Geometry of quadrilateral nets: second Hamiltonian form

GEOMETR Y OF QUADRILA TERAL NETS: SECOND HAMIL TONI AN F ORM. SERGEY M. S ERGEEV Abstra ct. Discrete Darboux- Manako v-Zakharo v s ystems p ossess tw o distinct Hamiltonian forms. In the framew ork of discrete-differential geometry on e Hamiltonian form appears in a geometry of circular net. In t his paper a geometry of second form is identified. The circular net [1] – a sp ecial t yp e of three-dimensional quadrilateral n et [2] – is an example of ge o metric al ly inte gr able (see [3] and references therein) system en d o wed by a discrete space-time Hamiltonian structure [4] w h at brings together geometrical ly integ rable and c ompletely inte g r able Hamiltonian systems. A class of analytical equations describ ing the thr ee-dimensional quadrilateral nets is usually refereed to as discrete Darb oux-Manak ov- Zakharo v sy s tems [5–7]. In this p ap er we d iscuss another sp ecial t yp e of qu ad r ilateral net whose geometry is describ ed b y th e second Hamiltonian form of DMZ systems [8]. F ollo wing [2], the 3 D quadr ilateral net is a Z 3 lattice imb edded into a multidimensional linear space, (1) ( n 1 , n 2 , n 3 ) ∈ Z 3 → x ( n 1 , n 2 , n 3 ) ∈ R M , M ≥ 3 , suc h that eac h quad r ilateral, e.g. (2) x = x ( n 1 , n 2 , n 3 ) , x 1 = x ( n 1 + 1 , n 2 , n 3 ) , x 2 = x ( n 1 , n 2 + 1 , n 3 ) , x 12 = x ( n 1 + 1 , n 2 + 1 , n 3 ) , is the planar one. A lo cal cell (hexahedron) of quadrilateral net is shown in Fig. 1. Geometric in tegrabilit y is based on the axiomatic state men t [2]: giv en the p oint s x 1 , x 2 , x 3 , x 12 , x 13 , x 23 of the h exahedr on, its corners x and x 123 can b e obtained uniqu ely by a t wo-dimensional ruler (ruler whic h dr a ws a plane via thr ee non-collinear p oin ts). The circular net is the quadrilateral net suc h that eac h its hexahedr on can b e inscrib ed in to a sphere. In this pap er, instead of the circular condition, sup p ose firstly that the target space is four-dimensional Euclidean space, (3) Q. net: Z 3 → E 4 . Eac h h exahedr on is an elemen t of a three-dimensional h yp erplane. In the framework of discrete-differen tial geometry [3], the quadr ilateral net can b e view ed as a planar mesh of three-dimensional manifold em b edded int o four-dimensional space. The hyp erplanes are more general ob jects then quadrilaterals since suc h net is not necessarily quadrilateral. 1991 Mathematics Subje ct Classific ation. 37K15. Key wor ds and phr ases. Quadrilateral net, discrete-differential geome try , discrete Hamiltonian structure. 