DILOGARITHM IDENTITIES, q-DIFFERENCE EQUATIONS AND THE VIRASORO ALGEBRA

EDWARD FRENKEL AND ANDRÁS SZENES

December, 1992

1. Introduction

In this paper we give a new proof of the identity

∑_{j=1}^{n−1} L(δ_{j}) = \frac{π^{2}}{6} \frac{2n − 2}{2n + 1}, (1)

where L(z) is the Rogers dilogarithm function:

L(z) = − ∫_{0}^{z} log(1 − w) d log w + \frac{1}{2} log z log(1 − z), 0 ≤ z ≤ 1,

and δ_{j} = sin^{2}(\frac{π}{2n+1})/ sin^{2}(π \frac{j+1}{2n+1}).

This equality was proved by Richmond and Szekeres from the asymptotic analysis of the Gordon identities [1] and also by Kirillov using analytic methods [2]. There is a wide class of identities of this type, which emerged in recent works on two-dimensional quantum field theories and statistical mechanics. They appear in the calculation of the critical behavior of integrable models, using the so-called Thermodynamic Bethe Ansatz equation ([3, 4, 5, 6, 7, 8]). This is an elegant, but mathematically nonrigorous method.

In this paper, we present a new approach to the identities, which relies on the concepts of quantum field theory, and is rigorous at the same time. We hope this will lead to a better understanding of the mathematical structures behind general integrable two-dimensional quantum field theories.

The idea goes back to Feigin [9]: There is a certain increasing filtration on each irreducible minimal representation of the Virasoro algebra (or another conformal algebra) by finite-dimensional subspaces, whose characters are connected by a q-difference equation. This equation can be used to obtain an expression for the asymptotics of the character of this representation in terms of values of the dilogarithm. On the other hand, it is known that this asymptotics is determined by the (effective) central charge c_{eff} (cf. [10]), and this gives us an identity.

Research of the first author was supported by a Junior Fellowship from the Harvard Society of Fellows and by NSF grant DMS-9205303

E-mail addresses: frenkel@math.harvard.edu, szenes@math.mit.edu

To illustrate that the values of the dilogarithm should appear in the asymptotics of the solutions to q-difference equations, consider the equation

f(qx) = (1 − aq)f(x), 0 < q < 1.

Then

lim_{n→∞} f(q^{n}) = f(1) ∏_{n=1}^{∞}(1 − aq^{n}). (2)

Let us calculate the asymptotics of this solution as q → 1. We have

log q log f(q^{n}) = log q(log f(1) + ∑_{n=1}^{∞} log(1 − aq^{n})).

This expression can be interpreted as a Riemann sum. Thus in the limit q → 1 it gives

− ∫_{0}^{1} log(1 − ax) d log x = L(a) − \frac{1}{2} log a log(1 − a).

We will show that this approach can be carried out for the irreducible representations of the (2, 2n + 1) models of the Virasoro algebra. In this case c_{eff} = (2n − 2)/(2n + 1), and the corresponding identity is (1).

Let us remark that according to Goncharov [11], the dilogarithm identities of this type define elements of torsion in K_{3}(ℝ).

2. (2, 2n + 1) minimal models of the Virasoro algebra

The Virasoro algebra is the Lie algebra generated by elements {L_{i}, i ∈ ℤ} and C, with the relations

[L_{i}, L_{j}] = (i − j)L_{i+j} + \frac{1}{12}(i^{3} − i)δ_{i,−j}C and [L_{i}, C] = 0.

Fix an integer n > 1. Let 1_{n} be the one-dimensional representation of the subalgebra generated by {C, L_{i}, i ≥ −1}, on which the elements L_{i} act by 0, and C acts by multiplication by 1 − 3(2n − 1)^{2}/(2n + 1). Denote by Ṽ_{n} the induced module of the Virasoro algebra. It is ℤ-graded with respect to the action of L_{0}, deg(L_{−i}) = i. By the Poincaré-Birkhoff-Witt theorem this module has a ℤ-graded basis consisting of the monomials {L_{−m_{1}} … L_{−m_{p}}v| m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{p} > 1}, where v is the generator of 1_{n}. The module Ṽ_{n} has a unique vector u of degree 2n + 1, such that L_{i}u = 0 for i > 0. The quotient V_{n} of the module Ṽ_{n} by the submodule generated by u inherits the ℤ-grading, and is irreducible [12, 13]. It is called the vacuum representation of the (2, 2n + 1) minimal model of the Virasoro algebra [14].

