Abstract
Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems. This can be used to compute spectral traces, both exactly and as a semiclassical expansion. As a main example, we consider the system related to the quantized mirror curve of local $\mathbb P^2$. According to a recent proposal, the traces of this operator are determined by the refined BPS indices of the underlying CY. We use our non-linear integral equations to test that proposal.
Highlights
The solution of the non-linear Thermodynamic Bethe Ansatz (TBA) integral system (1.1) is characterized by some spectral function of the operator (1.2)
We show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems
Let us emphasize the two most important features of the generalization, which are the decomposition of the resolvent function according to the ZD symmetry, and the generalized quantum Wronskian relations
Summary
In the case α = 1 (ω = −1), there is a natural splitting of the resolvent into an odd and an even part [2, 3], which do not enter symmetrically in the TBA system This comes from the fact that ω is a square-root of 1, the non-trivial element of Z2. Because our operator is of trace class, we expect the Rk(p) to be well defined functions with finite integral over real p. The denominator of R2 in (2.16) vanishes when p → p , which implies that the numerator should vanish in this limit: ωrΦr(p)Φ 2−r(p) = 0 Taking this into account, we end up with the following expressions for the diagonal resolvent kernels:. The expressions (2.20) are to be taken as the direct equivalents of Lemma 1 in [3]
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