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Dilogarithm Identities in Conformal Field Theory and Group Homology∗)

arXiv:hep-th/9303111v1 19 Mar 1993hep-th/9303111Dilogarithm Identities in Conformal Field Theory and Group Homology∗)Johan L. DupontMatematisk InstitutAarhus UniversitetNy Munkgade, DK-8000 CAarhus, DenmarkandChih-Han SahDepartment of MathematicsSUNY at Stony BrookNew York 11794-3651, USAAbstract. Recently, Rogers’ dilogarithm identities have attracted much attentionin the setting of conformal field theory as well as lattice model calculations.

One of theconnecting threads is an identity of Richmond-Szekeres that appeared in the computationof central charges in conformal field theory. We show that the Richmond-Szekeres identityand its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator ofthe third integral homology of a finite cyclic subgroup sitting inside the projective speciallinear group of all 2×2 real matrices viewed as a discrete group.

This connection allows usto clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston’s program on hyperbolic3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but ismore appropriately related to the group manifold of the universal covering group of theprojective special linear group of all 2×2 real matrices viewed as a topological group.

Thisalso resolves the weaker version of the conjecture as formulated by Kirillov. We end withthe summary of a number of open conjectures on the mathematical side.∗)This work was partially supported by grants from Statens Naturvidenskabelige Forskn-ingsraad, and the Paul and Gabriella Rosenbaum Foundation.1

§0. Introduction.Very recently, much has been written about the Rogers’ dilogarithm identities and itsrole in conformal field theory, see [BR], [KKMM], [FS], [K], [KR], [KP], [KN], [KNS], [NRT].For an excellent general survey for mathematicians concerning hypergeometric functions,algebraic K-theory, algebraic geometry and conformal field theory, see [V] and its extensivesection of references.

For a recent review from the physics side, see [DKKMM]. In thepresent work, we limit our attention to the special case of dilogarithm identities.

In spirit,it fits into the program surveyed by Varchenko [V]. Some, though not all, of the relevantcalculations have been carried out on both sides of the fence.

Conjectures abound evenin this case.Most of our task consists of pulling together items that are scattered inthe literature invarious forms. The new ingredient is to give a direct interpretation interms of group homology to account for the Richmond-Szekeres identity, see [RS], and itsextension by Kirillov-Reshetikhin, see [KR, II, (2.33) and Appendix 2].

What we show isthat the basic identities are the ones found by Rogers in [R]. Rogers’ dilogarithm functionthen leads to a real valued cohomology class defined on the third integral homology ofthe universal covering group of PSL(2, R), viewed as a discrete group.

The Richmond-Szekeres identities, see [RS], and the Kirillov-Reshetikhin identities, see [KR II, (2.33) andAppendix 2], are the results of restricting the evaluation of this cohomology class (thereal part of the second Cheeger-Chern-Simons class) to the inverse image of a suitablehomology class that covered a generator of suitable finite cyclic subgroup. This will thenprovide partial clarifications of some of the assertions and conjectures made by Nahm-Recknagel-Terhoven [NRT] related to algebraic K-theory [Bl] and Thurston’s program onhyperbolic 3-manifolds [Th2].

Specifically, we show that it is more appropriately relatedto the group manifold underlying the universal covering group of PSL(2, R).§1. Rogers’ Dilogarithm.Rogers’ dilogarithm (also called Rogers’ L-function) was defined in [R] :(1.1)L(x) = −12{Z x0log x1 −x dx +Z x0log(1 −x)xdx}.=Xn>0xnn2 + (log x) · (log(1 −x))2, 0 < x < 1.L(x)is real analytic, strictly increasing andlimx→1 L(x) = π2/6.Rogers showed that L satisfied the following two basic identities:(1.2)L(x) + L(1 −x) = π2/6, 0 < x < 1.

(1.3)L(x) + L(y) = L(xy) + L(x −xy1 −xy ) + L(y −xy1 −xy ), 0 < x, y < 1.2

If we use (1.2), take s1 = (1 −x)/(1 −xy) and s2 = y(1 −x)/(1 −xy) so that y = s2/s1and x = (1 −s1)/(1 −s2) with 0 < s2 < s1 < 1, then (1.3) is seen to be equivalent to:(1.4)L(s1) −L(s2) + L(s2s1) −L(1 −s−111 −s−12) + L(1 −s11 −s2) = π26 , 0 < s2 < s1 < 1.If we set ri = s−1i , then (1.4) can be rewritten in the form:(1.5)L(r1) −L(r2) + L(r2r1) −L(r2 −1r1 −1) + L(1 −r−121 −r−11) = π26 , 1 < r1 < r2.Motivated by [DS1], Rogers’ dilogarithm was shifted in [PS] to:(1.6)LP S(x) = L(x) −π26 = −L(1 −x), 0 < x < 1.If we replace L by LP S throughout, then (1.2) and (1.4) become:(1.7)LP S(x) + LP S(1 −x) = −pi26 ,(1.8)LP S(x) −LP S(y) + LP S(yx) −LP S(1 −x−11 −y−1 ) + LP S(1 −x1 −y ) = 0, 0 < y < x < 1.A huge number of identities have been found in connection with Rogers’ dilogarithm.The situation is somewhat similar, and is often, related to trigonometry, where the basicidentities are the two additional formulae for the sine and cosine function, which are justthe coordinate description of the group law for SO(2) or U(1).This analogy can bemade more precise. Namely, U(1), more appropriately, GL(1, C) ∼= C× is just the firstCheeger-Chern-Simons characteristic class in disguise.