Let C_{n} = {(m_{1}, …, m_{p})| m_{1} ≥ ⋯ ≥ m_{p} > 1, m_{i} ≥ m_{i+n−1} + 2}.

Proposition 1 ([15]). The image of the set

{L_{−m_{1}} … L_{−m_{p}}v| (m_{1}, …, m_{p}) ∈ C_{n}}

under the homomorphism Ṽ_{n} → V_{n} gives a linear basis of V_{n}.

Remark 1. While the argument in [15] relied on the Gordon identities, this statement can be proved directly, using only the representation theory of the Virasoro algebra (see [16]).

For every integer N > 0, introduce the subspaces W_{N}^{r}, 0 ≤ r < n, of the module V_{n}, which are linearly spanned by the vectors

{L_{−m_{1}} … L_{−m_{p}}v| (m_{1}, …, m_{p}) ∈ C_{n}, m_{1} ≤ N, m_{r+1} ≤ N − 1}.

For any ℤ-graded vector space V = ⊕_{m=0}^{∞}V (m) with dim V (m) < ∞, let ch V = ∑_{m=0}^{∞} dim V (m)q^{m}, which is called the character of V. Denote by w_{N}^{r} the character of W_{N}^{r}, and introduce w_{N} = (w_{N}^{0}, …, w_{N}^{n−1}) and w_{0} = (0, …, 1).

Lemma 2. We have the following recursion relation:

w_{N} = M_{n}(q^{N})w_{N−1},

where

M_{n}(x) = (\begin{matrix}0 & & 0 & & … & & 0 & & 1 \\ 0 & & 0 & & … & & x & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & x^{n−2} & & … & & x & & 1 \\ x^{n−1} & & x^{n−2} & & … & & x & & 1 \end{matrix}) .

This gives us the following formula for the character of V_{n}:

ch V_{n} = w_{0}^{t} ∏_{N=1}^{∞} M_{n}(q^{N})w_{0} = w_{0}^{t} … M_{n}(q^{N}) … M_{n}(q^{2})M_{n}(q)w_{0}. (3)

Thus ch V_{n} is equal to w_{0}^{t} lim_{n→∞} f(q^{n}), where f(x) is the solution to the q-difference equation f(qx) = M_{n}(x)f(x) with the initial condition f(1) = w_{0}.

The character of V_{n} is a function in q for 0 < q < 1. We will study its asymptotics as q → 1.

The following result was proved by Kac and Wakimoto, and follows from the modular properties of ch V_{n}.

Proposition 3 ([10]).

− lim_{q→1} log q log ch V_{n} = \frac{π^{2}}{6} \frac{2n − 2}{2n + 1}.

Remark 2. The number \frac{2n−2}{2n+1} is called the effective central charge of the (2, 2n + 1) minimal model.

In the next section, we will use (3) to derive another expression for this asymptotics.

3. Asymptotics of the infinite product

The following lemma relates the asymptotics of (3) to the asymptotics of the infinite product of the highest eigenvalues of M_{n}(q^{N}).

Lemma 4. Denote by d_{n}(x) the eigenvalue of M_{n}(x) of maximal absolute value.

(a) For any x, 0 < x ≤ 1, the matrix M_{n}(x) has n different real eigenvalues and d_{n}(x) > 0.

(b) The asymptotic behavior of ch V_{n} is the same as that of the infinite product of the d_{n}(q^{N}), more precisely

(4) − lim_{q→1} log q log ch V_{n} = − lim_{q→1} log q log ∏_{N=1}^{∞} d_{n}(q^{N}).

Similarly to the calculation of (2), the right hand side of (4) can be written as the following integral:

∫_{0}^{1} log d_{n}(x) d log x. (5)

The rest of this section is devoted to the calculation of this integral.

Denote by Q_{n}(λ, x) the characteristic polynomial of M_{n}(x). Our computation is based on an explicit rational parametrization of the curve Q_{n}(λ, x) = 0.

Remark 3. On this curve, λ and x define two algebraic functions. Our integral (5) is the integral of log λ d log x along a path on the curve. We would like to stress that it is the rationality of this curve, that allows us to express this integral in terms of values of the dilogarithm function.

