Main Session

We study path integrals of M-theory beyond the semi-classical limit using the AdS/CFT correspondence, which equates them to supersymmetric partition functions of 3d \(\mathcal{N}=2\) holographic SCFTs arising on the worldvolume of \(N\) coincident M2-branes. Focusing on the squashed \(S^3_b\) partition function of Chern-Simons-matter quiver gauge theories dual to M-theory on \(\text{AlEAdS}_4 \times Y_7\), we conjecture that the all-order large \(N\) perturbative expansion resums into a compact expression in terms of an Airy function, valid for generic values of the squashing parameter \(b\) and real mass deformations \(\xi\). This Airy conjecture generalizes previous results known only for special examples to a broad class of M2-brane theories. We discuss several motivations: the conjecture encodes integrated CFT correlators including the stress-tensor two-point coefficient \(C_T\) to all orders in \(1/N\), serves as a building block for other supersymmetric partition functions via holomorphic factorization, and admits a reorganization as a genus expansion probing IIA string free energies in the 't Hooft limit. We review existing analytic derivations based on topological strings and the Fermi gas formalism, and present new evidence through direct numerical integration of matrix models, saddle point analysis, and the relation to the Cardy-like limit of the superconformal index.

I will discuss how the Coulomb branch solution of the effective 4d theory can be used to classify 4d \(\mathcal{N}=2\), 5d \(\mathcal{N}=1\), and 6d \((1,0)\) theories. Applications to the classification of rank-one and rank-two theories will be discussed in detail.

Certain 4d and 5d supersymmetric gauge theories with eight supercharges admit a correspondence with Painlevé-type integrable systems, including q-Painlevé and elliptic Painlevé equations. In this framework, the associated tau functions can be expressed in terms of dual partition functions of the gauge theories. We propose a bootstrap approach that determines these tau functions and their bilinear relations using only perturbative data and symmetries. Applying this method, we obtain the bilinear relations for the tau functions associated with q-Painlevé and elliptic Painlevé equations.

The Schur operators of 4d \(\mathcal{N}=2\) SCFTs form a 2d vertex operator algebra (VOA). Recently, it has been proposed that this 4d SCFT/2d VOA correspondence can be understood via topologically twisted 3d \(\mathcal{N}=4\) SCFTs obtained by twisted compactification of the 4d SCFTs. In this talk, we discuss systematic methods for constructing these 3d topological field theories (TFTs) using BPS spectra and wall-crossing invariants of the 4d theories, focusing on Argyres-Douglas theories. We also extract the modular data of the VOAs supported on the boundary of the 3d theories, which are related to the Schur sectors of the 4d theories. This establishes a useful computational bridge between 4d SCFTs and 2d VOAs via interpolating 3d TFTs.

In this talk I will present examples of SUSY partition functions in various dimensions that, despite appearances, are not ordinary functions but are naturally distributional. I will show how taking this point of view seriously allows us to interpret singularities and better understand what these objects are really telling us.

In this short talk I will discuss the process of channel decompositions of 4d \(\mathcal{N}=2\) class S theories of type \(\mathfrak{su}(N)\), from the perspective of the 3d \(\mathcal{N}=4\) mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological 4d configurations, corresponding to spheres with non maximal punctures, whose 3d \(\mathcal{N}=4\) mirror star shaped quivers are bad theories. I will explain how in these cases the weakly-coupled description arises dynamically through Higgs mechanisms, triggered by operators acquiring VEVs. I will introduce a novel family of bad theories characterized by a partition function with a particular distributional nature, labelled broken.

In this talk, I will present a framework relating torus knots to characters of 2d RCFTs. Using the 3d-3d correspondence, we associate a 3d \(\mathcal{N}=2\) theory to the complement of a torus knot. In the IR, this theory flows to a unitary TQFT or a 3d \(\mathcal{N}=4\) rank-0 SCFT, which can realize a non-unitary TQFT. By computing the half-index, we can find a character of the RCFT on the boundary. We find that the half index of a theory associated with a torus knot complement is a character of Virasoro minimal model or a related RCFT.

