The tetrahedron equation and the four-simplex equation

Volume 134, number 4

PHYSICS LETTERS A

2 January 1989

THE TETRAHEDRON EQUATION AND THE FOUR-SIMPLEX EQUATION Jean Michel MAILLET’ Fermi NationalAccelerator Laboratory,

P.O. Box 500, Batavia,

IL 60510, USA

and Frank NIJHOFF Department of Mathematics and Computer Sciences, Clarkson University, Potsdam, NY 136 76, USA Received 28 September 1988; accepted forpublication 25 October 1988 Communicated by A.P. Fordy

The tetrahedron equation and the four-simplex equation are multidimensional generalizations of the Yang—Baxter or triangle equations. We discuss common features of thesemembers of the family of “simplex equations”. Zamolodchikov’s solution of the tetrahedron equation is rewritten in an algebraic form and a generalization of it to the four-simplex case is proposed. Relevance of the simplex equation for the understanding ofmultidimensional integrability is briefly discussed.

1. It is well known that the concept of quantum and classical integrability in one- and two-dimensional systems is intimately linked to the notion of triangle or Yang—Baxter equations [1—6].They emerged on the one hand as the conditions for the commutation of transfer matrices in statistical mechanical spin systems, and on the other hand as the consistency condition of factorizability for the S-matrix in two-dimensional quantum field theory. Thus, the triangle equation (1)

~

is the compatibility condition for the factorization of the three-particle S-matrix in terms of the two-particle scattering amplitudes (0, 0’, 0+0’, are the difference of rapidity between the three incident particles and the indices ~a, stand for the internal quantum numbers of the particles (a = 1, N)), pictorially depicted in fig. 1. In eq. (1), we have taken into account the constraint on the, in general, three parameters 0~,02, 03 en~

...,

Address after October 1, 1988: CERN, Theory Group, CH-121 I Geneve 23, Switzerland. 13

1~~

i~

/

I3~I

k3

=

2

‘1

3

‘2

Fig. 1. Condition for factorization of three-particle scattering in terms of two-particle scattering amplitudes.

0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

221

Volume 134, number 4

PHYSICS LETTERS A

2 January 1989

tering the equation, coming from the well-known relation of the three angles of a triangle in a plane. Nowadays, there exists a great deal of insight into the algebraic structure of the Yang—Baxter equations, as well as an almost exhaustive classification of solutions within the context of simple Lie algebras at the classical level and to some extent also at the quantum level [6—12]. An equivalent way of writing eq. (1) is in terms of Boltzmann weights w(aI bcjd), depending on the values ofspins around a plaquette on a square two-dimensional lattice denoted here by a, b, c, dwhich may take values ±,or, more generally, any integer number. In terms of these objects, the Yang—Baxter equation reads [2,4] ~ w(a~b1,b2 c)w’(b2jc, b’1 Ia’)w”(clb, a’Ib’2)—~w”(b2 a, b’1 Ic)w’(alb, cIb’,)w(c~b’T,,b’, a’),

(2)

the sum being performed over an internal spin value c. Eq. (2) which is also referred to as generalized star— triangle relation, implying the commutation of transfer matrices for the associated lattice spin model, is pictorially given in fig. 2. The prime attached to the win eq. (2) stands for different values of internal continuous parameters. A multidimensional generalization of the triangle equations was proposed in 1980 by Zamolodchikov, who wrote down the conditions for the factorizability of the scattering amplitudes ofstraight strings in a plane [13]. He also proposed what is up to now the one and only non-trivial solution of these equations [141. It is equivalent to a free fermion system on a cubic three-dimensional lattice [15]. We shall not write down here the original Zamolodchikov equations because he used a rather complicated representation. A more convenient representation is obtained by generalizing eq. (2), cf. refs. [16,17], in terms of Boltzmann weights which no longer live on faces of a two-dimensional square latice, but on cubes of a three-dimensional cubic lattice, depending hence on eight spins values, i.e., w(aI efgl bcdf h). In terms ofthis description, Baxter was able to prove that Zamolodchikov’s proposal indeed solves the so-called tetrahedron equation, which we now state in full:

~ ~ d

d

(3) the sum being performed over a central spin value d. Eq. (3) which describes the conditions for commutativity of the layer-to-layer transfer matrices of a three-dimensional lattice model is given pictorially in fig. 3. Again, the primes attached to the win eq. (3) denote the dependence on (continuous) parameters, on which we will comment below. Note that the tetrahedron equations are invariant under a multiplication of w by

b2

b~

a

b2

a’

b1

b’2

=

b~

~

a’

b1

b~

Fig. 2. Generalized star—triangle equation for Boltzmann weights w, w, w’ of a lattice spin model.

222

Volume 134, number 4

~

_~1~

PHYSICS LE1’TERS A

O’~b—~~

W”

2 January 1989

=

Fig. 3. Pictorial representation ofthe tetrahedron equation.

—

y(eic,dih)y(J1b,d~h)y(gIb,cIh) y(a~e,~d)y(aIe,gIc)y(aI~g~b)’

in which y(ajb, cid) =y(aic, bid)=y(dib, cia) =y(cia, dib) is independent of the parameters. 2. In ref. [18] it was argued that both the triangle equations as well as the tetrahedron equations are part of a larger family of equations corresponding to increasing dimensionalities, the so-called d-simplex equations. We give here the first members of this family (summation over repeated indices is understood): d= 1 (commutativity condition): A~B~=B~A~.

(4)

d= 2 (Yang—Baxter equation): A~

C1?~= ~

(5)

d= 3 (Zamolodchikov equation): A klk2k3Bj~Ik4k5C~2D~.= D~~c6C~~6B~I1~44/~A ~

(6)

d= 4 (Bazhanov—Stroganov equation): A iI121314 k1 k2k3k4BjI k5koki C~2j~8/(I — Ek4.k?k9k k3k6k8jIOCk2k5jSJ9BkIj5j6j7A jIf2J~J4 k11516i7 k2k5,819 D~3j6j8I~.b0E~4j719ub0 k3k6kglio k4k7koklO — 141719110 °D 13,618kb 120k8k9 11k5k6k7 kik2k3k4•

(7

It is easy to see from eqs. (4)— (7) how to generalize them to arbitrary dimension d leading to the d-simplex equations. These equations can be considered to be generalizations of matrix commutativity conditions for objects depending on multiple (2d) indices. The (d+ 1) quantities A, B, C, entering the d-simplex equation are assumed to be respectively equal to the different values of one object, say, S~’j~(D~) corresponding to (d+ 1) choices of the set of parameters D,~,i, j= 1, d. The matrix of parameters D,, is a dxd symmetric matrix that we construct as follows. Let us first consider a d-simplex defined in a d-dimensional affine euclidean space by giving (d+ 1) points, F1, which form a complete coordinate system for this space. The (d+ 1) faces of the d-simplex are respectively the (d+ affine hyper-planes of codimension onetoconstructed out 1k J. To each face1)II~ we associate a unitII~ vector u~orthogonal it and pointing of the d points ~bk’ k= 1 d, outside the d-simplex. Now, each point P, belongs to the d faces H~,j i. Hence, to each vertex P, of the dsimplex we can associate the d unit vectors u 1, j ~ i, and the matrix D~ of their scalar products, i.e., DJ,.J) = Uk’ U1. We also associate to each vertex F, the tensor S~j~(D~J~) where the indices ~k and .lk are, for each k, respectively characterizing a line joining the point P, to another point P1, 1# i, of the d-simplex. If now, we consider this tensor as a statistical weight attached to the vertex P1, we can compute the partition function associated ...

...,

...,

223

Volume 134, number 4

PHYSICS LETTERS A

2 January 1989

d+ 1, the sum being performed on the indices to the d-simplex; it is just the product of the tensors St’~,i= 1 1k orjk attached to the internal edges of the d-simplex. The d-simplex equation expresses now that this partition function is the same for the d-simplex we started with (for which all the unit vectors u 1 are pointing outside of it) and for the d-simplex obtained by a translation of the hyper-planes H1 (conserving their orientation) in such a way that all the above u1 are now pointing inside the new d-simplex. The first examples of these equations for d= 1, 2, 3, 4 are given in eqs. (4)—(7). Let us note that an equivalent description of the set of parameters D~>is given in terms of the ~d(d+ 1) relatives angles O,~between the (d+ I) unit vectors u1, u1. However, not all these angles are independent, since (d+ 1) vectors in d dimensions are not independent. If we write u,’u,=cos(ir—O,,), we obtain one relation between these ~d(d+ 1) angles, given by the vanishing of the determinant of the (d+ 1) x (d+ 1) matrix =

u,~u1.

In the case of the tetrahedron equation (d= 3), the tensor S depends on three parameters, e.g., the respective angles between the three planes defining each vertex P•, i= 1 4. Hence we have ~

02, 03)S~’~.~~(01, 04, 05)S __S~~<6(03, 05, 06)S~,~6(92,04,

1~’~(02, 04,

06)S ~

O6)S~3~’k6(O3, 05, 06) 04, 05)SL’,-~’,~(O1, 02, 03),

(8)

where the six parameters °k, k= 1 6, which are a relabeling of the angles ObJ~are not all independent as we have seen before. In fact, if cos(,t—01)=c,, i=l, 6, we have the relation: ...,

1 det c c., c4

c

c2

c4

1 c3 c5

C3

C5

1 c6

c6 I

=0.

(9)

Note that the angles 0, and the angles a~between the lines in the tetrahedron of fig. 4 are related by spherical trigonometry [19]. The relation between the tetrahedron equations (6), (8) and eq. (3) in terms of Boltzmann weights is given by the following correspondence: w(a~efgibcdIh)~ assuming that any one of the indices of S in eq. (8) correspond to a set of four spins on any one of the faces of the cube with which we associate the Boltzmann weight w.

~

Fig. 4. The tetrahedron equation in the representation of eq. (4).

224

12

Volume 134, number 4

2 January 1989

PHYSICS LETTERS A

In a similar fashion as before, the four-simplex equation is given by eq. (7) where the quantities A, B, in the four-simplex equation (7) are given by SJ~j~ (0~, 06) for prescribed values of the angles °k• The foursimplex is described by the 10 relative angles between its 5 faces, 9 of them being independent, the last one being constrained by the vanishing of the matrix determinant, det (Q~)= 0; i, j= 1, 5. Again, we can associate, in a similar manner as before, with eq. (7) a representation of the four-simplex equation in terms of Boltzmann weights depending on spins on a hypercubic four-dimensional lattice. Let a, b~, b4, c,, c6, d,, d4, e denote the 16 spin values corresponding to the 16 vertices of an elementary ...

...,

...,

...,

...,

...,

hypercube of such a lattice (see fig. 5), then we can introduce the following correspondence: w(aIbi,...,b4ic~,...,c6ld4,...,diie)~—’S~j, where every index of S corresponds to a face of the hypercube, i.e., a cube with 8 spin values that determine the respective values of the indices ~k and Jk~k= 1, 4 in the following way: ...,

~

(b1c4c5c6d2d3d4e)j1

1—8

(ab2b3b4c~c2c3d1),

i2~—~ (b2c2c3c6d1d3d4e)j2~—’(ab1b3b4c~c4c5d2), i3~—’(b3c~c3c5d~d2d4e)j3*-~ (ab,b2b4c2c4c6d3), j48—)

(b4c,c2c4d1d2d3e)j44—8 (ab~b2b3c3c5c6d4).