Introduce the n × n matrix

L_{n}(x) = (\begin{matrix}1 + x & & −1 & & & 0 & & & … & & 0 & & 0 & & & 0 \\ −x & & 1 + x & & & −1 & & & … & & 0 & & 0 & & & 0 \\ 0 & & −x & & & 1 + x & & & … & & 0 & & 0 & & & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & 0 & & & 0 & & & … & & 1 + x & & −1 & & & 0 \\ 0 & & 0 & & & 0 & & & … & & −x & & 1 + x & & & −1 \\ 0 & & 0 & & & 0 & & & … & & 0 & & −x & & & x \end{matrix}) ,

and observe that M_{n}(x)^{−2} = x^{−n}L_{n}(x). Then the roots of the characteristic polynomial P_{n}(μ, x) = det(L_{n}(x)−μ) are connected to the roots of Q_{n}(λ, x) via the formula λ^{2} = x^{n}/μ. On the other hand, we have the following recursion for P_{n}(x):

P_{n}(μ, x) = (1 + x − μ)P_{n−1}(μ, x) − xP_{n−2}(μ, x),

with initial conditions P_{0}(μ, x) = 1 and P_{1}(μ, x) = x − μ.

Introduce t = x/(x − μ + 1)^{2} and u = 1/(x − μ + 1). In these new variables our recursion can be written as

u^{−1} \frac{P_{n−1}}{P_{n}} = \frac{1}{1 − t (u^{−1} \frac{P_{n−2}}{P_{n−1}} )}.

This can be solved by the continuous fraction

u^{−1} \frac{P_{n−1}}{P_{n}} = \frac{1}{1 − \frac{t}{1 − \frac{t}{… \frac{…}{1 − \frac{t}{1 − u}}}}}.

As a consequence the equation P_{n} = 0 is equivalent setting the denominator of this continuous fraction to 0. This leads to the following parametrization for u:

u = A_{n}(t), where A_{n}(t) = 1 − \frac{t}{1 − \frac{t}{… \frac{…}{1 − \frac{t}{1 − t}}}}.

Naturally, A_{n}(t) satisfies the recursion A_{n}(t) = 1 − t/A_{n−1}(t) with the initial condition A_{1}(t) = 1. A quick calculation shows that as functions of the parameter t our original variables λ and x take the form

x_{n}(t) = \frac{t}{A_{n}^{2}(t)} and λ_{n}(t) = \frac{A_{1}(t)A_{2}(t) … A_{n−1}(t)}{A_{n}^{n−1}(t)}.

Lemma 5. For any α, 0 < α ≤ 1, there are n real positive solutions t_{1}(α) < ⋯ < t_{n}(α) to the equation x_{n}(t) = α. The numbers λ_{n}(t_{1}(α)), … λ_{n}(t_{n}(α)) are the eigenvalues of the matrix M_{n}(α), and

d_{n}(α) = λ_{n}(t_{1}(α)).

The lemma implies that our integral (5) can be written as

∫_{t=0}^{t=t_{1}(1)} log λ_{n}(t) d log x_{n}(t). (6)

Introduce the rational functions f_{i}(t) = 1−A_{i+1}(t)/A_{i}(t). The following key formula can be proved by induction:

f_{i}(t) = \frac{t^{i}}{(1 − f_{1}(t))^{2(i−1)} ⋯ (1 − f_{i−1}(t))^{2}} . (7)

In terms of the these functions our variables can be expressed as follows:

x_{n} = \frac{t}{(1 − f_{1})^{2}(1 − f_{2})^{2} … (1 − f_{n−1})^{2}}

λ_{n} = \frac{1}{(1 − f_{1})(1 − f_{2})^{2} … (1 − f_{n−1})^{n−1}}

Now we can calculate the integral (6). We begin with a more general integral

∫_{0}^{γ} log λ_{n}(t) d log x_{n}(t),

where γ > 0 is such that λ_{n}(t) and x_{n}(t) have no poles on the interval [0, γ]. Substituting the formulas above, we obtain

− ∫_{0}^{γ} ∑_{j=1}^{n−1} j log(1 − f_{j}(t)) d (log t − ∑_{i=1}^{n−1} 2 log(1 − f_{i}(t))) =

− ∫_{0}^{γ} ∑_{j=1}^{n−1} log(1 − f_{j}(t)) d log (\frac{t^{j}}{∏_{i=1}^{j−1}(1 − f_{i}(t))^{2(j−i)}} ) +

∫_{0}^{γ} ∑_{i,j=1}^{n−1} 2 min(i, j) log(1 − f_{j}(t)) d log(1 − f_{i}(t)).

