Abstract
The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations — or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type — which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. We demonstrate the efficiency of our formulation through the numerical computation and compare the results in the UV limit with the Liouville CFT. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.
Highlights
We reformulated the g-functions in the sinh-Gordon theory in terms of the thermodynamic Bethe ansatz (TBA)-like integral equation, which we called the Tracy-Widom TBA
The resulting integral equation is more efficient than the results based on the Fredholm determinants
The result is given by a generalization of the Fredholm determinant which involves both integrals and sums
Summary
To discuss the g-function in integrable theories, we consider a partition function of a cylinder with circumference L and length R whose boundaries are contracted with boundary states B| and |B (see figure 1). We assume that |B belongs to a class of integrable boundary states introduced in [36] In such a case, we can apply the standard arguments of the thermodynamic Bethe ansatz and compute the thermal partition function. Where the parameter p is related to the coupling constant b by the relation p = b2(1 + b2)−1 and throughout this work we will focus on the self-dual point for which b = 1 We will consider this model defined on geometries with specific boundaries that preserve integrability. The g-function is fully determined through (2.7)
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