Using this correspondence we can write the following representation of the four-simplex equation in terms of Boltzmann weights w: ~ w(aIb,b2b3b4ic~c2c3c4c5c6id4d3d2diIe)w’(bi id’~c6c5c4Id2d3d4c~c~c’~ Ib’4b’3b’2eia’)

xw”(c6Ic~b2d4d3Iec2c3b’3b’4d’2Ic’4c’5a’di~

ic’6)

xw”(eia’d~d2d3ic4c2c~b’~b’~b’, ic~c~c~b4 id’4) ~

=~

xw”(b, id’1ac5c4Id2b4b3c’3c’2eid’3d’4b’2c1 ~ ~

ib’~)

ib~b’3b~b; Ia’),

(10)

where the summation is over a center spin e, and the primes attached to the w stand for different values of the parameters. As for the tetrahedron equation, the four-dimplex equation is invariant under the multiplication of w by the quantity

b2

a

~ ~

4~ d

/ b1

Fig. 5. Boltzmann weight w(a I b~b 2b3b4I c1c2c3c4c5c6 I d4d3d2d~I e) for a four-dimensional lattice spin model.

225

Volume 134, number 4

PHYSICS LETTERS A

2 January 1989

~ xy~’(b

4IC1C2C3id3d2d, e)y’(b3IC,C4C5 Id4d2d, e)v’(b2 C2C3C6Id4d3d je)y~(b,1c4c5c61d4d3d2 le),

where y(ajbib,b3IC,C2C3Id) is a function, depending on the cube of spin values (a,b,b7b3c,C,c3d) and which is invariant by the various permutations of the spins values describing geometrically the same cube. In addition, it is independent of the parameters 3. We believe that the representation of the simplex equation in terms of the quantities S, such as in eq. (1) and eqs. (4)—(8), is the most simple one to get an insight into their algebraic structure and also for studying their solutions. Using now the correspondence between the Boltzmann weights w and the quantity S we may obtain an explicit form of Zamolodchikov’s solution of the tetrahedron equations, which has the advantage above previous descriptions [13—171that it is more algebraic in nature. Zamolodchikov’s solution corresponds to the case where the spin values in eq. (3) and consequently the indices in eqs. (6), (8) which are the products of 4 spin values along a face of a cube of the three-dimensional lattice take only two values Furthermore, we impose the symmetry conditions: ±.

(i)

5~/~3 (01, 02,

03) invariant under any permutation of (1, 2, 3).

(ii)

S~,’/~~(0 02, 03) =S~(0 0~,03) =S~’(0 it—0,, ,

,

,

(11 a) (1 Ib)

it03).

Then, Zamolodchikov’s solution is given by the following choice of matrices (S~,’/~) with matrix elements

(SL’J~~

it,, j~,=±:

S±±~~°~ ++ I~0 xj’

~

~

~

~

(0 ~R

(0 ~R,7

R~ 0)’

0

~ y[

~_~j~’2 ~ \. o

0~ y1)’

R0~ 0)’

s-~~R2 (0

R3~

o)’

~

~±_

-±

\.

0

(0 ~R3

0 y~, R2~ 0)’

P -)

in which X,=P1+Q,, Y,=P1—Q, and R~are functions of the angles 0,, 02, 03. Inserting (12) into eq. (6) we essentially get two types of equations for these functions: X~X~X~’R’0”+R~R~R~Xg’ =R~’X~jX~X0 +Xg’R’1’R~R0,

(1 3a)

XOX~RgR~” +R0R’2 Y~’Y~”=Xg’XgR~R2+R~çR~’ V’2 V2.

(1 3b)

Eqs. (1 3a), (1 3b) and similar equations which are obtained either by inverting all spins, leading to X ~ V. or permutating the indices 1, 2, 3, allow for a parametrization in terms of functions of spherical angles, as was shown by Baxter [17]. In fact, let 0~,02, 03 be the interior angles of a spherical triangle and a23, a ~ a,2 the respective opposite sides, then we have P0 = 1, R0=

P

Qo = 10t1 t2 t3, S0

C,C2C3

, R1=

=

tj tk

(cyclic),

(cyclic), Q1=t0t,, 2,c,(cos ~a,)1~2,t,=(tan ~a

(14)

S1

COCJCA

2 and 2a

where S,zr (sin ~a,)l~z 1)” 0=0~+02+03—it,a.=a0—0,+x. Another way of writing (12) in a more algebraic and compact way, is in terms of a threefold tensor product of SL,. Namely, let us define the following tensors: Q0 =.~{1®1 ®l+1 ®rn.®a +a..®1 ®o~+ci®a®1}, 226

Volume

134, number 4

2 January 1989

PHYSICS LETTERS A

Q1 = (1 ®1 ®a~)Q0(1®1 ®a~), Q2=(a~®1®1)Qo(a~®1®1), Qs=(1®cr~®1)Qo(1®o~®1), with the property that Q0+Q1 +Q2+Q3= 1 ®1 ®1, and where a~.(.o~,o~are the usual Pauli matrices. Then Zamolodchikov’s solution (12) can be written as (15)

SQaPa(az®O~®t7z)Qc~Qa+(Cx®ax®ax)QaRa,

where the sum over a=0, 1, 2, 3 is understood. 4. Rewriting Zamolodchikov’s solution of the tetrahedron equation in terms ofthe quantities S as in eq. (12) immediately suggests a possible form ofthe solution of the four-simplex equation, which should have been hard 3denote to guess from the representation of it in terms of Boltzmann weights as in eq. (10). In fact, let S1~/~73 the matrices with components (S1~I~~ )~ ~ with ~a, Ja = ±;we make the following ansatz for a possible solution of the four-simplex equation (7):

(x

~ ++±

00

~

~

0 ~ x4)’

(y~ 0 ~ ü ye)’

~w00 ~ ~

0

~

(x30 z12)’ 0 ~

~÷~± ~

(z34

~

\.

~

~__+

~

w~ 0 )‘

~

w~ 0 )‘

~~

0 ~ y3)’

0 0

\\~

V~2~

0 )‘

~ ~

~

(z24

~~÷(~ ~÷( ~

(x~

0

(z~

0 Y~

\. ~

0 ~ y2)’

~

0 0

V~3~

s-~÷—( ~ ~_-—\~

-~-

w3~ 0 )‘

~V1~0

(x2 0 ~ 0 z13)’ \.

~VI~0

0 )‘

~

w~ 0 )‘ 0a~ a =

0

0

~( ~v2~ 0 w1 0 1,

...,

‘

~

16

6. In addition, we impose

where the functions X,, Y~,W~, ~ and Z,~are depending on six angles the following symmetry properties on S:

4(0~, 02, 03, 04~05, 0

S~/

4(0 6)=S~/,’~,~ 04~05, 1, 02, 03, 06)=S ~ 0,~0~,04, 06, 05) 4(0 03~02, 05, 04~06) =S~/~~ 02, 1,03~7t—04, it05, ~t06).

2~~ ~

(17)

Some typical equations obeyed by S are X0 W’~W’~Z’~”4Z’~ + W0X’~X’~’W’1” V’2”3’ =

W’0”

W’~’Z’1.’2Z’~2 V~2+X’~”X~” Wg W’2Z3~

(l8a)

and ZI2X~V’~Z’~Z’~’~’ + V12 W’~Y’~ W’~”V’~’=

Y’~”Y’2” V’~’~X’2 Y, + W’2” V’~”3X~’ V’~2W1,

(18b)

where the primes in these equations stand for the different prescribed values of six angles O,~out of the ten angles 0!, ~ which obey in addition the constraint ...,

det

1 c6

C6

C5

C3

C9

1

C4

C2

C8

C5

C4

1

C3

C2

C1

c1 1

C10

C9

C8

C7

C10

1

c7

=0,

(19)

227

Volume 134, number 4

PHYSICS LETTERS A

2 January 1989

where c1=cos(it—01). It, of course, remains now to find the parametrization of the functions X1, V,, W,, V,, and Z,, in terms of these (hyper) spherical angles. 5. In this Letter, we have tried to put forward the idea that the triangle equations and the tetrahedron equations are members of a hierarchy of equations, called the simplex equations, and that it is useful to consider these equations as such. An insight into how this hierarchy develops as we increase the dimensionality might lead to a systematic understanding of the algebraic structure of its individual members and hopefully to the development of solution methods. The recent interest in the theory of quantum groups [11,12], which is until now linked solely to the quantum Yang—Baxter equations, might also gain from a broadening of horizon by taking other members of the hierarchy of simplex equations into consideration. Another field of application is the study of multidimensional integrability. It is well known that the classical and quantum Yang—Baxter equation is one of the fundamental building blocks in the theory of two-dimensional integrability [1—9].The development of a genuinely multidimensional notion of integrability is one of the major problems in the theory of integrable systems. In a previous publication [20] we have addressed ourselves more explicitly to these problems. The main idea is that the notion of integrability is intimately linked to the question of the possibility of posing an overdetermined, but solvable, system of equations which does not trivially reduce to a dimensionally smaller system. The simplex equations can be of help in the search of such systems, because these equations themselves can be shown to emerge as the consistency conditions of a related system of equations, namely those involving the permutations of transfer matrices. In a future publication [21], we shall treat these results in more detail and show that by analogy with the two-dimensional case, this leads in a natural way to a notion of classical integrability and generalized Lax equations on multidimensional lattices.

References [I] L.D. Faddeev, in: Recent advances in field theory and statistical mechanics. eds. R. Stora andJ.B. Zuber (North-Holland, Amsterdam, 1983). [2] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press. London, 1982). [3] A.B. Zamolodchikov, Soy. Sci. Rev. A 2 (1980) 2. [4]C.N. Yang, Phys. Rev. Lett.29 (1967) 1312: R.J. Baxter, Ann. Phys. 70 (1972)193. [5]L.D. Faddeev, Soy. Sci. Rev. C 1(1980)107. [6]A.A. Belavin, NucI. Phys. B 180 (1981) 189. [7] A.A. Belavin and V.G. Drinfeld, Funct. Anal. AppI. 16 (1982) 1: 17 (1983) 69. [8] F. Ogievetsky and P. Wiegmann, Phys. Lett. B 168(1986) 360; N.Yu. Reshetikhin and P. Wiegmann, Phys. Lett. B 189 (1987) 125: V.V. Bazhanov, Phys. Lett. B 159 (1985) 321. [9]M.A. Semenov-Tian-Shanskii, Funct. Anal. AppI. 17 (1983)19. [10] V.V. Bazhanov and Yu.G. Stroganov, NucI. Phys. B 205 (1982) 505. [II] V.G. Drinfeld, Quantum groups, Berkeley lectures (1986). [12] M. Jimbo, Commun. Math. Phys. 102 (1986) 537; Lett. Math. Phys. 10 (1985) 63; 11(1986) 247. [13] A.B. Zamolodchikov, Soy. Phys. JETP 52 (1980) 325. [14] A.B. Zamolodchikov, Commun. Math. Phys. 79 (1981) 489. [IS] V.V. Bazhanov and Yu.G. Stroganov, NucI. Phys. B 230 IFS 10] (1984) 435. [16] Ml. Jaeckel and J.M. Maillard, J. Phys. A 15 (1982)1309. [171 R.J. Baxter, Commun. Math. Phys. 88 (1983) 185. [18] V.V. Bazhanov and Yu.G. Stroganov. Theor. Math. Phys. 52 (1982) 685. [19]1. Todhunter and J.G. Leathem, Spherical trigonometry (MacMillan, London, 1949). [20] F.W. Nijhoff and J.M. Maillet, in: Proc. IV mt. Workshop on Nonlinear evolution equations and dynamical systems, ed. J. Leon (World Scientific, Singapore), to be published. [21] J.M. Maillet and F.W. Nijhoff. to be published.

228