L4

Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.

Joint with Sean Jaffe.

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.

 

The ’t Hooft expansion is a powerful organizational framework for understanding QFTs as perturbations away from the large N limit and has deep connections to string theory and holography. In this talk, I will discuss categorical aspects of the ’t Hooft expansion, i.e. what one learns about topological defects from the ’t Hooft expansion and, correspondingly, topological strings and twisted holography. This talk is based off the paper arXiv:2411.00760 from last year as well as the more recent review paper arXiv:2511.19776.

I will discuss some of my work on thermal correlators in AdS/CFT. In particular, given a thermal correlator, how are the characteristic properties of bulk black holes encoded in such correlators? This includes exploring the spectrum of QNMs, the so-called thermal product formula, the photon ring, and geodesics bouncing off the black hole singularity. I will discuss how the latter might change when finite string effects are considered.

There will be no journal club this week to avoid conflicting with FPUK.

I will discuss a two-dimensional dilaton gravity theory with a sine potential. At the disk level, this theory admits a microscopic holographic realization as the double-scaled SYK model. Remarkably, in the open channel canonical quantization of the theory, the momentum conjugate to the length of two-sided Cauchy slices becomes periodic. As a result, the ERB length in sine dilaton gravity is discretized upon gauging this symmetry. For closed Cauchy slices, a similar discretization occurs in the physical Hilbert space, corresponding to a discrete spectrum for the length of the necks of trumpet geometries. By appropriately gluing two such trumpets together, one can then construct a wormhole geometry in sine dilaton gravity, whose amplitude matches the spectral correlation functions of a one-cut matrix integral. This correspondence suggests that the theory provides a path integral formulation of q-deformed JT gravity, where the matrix size is large but finite. Finally, I will describe how this theory of gravity can be regarded as a realization of q-deformed holography and propose a possible implementation of this framework to study the near-horizon dynamics of near-extremal de Sitter black holes.

The organisers asked me to give a brief talk on what I’ve been thinking about lately. So, I’ll tell you about Schwinger-Keldysh EFTs: an EFT framework for non-equilibrium dissipative systems such as hydrodynamics. These are built on a closed-time contour that runs forward and backward in time, allowing access to a variety of non-equilibrium observables. However, these EFTs fundamentally miss a wider class of observables, called out-of-time-ordered correlators (OTOCs), which are closely tied to quantum chaos. In this talk, I’ll share some thoughts on extending Schwinger-Keldysh EFTs to multifold contours that capture such observables. I’ll also touch on the discrete KMS symmetry of thermal systems, which generalises from Z_2 in the single-fold case to the dihedral group D_n in the n-fold case. With any luck, I’ll reach the point where I’m stuck and you can help me figure it out.