1 2 S. SER GEEV e 1 e 3 e 2 x 13 x 3 x x 23 x 1 x 123 x 12 x 2 Figure 1. A cell of quadrilateral net. Let e 1 , e 2 , e 3 , (4) e 1 = x 123 − x 23 | x 123 − x 23 | , etc., b e unit v ectors d efining the orien tation of hexahedron in Fig. 1, and let (5) n = ∗ ( e 1 ∧ e 2 ∧ e 3 ) V ( e 1 , e 2 , e 3 ) , in indices: ( n ) α = ǫ αβ γ δ ( e 1 ) β ( e 2 ) γ ( e 3 ) δ V ( e 1 , e 2 , e 3 ) , b e the unit normal v ector to the hyp erplane ( e 1 , e 2 , e 3 ). Here (6) V ( e 1 , e 2 , e 3 ) = v olume of parallelipip ed with the edges ( e 1 , e 2 , e 3 ) . Consider now no de x 123 of the net: the jun ction of eight hyp erplanes sho wn on the left of Fig. 2 . T his junction is the sub ject of an extra “orthogo nalit y” condition: e ′ 1 e ′ 2 e 3 e 1 e 2 e ′ 3 f d h e b a g c Figure 2. O n the left: no de x 123 from Fig. 1, the ju nction of eigh t hyper- planes. On the r igh t: dual graph to the left vertex, corners a, ..., h lab el the h yp erp lanes. 3 (7) V ( e 1 , e 2 , e 3 ) V ( e 1 , e ′ 2 , e ′ 3 ) V ( e ′ 1 , e 2 , e ′ 3 ) V ( e ′ 1 , e ′ 2 , e 3 ) V ( e ′ 1 , e ′ 2 , e ′ 3 ) V ( e ′ 1 , e 2 , e 3 ) V ( e 1 , e ′ 2 , e 3 ) V ( e 1 , e 2 , e ′ 3 ) = 1 . In some sense th is condition is analogues to extra condition for the circular n et. It is con v enien t to lab el the “o ctants” on the left of Fig. 2 by corners of du al cub e, see righ t part of Fig . 2 . F or in stance, (8) n h ∼ ∗ ( e 1 ∧ e 2 ∧ e 3 ) , n d ∼ ∗ ( e 1 ∧ e 2 ∧ e ′ 3 ) , n e ∼ ∗ ( e 1 ∧ e ′ 2 ∧ e ′ 3 ) , etc. Orien tation of eac h hyp erplane is e # 1 ∧ e # 2 ∧ e # 3 . Consider no w four h yp erplanes n c , n e , n h and n d surroun ding the edge e 1 . Evidently , four h yp erp lanes in four-dimensional linear s pace ha ve a common edge if their normal v ectors are linearly dep end en t, (9) n c − u 1 · n e − w 1 · n h + κ 1 u 1 w 1 · n d = 0 . Numerical co efficien ts u 1 , w 1 , κ 1 in (9) are associated w ith the edge e 1 whic h is orthogonal to all n c , n e , n h and n d . Analogous relations for edges e 2 and e 3 are resp ectiv ely (10) n h − u 2 · n d − w 2 · n b + κ 2 u 2 w 2 · n f = 0 , n c − u 3 · n h − w 3 · n g + κ 3 u 3 w 3 · n b = 0 , and suc h equations f or outgoing ed ges e ′ i are (11) n g − u ′ 1 · n a − w ′ 1 · n b + κ ′ 1 u ′ 1 w ′ 1 · n f = 0 , n c − u ′ 2 · n e − w ′ 2 · n g + κ ′ 2 u ′ 2 w ′ 2 · n a = 0 , n e − u ′ 3 · n d − w ′ 3 · n a + κ ′ 3 u ′ 3 w ′ 3 · n f = 0 . All n umerical co efficien ts u # i , w # i and κ # i can b e expressed in terms of an gu lar data as follo w s . Let θ ce b e an angle b et ween n c and n e , (12) ( n c , n e ) = cos θ ce . Let further ϕ 1 ,e b e a dih ed ral an gle of h yp erplane n e for the edge e 1 . In terms of un it v ectors of Fig. 2 , ϕ 1 ,e is the dihedral angle b etw een p lanes ( e 1 , e ′ 2 ) an d ( e 1 , e ′ 3 ). W e extend straigh tforwardly these self-explanatory notations to whole dual graph of the junction, Fig. 2. Then the coefficien ts in relation (9) are giv en by (13) u 1 = sin ϕ 1 ,h sin ϕ 1 ,d sin θ ch sin θ ed , w 1 = sin ϕ 1 ,e sin ϕ 1 ,d sin θ ce sin θ dh , κ 1 = sin ϕ 1 ,c sin ϕ 1 ,d sin ϕ 1 ,e sin ϕ 1 ,h , and similarly f or all other relations and their co efficien ts. The geometry of ju nction without