A Graphical Coaction for FRW Wavefunction Coefficients

A Graphical Coaction for FR W W a v efunction Co efficien ts Andrew J. McLeod , 1 , ∗ Andrzej P okraka , 2 , † and Lec heng Ren 3 , ‡ 1 Higgs Centr e for The oretic al Physics, School of Physics and A str onomy, The University of Edinbur gh, Edinbur gh EH9 3FD, Sc otland, UK 2 Institute of The oretic al Physics, University of A mster dam, A mster dam, 1098 XH, The Netherlands 3 Centr e for The or etic al Physics, Dep artment of Physics and A str onomy, Que en Mary University of L ondon, E1 4NS, UK W e sho w that the w av efunction of the universe in theories of conformally coupled scalars in pow er- la w F riedmann-Rob ertson-W alk er (FR W) cosmologies satisfies a graphical coaction, by means of whic h w e can understand its complete analytic structure in terms of the acyclic minors of F eynman graphs. Our construction extends to all particle multiplicities and any loop order, and if we isolate certain w eigh t-one con tributions, it reproduces the “kinematic flo w” that enco des the differential equation of the wa vefunction co efficients. In a similar manner, the (sequen tial) discon tinuities of w av efunction co efficien ts can also b e extracted from the coaction. I. INTR ODUCTION Conformally-coupled scalar field theories in FR W cos- mologies pro vide enlightening to y models with a rich mathematical structure and a connection to inflationary ph ysics. These theories are usually studied in the pres- ence of (non-conformal) p olynomial interactions where d s 2 = a 2 ( η ) h − d η 2 + P d i =1 d x i d x i i is the FR W metric, η ∈ ( −∞ , 0] is the conformal time, and the scale factor is a p ow er-la w a ( η ) = ( η /η 0 ) − (1+ ε ) c haracterized by the cosmological parameter ε . This scale factor specializes to sev eral well-studied cosmolog- ical scenarios including de Sitter ( ε = 0), flat ( ε = − 1), radiation-dominated ( ε = − 2), and matter-dominated ( ε = − 3) universes [1, 2]. Moreov er, for 0 < ε ≪ 1, this mo del is expected to capture the essen tial dynamics of near de Sitter inflationary cosmology . In de Sitter ( ε = 0) theories, equal-time correla- tion functions enco de the quantum fluctuations at the end of inflation that ultimately seed the temp erature and matter-density v ariations observ ed to day . These correlators can b e expressed as linear combinations of more primitiv e ob jects called w av efunction coefficients, whic h exhibit fascinating mathematical structures, re- lated to algebraic and positive geometry [3 – 5], twisted (co)homology [1, 2, 4, 6], com binatorics [7, 8], and am- plitudes [9, 10]. A uspiciously , w a vefunction coefficients take a univer- sal form in the class of theories under consideration. The flat-space w av efunction co efficient ψ (flat) G asso ciated to a F eynman diagram G can b e upgraded to a p ow er-law FR W wa v efunction co efficient ψ G ( ε ∈ C \ Q ) 1 b y in- ∗ andrew.mcleod@ed.ac.uk † a.m.pokraka@uva.nl ‡ lecheng.ren@qm ul.ac.uk 1 When ε ∈ C \ Q the integral (1) is a twisted integral and w e hav e goo d mathematical con trol ov er the resulting space of integrals. More sp ecifically , easy access to a coaction [11]. tegrating against a kernel u G called the twist: ψ G = Z ∞ 0 u G φ G , φ G = ψ (flat) G ( X + x , Y ) d n x . (1) They ev aluate to h yp ergeometric functions that depend on the external energies X v that flo w from eac h in ter- action vertex v to the b oundary , collectively denoted by X , and the energies Y e that are exchanged through each in ternal edge e , collectiv ely denoted by Y . F or example: X 1 X 3 Y 12 Y 23 k 1 k n 1 k n 3 k n 2 X 2 k 1 k n 1 k n 2 X 1 X 2 Y 12 Y 21 . With V G the set of vertices of G , the