Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantum curves and $q$-deformed Painlev\'e equations   We propose that the grand canonical topological string partition functions
satisfy finite-difference equations in the closed string moduli. In the case of
genus one mirror curve these are conjectured to be the q-difference Painlev\'e
equations as in Sakai's classification. More precisely, we propose that the
tau-functions of q-Painlev\'e equations are related to the grand canonical
topological string partition functions on the corresponding geometry. In the
toric cases we use topological string/spectral theory duality to give a
Fredholm determinant representation for the above tau-functions in terms of the
underlying quantum mirror curve. As a consequence, the zeroes of the
tau-functions compute the exact spectrum of the associated quantum integrable
systems. We provide details of this construction for the local
$\mathbb{P}^1\times \mathbb{P}^1$ case, which is related to q-difference
Painlev\'e with affine $A_1$ symmetry, to $SU(2)$ Super Yang-Mills in five
dimensions and to relativistic Toda system.

31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys