Beijing Institute of Mathematical Sciences and Applications

We will show how the formalism of resurgence monomials can yield proofs of resurgence properties for normalizing transformations or transseries solutions of dynamical systems, with one or (time permitting) multiple “critical times”.

The TBA/WKB correspondence describes a mysterious correspondence between the TBA equations of the quantum integrable model and the exact WKB method of the Schrodinger equation. In this talk, we will first provide an overview on the TBA/WKB correspondence, and then apply this framework to the Schrodinger equation for deformed supersymmetric quantum mechanics. The TBA equations and the corresponding wall-crossing will be shown.

We study the resurgent structure of the topological string dual to 2d $U(N)$ Yang-Mills on torus. We find closed form formulas for instanton amplitudes up to arbitrarily high instanton orders, based on which we propose the non-perturbative partition function including contributions from all the real instantons, which is real for positive modulus and string coupling. We also explore complex instantons and find two infinite towers of them. We expect them to correspond to BPS states in type II string.

During the 1980s, Ecalle formulated the framework of resurgence theory, which in recent years has undergone significant development and found broad applications in mathematical physics. In recent work on topological string theory, nonlinear operators acting on formal power series have emerged as central to mathematical physics, enabling asymptotic limits arising in distinct regimes to be understood within a unified framework. In this talk, we examine the relations among the free energies of the B-model topological string in distinct asymptotic limits.

We study two types of regularizations of the determinant of Laplace operator on Riemannian manifold from the viewpoint of resurgence theory. One is the formal logarithmic derivative of the determinant, and the other is its exponential deformation. Under appropriate conditions, the close formula for both regularized determinant are established through resurgence theory which can be viewed as the summation of the singularities along the analytic continuation of theta series $\hat{Theta}_{D_X}$. The series resembles the trace of the heat kernel, but is defined via the spectrum of the square-root of the Laplacian. As applications, we revisit the well known cases of the determinant of $S^1$ and compact Riemann surface with genus $\geq 2$, which correspond to the Poisson summation formula and Selberg trace formula respectively. Furthermore, the 1-Gevrey asymptotic behavior of the second regularization at infinity is considered whose asymptotic coefficients are determined by the trace of the heat kernel. In the end, we establish the relationship between the two regularized determinants. In fact, they have the same derivatives if we take the deformation parameter tends to 0 in exponentially deformed regularization.

We consider the Ai-Bender-Sarkar conjecture for the massless QM given by defined by a negative potential $V(x)=gx^2(ix)^2$, which proposes a relation between the partition function of PT-symmetric QM and that of the analytic continuation of the Hermitian QM. There exists evidence suggesting that the original conjecture fails in the massless case, so our aim is to construct its modified formal relation. To address this problem, we focus on the simplest setting of the ODE/IM correspondence, the $A_1$ T-system and formulate a relation between the two partition functions by constructing the zeta generating formula (ZGF) which provides a direct mapping between the spectral zeta functions of the PT-symmetric and Hermitian QMs. Our approach is based on the spectral-zeta aspect of the $A_1$ T-system and also can reproduce relations among zeta functions known as exact sum rules (ESR) and spectral sums.

Joint work with J. A. Andersen, L. Han, Y. Li, W. E. Mistegard and S.Sun. Consider a general Seifert fibered integral homology 3-sphere with $r\ge 3$ exceptional fibers. We show that its $SU(2)$ Witten-Reshetikhin-Turaev invariant (WRT) evaluated at any root of unity $\zeta$ is (up to an elementary factor) the non-tangential limit of its Gukov-Pei-Putrov-Vafa invariant (GPPV) as $q$ tends to $\zeta$, thereby generalizing a result from Andersen-Mistegard [JLMS 2022]. The quantum modularity results developed by Han-Li-Sauzin-Sun for functions like the GPPV invariant [FAA 2023], based on Ecalle’s resurgence theory and median summation, then help us to prove Witten’s asymptotic expansion conjecture [CMP 1989] for such a manifold: the asymptotic behavior of the WRT invariant at $e^{2\pi i/k}$ as $k$ tends to infinity is dictated by the $SU(2)$ Chern-Simons critical values. The GPPV invariant gives rise to infinitely many resurgent- summable series, one formal series of $q−\zeta$ for each root of unity $\zeta$, which are related to the expansion at $\zeta$ of the unified Habiro invariant; equivalently, when going to the variable $\tau$ defined by $q=e^{2\pi i\tau}$, one formal series of $\tau-\alpha$ for each rational $\alpha$; in the variable $\tau$ they make up a higher depth strong quantum modular form in the sense of D. Zagier.