twist u G = Y v ∈V G x α v v (2) is a multi-v alued function whose exponents α v ∈ C \ Z are determined b y the underlying p ow er-law cosmology (sp ecified by ε ), the num b er of spatial dimensions d , and the v alency p v of eac h v ertex v [2]. The simplest case o ccurs when p v = 3 and d = 3, since then α v = ε . The flat-space integrand φ G is itself given b y a simple com binatorial formula: φ G = d |V G | x X T G Y τ ∈ T G 1 B τ , (3) where T G denotes the set of (non-intersecting) complete tubings of G . Eac h tub e in a tubing is sp ecified by a pair of sets τ = {V τ , E τ } , where V τ is the set of vertices of G that are encircled b y the tube τ , while E τ is the set of edges that cross the tub e τ . Given this data, w e define the “propagators” 2 B τ = X v ∈V τ x v + X v ∈V τ X v + X e ∈E τ Y e . (4) 2 T echnically , the B τ are the square ro ots of propagators. How- ever, w e wan t the reader to approach the B τ with the same intuition as if they were true propagators. 2 F or more details on this com binatorial construction of φ G , see [3]. In this letter, we sho w that the wa vefunction of the univ erse—or, more directly , the w a vefunction co ef- ficien ts (1)—satisfy a graphical coaction, by means of whic h their analytic structure can b e easily understo o d. Our construction is similar in spirit to the diagrammatic coaction for F eynman in tegrals [11 – 16], but makes use of a slightly richer graphical language that keeps trac k of the direction of time (or energy flo w). Roughly sp eaking, our graphical coaction, ∆, takes the form ∆ ψ G = X decorations g of G  rational function of α v   g ⊗ C g ( G )  , (5) where the sum is o v er all (acyclic) decorations g of the graph G (detailed in section I I), and C g ( G ) represents the tubing of G that corresp onds to g . As we will show, each of the (decorated) graphs that appear on the right side of this formula ha v e theory-indep enden t integral in ter- pretations; the integral asso ciated with g enco des part of the differential equation satisfied by ψ G , while the graph C g ( G ) is a cut integral that computes one of the (sequen- tial) discontin uities of ψ G . In the α v → 0 limit, our coaction reduces to the w ell-studied coaction on m ulti- ple polylogarithms (MPLs) [17–30], which maps MPLs to their oft-utilized symbol when applied rep eatedly [26]. Note that, unl ike [11, 16], w e set all propagator p ow ers to in tegers form the outset; thus, w e w ork with ph ysically relev ant ob jects throughout. I I. PHYSICAL CUTS AND A CYCLIC MINORS F or de Sitter universes inv olving only cubic interac- tions, wa vefunction coefficients ha ve an especially sim- ple analytic structure, since (in three spatial dimensions) all α v = ε → 0. Under these conditions, wa v efunction co efficien ts ev aluate to multiple p olylogarithms (MPLs) of uniform transcenden tal w eight |V G | , with logarithmic branc h p oints generated by the p oles in the in tegrand where the propagators B τ v anish. Therefore, computing a residue with resp ect to any of the p olynomials B τ iso- lates the discontin uit y across a logarithmic branc h cut. More generally , trading the physical integration contour in (1) for one that computes residues with resp ect to m ul- tiple propagators is equiv alent to computing a sequence of discontin uities. These types of mo dified integrals— in which the in tegration contour wraps around the v an- ishing lo cus of a subset of the denominator factors—are often referred to as cut inte gr als . Only certain cut integrals are nonzero. This can al- ready be seen in the structure of (3), where pairs of prop- agators only app ear in the same term if the corresp ond- ing tubes do