Classical and Quantum Dilogarithm Identities

Kashaev, Rinat; Наканиши, Томоки

doi:10.3842/sigma.2011.102

Cited by 42 publications

(83 citation statements)

References 72 publications

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“…In another direction, an exponentiated quantum deformation of the dilogarithm, the quantum dilograrithm, which satisfies a deformed functional equation, was proposed by Faddeev and Kashaev [23]) in 1994. It has since been much studied, see the survey of Kashaev and Nakanishi [40]. Certain dilogarithm identities play a role in integrable models and in conformal field theory (Nahm et al [58], Kirillov [41,42]).…”

Section: Prior Workmentioning

confidence: 99%

The Lerch Zeta function III. Polylogarithms and special values

Lagarias

2016

Res Math Sci

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This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent (s, z, c) := ∞ n=0 z n (n+c) s is obtained from the Lerch zeta function ζ (s, a, c) by the change of variable z = e 2πia . We show that it analytically continues to a maximal domain of holomorphy in three complex variables (s, z, c), as a multivalued function defined over the base manifold C × (P 1 (C) {0, 1, ∞}) × (C Z) and compute the monodromy functions describing the multivaluedness. For positive integer values s = m and c = 1 this function is closely related to the classical m-th order polylogarithm Li m (z). We study its behavior as a function of two variables (z, c) for "special values" where s = m is an integer. For m ≥ 1 we show that it is a one-parameter deformation of Li m (z), which satisfies a linear ODE, depending on c ∈ C, of order m + 1 of Fuchsian type on the Riemann sphere. We determine the associated (m + 1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of Li m (z).

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Section: Prior Workmentioning

confidence: 99%

The Lerch Zeta function III. Polylogarithms and special values

Lagarias

2016

Res Math Sci

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“…This is our second main result. We note that a quantum dilogarithm identity is also associated with the same periodicity of the cluster algebra [Kel11,Nag11,KN11]. For example, the quantum dilogarithm identity associated with the same period of a pentagon gives the celebrated pentagon identity by [FK94], and it looks as follows:…”

Section: Introductionmentioning

confidence: 99%

Exact WKB analysis and cluster algebras

Iwaki

Наканиши

2014

J. Phys. A: Math. Theor.

118

189

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Abstract. We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrödinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts.

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“…The action of the operatorsμ k and P k defined before, extends naturally to an action on p and u. In the following we will only need explicitly the action of P k on u, which is linear [11]:…”

Section: Cluster Partition Function At Level (K S) = (1 S)mentioning

confidence: 99%

“…Moreover, we will actually see that Z cluster G (M ) is more well suited for the case that M can be obtained from a mapping torus construction. The cluster partition function was originally proposed in [10] based on ideas of [11], however important modifications were done in [3] to get it to the form we will present here. It is not immediately obvious that the function Z cluster G (M ) we obtain is well defined as a nonperturbative invariant but we can propose perturbative topological invariants of M starting from Z cluster G (M ).…”

Section: Introductionmentioning

confidence: 99%

Cluster partition function and invariants of 3-manifolds

Romo

2017

Chin. Ann. Math. Ser. B

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We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. We focus on the case G = SL(N, C) and with M a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the ChernSimons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory.

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Classical and Quantum Dilogarithm Identities

Cited by 42 publications

References 72 publications

The Lerch Zeta function III. Polylogarithms and special values

The Lerch Zeta function III. Polylogarithms and special values

Exact WKB analysis and cluster algebras

Cluster partition function and invariants of 3-manifolds

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