condition (7) pro vid es (14) κ ′ 1 κ ′ 2 = κ 1 κ 2 , κ ′ 2 κ ′ 3 = κ 2 κ 3 . Since there are at most four linearly indep endent ve ctors among eight n a , ..., n h , the con- sistency of equations (9-11) relates the fields u ′ i , w ′ i on outgoing edges and fields u i , w i on 4 S. SER GEEV incoming edges of Fig. 2 as follo ws (see e.g. [9]): (15) u ′ 1 = Λ − 1 2 w − 1 3 , u ′ 2 = Λ − 1 1 u 3 , u ′ 3 = Λ 1 u 2 , w ′ 1 = Λ 3 w 2 , w ′ 2 = Λ − 1 3 w 1 , w ′ 3 = Λ − 1 2 u − 1 1 , where (16) Λ 1 = u − 1 1 u 3 − u − 1 1 w 1 + κ 1 w 1 u − 1 2 , Λ 2 = κ 1 κ ′ 2 u − 1 2 w − 1 3 + κ 3 κ ′ 2 u − 1 1 w − 1 2 − κ 1 κ 3 κ ′ 2 u − 1 2 w − 1 2 , Λ 3 = w 1 w − 1 3 − u 3 w − 1 3 + κ 3 w − 1 2 u 3 . The “orthogonalit y” condition (7) p ro vides (17) κ i = κ ′ i , i = 1 , 2 , 3 , so that κ i b ecome in v arian ts. Map (15) is th e Hamiltonian one, it pr eserves the lo cal sym- plectic form (18) 3 X i =1 d u i ∧ d w i u i w i = 3 X i =1 d u ′ i ∧ d w ′ i u ′ i w ′ i , and w ith the orthogonalit y condition (17 ) it satisfies the functional tetrahedron equation [10]. In what follo ws, condition (7,17) is implied. Th us, due to (18), there exists a generati on f u nction, (19) dG ( u ; u ′ ) = 3 X i =1  log w ′ i d log u ′ i − log w i d log u i  u 2 u 3 = u ′ 2 u ′ 3 , where u # i and w # i are related by (15,16). In the definition of generating fu nction u i , u ′ i are c h osen as indep endent v ariables b oun ded b y condition u 2 u 3 = u ′ 2 u ′ 3 follo wing from (15). Let L ( z ) b e Roger’s diloga rithm, (20) L ( z ) = Z z 0 log(1 − x ) d log x , with the branc h cut z ≥ 1. Then the generating fun ction is giv en by (21) G ( u ; u ′ ) = log u ′ 3 κ 1 log u ′ 1 u 1 + log κ 3 log u ′ 1 u 2 + L ( κ 2 κ 1 u 2 u ′ 1 ) + L ( u ′ 2 u 1 ) − L ( κ 2 u ′ 2 u ′ 1 ) − L ( 1 κ 1 u 2 u 1 ) = log u 3 log u ′ 1 u 1 + log κ 3 log u ′ 1 u 2 + log κ 2 log u 2 u ′ 2 − L ( κ 1 κ 2 u ′ 1 u 2 ) − L ( u 1 u ′ 2 ) + L ( 1 κ 2 u ′ 1 u ′ 2 ) + L ( κ 1 u 1 u 2 ) . P ositiv eness of w # i guaran tees that arguments of all dilogarithms for one of the lines of (21) are out of the b ranc h cut and therefore the generation function is real. Quant ization of lo cal symp lectic stru cture (18) { u, w } = uw p ro duces th e lo cal W eyl algebra uw = q 2 w u . Quan tum coun terpart of Hamiltonia n form of (15) is an int ert winer R 123 in the 5 tensor cub e of prop er represen tations of local W eyl algebras suc h that (22) u ′ i = R 123 u i R − 1 123 , w ′ i = R 123 w i R − 1 123 , i = 1 , 2 , 3 . F or instance, the mo dular represent ation [11] of the local W eyl algebra is giv en b y (23) u = e 2 π b x , w = − e 2 π b p , κ = − e 2 π bλ , where x , p is the self-conjugated Heise nberg p air (24) [ x , p ] = i 2 π ⇒ q = e i π b 2 , and “ph ysical” regime for b is (25) η def = b + b − 1 2 > 0 . Mo dular partner to (23) is (26) ˜ u = e 2 π b − 1 x , ˜ w = − e 2 π b − 1 p , ˜ κ = − e 2 π b − 1 λ . F orm