not intersect. A dditionally , the num b er of denominator factors in eac h term generally exceeds the n umber of in tegration v ariables |V G | , implying we can- not compute residues with resp ect to all of them. Even so, the set of nonzero cuts of ψ G are easily en umerated; they are in one-to-one correspondence with the acyclic minors of G [4]. Each acyclic minor corresp onds to a decorated graph in whic h eac h edge in G is replaced by an oriente d e dge ( or ), a pinche d e dge ( ), or a disc onne cte d/br oken e dge ( ), sub ject to the con- dition that there are no directed cycles in the resulting graph when all pinched edges are contracted. F rom a bulk ph ysics p oint of view, we can think of the oriented edges as ha ving a definite direction of time or energy flo w, while the pinc hed and brok en edges correspond to shrinking or eliminating the edge, resp ectively [9, 10]. T o each acyclic minor g , we asso ciate a set of cut tub- ings C g . These cut tubings c ∈ C g are maximal collec- tions of non-crossing/compatible tub es such that (C1) no τ ∈ c crosses a pinched edges of g , (C2) no τ ∈ c encircles a broken edge of g , (C3) an y τ ∈ c crosses an edge e only if e is a brok en edge of g or e is an orien ted edge AAAEJnicdZPdbtMwFMfTho9Rvja43E1EN6mT2pIUNCYh0NZxgeBmSHQbSqPJcZzEqmNH9klHifI+PADPwR1C3PEoOE0GtF3P1d/n/JycL/spowps+1ejad64eev2xp3W3Xv3Hzzc3Hp0qkQmMRlhwYQ895EijHIyAgqMnKeSoMRn5MyfHJfxsymRigr+EWYp8RIUcRpSjEC7LrYa33anTu/1dNAa+ySiPAc6+ZJSDJkkhaswYuSV47Ws2sZYCBlQjoBYnaM9C4HVsbv23svriWFFOE8H65njmllPvKmIZ8tfCSS6dCGmeNLt5QoIYhAX3jyvXq/kD9bx/6Dh/0BIGXN9hq4ATCVmOoFBCmux4VqMi4C4jITg/W2Tle9ML5ydYpmSNIprrOrVHBxcgWPCg4XBXGy27b49N2tVOLVoG7Wd6Dl/GgcCZwnhgBlSynXsFLwcSaA696I1zhRJdUUoIq6WHCVEdYMpTdVcevnn+a4tgDlKlJolfmHtJghitRwrndfF3AzCAy+nPM2AcKwRHQszZoGwyh21AioJBjbTAmFJdYoWjpFEGPQmL/wlUKHgsJhV2ahVTw8HlbM8MOpLJGe6fCkuVVfFKNXlpkLR8lVQHq2iFdOPiEgISIq7VnW5aOlhOMutXxWng76z39//8Lx9OKzHsmFsG0+MjuEYL4xD461xYowM3NxuHjXfNd+bX83v5g/zZ4U2G/Wdx8aCmb//AKSmOxY= v 1 v 2 with v 1 ∈ V τ and v 2 ∈ V τ . F or man y acyclic minors, there is a unique tubing that satisfies these conditions; for example, C = n o . (6) Ho wev er, some acyclic graphs ha v e multiple tubings c , eac h of whic h satisfies (C1-C3), such as C = n , o . (7) Imp ortan tly , ev en when m ultiple cut tubings b e- long to C g , the geometry defined b y the intersection T τ ∈ c { B τ =0 } is unique (different c ∈ C g just corresp ond to differen t wa ys of writing the same linear system of p olynomial equations). This means that w e can asso ciate a unique “physical” residue operator with each acyclic minor g : Res g = X c ∈ C g sgn c Res c , (8) where the op erator Res c computes a residue with resp ect to B τ for all τ ∈ c , and sgn c ∈ { 1 , − 1 } is given in (A1). Some examples include Res = Res , , , (9) Res = Res , , − Res , , . (10) Sp elling out the ab o ve notation in terms of propagators, Res τ 1 ,...,τ m := Res B τ 2 ,...,τ m ◦ · · · ◦ Res B τ 1 , (11) where B τ 1 ,...,τ k = T k i =1 { B τ i = 0 } is an intersection of hy- p erplanes. Imp ortantly , w e adopt a conv ention in whic h residues corresp onding to lar ger tub es (those that en- close more v ertices) are computed before those of smaller 3 tub es; geometrically , this amounts to a c hoice of the rel- ativ e rate at which the corresp onding hyperplanes are approac hed. Note that we also need to choose a conv en- tion for what region to integrate ov er once all residues in (8) ha ve b een computed; we will hav e more to say ab out this in the next section. The virtue of this acyclic minor construction is that it automatically tak es into accoun t all linear relations b et w een partially-ov erlapping propagators, namely those generated by the relation B τ 1 + B τ 2 = B τ 1 ∪ τ 2 + B τ 1 ∩ τ 2 . (12) In other words, the residue op erators Res g asso ciated with the acyclic minors of G span the space of op erators that do not annihilate the physical w av efunction coeffi- cien t; all other residue op erators either annihilate φ G , or are expressible as a linear com bination of those sp ecified b y (8). The same acyclic graph construction generalizes b e- y ond de Sitter cosmologies, and to all p olynomial inter- actions and spatial dimensions. F or generic ε ∈ C \ Q , the co ordinate hyperplanes x v = 0 b ecome singular sur- faces of u G in addition to playing the role of integration endp oin ts. While this c hanges the (co)homological un- derpinnings of the integral ψ G , the residue op erators (8) still span the space of cut integrals. The upshot is that acyclic graphs still pro vide a gr aphic al r epr esentation of the set of cuts—and b y extension, the (sequen tial) discon tinuities—of wa vefunction co efficients for the full class of theories under consideration. 3 I II. FROM CUTS TO THE COA CTION The full analytic structure of ψ G is c haracterized not only by its sequential discontin uities, but also b y its deriv atives. In fact, more attention in the literature has b een paid to the differential equations that describ e w av efunction co efficien ts, and their form ulation in terms of so-called kinematic flow [1, 2, 4 – 6, 9, 10]. Con venien tly , a structure that combines knowledge of an in tegral’s discontin uities and deriv atives is known. The c o action , b est dev eloped for MPLs [25 – 28, 31], can b e used to formally decompose twisted p erio d in tegrals in to (often simpler) building blocks whose anal ytic struc- ture is already under go o d control. F or wa v efunction co- efficien ts (1) (and the integrals that enco de their discon- 3 The full space of integrands/con tours—the (co)homology—is muc h larger than the physical subspace indexed by the acyclic minors. While our graphical coaction applies only to this sub- space, one can construct a non-graphical coaction on the full (co)homology from a generalization of (13). tin uities and deriv atives), this op eration takes the form ∆ Z γ u G φ = X g , h C − 1 gh ∂ kin Z γ u G φ g ⊗ Disc kin Z γ h u G φ , (13) where the second entry is implicitly understoo d to b e mo d iπ , { γ g } is the set of cut integration con tours dis- cussed in the last section, { φ h } is a basis of differen tial forms that hav e p oles at most where φ G has poles, and C gh is the intersection matrix asso ciated with these con- tours and forms [11–16]. 4 As indicated by the labeling of this formula, the first en try of the coaction encodes the kinematic deriv ativ es of the original in tegral, while the second en try enco des its kinematic discontin uities. The upshot is that, if the coaction of ψ G is kno wn, its differen- tial equation and discon tin uities can b e simply extracted. W e ha v e already seen that the discontin uities of the w av efunction co efficien t ψ G (or more sp ecifically the set of cut contours { γ g } ) can b e given a graphical interpre- tation. After specifying these contours in more detail, w e will now show that the remaining ingredients in (13) can also b e describ ed graphically . This facilitates the construction of a graphical coaction for the wa v efunction of the universe akin to the one for flat-space F eynman in tegrals [12, 14, 33]. A. A basis of contours from cuts Non trivially , the set of residue o perators (8) can b e used to build a basis of the homology on whic h ψ G and its discon tinuities are defined. Consequen tly , the physi- cal in tegration con tour γ phys = T i ∈V G { x i ≥ 0 } can b