I will present a new fermionic sum representation of the Macdonald index for a family of Argyres-Douglas theories. This expression emerges naturally from the three-dimensional theory produced by twisted compactification, providing strong support for the proposed formula and also giving a hint towards the IR formula for the Macdonald index in terms of 4d Coulomb branch BPS particle data.

BPS quivers of 4d \(\mathcal{N}=2\) QFTs are graphs described by a skew-symmetric adjacency matrix that encodes the worldvolume quantum mechanics of BPS particles, allowing for the computation of stable states in different chambers of the Coulomb branch related by wall-crossing phenomena. A rather different type of quivers, with symmetric adjacency matrix and trivial potential, describe BPS states of a certain class of 3d \(\mathcal{N}=2\) QFTs. In this talk I will review how each of these descriptions arises in M-theory, and will discuss how these two seemingly very different types of quivers are actually related, both by geometric constructions of 4d and 3d \(\mathcal{N}=2\) QFTs, and by well-known effects in gauge theories with a theta term.

The SymTFT construction relates different global forms of \(d\)-dimensional QFTs with discrete symmetry: realizing different global forms as a \((d+1)\)-dimensional TFT on an interval, with a common "physical" boundary condition on one side, and different topological boundary conditions on the other. In my talk, I will describe an analogue of the SymTFT which relates theories with the same "perturbative equations of motion," but different non-perturbative completions. I will start with a lightning overview of conformal blocks and relative QFT, then explain the BV formalism in some detail -- focusing on the elegant (super)geometric story in 0d for simplicity. I will argue that there is a natural 1d cohomological TFT (called the ThimTFT) associated to the classical action \(S\), with different topological boundary conditions described by convergent path-integral contours in a complexified field space. Time permitting, I will discuss extensions to higher dimensions. Based on WIP.

The most basic invariant of 2d \(\mathcal{N}=(0,1)\) theories is the elliptic genus. I will talk about subtler invariants, and their implications in heterotic string theory and cobordism conjecture.

We classify all large \(N\) supersymmetric gauge theories with a simple gauge group that flow to interacting 4d SCFTs, finding three universal classes: Type I, with \(a \neq c\), scaling linearly in \(N\) and a dense operator spectrum, and Type II/III, with \(a = c\), quadratic scaling, and a sparse spectrum. For Type II/III, the large \(N\) superconformal index exhibits a universal form determined by 't Hooft anomalies, reproducing the Bekenstein–Hawking entropy of AdS black holes. We also introduce an index-analog of the spectral form factor, providing a toy model for the black hole information paradox in higher dimensions.

In the first half of talk, I will review some basic properties of giant gravitons, previous connections with holography through backreacted SUGRA background, aspects of toric Calabi-Yau geometries, and finally giant graviton expansion of superconformal indices. In the second half, we will use these ingredients to establish a couple of new exact results based on arXiv:2412.05357 and arXiv:2601.17114. The first is a precise correspondence between the giant graviton expansion of the superconformal index of field theories in \(D \leq 4\), and the master volume formalism of Gauntlett, Martelli and Sparks (GMS) which determines the near horizon geometries of certain BPS black holes and black strings in supergravity. The wrapping numbers of a generic giant graviton configuration are identified with Kähler moduli of the corresponding metric and we will show large \(N\) the entropy function of the relevant \(\text{AdS}_5 \times \text{SE}_5\) BPS rotating black hole is recovered by extremising over these moduli. The other one result is a precise relation between the superconformal indices of two theories defined in different spacetime dimensions. The first is the 3d ABJM SCFT arising from parallel M2 membranes, the other is the 6d \((2,0)\) theory describing the world volume of parallel M5-branes in the same background. Such a relation is closely related to Imamura's proposal for the giant graviton expansion of the three-dimensional index. We check our conjecture against various results for the two indices available in the literature.