of the map (22) for ˜ u i , ˜ w i , ˜ κ i coincides with that for u i , w , κ i ; in the strong coupling regime 0 < η < 1 p artner equations are Hermitia n conjugated. Kernel of the in tert win er (22) in the coordin ate represen tation of Heisen b erg pairs (24) is (27) h x 1 x 2 x 3 | R | x ′ 1 x ′ 2 x ′ 3 i = δ ( x 2 + x 3 = x ′ 2 + x ′ 3 ) e 2 π i { ( x ′ 3 − λ 1 )( x 1 − x ′ 1 )+( λ 3 − i η )( x 2 − x ′ 1 ) } ϕ ( x 2 − x 1 − λ 1 ) ϕ ( x ′ 2 − x ′ 1 + λ 2 ) ϕ ( x ′ 2 − x 1 − i η + i ǫ ) ϕ ( x 2 − x ′ 1 + λ 2 − λ 1 − i η + i ǫ ) , where function ϕ is the n on-compact quan tum dilogarithm [11] (28) ϕ ( z ) def = exp  1 4 Z R + i 0 e − 2 i z w sinh( wb )sin h ( w /b ) dw w  . Sym b ols i ǫ in denominator of (27) define circum v en tions of p oles. Op erator (27) satisfies the quan tum tetrahedron equation with free λ i . The c hoice of n egativ e signs near w and κ in (23) pro vid es the un itarit y of op erator (27) for real λ i , R − 1 123 = R † 123 . Positiv e “geometric” signs can b e ob tained by th e analytical con tinuatio n λ i → λ i + i η and n on-unitary gauge transf ormation w → e − 2 π η x w e 2 π η x = − q − 1 w . In th at case the k ern el of R -matrix (27) h as the semi-classical ( b → 0 and e 2 π b x → u ) asymptotic (29) log  e − 2 π ηx 1 h x | R | x ′ i e 2 π ηx ′ 1  − → b → 0 − G ( u ; u ′ ) 2 π i b 2 , where the generating function is giv en by (21). It w orth mentio ning the cyclic represent ations of W eyl algebra w ith q 2 N = 1. The cyclic represent ation is a Z N fib er ov er the base of cente rs ˜ u = u N , ˜ w = w N [12]. Equations of motion for C -v alued cen ters follo w f rom quantum map (15), they ju s t coincide with classical 6 S. SER GEEV equations of m otion. It is n atural then to iden tify the ev oluting centers ˜ u i , ˜ w i directly with the geome tric data (13) and p ose quan tum pr oblems in Hilb ert space (30) H = Z ⊗ (size of net’s s ection) N in the presence of external classical ge ometry . The structure of Z ⊗ 3 N in tert win ers and mo d ified tetrahedron equations are discussed in more details in e.g. [13, 14]. Homogeneous p oint ˜ u ′ i = ˜ u i , ˜ w ′ i = ˜ w i of the Zamolod c h iko v-Bazhano v-Baxter mo del [15] is complex one, it is not a geometrica l regime. Ac knowledgemen ts. I am grateful to V. Bazhano v and V. Mangazeev for v aluable discus- sions and fruitful collab oration. Also I w ould lik e to thank M. Hewett , P . V assiliou and J. Ascione for an en cour agemen t. Referen ces [1] Konop elchenko , B. G. and Schief, W. K . Thr e e-dimensional inte gr able lattic es in Euclide an sp ac es: c onju- gacy and ortho gonality. R. Soc. Lond. Pro c. Ser. A Math. Ph ys. Eng. Sci. 454 (1998) 3075–3 104. 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F a cul ty of Informa tional Sciences and Enginee ring, Unive rsity of Can berra, Bruce ACT 2601 E-mail addr ess : sergey.ser geev@canberr a.edu.au