e ex- pressed as a linear combination of cut contours. In order to fully specify these contours, ho w ever, w e m ust iden tify the domain ov er which the remaining v ariables should b e in tegrated, after all the residues in Res g ha ve b een com- puted. T o do so, w e note that the twisted coordinate h yp erplanes { x i = 0 } that b ound the original in tegra- tion region alw ays define a unique b ounded region once a set of propagators ha ve b een set to zero. 5 Therefore, for every acyclic minor g we define a cut contour γ g via Z γ g u G φ := Z ∆ g Res g [ u G φ ] , (14) 4 W e provide an explicit formula for the intersection num b ers in (25); more details will b e provided in [32]. 5 F or twisted in tegrals ( ϵ ∈ C \ Q = ⇒ α v ∈ C \ Z ), the b oundaries at { x i = 0 } are in the kernel of the twisted b oundary op erator. Therefore, twisted cycles are allowed to end on the co ordinate hyperplanes. One way to think ab out why contours are allowed to end on the twisted hyperplanes is that the exp onents α v reg- ulate the integral; there exist a region in parameters where the integral conv erges and aw ay from this region the integral is de- fined by analytic contin uation. 4 where ∆ g is the unique b ounded region that has supp ort on this cut. Explicitly , ∆ g := n ( x 1 , . . . , x |V G | ) ∈ C |V G |    B τ =0 ∀ τ ∈ C g , x v ≤ 0 ∀ v ∈ V g o , (15) where V g is the set of all v ertices in g connected to at least one pinche d edge. B. A canonical basis of forms from cuts Ha ving defined a basis of in tegration cycles { γ g } , w e no w identify a basis of differen tial forms { φ h } that are dual to these cycles via integration. In fact, an appro- priate basis of forms was constructed in [4], which are also in one-to-one corresp ondence with the acyclic mi- nors of G . In particular, the form asso ciated with the acyclic minor h is defined b y having dlog singularities on all the propagators B τ for τ ∈ C , as well as on the co ordi- nate hyperplanes x v = 0 that b ound the region ∆ h . One w ay to construct such a form is by finding a region that is bounded b y all these h yp erplanes, and computing its canonical form. In fact, there exist many suc h regions, and any choice would w ork. In practice, w e find that the optimal c hoice is to set φ g = Ω[ ˇ Γ g ] equal to the canonical form of the (un- b ounded) region ˇ Γ g := n ( x 1 , . . ., x |V G | ) ∈ C |V G |    B τ ≤ 0 ∀ τ ∈ C g , x v ≤ 0 ∀ v ∈ V g o , (16) since it results in a diagonal intersection matrix C gh . F ur- ther, note that since the maximal co dimension b oundary of ˇ Γ g is ∆ g , the maximal cut of these differential forms is the canonical form of ∆ g Res g [ φ g ] = | C g | Ω[∆ g ] , (17) where | C g | is the cardinality of the set C g . 6 C. Graphical FR W p eriods and their cuts Recalling (14), the p erio ds P gh of our basis are P gh := Z γ g u G φ h = Z ∆ g Res g [ u G φ h ] . (18) Graphically , we denote the p eriod P gh b y sup erimposing all cut-tubings of C g (represen ting the cut contour γ g ) on to the acyclic minor h (representing the FR W-form φ h ). F or example, in this graphical language, the “di- agonal” p erio ds of the tw o-site c hain are Z ∆ Res [ u G φ ]= = Γ( α 1 )Γ( α 2 ) Γ( α 1 + α 2 ) f α 1 + α 2 , 6 One can think of (17) as fixing the relative orientation b etw een γ g and ˇ Γ g . Z ∆ Res [ u G φ ]= = f α 1 f α 2 , Z ∆ Res [ u G φ ]= = f α 1 f α 2 , Z ∆ Res [ u G φ ]= = f α 1 f α 2 , (19) where f = X 1 + X 2 , f = X 2 − Y 12 , f = X 1 − Y 12 , f = X 1 + Y 12 = f , f = X 2 + Y 12 = f , (20) are the symbol letters that app ear in the differential equations (see [4] for more on this notation). The re- maining non-zero p erio ds for the tw o-site chain