once around the Equator ψ S N | Eq = A ( m ) . 3. Then the ev aluation of the HLGF A H L = c X ( A ) on [ m ] the 1-glob e of paths in X (up to thin homotop y) is precisely the relative homotop y class of A ( ψ S N | Eq ) [ A ( ψ S N | Eq )] = ( c X ( A ))([ m ]) . 4. Since d − 1 [ m ] = d + 1 [ m ] the ev aluation ( c X ( A ))([ m ]) is a 1-globe in G = U (1) with its source equal to its target, and its homotop y class relativ e to endp oin ts is characterized by its winding num b er W ( A H L ([ m ])) = W (( c X ( A ))([ m ])) = W ( ψ S N | Eq ( A )) = Q ( A ) . This result captures the essence of what HLGFs can do that standard lattice gauge ﬁelds can’t. Belo w we present a more abstract version of the same idea, which sho ws the p o w er and v ersatility of the algebraic mac hinery presented in this article. A simple observ ation lets us change the p ersp ectiv e of the calculation to base it on higher parallel transp ort. A H L determines parallel transp ort along 1- glob es of paths transp orting 1-glob es of initial conditions on the 0-source of the glob e of paths. Th us, a constant initial condition s 0 , 1 u ∈ ρ  1 F S is transp orted along [ m ] from the South Pole to the North Pole. The result is a 1-glob e A H L ([ m ])( s 0 , 1 u ) ∈ ρ  1 F N with equal source and target; its homotop y class is determined b y its winding num b er, and this calculation coincides with the top ological charge W ( A H L ([ m ])( s 0 , 1 u )) = Q ( A H L ) in the sense that if A H L = c X ( A ) then Q ( A H L ) = Q ( A ). 40 In a more abstract view, notice that π 2 ( S 2 , x ) ⊂ ρ  X ∗ is a subgroup for an y vertex x ∈ X 0 . Then the restriction of any HLGF A H L induces a group morphism Q ( A H L ) : π 2 ( S 2 , x ) ≃ Z → π 1 ( U (1) , id ) ≃ Z (9) whic h in this case is c haracterized b y the m ultiplication by an in teger. F or other gauge groups, lik e G = S O (3), this deﬁnition has a clear extension. The relation b et w een this deﬁnition and the previous paragraphs, for which the 1-globe of paths is not an element of π 2 ( S 2 , x ) ⊂ ρ  X ∗ for an y x is a simple e xercise in globular algebra left to the reader. Notice that our deﬁnition of Q ( A H L ) implies a collection of equiv alent for- m ulas to ev aluate the top ological charge in terms of local data. An ELGF A E L is lo cal data characterizing a HLGF A H L ( A E L ), and the globular algebra gives us precise form ulas to ev aluate Q ( A H L ). In Section 2.4 we work ed out the exam- ple of A H L LC the S O (2) ≃ U (1) gauge ﬁeld induced by the Levi-Civita connection on S 2 restricted to parallel transport unit v ectors. W e found that QA H L LC = 2. It is easy to show that for any given n ∈ π 1 S O (2) = Z there are HLGFs A H L with Q ( A H L ) = n . 4.2 The bundle induced by a HLGF o v er 2 or 3 dimen- sional base spaces The bund le induc e d by a HLGF W e sa w that a HLGF A H L determines an extended lattice gauge ﬁeld A E L ( A H L ) simply b y ev aluating on simplices of paths. On the other hand, on triangulated base space of dimension 2 or 3 an extended lattice gauge ﬁeld with gauge group G induces a G -bundle π A E L [2, 1] . Therefore, for base space of dimension 2 or 3, a HLGF A H L induces a G -bundle A H L 7− → π A H L . The pro of given in [2] explicitly constructs lo cal trivializations and transition functions. The transition functions are determined b y the parallel transp ort map induced by the gauge ﬁeld in the con tinuum; then, as shown in Section 3.1, for base spaces of dimension 2 or 3 the HLGF captures the information of the parallel transp ort up to homotopy rel. vertices. Here w e sk etch its basic idea: Consider M triangulated by a simplicial set X . Each simplex ν of the triangu- lation ma y b e considered as a c hart, and with the aid of A a gauge ﬁeld in the con tinuum, a local trivialization is constructed: A path system on ν connects the maximal v ertex of ν to eac h p oin t x ∈ ν b y a path γ x ν . Then, once an initial condition u 0 in the ﬁber ov er the maximal v ertex is chosen, A ( γ x ν ) transports u 0 to a ﬁb er ov er x deﬁning the local trivialization. F or p oin ts that lie in the b oundary of a simplex x ∈ τ ⊂ ν (with τ not con taining the maximal vertex of ν ) the parallel transport maps A ( γ x ν ) , A ( γ x τ ) tell us how to glue the tw o lo cal trivializations. In that wa y , a bundle ov er M is determined by gluing the trivial bundles ov er the simplices of X . The standard lattice cutoﬀ tells us how to glue the men tioned trivial bundles ov er simplices at the vertices only . The homotopy 41 lattice cutoﬀ reﬁnes the standard lattice cutoﬀ in that (for base spaces of di- mension 2 o 3) A H L ( A ) = c X ( A ) k eeps the homotopy class of the gluing maps relativ e to the ev aluation at vertices (see Figure 21), and this information is enough to determine the bundle (up to a bundle equiv alence maps determined b y the homotopies). Thus, for base s paces of dimension 2 or 3, A H L ( A ) induces a G -bundle; the same G -bundle that A induces. Figure 21 5 Summary and discussion In this article we introduced HLGFs providing a geometrical motiv ation and a precise algebraic deﬁnition. W e also p edagogically presented the necessary algebraic framework backing up our proposal. W e sho wed that, for base spaces of dimension 2 or 3, a HLGF determines a principal bundle. W e also gav e exact explicit formulas for the topological charge of a HLGF in dimension 2. In a base space triangulated b y a simplicial set, HLGFs ev aluated on sim- plices of paths determine a set of generators for HLGFs. In previous work of one of the authors in collab oration with Claudio Meneses, these generators w ere called extended lattice gauge ﬁelds [1, 2]. This work is the direct precursor of what we presen ted here. W e sa w that in dimension 2 HLGFs repro duce a previous prop osal giv en b y Pfeiﬀer [3]. Let us brieﬂy mention the reason making our claim of characterizing a prin- cipal bundle ov er the base manifold b e v alid only for bases of dimension 2 or 3. The obstacle for a stronger result comes from our use of a trivial ﬁltration in the gauge group G . the mathematical framew ork mixing homotop y with the group structure in G requires that the ﬁrst level of the ﬁltration G 0 b e a sub- group of G . Mathematically sp eaking w e could w ork with a discrete subgroup G 0 ⊂ G , but physics prefers the choice G 0 = G , and we stick with it. Questions regarding π k Y in volv e maps S k → Y . At the algebraic lev el, the framework that we used to talk ab out higher homotopy is based on maps G k → Y whose domain is the standard k-glob e, which has the top ology of a disk, instead of the sphere. Since the maps are considered up to hopmotop y relative to vertices in the case k = 1 when the image of the tw o vertices of G 1 coincide, we obtain the information ab out the fundamen tal group. F or higher dimensional glob es, if the whole b oundary of G k where ﬁxed during the homotopy , we could capture 42 the information ab out π k Y . This happ ens when we restrict to ﬁltered maps G k → Y and the ﬁltration in Y do es not allo w homotopies unwinding ∂ G k . This is what fails due to our use of a trivial ﬁltration in G . Then w e only see π k G for k = 1. F ortunately for any Lie group π 2 G is trivial. A detailed explanation will be included in Iv an Sanchez’ PhD thesis (in preparation). F or purely mathematical interests, we could use a v ariant of our HLGFs using a nontrivial ﬁltration in the gauge group to calculate c haracteristic classes. In the pap er describing second part of this work w e will give structure to the space of ﬁelds that le t us deﬁne measures and amplitudes. W e apply the ideas b ehind the men tioned results solving the lo cal to global problem in higher dimensional homotopy [4] to construct the space of ﬁelds (and the Hilb ert space asso ciated to such spaces) from the space of ﬁelds asso ciated to elementary blo c ks. W e also use general theorems in higher algebra to justify our deﬁnition of the spaces corresp onding to the contin uum limit as the appropriate inv erse limits. All these results announced in the conference 17th Marcel Grossmann Meeting, and the pro ceedings pap er [14] has ben accepted for publication. W e also sho w that our framework reﬁnes standard lattice gauge theory in the sense that it agrees in its predictions regarding observ ables following from their 1- dimensional parallel transp ort, but we hav e access to a wider algebra of observ- ables following from our higher dimensional parallel transport. In the case of 2d Y ang-Mills theory , for example, w e hav e a lattice version based on the heat k er- nel measure that is quan tum p erfect. Since we can also calculate the topological c harge exactly , we can calculate the topological susceptibility exactly [14]. 