Applying formula (7) to the first summand, and partial integration to the second, our integral takes the form

− ∫_{0}^{γ} ∑_{j=1}^{n−1} log(1 − f_{j}(t)) d log f_{j}(t) + ∑_{i,j=1}^{n−1} min(i, j) log(1 − f_{j}(γ)) log(1 − f_{i}(γ)).

Now by the definition of the Rogers dilogarithm, this can be written as

(8) ∑_{j=1}^{n−1} L(f_{j}(γ)) − \frac{1}{2} ∑_{j=1}^{n−1} log f_{j}(γ) log(1 − f_{j}(γ)) +

∑_{i,j}^{n−1} min(i, j) log(1 − f_{j}(γ)) log(1 − f_{i}(γ)).

To complete our proof, we need to describe some properties of the numbers δ_{i} of (1).

Lemma 6. Fix an integer n > 1. The numbers δ_{i}, 1 ≤ i ≤ n−1, satisfy the following properties:

(a) f_{i}(δ_{1}) = δ_{i},

(b) δ_{i} = ∏_{i=1}^{n−1}(1 − δ_{j})^{2 min(i,j)},

(c) δ_{1} = A_{n}(δ_{1})^{2}, thus x_{n}(δ_{1}) = 1. Moreover, t_{1}(1) = δ_{1}.

Combining the calculation above with property (c), we see that the integral (6) equals to the expression (8), with γ = δ_{1}. Using property (a) this is simply

∑_{j=1}^{n−1} L(δ_{j}) − \frac{1}{2} ∑_{j=1}^{n−1} log δ_{j} log(1 − δ_{j}) + ∑_{i,j}^{n−1} min(i, j) log(1 − δ_{i}) log(1 − δ_{j}).

Finally, according to property (b), the last two terms cancel. Hence we obtain the following

Proposition 7.

∫_{0}^{1} log d_{n}(x) d log x = ∑_{j=1}^{n−1} L(δ_{j}).

This result, together with Lemma 4 and Proposition 3, proves identity (1).

Remark 4. We are certain that this method is applicable to other models of conformal field theory. Acknowledgements. We are grateful to B. Feigin for sharing his ideas with us. We would like to thank A. Goncharov, V. Kac, A. Levin and T. Nakanishi for valuable discussions.

References

1. B.Richmond, G.Szekeres, Some formulas related to dilogarithms, the zeta function and the Andrews-Gordon identities, J. Australian Math. Soc. (Series A) 31 (1981) 362-373

2. A.N.Kirillov, On identities for the Rogers dilogarithm function related to simple Lie algebras, Zap. Nauch. Sem. LOMI, 164 (1987) 121-133

3. A.N.Kirillov, N.Reshetikhin, Exact solution of the XXZ Heisenberg model of spin S, Zap. Nauch. Sem. LOMI, 160 (1987) 211-222

4. V.Bazhanov, N.Reshetikhin, Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A 23 (1990) 1477-1492

5. A.Kuniba, T.Nakanishi, Spectra in conformal field theory and the Rogers dilogarithm, Preprint SMS-042-92, June 1992, to appear in Mod. Phys. Lett. A

6. W.Nahm, A.Recknagel, M.Terhoeven, Dilogarithm identities in conformal field theory, Preprint BONN-HE-92-35, November 1992

7. Al.Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models: scaling 3-state Potts and Lee-Yang models, Nucl. Phys. B342 (1990) 695-720

8. T.Klassen, E.Meltzer, Purely elastic scattering theories and their ultraviolet limit, Nucl. Phys. B338 (1990) 485-528

9. B.Feigin, private communication

10. V.Kac, M.Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Natl. Acad. Sci. USA, 85 (1988) 4956-4960

11. A.Goncharov, private communication

12. V.Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras, Lect. Notes in Phys. 94 (1979) 441-445 13. B.Feigin, D.Fuchs, Representations of the Virasoro algebra, in “Representations of infinite-

dimensional Lie groups and Lie algebras”, eds. A.M.Vershik and D.P.Zhelobenko, Gordon and Breach, 1990, 465-554

14. A.Belavin, A.Polyakov, A.Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333-380

15. B.Feigin, T.Nakanishi, H.Ooguri, The annihilating ideals of minimal models, Int. J. Mod. Phys. Suppl. 1A (1992) 217-238

16. B.Feigin, E.Frenkel, in preparation.

Department of Mathematics, Harvard University, Cambridge, MA 02138

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139