ev alu- ate to hypergeometric functions. In the kinematic region where X 1 , X 2 > Y 12 > 0, = c 2 F 1  1 , 1+ α 1 ; 2+ α 1 + α 2 ; f f  f f , = c 2 F 1  1 , 1+ α 2 ; 2+ α 1 + α 2 ; f f  f f , (21) for c = − α 1 α 2 Γ( α 1 + α 2 ) / Γ(2+ α 1 + α 2 ). Note that p erio ds with tub es that cross a pinched edge, or an oriented edge that p oin ts the wrong wa y , v anish: 0 = = = = = = = = = = . (22) This is the graphical translation of the fact that Res g [ φ h ] = 0 ∀ g / ∈ pinch( h ) , (23) where pinc h( g ) is the set of all acyclic minors obtained from g by turning an y n umber of oriented edges into pinc hed edges (including no edges). In other words, P gh  = 0 if and only if g is a pinch of h . IV. THE GRAPHICAL CO ACTION Rephrasing the coaction (13) in terms of the contours and forms defined in the last section, w e get ∆ Z γ g u G φ h = X f C − 1 ff Z γ g u G φ f ⊗ Z γ f u G φ h , (24) where the right entry is again understo o d mo dulo iπ , while [4, 32] C gh := δ gh | C g | Y r ∈ Sub g P v ∈V r α v Q v ∈V r α v , (25) and δ gh is the Kronec k er delta sym b ol. Here, C gg are in tersection n umbers associated to our basis choice and Sub g is the set of all connected subgraphs of g with pinc hed edges. Due to (23) only the terms in the sum with g ∈ pinch( f ) and f ∈ pinch( h ) are non-v anishing. 5 A. The t wo-site chain graph The coaction of an y p ow er function (suc h as seen in (19)) is trivial; b oth their deriv ative and discontin uity are prop ortional to themselves. F or example, ∆ h i = α 1 α 2 α 1 + α 2 ⊗ , ∆ h i = ⊗ . (26) Con versely , the hypergeometric functions (21) enjo y a more interesting coaction. F or example, ∆ = α 1 α 2 α 1 + α 2 ⊗ + ⊗ . (27) Indeed, the deriv ative/discon tin uity of a 2 F 1 can b e pro- p ortional to either itself or to a p o w er function. F or ex- ample, extracting the w eigh t-one component of the sec- ond entry in (27) yields the differential equation satisfied b y : d[ ]= α 1 α 2 α 1 + α 2 d W 1 [ ]  α 1 α 2 α 1 + α 2 dlog f f  + ( α 1 dlog f + α 2 dlog f ) d W 1 [ ] , (28) This agrees with [4]. Similarly , any discontin uity of can b e obtained by extracting the weigh t-one part of the first entry: Disc[ ]= α 1 α 2 α 1 + α 2 Disc W 1 [ ] ( α 1 + α 2 ) (Disc log f ) +  Disc log f f  Disc W 1 [ ] . (29) Here, Disc is a stand-in for a sp ecific c hoice of discon ti- n uity , for instance Disc f =0 . Note that, to make use of (24) for the full wa vefunc- tion co efficien t ψ G , we should first decomp ose the ph ys- ical contour γ phys and physical differential form φ G in to our chosen bases { γ g } and { φ g } . This decomp osition is provided for the t wo-site c hain in app endix B. 7 F rom this, the coaction on the physical integral is deduced. B. A one-lo op example T o illustrate the structure of the graphical coaction b e- y ond tree-level, consider the following one-lo op example ∆ = ⊗ + α 12 ⊗ 7 The decomp osition of any φ G into our basis is trivial—a closed form formula exists [4, 32, 34]. On the other hand, the decom- position of γ phys straightforw ard but tedious. + α 23 ⊗ + α 34 ⊗ (30) + α 123 ⊗ + α 234 ⊗ + α 12 α 34 ⊗ + α 1234 ⊗ , where α ij ··· := α i α j ··· α i + α j + ··· . Note that with loop graphs, the acyclic condition actually matters. The decorated minors  , , , , , , ,  , (31) do not app ear b ecause they are cyclic once the pinched edges are contracted. C. F urther Examples and Cross-Checks A more complete deriv ation of (24), as well as fur- ther examples, will b e provided in a forthcoming publi- cation [32]. In particular, we hav e c heck ed (in a n um- b er of nontrivial examples) that our coaction formula comm utes with the well-kno wn MPL coaction, when ex- panded in α