6 App endix Here we write the structural relations of ω -groupoids. The original reference is [13], but the notation follows [12], which is compatible with their notation for ∞ -goup oids that we use here. W e hav e added an extra subindex to the degeneracy op erator because that “extended notation” makes our form ula for gauge transformations more explicit. X is a cellular decomp osition of M , and X ∗ is the ﬁltered top ological space determined by the skeleta of the cellular decomp osition. d ± j : ρ  n X ∗ − → ρ  j X ∗ s i,n : ρ  i X ∗ − → ρ  n X ∗ d ± j s i,n : ρ  i X ∗ − → ρ  n X ∗ − → ρ  j X ∗ d ± j s i,n =            d ± j si j < i I d i si j = i s i,j si j > i s i,j : ρ  i X ∗ − → ρ  j X ∗ s j,n : ρ  j X ∗ − → ρ  n X ∗ s j,n s i,j = s i,n para i < j 43 ∂ α i ∂ β j = ∂ β j − 1 ∂ α i i < j ε i ε j = ε j +1 ε i i ≤ j (10) ∂ ± i ε j =            ε j − 1 ∂ ± i si i < j I d si j = i ε j ∂ ± i − 1 si i > j (11) Γ α i Γ β j = Γ β j +1 Γ α i i < j Γ α i Γ α i = Γ α i +1 Γ α i (12) Γ ± i ε j =    ε j − 1 Γ ± i si i < j ε j Γ ± i − 1 si i > j (13) Γ α j ε j = ε 2 j = ε j +1 ε j (14) ∂ α i Γ β j =    Γ β j − 1 ∂ α i si i < j Γ β j ∂ α i − 1 si i > j + 1 (15) ∂ α j Γ α j = ∂ α j +1 Γ α j = I d ∂ α j Γ − α j = ∂ α j +1 Γ − α j = ε j ∂ α j (16) If a, b ∈ K n , then a ◦ j b is deﬁned if and only if ∂ − i b = ∂ + i a and then    ∂ − j ( a ◦ j b ) = ∂ − j a ∂ + j ( a ◦ j b ) = ∂ + j b (17) ∂ α i ( a ◦ j b ) =    ∂ α i a ◦ j − 1 ∂ α i b si i < j ∂ α i a ◦ j ∂ α i b si i > j (18) The inter change laws . If i  = j then ( a ◦ i b ) ◦ j ( c ◦ i d ) = ( a ◦ j c ) ◦ i ( b ◦ j d ) (19)  a b c d  / /   i j 44 ε i ( a ◦ j b ) =    ε i a ◦ j +1 ε i b si i ≤ j ε i a ◦ j ε i b si i > j (20) Γ α i ( a ◦ j b ) =    Γ α i a ◦ j +1 Γ α i b si i < j Γ α i a ◦ j Γ α i b si i > j (21) Γ + j ( a ◦ j b ) =  Γ + j a ε j a ε j +1 a Γ + j b  / /   j j + 1 Γ − j ( a ◦ j b ) =  Γ − j a ε j +1 b ε j b Γ − j b  / /   j j + 1 Supp orted by grant P APITT-UNAM IN114723 References [1] Claudio Meneses and Jose A. Zapata. Homotopy classes of gauge ﬁelds and the lattice. A dv. The or. Math. Phys. , 23(8):2207–2254, 2019. [2] Claudio Meneses and Jose A. Zapata. Macroscopic observ ables from the comparison of lo cal reference systems. Class. Quant. Gr av. , 36(23):235011, 2019. [3] Hendryk Pfeiﬀer. Higher gauge theory and a nonAbe lian generalization of 2-form electro dynamics. Annals Phys. , 308:447–477, 2003. [4] Ronald Bro wn, Philip J. Higgins, and Rafael Sivera. Nonab elian A lge- br aic T op olo gy: Filter e d Sp ac es, Cr osse d Complexes, Cubic al Homotopy Gr oup oids , v olume 15. Elemen tary T racts in Mathematics, EMS Press, 2012. [5] JHC Whitehead. On op erators in relative homotopy groups. A nnals of Mathematics , 49(3):610–640, 1948. [6] Ronald Bro wn and Philip J Higgins. Colimit the or ems for r elative homotopy gr oups . Universit y of W ales. Sc ho ol of Mathematics and Computer Science, 1979. [7] Juan Orendain and Jose Antonio Zapata. Higher homotopy and lattice gauge ﬁelds. 11 2023. 45 [8] Urs Schreiber and Konrad W aldorf. Parallel transp ort and functors. arXiv pr eprint arXiv:0705.0452 , 2007. [9] Shoshichi Kobay ashi and Katsumi Nomizu. F oundations of diﬀer ential ge- ometry, volume 1 , volume 61. John Wiley & Sons, 1996. [10] Patric k Iglesias-Zemmour. Diﬀe olo gy , v olume 185. American Mathematical So c., 2013. [11] John W. Barrett. Holonom y and path structures in general relativity and Y ang-Mills theory. Int. J. The or. Phys. , 30:1171–1215, 1991. [12] Ronald Brown. A new higher homotopy group oid: the fundamental globu- lar omega-groupoid of a ﬁltered space. arXiv pr eprint math/0702677 , 2007. [13] R. Brown and P . H. Higgins. On the algebra of cub es. Journal of Pur e and Applie d A lgebr a , 21:133–260, 1981. [14] Juan Orendain and Jose A. Zapata. A b etter space of generalized connec- tions. 7 2024. 46

Homotopy lattice gauge fields 1: The fields and their properties

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