This is wellknown and tends to beoverlooked.Richmond-Szekeres [RS] obtained the following identity (in a slightly different form)from evaluating the coefficients of certain Rogers-Ramanujan partition identities as gener-alized by Andrews-Gordon:(1.9)X1≤i≤rL(di) = π26 ·2r2r + 3, dj =sin2θsin2(j + 1)θ, θ =π2r + 3.This has been extended by Kirillov-Reshetikhin [KR] to:(1.10)X1≤j≤n−2L(dj) = π26 · 3(n −2)n, dj =sin2θsin2(j + 1)θ, θ = πn.3

Apparently, identity (1.9) arose in the study of low-temperature asymptotics of entropy inthe RSOS-models, see [ABF], [BR], and [KP] while (1.10) arose in the calculation of mag-netic susceptibility in the XXZ model at small magnetic field, see [KR]. They are connectedto conformal theory in terms of the identification of the right hand sides as the effectivecentral charges of the non-unitary Virasoro minimal model and with the level ℓA(1)1WZWmodel respectively, see [BPZ], [Z2], [K], [KN], [KNS], [DKKMM], [KKMM] [Te],· · ·.

Ourgoal is to show that these identities can be understood in terms of the evaluation of aCheeger-Chern-Simons characteristic class on a generator of the third integral homologyof a finite cyclic group of order 2r + 3 and n respectively.§2. Geometry and algebra of volume calculations.In any sort of volume computation, the volume is additive with respect to division ofthe domain into a finite number of admissible pieces.

Depending on the coordinates usedto describe the domain the volume function must then satisfy some sort of “functionalequation”. This is the geometric content behind the Rogers’ dilogarithm identity.

Thegeometric aspect was described in [D1] while some of the relevant algebraic manipulationswere carried out in [PS] (up to some sign factors that only became important in [D1]). Toget a precise description, it is necessary to examine [DS1], [DPS] and [Sa3].

These usedalgebraic K-theory. We review the ideas and results but omit the technical details.To begin the review, we recall the definition of some commutative groups (called the“scissors congruence groups”, cf.

[DPS]. Let F denote a division ring (we are only interestedin three classical cases: R = real number, C = complex numbers, H = quaternions.

).The abelian group PF is generated by symbols: [x], x in F, x ̸= 0, 1 and satisfies thefollowing identity for x ̸= y :(2.1a)[xyx−1] = [y], (this is automatic for fields)(2.1b)[x] −[y] + [x−1y] −[(x −1)−1(y −1)] + [(x−1 −1)−1(y−1 −1)] = 0This group was studied in [DS1] for the case of F = C. It is closely related to, but notidentical to, the Bloch group that was studied in [Bl]. A second abelian group P(F) isdefined by using generating symbols [[x]], x in F −{0, 1}, with defining relations:(2.2)same as (2.1) with [[z]] in place of [z].

(2.3)[[x]] + [[x−1]] = 0. (2.4)[[x]] + [[1 −x]] = cons(F) (a constant depending on F).4

The following result can be found in [DPS]:(2.5)0 →F ×/(F ×)2 →PF →P(F) →0 is exact for F = R, C, H.The first map in (2.5) is defined by sending x in F −{0, 1} to [x] + [x−1]. The second mapthen sends [x] to [[x]].

In particular, when F = C, we may set [x] = 0 for x = ∞, 0, 1 andremove the restriction x ̸= y in (2.1) by adopting the convention: meaningless symbols aretaken to be zero, see [DS1]. For the division ring H, we observe that every element of His conjugate to an element of C, thus P(H) is a quotient of P(C).The geometric content of (2.1b) is best seen by thinking in terms of a Euclideanpicture.

Suppose we have 5 points in Euclidean 3-space so that p1, p2, p3 form a horizontaltriangle while p0, p4 are respectively above and below the triangle. The convex closure isdivided by the triangle into two tetrahedra and also divided into three tetrahedra by theline joining p0 and p4, see (Fig.1).(Fig.

1)Thus, if any function of a tetrahedron is additive with respect to finite decompositions, itwould follow from (Fig. 1) that there should be a 5 term identity to be satisfied by sucha function.We examine the special case of F = C. Here PC = P(C) is known to be a Q-vectorspace of continuum dimension, see [DS1].

It is best to consider the (-1)-eigenspace P(C)−of P(C) under the action of complex conjugation. It is classically known that the projectiveline P 1(C) can be viewed as the boundary of the hyperbolic 3-space.

An ordered set of 4n on-coplanar points on P 1(C) (in terms of the extended hyperbolic 3-space) determinesa unique ideal (or totally asymptotic) tetrahedron of finite invariant volume (by usingthe constant negative curvature of hyperbolic 3-space). Since the orientation preservingisometry group is PSL(2, C), we can take 3 of the 4 vertices to be ∞, 0, 1, the 4-th pointis then defined to be the “cross-ratio” of the 4 distinct points (which may determine adegenerate tetrahedron when they are coplanar).

(2.1b) is the result of taking 5 distinctpoints: ∞, 0, 1, x and y as pictured in (Fig. 1).

For a general division ring F, PF merelyformalizes the discussion. The difference between P(F) and PF amounts to permittingsome of the vertices to be duplicated.

(2.3) and (2.4) express the fact that oriented volumechanges sign when the exchange of two vertices reverses the orientation. The equalityPC = P(C) simply means that the introduction of degenerate tetrahedra with duplicatedvertices does not make any difference (it does make a difference in the case of F = R).With (2.3) in place, it is now evident that (1.7) and (1.8) are directly related to (2.4) and(2.1b).

The problem is that our explanation so far is based on F = C while LP S dealtwith F = R. This will be reviewed in the next section. It should be noted that the volumecalculation makes perfectly good sense for tetrahedra with vertices in the finite part of thehyperbolic 3-space.

It is known that any such tetrahedron can be written in many differentways as a sum and difference of ideal tetrahedra, see [DS1]. A general volume formula fora tetrahedron is quite complicated.

However, the volume of an ideal tetrahedron is quitesimple. It is given by the imaginary part of the complexified Rogers’ dilogarithm function(up to normalization) evaluated at the cross-ratio.5

We end the present section by giving the structures and inter-relations of the groupsP(F), F = R, C, H, with R ⊂C ⊂H. The details can be found in [DPS] and [Sa3].

(2.6)P(C) = P(C)+ ⊕P(C)−.This is a Q-vector space direct sum in terms of its ±1 eigenspaces under the action ofcomplex conjugation. Both summands have continuum dimension.

(2.7)0 →Q/Z →P(R) →P(C)+ →Λ2Z(R/Z) →0 is exact.P(R) is the direct sum of Q/Z and a Q-vector space of continuum dimension. (2.8)P(C)+ →P(H) →0 is exact and P(H) ∼= Λ2Z(R+).The group P(C)−is the “scissors congruence group” in hyperbolic 3-space, see [DS1].The kernel of the homomorphism in (2.8) is related to the “scissors congruence groupmodulo decomposables” in spherical 3-space and is conjecturally equal to it, see [DPS].These results depend on algebraic K-theory and use, in particular, a special case of Suslin’scelebrated solution of the conjecture of Lichtenbaum-Quillen, see [Su2].§3.

Rogers’ Dilogarithm and Characteristic Classes.As reviewed in preceding sections, there is a formal resemblance between the Rogers’dilogarithm identities and volume calculation in hyperbolic 3-space. In fact, the underlyingspace is quite different.

The explanations were carried out in [D1]. For the convenienceof the reader, we review the results.

The relevant characteristic class is that of Cheeger-Chern-Simons characteristic class ˆc2 which lies in the third cohomology of SL(2, C) viewedas a discrete group and where the coefficients lie in C/Z. In general, one has ˆcn which liesin the (2n-1)-th cohomology of GL(m, C), m ≥n, viewed as a discrete group, where thecoefficients lie in C/Z.

The standard mathematical notation for this cohomology groupis H2n−1(BGL(m, C)δ, C/Z), this is the group cohomology where GL(m, C) is given thediscrete topology (the superscript δ emphasizes this fact). ˆc1 is nothing more than thelogarithm of the determinant map with kernel SL(m, C).

With the replacement of GLby SL, ˆc1 becomes 0. The replacement of GL(m, C) by GL(n, C) arises from homologicalstability theorems, see [Su1] (a simplified version can be found in [Sa2]).

In general, ˆcnis conjectured to be connected to the n-polylogarithm, see [D2 and D3]. Although weare only interested in ˆc2, we will state the results for general n. The construction arisesby starting with the Chern form cn (a 2n-form) which represents an integral cohomologyclass of the classifying space BGL(n, C) where GL(n, C) is now given the usual topology.Since we have replaced the usual topology by the discrete topology (this amounts to “zerocurvature condition”), it follows from Chern-Weil theory (where closed forms are viewedas complex cohomology classes) that cn can be written as the differential of a (2n-1)-form,(for n = 2 this is the Chern-Simons form that appears ubiquitously in physics).

When6

the coefficients are taken in C/Z, this (2n-1)-form is closed and leads to the class ˆcn inH2n−1(BGL(n, C)δ, C/Z) through the exact sequence:(3.1)0 →Z →C →C/Z →0.We now concentrate on n = 2. If we take the coefficients to be C/Z, then the characteristicclass ˆc2 has a purely imaginary part and a real part.

The purely imaginary part has valuesin R and is related to volume calculation in hyperbolic 3 space while the real part lies inR/Z and is related to volume calculations in spherical 3-space. These volume calculationsare classically known to involve the dilogarithm function.

See [C] for the details related tothe work of Lobatchevskii and Schl¨afli respectively. The integer ambiguity in the sphericalcase arises from the fact that a large tetrahedron can be viewed as a small tetrahedron onthe “back side” of the sphere with a reversed orientation.

Thus its volume is only uniqueup to an integer multiple of the total volume of the spherical 3-space.For the Rogers’ dilogarithm, the space is actually the group-space eS of the universalcovering groupgPSL(2, R). The task of defining a tetrahedron and calculating its volumebecomes more delicate.

If we select a base point p in eS, then any point can be writtenas g(p) for a uniquely determined group element g ofgPSL(2, R). We first define a leftinvariant “geodesic” in the group that joins 1 to g (this definition is asymmetric).

Thiscan be accomplished by exponentiating a Cartan decomposition of the Lie algebra ofgPSL(2, R).In essence, we coordinatizegPSL(2, R) by R × H2 where H2 denotes thehyperbolic plane. Inductively, we can then define a “geodesic cone” for any ordered setof n + 1 points, n ≥0, see (Fig.

2). This is similar to [GM] where Rogers’ dilogarithmappeared in terms of volumes in Grassmann manifolds of 2-planes in R4.

Our interpretationis dual to [GM] since the transpose of a 4 × 2 matrix is a 2 × 4 matrix. Namely, for theordered set (p0, ···, pn), the cone is the collection of all points on the “geodesics” from p0 tothe “geodesic cone” inductively defined for (p1, ···, pn).

For the definition of volume (n = 3),the next step is to show that it is enough to consider the case where the 4 vertices areclose to each other. In fact, in terms of the Cartan coordinates of the group elements, onemay assume that the θ-coordinates are strictly positive and small (this involves changingby a boundary which causes no problem because the volume is obtained by evaluating a3-cocycle on the chain, in essence we invoke Stoke’s Theorem).

We next form the boundaryR × ∂H2 where ∂H2 = P 1(R) is the projective line over the real numbers (which canbe identified with {−∞} ∪R by using the slopes in the right half plane as in [PS]). Atthis point, we begin to mimic the hyperbolic 3-space and move p continuously towards{0} × P 1(R) (this amounts to right multiplication).

When p lands on {0} × P 1(R), sowill all four vertices so that we have the analog of an ideal hyperbolic tetrahedron. Thevolume (up to a normalizing factor) is just the value of the Rogers’ dilogarithm evaluatedon the “cross ratio” of the ordered set of vertices viewed as points of P 1(R) (adjustmentsare needed for the degenerate cases).

The situation now resembles the case of spherical3-space. Namely, the final volume will involve an integer (after normalization) ambiguitywhich depends on the path of p.We ignore the question of representing the originaltetrahedron as sums and differences of these “ideal tetrahedra” since our concern is tointerpret the value of the Rogers’ dilogarithm as a volume.7

(Fig. 2)We summarize this discussion in the form, cf.

[D1, Th. 1.11]:Theorem 3.2.

The restriction of the second Cheeger-Chern-Simons characteristicclass ˆc2 to PSL(2, R) can be lifted to a real cohomology class on the universal coveringgroupgPSL(2, R) and is then given by the Rogers’ dilogarithm (more precisely, by LP Sthrough L).A more detailed discussion will be given in the following sections.§4. Homology of Abstract Groups.The basic reference is [Br].Let G be an abstract group.We consider the non-homogeneous formulation of the integral homology of G with integer coefficients Z. Thej −th chain group Cj(G) is the free abelian group generated by all j-tuples [g1|···|gj] withgi ranging over G, j ≥1.

C0(G) is the infinite cyclic group generated by [·]. Such a j-cellshould be identified with each of the formal j-simplices (g0, g0g1, g0g1g2, · · ·, g0g1 · · · gj)as g0 ranges over G. The boundary homomorphism: ∂j : Cj(G) →Cj−1(G) is definedby translating the usual boundary of the formal j-simplex.

For example, ∂3[g1|g2|g3] =[g2|g3]−[g1g2|g3]+[g1|g2g3]−[g1|g2]. The j−th integral homology group of G, Hj(G, Z), orsimply Hj(G), is defined to be ker ∂j/im ∂j+1.

H0(G) is just Z while H1(G) is canonicallythe commutator quotient group of G with the class of [g] mapped onto the coset of g in thecommutator quotient group. We note that homology groups can also be defined for anyG-module M (e.g.

any vector space on which G acts by means of linear transformations).This generalization is often needed for computational purposes and requires more care.In general, the procedure described in the preceding paragraph is not very revealing.Somewhat more revealing is to use the action of G of a suitably selected set X. Typically,we end up describing the homology groups through a spectral sequence that reveals acomposition series. If X is the underlying set of G under the left multiplication action andthe spectral sequence “degenerates”.

In the case ofgPSL(2, R), we can take the space X tobe that of P 1(R) = {0}×P 1(R) which is viewed as part of the boundary of the group spaceeS. The spectral sequence is the algebraic procedure to keep track of the geometry.

If p isa base point in the group space eS, the 3-cell [g1|g2|g3] is an abstraction of the “geodesic”3-simplex (p, g1(p), g1g2(p), g1g2g3(p)) in the group space eS. If p is moved to ∞= R(10)in P 1(R), then we have an “ideal” 3-simplex.

Although the action ofgPSL(2, R) on eS isfaithful, its action on P 1(R) is not. In fact, it factors through PSL(2, R) by way of thefollowing exact sequence:(4.1)0 →Z · c →gPSL(2, R) →PSL(2, R) →1.The results in [PS] and [DPS] can be recast and summed up by the following commutativediagram of maps where the rows and columns are exact:8

(4.2)00yyZ∼=−→Z · cyy0−→H3( ˜S, Z)−→PS(R)yyη0−→Z2−→H3(S, Z)σ−→P(R)d2−→Λ2(R+)yy00In (4.2), we abuse the notation and set S = PSL(2, R). PS(R) is the abelian groupgenerated by all cross-ratio symbols {r} = (∞, 0, 1, r), r ∈R× ∪{∞}, and subjected tothe defining relations, cf.

(1.5), (1.8):(4.3){r1} −{r2} + {r2r1} −{r2 −1r1 −1} + {1 −r−121 −r−11} = 0, 1 < r1 < r2,(4.4){r} + {r−1} = 0, r > 1,(4.5){∞} = 2{2} = −2{1/2} and {1} = 0,(4.6){−r} = {1 + r−1} + {∞}, r > 0.These involve slight modifications of the results in [PS]. The group PS(R) is isomorphicto the group H3(W/S) of [PS] if we simply view (4.4) through (4.6) as the definition of{s} for 0 < s < 1, s = ∞or 1 and s < 0 respectively.

More precisely, we take as j-cellsthe ordered (j + 1)-tuples of elements of the universal covering group R of PSO(2, R) sothat the convex closure of these points cover an interval of length less than π (length ofPSO(2, R)). Moreover, we also enlarge the action to the ”universal covering group” ofPGL(2, R).

We note that in general, the universal covering group of a disconnected Liegroup is not well defined. In the present case, it is well defined and happens to be a semi-direct product of the universal covering group of PSL(2, R) by an element of order 2 thatinverts its infinite cyclic center.

The later results in [DPS] and [Sa3] showed that H3(W/S)is a Q-vector space. In [PS], it was shown that H3(W/S)/Z ·48{2} ⊃H3(SL(2, R), Z) and9

H3(W/S)/Z · 12{2} ∼= PR ⊃H3(PSL(2, R), Z). The first arose by showing that a certainelement c(−1, −1) = 8c is mapped onto ±48{2} (with a little care, the image is −48{2}).The second involves a direct argument.

