We adapt Joyce's theory of wall-crossing for enumerative invariants of $\mathbb C$-linear additive categories to Pandharipande-Thomas stable pairs on smooth projective Fano 3-folds of "type C or D", and investigate implications for Pandharipande-Thomas generating functions with descendent insertions.

By analyzing the wall-crossing behavior from a stability condition where pairs are unstable to the standard stability condition for PT stable pairs, we derive an explicit formula expressing the PT stable pair invariants $[P_n(X,\beta)]^{virt}$ in terms of sheaf-theoretic invariants $[\mathcal M^{ss}_{(0,0,\beta_i, n_i - \beta_i.c_1/2)}(\tau_-)]_{\rm inv}$ for the moduli space of Gieseker semistable coherent sheaves on $X$ with Chern character $(0,0,\beta_i, n_i - \beta_i.c_1/2)$.

These enumerative invariants are defined as elements in the Lie algebra on the rational Betti homology of the piecewise-linear rigidified higher moduli stack of objects in the bounded derived category of X. Under tensoring by a line bundle, we exhibit a control over the periodicity of sheaf-theoretic invariants with respect to the Euler characteristic $n_i$, which we use to show that the sheaf-theoretic invariants form a quasi-polynomial in $n_i$ of degree $2$ with period given by the divisibility of $\beta_i$ in the lattice $H_2(X,\mathbb Z)/\text{torsion}$.

We use this periodicity in the sheaf-theoretic invariants to show that the descendent generating series for Pandharipande-Thomas stable pairs is the Laurent expansion of a rational function over $\mathbb Q$ in this setting, thus confirming a conjecture due to Pandharipande-Thomas from 2007. Furthermore, we construct a counterexample to a conjecture due to Pandharipande from 2017 on the location of the poles of the descendent generating series, and give a direct proof of a slightly modified conjecture on the location of these poles using wall-crossing techniques.

 

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

In the first half of this talk, I will provide a brief introduction to Isomonodromic deformations with the one-point torus as my main example, and show the relation to the elliptic form of Painlevé VI equation as well as the Lamé equation. In the second half of this talk, I will present an overview of my results in the past few years concerning the associated tau-functions, conformal blocks, and accessory parameters. Finally, I will motivate how probabilistic methods in conformal field theory help us understand the data within Lamé type equations.