v . In [32], we also deriv e an explicit formula for the w eigh t-one part of an y p erio d P gh , allowing us to sho w that the kinematic flow construction follows di- rectly from (24). V. CONCLUSION In this letter, we hav e form ulated a graphical coac- tion for the wa v efunction of the universe in theories of conformally coupled scalars in p ow er-law FR W cosmolo- gies, v alid at all particle multiplicities and arbitrary lo op orders. By lev eraging the t wisted (co)homology of the as- so ciated in tegral families, the coaction decomp oses w av e- function coefficients into tensor pro ducts of simpler FR W p eriods, eac h represented by a decoration of the original F eynman diagram. This yields a unified, graphical lan- guage that encapsulates both the differential equations and the sequen tial discontin uities of cosmological correla- tors. Our construction is analogous to (and may provide hin ts for studying) the diagrammatic coaction for flat- space F eynman integrals, but in volv es a ric her graphical language that incorp orates the direction of time. Sev eral natural extensions remain. It w ould b e inter- esting to connect this construction more explicitly to the graphical coaction for flat-space F eynman in tegrals in the ε → − 1 limit, and to explore whether the coaction struc- ture p ersists for theories with spinning fields, non-p ow er- la w scale factors or non-conformally coupled scalars. W e lea ve these fascinating directions for future w ork. 6 A CKNOWLEDGMENTS The authors would like to thank D. Baumann, R. Britto, and C. Dupont for stim ulating discussions. W e also thank C. Duhr, whose original suggestion led to this work. AJM is supp orted b y the Roy al So ciet y grant URF \ R1 \ 221233, and additionally ackno wledges supp ort from the Europ ean Research Council (ER C) under the Europ ean Union’s Horizon Europ e researc h and innov a- tion program grant agreement 101163627 (ERC Starting Gran t “AmpBo ot”). AP is supported by the Europ ean Union (ER C, UNIVERSE PLUS, 101118787). LR is sup- p orted by the Roy al So ciet y via a Newton International F ello wship. App endix A: Definition of sgn c In this appendix, w e define sgn c . While easy to com- pute, the definition requires the in tro duction of new ideas that are not needed for the main text. Let σ be the p ermutation that orders the tub es τ ∈ c according to size. Also let R g = Sub g ∪ Sub • g where Sub g is the collection of all connected subgraphs of g with only pinche d edges and Sub • g is the set of all 1- v ertex subgraphs attached to no pinched edges. Then, with r i ∈ R g , sgn c := sig  ⟨ ¯ r 1 ⟩ c , . . . , ⟨ ¯ r | R g | ⟩ c  sig  σ ( ⟨ ¯ r 1 ⟩ c ) , . . . , σ  ⟨ ¯ r | R g | ⟩ c  , (A1) where ¯ r i := min V r is minimal vertex of the subgraph r i . W e also define ⟨ v ⟩ c as the smallest tub e τ ∈ c that con tains the subgraph r that also contains the v ertex v : ⟨ v ⟩ c := τ ∈ c suc h that v ∈ V r and |V r | > |V r ′ | ∀ r ′ ∈ τ with r ′  = r . (A2) Note that the map ⟨ v ⟩ c : V g → c is many-to-one whenever the asso ciated minor g contains pinc hed edges. App endix B: The t wo-site chain wa v efunction co efficien t in terms of FR W p erio ds The ph ysical wa v efunction co efficien t ψ G can b e writ- ten in terms of the FR W perio ds (19) and (21). First, one decomp oses the ph ysical differen tial form in to the φ g using partial fractions φ 2 − chain = φ + φ − φ . (B1) The general partial fraction decomposition has a closed form expression (see [4]). W e can also decomp ose the ph ysical con tour into our basis of cycles. While w e do not ha ve a general formula, it can be done case-by-case when needed γ 2 − chain phys = − π e − π i ( α 1 + α 2 ) γ sin  π ( α 1 + α 2 )  + π 2 e − π iα 2 γ sin  π α 1  sin  π ( α 1 + α 2 )  + π 2 e − π iα 1 γ sin( π α 2 ) sin  π ( α 1 + α 2 )  + π 2 e − π i ( α 1 + α 2 ) γ sin( π α 1 ) sin( π α 2 ) (B2) Com bining (B2) with (B1), yields ψ 2 − chain = − π e − π i ( α 1 + α 2 ) sin  π ( α 1 + α 2 )  h + i + π 2 e − π iα 2 sin  π α 1  sin  π ( α 1 + α 2 )  + π 2 e − π iα 1 sin( π α 2 ) sin  π ( α 1 + α 2 )  − π 2 e − π i ( α 1 + α 2 ) sin( π α 1 ) sin( π α 2 ) . (B3) [1] S. De and A. Pokraka, JHEP 03 , 156 (2024), arXiv:2308.03753 [hep-th]. [2] N. Arkani-Hamed, D. Baumann, A. Hillman, A. Jo yce, H. Lee, and G. L. Pimentel, (2023), [hep-th]. [3] N. Arkani-Hamed, P . Benincasa, and A. Postnik ov, (2017), arXiv:1709.02813 [hep-th]. [4] R. Glew and A. Pokraka, (2025), arXiv:2508.11568 [hep- th]. [5] M. Capuano, L. F erro, T. Luko wski, and A. Palazio, (2025), arXiv:2505.14609 [hep-th]. [6] S. De and A. Pokraka, JHEP 03 , 040 (2025), arXiv:2411.09695 [hep-th]. [7] C. F ev ola, G. L. Pimentel, A.-L. Sattelb erger, and T. W esterdijk, Matematic he 80 , 303 (2025), arXiv:2410.14757 [math.AG]. [8] A. Birk emey er, T. Donzelmann, M. Fink, and M. Juhnke, (2026), arXiv:2603.03894 [math.CO]. [9] S. He, X. Jiang, J. Liu, Q. Y ang, and Y.-Q. Zhang, (2024), arXiv:2407.17715 [hep-th]. [10] D. Baumann, H. Go o dhew, A. Joyce, H. Lee, G. L. Pi- men tel, and T. W esterdijk, (2025), [hep-th]. [11] S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, JHEP 02 , 122 (2020), arXiv:1910.08358 [hep-th]. [12] S. Abreu, R. Britto, C. Duhr, and E. Gardi, Phys. Rev. Lett. 119 , 051601 (2017), arXiv:1703.05064 [hep-th]. [13] S. Abreu, R. Britto, C. Duhr, and E. Gardi, JHEP 06 , 114 (2017), arXiv:1702.03163 [hep-th]. 7 [14] S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, JHEP 10 , 131 (2021), arXiv:2106.01280 [hep-th]. [15] F. Bro wn, Commun. Num. Theor. Phys. 11 , 453 (2017), arXiv:1512.06409 [math-ph]. [16] F. Brown and C. Dup on t, Nagoy a Math. J. 249 , 148 (2023), arXiv:1907.06603 [math.AG]. [17] K.-T. Chen, Bull. Amer. Math. So c. 83 , 831 (1977). [18] A. B. Goncharo v, Adv. Math. 114 , 197 (1995). [19] A. B. Gonc harov, Math. Res. Lett. 5 , 497 (1998), arXiv:1105.2076 [math.AG]. [20] E. Remiddi and J. V ermaseren, Int.J.Mod.Phys. A15 , 725 (2000), arXiv:hep-ph/9905237 [hep-ph]. [21] J. M. Borw ein, D. M. Bradley , D. J. Broadh urst, and P . Lisonek, T rans. Am. Math. Soc. 353 , 907 (2001), arXiv:math/9910045 [math-ca]. [22] S. Mo ch, P . Uwer, and S. W einzierl, J.Math.Ph ys. 43 , 3363 (2002), arXiv:hep-ph/0110083 [hep-ph]. [23] V. Del Duca, C. Duhr, and V. A. Smirno v, JHEP 03 , 099 (2010), arXiv:0911.5332 [hep-ph]. [24] V. Del Duca, C. Duhr, and V. A. Smirno v, JHEP 05 , 084 (2010), arXiv:1003.1702 [hep-th]. [25] A. B. Goncharo v, Duke Math. J. 128 , 209 (2005), arXiv:math/0208144 [math.AG]. [26] A. B. Goncharo v, M. Spradlin, C. V ergu, and A. V olo vich, Phys.Rev.Lett. 105 , 151605 (2010), arXiv:1006.5703 [hep-th]. [27] F. Bro wn, A dv. Studies in Pure Math. 63 , 31 (2012), arXiv:1102.1310 [math.NT]. [28] F. Bro wn, Ann. of Math. (2) 175 , 949 (2012), arXiv:1102.1312 [math.AG]. [29] H. F rost, M. Hidding, D. Kamlesh, C. Ro driguez, O. Schlotterer, and B. V erb eek, J. Phys. A 57 , 31L T01 (2024), arXiv:2312.00697 [hep-th]. [30] H. F rost, M. Hidding, D. Kamlesh, C. Ro- driguez, O. Schlotterer, and B. V erb eek, (2025), arXiv:2503.02096 [hep-th]. [31] C. Duhr, JHEP 1208 , 043 (2012), arXiv:1203.0454 [hep- ph]. [32] A. McLeo d, A. Pokraka, and L. Ren, A Graphi- cal Coaction for FR W In tegrals from Relativ e T wisted (Co)homology , to app ear. [33] S. Abreu, R. Britto, C. Duhr, and E. Gardi, JHEP 12 , 090 (2017), arXiv:1704.07931 [hep-th]. [34] R. Glew, (2025), arXiv:2507.07199 [hep-th].