We note that H3(SL(2, R), Z) maps surjectivelyto H3(PSL(2, R), Z) with kernel Z4. This accounts for various Z2’s.

(4.2) now resultsfrom (2.5) with c mapped by η onto -6[[2]] in P(R), namely, P(R) ∼= H3(W/S)/Z · 6{2}.From section 1, we have a surjective homomorphism:(4.7)LP S : PS(R) →R, where LP S({s}) = L(s) −π26 = −L(1 −s), 0 < s ≤1.In particular, LP S({1/2}) = −π2/12 and LP S({r}) = L(1 −r−1), for 1 ≤r ≤∞.This leads to surjective homomorphisms:(4.8)LP SR: PR →R mod Z · (π2)LP S(R) : P(R) →R mod Z · (π22 ).Using (2.5) and (2.7) we then have:(4.9)LP SR: H3(PSL(2, R), Z) →R mod Z · (π2)LP S(R) : H3(PSL(2, R), Z) →R mod Z · (π22 ).LP SRis injective on torsion elements and LP S(R) maps an element of order m to one oforder m or m/2 according to m is odd or even.Remarks 4.10. (i) In using the extension to PGL(2, R) and its universal coveringgroup, [[r]] is the usual cross-ratio symbol associated to (∞, 0, 1, r) for r in R −{0, 1},see [PS].

Thus, {r} is mapped to [[r]]. (ii) H3(PSL(2, R), Z) is conjectured to be equalto H3(PSL(2, Ralg), Z) where Ralg denote the field of all real algebraic numbers.

Thisfollows from a similar conjecture for C in place of R. Thus, the two maps in (4.9) are notexpected to be surjective. So far, all the non-trivial elements in the image are obtainedby using algebraic numbers.

(iii) It is both convenient and essential to consider the groupH3(PSL(2, C), Z) or H3(SL(2, C), Z). Namely, C admits a huge group of automorphismswhile R has only the trivial automorphism.

While we do not know the injectivity of ˆc2 :H3(SL(2, C), Z) →C/Z, we do know that a non-zero element of H3(SL(2, Calg), Z) can bedetected by a composition ˆc2 ◦τ for a suitable automorphism τ of C. This is a theorem ofBorel, see [Bo]. Except when τ is the identity or the complex conjugation map, the imageτ(R) is everywhere dense in C. It is the use of the hyperbolic volume interpretation thatultimately leads to conclusion that H3(SL(2, C), Z) and H3(SL(2, R), Z) both contain aQ-vector subspace of infinite dimension.10

§5. Connection with Richmond-Szekeres and Kirillov-Reshetikhin Identi-ties.Granting the assertions in the preceding reviews, we can now describe the relationof the above discussions with the Richmond-Szekeres identity (1.9) and the extensionby Kirillov-Reshetikhin (1.10).As described in [PS], if G is a cyclic group of orderm with generator g, then the following chain is a (2j −1)-cycle and its class generatesH2j−1(G, Z) ∼= Zm, j > 0 :(5.1)c(j)m =X[g|x1|g| · · · |xj−1|g], xi range over G independently.More generally, P[gi(1)|x1|···|xj−1|gi(j)] is homologous to i(1)···i(j)·c(j)m .

The superscriptis used to remind us that the class behaves as a j −th power character on the cyclic groups.We now map G into S = PSL(2, R) by sending g to the following matrix:cos θ−sin θsin θcos θ, θ = π/m.The map σ in (4.2) sending H3(S, Z) into P(R) is obtained by sending the 3-cell [g1|g2|g3]to the cross-ratio symbol of (∞, g1(∞), g1g2(∞), g1g2g3(∞)). Here ∞= R(01), r = R(1r),more generally, y/x = R(xy), x ≥0 and PGL(2, R) acts on these lines through matrixmultiplication.

However, as discussed in section 3, in the evaluation of volume, chains maybe modified by boundaries. For the special form of the 3-cells that appears in c(2)m , thisis not a serious problem.

In any event, we have a canonical identification of the torsionsubgroup:(5.2)tor(H3(PSL(2, R), Z)) ∼= Qπ/Zπ, the rational rotations in PSO(2, R).We now consider cm = c(2)mand note that σ(cm) is of order m or m/2 in P(R)according to m is odd or even. Thus, we will restrict ourselves to m > 2.

[g|gj|g] corre-sponds to (∞, g(∞), gj+1(∞), gj+2(∞)). Except when j = 0, m −2, m −1, this is just[[Q2j/Qj−1Qj+1]], where Qj = Qj(θ) = sin(j + 1)θ/ sinθ, θ = π/m.When j = 0.

[g|1|g] is 0 under the usual normalization. The corresponding formal3-cell has two identical adjacent vertices and represents 0.When j = m −2 > 0.

We have the formal 3-cell (∞, −1, 1, ∞) independent of m.It is the same as (∞, 0, 1, ∞) and is assigned the cross ratio symbol {∞}. By taking theboundary of (∞, 0, 1, 2, ∞), {∞} is homologous to 2{2} = −2{1/2} as in (4.5).When j = m −1 ≥2.

We have the formal 3-cell (∞, 0, ∞, 0) independent of m. It isthe boundary of (∞, 0, ∞, 0, 1). Thus, we set it to 0.To see how the preceding assignments work, we consider the cases: m = 3 and 4.11

When m = 3, σ(c3) = [[∞]] and LP S({∞}) = π2/6. This represents an element oforder 3 in R mod Z · (π2/2).When m = 4, σ(c4) = [[∞]] + [[2]] and LP S({∞}) + LP S({2}) = π2/6 + π2/12 = π2/4.

This represents an element of order 2 in R mod Z(π2/2).We now go to the general case. For m > 2, we have:(5.3)σ(cm) = [[∞]] +X1≤j≤m−3[[Q2jQj−1Qj+1]],Qj = Qj(θ) = sin(j + 1)θsin θ, 1 ≤j ≤m −3, θ = π/m.The above calculation is purely formal and the only reason that θ is chosen to be π/marises from the fact that the expression in (5.1) represents the image of an element of orderm or m/2 in H3(S, Z).

The expression for Qj is well known in terms of representationtheory. Namely, consider the irreducible representations of SL(2, C) of finite dimension.It is well known that there is exactly one in each dimension n + 1 ≥1.

It is realized in then−th symmetric powers of the fundamental representation of SL(2, C) on C2. This is thespin n/2 representation in physics.

Evidently, the matrix diag(z, z−1) is represented bydiag(zn, zn−2, ···, z−n). Qj(θ) is just the trace of diag(z, z−1) in the spin j/2 representationwhere z = exp(ιθ).The following lemma results from looking at the character of therepresentation theory of SL(2, C) :Lemma 5.4.

Let S(i) denote the i−th symmetric tensor representation of SL(2, C),i ≥0. Let j, p, q > 0.

Then S(p + j −1) ⊗S(q + j −1) ∼= S(p −1) ⊗S(q −1) ⊕S(p + q +j −1) ⊗S(j −1) holds. (Note: the representation S(i) has degree i + 1.

)For the proof, it is enough to looke at the trace of the matrix diag(z, z−1). If weconsider the special case of z = exp(ιθ), p = q = 1, we get Q2i = Qi−1Qi+1 + 1.

SinceQ2j = 1/dj by definition, we have:(5.5)σ(cm) = [[∞]] +X1≤j≤m−3[[(1 −dj)−1]].The right hand side of (5.5) is [[∞]] + 2 P1≤j≤k−1[[(1 −dj)−1]] for m = 2k + 1 and is[[∞]] + [[(1 −dk)−1]] + 2 P1≤j≤k−1[[1 −dj)−1]] for m = 2k + 2.We next have the following elementary result:Lemma 5.6. Let F : Q →Q be an additive homomorphism so that F(Z) ⊂Z andso that F : Q/Z ∼= Q/Z.

Then F = ±Id. If F(1/3) ≡−1/3 mod Z, then F = −Id.Proof.

Recall that F is just multiplication by a rational number because division byintegers is unique. The two restrictions on F force F to be multiplication by ±1.

The finalrestriction forces F to be minus identity.12

We can now apply Lemma 5.6 to obtain the following:Theorem 5.7. For m ≥3, LP S(R)(σ(cm)) ≡−π2/m mod Z · (π2/2).

In general, wehave the congruence Kirillov-Reshetikhin identity:X1≤j≤m−2L(sin2 πmsin2 (j+1)πm) = π26 · 3(m −2)m≡−π2mmod Z · (π22 ).In particular, we have the congruence Richmond-Szekeres identity for m = 2k+1:X1≤j≤k−1L(sin2π2k+1sin2 (j+1)π2k+1) ≡π2(2k −2)6(2k + 1) mod Z · (π24 ).Proof. We already know that Q{2} is the inverse image of the torsion subgroup ofP(R) in PS(R).Moreover, LP S : Q{2} →Qπ2 is an isomorphism that carries 6{2}onto π2/2.

The torsion subgroup of H3(PSL(2, R), Z) is identified with Qπ/Zπ where theelements cm arising from rotation by π/m in PSO(2, R) and σ(cm) has order m or m/2in P(R) according to m is odd or even. Since cm corresponds to π/m in Qπ/Zπ, Lemma5.6 shows that LP S(R)(σ(cm)) must be ± π2/m in Qπ2 mod Z · (π2/2).

When m = 3, wesaw that the image is π2/6 = π2/2 −π2/3. It follows that LP S(R)(σ(cm)) = −π2/m modZ · (π2/2).

This is just the general congruence identity. The more precise equality wasproved in [KR-II, (2.33) and Appendix 2.] by an analytic argument.Let m = 2k + 1.

By (4.4), (4.7), and sin(π −φ) = sin φ, LP S(R)(σ(cm)) = π2/6 +2 Pj L(dj), 1 ≤j ≤k −1. Next π2/2 −π2/(2k + 1) = (2k −1)π2/2(2k + 1) = π2/6 +(4k −4)π2/6(2k + 1).

The congruence immediately follows.If we use the fact LP S is injective on Q{2}, we have the corollary:Corollary 5.8. In PS(R), 4(m −3){2} = m · P1≤j≤m−3{(1 −dj)−1}, m > 2.

Equiv-alently, 6(m −2){2} = m · P1≤j≤m−2{(1 −dm)−1}, m > 2.We may obtain more congruence identities by computing the image in PS(R) of arepresentative for the class p · q · c(2)m , 0 < p, q < m. Namely, we take i(1) = p and i(2) = qin the extension of (5.1). There are at most 4 exceptional symbols to consider according toj mod m. When j = 0, we always have 0.

We therefore assume 0 < j < m. If j = −p or−q, depending on p = q or p ̸= q, we end up with either 0 or −{∞}. Finally, if j ≡−p −qmod m (this forces p+q ̸= m), then the symbol is {∞} as before.

The general congreunceidentity then takes on the folloing form:Theorem 5.9.Let LP S denote the shifted Rogers’ dilogarithm as in (4.7).Letm > 2, 0 < p, q < m. Letδj(p, q; m) = sin(p + j)θ · sin(q + j)θsinjθ · sin(p + q + j)θ , 0 < j < m, θ = πm.We then have the following congruence with the understanding that: the index j isto skip over the cases, −p, −q, −p −q mod m; and δa,b is the Kronecker delta mod m:13

X1≤j≤m−1LP S({δj(p, q; m)}) ≡−pqπ2m+ (δp,−q −δp,q) · π26 mod Z · (π22 ).We note that the number δj(p, q; m) lies in R −0, 1 after we exclude the exceptional cases.It is easy to see that sin(x+p)θsinxis strictly decreasing in x. Thus, δj(p, q; m) can be negative.In general, it is necessary to use the defining properties (4.3)-(4.7) of LP S in order toexpress the congruence in terms of L. If we use Lemma 5.4, it is easy to see that:δj(p, q; m)−1 = 1 −sinpθ · sinqθsin(p + j)θ · sin(q + j)θ.In the case of p = q = 1, the right hand side is strictly between 0 and 1 so that (4.4) and(4.7) recover the congruence in Theorem 5.7.

However, for general p, q, we do not have agood way to determine the ”integral ambiguity” implicit in lifting the congruence to anidentity. This resembles the classical treatment of Gauss’ quadratic reciprocity theorem innumber theory via the use of Gauss’ sums.Remark 5.10.

In Theorem 5.7, the rational numbers: (2k −2)/(2k + 1) are the“so-called” effective central charge of the (2, 2k + 1) non-unitary Virasoro minimal model.Similarly, the rational number 3ℓ/(ℓ+ 2) is the central charge of the level ℓA(1)1WZWmodel. Both are models in conformal field theory.

In our present setting, they are identifiedas specific values of the evaluation of the Cheeger-Chern-Simons characteristic class on thethird integral homology of the universal covering groupgPSL(2, R) of PSL(2, R) (viewed asa discrete group). These homology classes are the lifts of the torsion classes for PSL(2, R).In the recent work of Kirillov [K] concerning a conjecture of Nahm on the spectrumof rational conformal field theory [NRT], the following abelian subgroup W of Q wasconsidered:W = {XiniL(ai)/L(1)| ni ∈Z, ai ∈Ralg} ∩Q.From our discussion, it is clear that W contains both 1 as well as −1/m mod Z for everypositive integer m. Thus, W is simply Q.

In the conjecture of Nahm, one is more concernedwith the set of effective central charges and ni is assumed to be non-negative. This is closedunder addition because one can form tensor product of models.

Our discussion only pinsdown the fractional part of such central charges while the integral parts apparently spreadthe central charges out in a way that resembled the volume distribution of hyperbolic 3-manifolds. In the present approach, these effective central charges are volumes of certain3-cycles in a totally different space–the compactification of the universal covering groupof PSL(2, R).

These 3-cycles can be viewed as “orbifolds” since they arise from the finitecyclic subgroups of SL(2, R). It should also be noted that the central charge of the Virasoroalgebra is the value of a degree two cohomology class while our description is on the level14

of degree three group cohomology, but for the Lie groups viewed as discrete groups. Theprecise relation between these cohomologies is not too well understood.

On the level ofclassifying spaces of topological groups, there is the wellknown conjecture, see [M] and[FM]:Conjecture of Friedlander-Milnor. Let G be any Lie group and let p be a prime.Then Hi(BGδ, Zp) →Hi(BG, Zp) is an isomorphism (it is known to be surjective).§6.

The “beta map” and various conjectures.In the work of Nahm-Recknagel-Terhoeven, [NRT], speculations were made aboutthe relevance of algebraic K-theory, Bloch groups [Bl], geometry of hyperbolic 3-manifolds[Th1] as well as the “physical meaning” of a “beta map”. To some extent, we have clarifiedthe first three of these.

Namely, a connection between the effective central charge in rationalconformal field theory with algebraic K-theory and Bloch groups [Bl] can be made byway of the second characteristic class of Cheeger-Chern-Simons and its interpretation viavolume calculation in the universal covering group space of PSL(2, R). Specifically, itis not connected with the volume calculation in hyperbolic 3-space.

(Note: Accordingto Thurston’s work, [Th], volume of hyperbolic 3-manifolds is a topological invariant. )Roughly speaking, the difference rests with a missing factor of (−1)1/2.

We next clarifythe origin of the “beta map”. In terms of diagram (4.2), the “beta map” is denoted by:(6.1)d2 : P(R) →Λ2Z(R×), d2([[r]]) = r ∧(r −1), r > 1.d2 arises as the second differential in a spectral sequence.

It is defined by solving a “descentequation”. This is typical of the higher differential maps in a spectral sequence.Theexactness of the rows in (4.2) showed that ker d2 = imσ.

If we move up to the level ofPS(R), it is then clear that the vanishing of the d2-invariant characterizes the elements ofH3(gPSL(2, R), Z). The origin of d2 comes from the Dehn invariant in Euclidean 3-space.In 1900, Dehn used it to solve Hilbert’s Third Problem and extended it to hyperbolicand spherical 3-space, see [DS2].

By working with P(C), see [DS1] and [DPS], d2 thenincorporates both versions of the Dehn invariants. In the present case, we would interpretd2 in terms of “ideal polyhedra” in eS.

As pointed out in [PS], the following conjecture isstill open:Conjecture 6.2. LP S : H3(gPSL(2, R), Z) →R is injective.We already mentioned the following conjecture along this line:Conjecture 6.3.

H3(gPSL(2, Ralg), Z) →H3(gPSL(2, R), Z) is bijective.The preceding conjecture is a special case of the more general “folklore” conjecture:Conjecture 6.4. H3(SL(2, Calg), Z) →H3(SL(2, C), Z) is bijective.More precisely, Conjecture 6.3 is equivalent to any of the corresponding conjecture fora nontrivial quotient group ofgPSL(2, R), for example PSL(2, R).H3(SL(2, R), Z) is15

known to be isomorphic to the fixed point set of H3(SL(2, C), Z), see [Sa3]. The map inConjecture 6.4 is known to be injective, see [Su2].

Thus Conjectures 6.3 and 6.4 wouldfollow from:Conjecture 6.5. H3(SL(2, Calg), Z) →H3(SL(2, C), Z) is surjective.It should be mentioned that the map H3(SU(2), Z) →H3(SL(2, C), Z) has image equalto the image of H3(SL(2, R), Z).

In this connection, we have:Conjecture 6.6. H3(SU(2), Z) →H3(SL(2, C), Z) is injective.Conjecture 6.7.

ˆc2 : H3(SL(2, C), Z) →C/Z. is injective.Conjecture 6.7 is equivalent to the conjunction of conjecture 6.6 and the converse of theHilbert’s Third Problem for hyperbolic as well as spherical polytopes in dimension 3.Namely, the Dehn invariant together with volume detect the scissors congruence classes ofsuch polytopes.

The Euclidean case was solved by Dehn-Sydler, see [DS2] for discussions.The best result in this direction is the theorem of Borel, [Bo]:Borel’s Theorem. Suppose c is non-zero in H3(SL(2, Calg), Z), then ˆc2(τ(c)) isnon-zero for a suitable automorphism τ of C.We note that an illustration of the idea behind Borel’s Theorem was the proof given in[PS] that H3(SL(2, Ralg), Z) contains a rational vector space of infinite dimension.

Recall,we consider a real algebra number rp satisfying the equation Xp−X+1 = 0, p an odd prime.d2({rp}) is therefore 0 and [[rp]] then defines an element of H3(SL(2, Ralg), Z). Since LP Sis strictly monotone, there is no problem showing that we have distinct elements.

However,it is not obvious that these elements are Q-linearly independent. This stronger statementwas a combination of Galois theory together with the use of the hyperbolic volume.§7.

Concluding Remarks.In the present work, we showed that the effective central charges for certain modelsin conformal field theory can be connected to the evaluation of a real valued cohomologyclass on a suitable degree 3 homology class for the integral group homology of the universalcovering groupgPSL(2, R) of PSL(2, R). The important point is that we have replacedthe usual topology by the discrete topology.In addition, instead of the hyperbolic 3-space, we use the group space of this universal covering group.

The particular homologyclass is a suitable lift of a homology class of finite order that generates the third integralhomology of a finite cyclic subgroup of PSL(2, R). The lift is connected with the Rogers’dilogarithm identities due to Richmond-Szekeres [RS] and Kirillov-Reshetikhin [KR].

Allthese identities are shown to originate from the basic identities found by Rogers [R]. Ourroute ends in the central charge identification but there are no firm connections betweenany of the intermediate steps followed by us with the intermediate steps used in solvablemodels in conformal field theory.

A casual reading of [BPZ] and [Z2] does show the manyappearances of cross-ratios. However, instead of the complex numbers or the real numbers,we see meromorphic functions.

This is also the basic theme in the work of Bloch [Bl]. Onthe mathematical side, there are efforts to build up enormous structures to explain thesteps used in the physics side.

Our present effort does not do this.16

Another of the principal points in the present work is the fact that Rogers’ dilogarithmhas long been known to be connected with the second Cheeger-Chern-Simons character-istic class which is represented by the Chern-Simons form that appears in many currenttheoretical physics investigations. This connection is related to the interplay between the”continuous” picture and the ”discrete” picture.

On the mathematics side, we have a di-rect map on the level of classifying spaces for groups equipped with two topologies: onediscrete, the other continuous. The map is the one that goes from the discrete to the con-tinuous.

On the physics side, the passage from the discrete to the continuous is a subjectof debate since there does not appear to be a specific map (in the mathematical sense).However, there are still a large number of unresolved issues on the mathematical side. Forexample, the Virasoro algebra is typically viewed as the algebraic substitute for the dif-feomorphism group of the circle.

(More precisely, it may be viewed as the “pseudo-group”of holomorphic maps on the sphere with two punctures). This contains PSL(2, R) whichacts as a group of diffeomorphisms on the circle through the identifiction of the circlewith P 1(R).) Our procedure replaces these infinite dimensional (pseudo-) groups by thefinite dimensional subgroups.

However, it is also accompanied by the use of the discretetopology. Although the process of playing offone topology against another is familiar infoliation theory, it is not explored in the present work.In passing, we would like to indicate that Rogers’ dilogarithm has appeared in variousrelated works on the physics side.

Aside from the work [BR] that led Bazhanov to ask oneof us (CHS) about the connection between [BR] and [PS] in the summer of 1986-7, thereare the earlier works of Zamolodchikov [Z2] and Baxter [B]. Specifically, in the appendixof [B], Rogers dilogarithm appeared.

This has been extended recently in [BB] where theyhave shown that the 3-d models of Zamolodchikov can be related to the earlier 2-d chiralPotts models considered in [AMPTY], [MPST], and [BPA] after suitable generalizations.On the mathematics side, Atiyah and Murray [A] have identified the algebraic curves in[BPA] and [MPST] as the spectral curves of N magnetic monopoles arranged cyclicallyaround an axis in hyperbolic 3-space. In view of the fact that our present work indicatesthat the group manifoldgPSL(2, R) is more appropriate than the hyperbolic 3-space, onecan not help but ask if there might be an interesting mathematical theory of magneticmonopoles ingPSL(2, R).

Evidently, the present work raises many more questions than itanswers.Acknowledgements: It is evident that we have benefited from many, many col-leagues. Since it is impractical to list them all, we will, at the risk of insulting many,limit ourselves to Profs.

B. M. McCoy, C. N. Yang, I. M. Gelfand and L. Takhtajan forinspiring tutorial discussions. Special thanks are due to J. D. Stasheffand V. V. Bazhanovfor raising the crucial questions at the right time, these questions led to the present work.17

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