Program

Si Li

Heat kernel, Picard–Lefschetz and Alien 

We explain the resurgent structure of short-time heat kernel asymptotics from the viewpoint of Picard–Lefschetz theory. For a real analytic Riemannian manifold, the heat kernel admits a 1-Gevrey small-time expansion whose Borel transform detects complex-geometric data beyond the real geodesic sector. We formulate an infinite-dimensional Picard–Lefschetz problem of Morse–Floer type on the complexified path space, and propose a heat-kernel analogue of the Picard–Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat-kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. Joint work with Xinxing Tang and Yong Li.  

Sungjay Lee

Resurgence of the Thermal Transition between Bounce and Sphaleron

We study the thermal transition between the bounce and the sphaleron in quantum mechanics with a metastable vacuum from the viewpoint of Borel resurgence. For two models representing a second-order and a first-order transition, we compute the perturbative expansion of the thermal free energy to high orders and extract the leading Borel singularity data (A,b,S) as functions of temperature. The Borel singularity location A reproduces the on-shell action of the dominant saddle on both sides of the transition, joining smoothly in the second-order case and developing a kink in the first-order case. The characteristic exponent b jumps between 0 and 1/2 across the transition, counting the zero modes of the corresponding saddle. The Stokes constant S matches the one-loop determinant around the saddle. The perturbative expansion around the false vacuum thus determines the transition temperature, the order of the transition, and the decay rate including the one-loop prefactor without relying on semiclassical inputs. 

Wei Li

Structure of N=4 chiral algebra

The N=4 chiral algebra is a vertex operator algebra generated by the Schur subsector of 4D N=4 SYM, and it can be obtained by applying the holomorphic-topological twist to the theory in the presence of the Omega-background. I will view the N=4 chiral algebra as an abstract VOA and determine its structure, using only the associativity of the OPE. We find that the algebra is uniquely characterized by the central charge (which can take an arbitrary value), without any additional free parameter. This rigidity result can help prove the twisted holography. Furthermore, the truncation pattern of the OPE coefficients suggests that the algebra cannot directly arise from the symmetric orbifold. 

Yiwen Pan

Modularity in Schur index

In this talk we will discuss topics centering around the modularity of the (defect) Schur index of 4d N=2 SCFTs. We will review old and introduce new exact analytic methods computing the indices, and we are able to produce the representation theory of some quasi-lisse associated VOAs. The characters, the set of (un)flavored modular differential equations they satisfy, and the null states that generate the equations, transform under the modular group. Finally we relate the modular structure of the characters with the fixed-varieties of the Coulomb branch (circle compactified).

Heeyeon Kim

The Minimal 3d SCFT and an Exceptional Family of Vertex Algebras

I will discuss a three-dimensional field theory construction that uniformly realizes an exceptional family of intermediate vertex subalgebras and the related rational minimal W-algebras of the Deligne-Cvitanović series as the boundary algebras. The construction is based on the minimal three-dimensional N=4 superconformal field theory coupled to a topological field theory. 

Mikhail Bershtein

Coset decomposition and blowup equations

We revisit the classical Goddard-Kent-Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators between these vectors. This leads to relations on conformal blocks.  Due to the AGT relation, these relations are equivalent to blowup relations on Nekrasov partition functions with the presence of the surface defect. These relations can be used to prove Kyiv formulas for the Painlev\'e tau-functions  (following Nekrasov's method). 

Jean-Emile Bourgine

Chiral cluster seeds and deformed W-algebras

In the formalism of chiral cluster seeds, recently introduced by M. Bershtein, quantum cluster variables are replaced with vertex operators, and a decorated quiver encodes their operator product expansion. In this talk, I will show how this framework can be used to study (q,t)-deformed W-algebras. I will explain how a quiver can be associated with a given free field realization, and how quiver mutations relate different realizations, focusing on the examples of regular algebras W_{q,t}(gl(N|M)), subregular algebras W_{q,t}^{sub}(gl(N)), and quantum affine sl(N) algebras. This formalism can also be used to understand relations between different deformed W-algebras. As an example, I will present an embedding of the algebra W^{sub}_{q,t}(sl(N))$ into the algebra W_{q,t}(sl(N)) tensored with a rank-two Heisenberg algebra. This embedding can be viewed as a deformed analogue of inverse quantum Hamiltonian reduction. Finally, I will also show how the algebras W_{q,t}^{sub}(gl(N)) and W_{q,t}(gl(1|N)) are related in this framework. 

Yehao Zhou

Critical stable envelopes, Yangians, and cohomological Hall algebras

Given a symmetric quiver Q with potential W, I will describe in this talk a construction of a (shifted) Yangian Y(Q,W) and its underlying Lie algebra g(Q,W) using the critical stable envelopes and R-matrices. I will also talk about relation between Y(Q,W) and the cohomological Hall algebra, and between g(Q,W) and the BPS Lie algebra. This is based on works 2512.23929 and 2601.01518 with Cao, Okounkov, and Zhou, and work in progress with Botta, Davison, and Zhu. 

Wenjun Niu

New classical and quantum R-matrices on higher genus curves

A spectral (dynamical) R-matrix is a solution to the spectral (dynamical) Yang-Baxter equation. In the case of simple Lie algebras and no dynamical parameters, Belavin-Drinfeld classification restricts the space of spectral parameters to be a curve of genus at most one. In the dynamical case, one usually classifies solutions where the dynamical parameters are restricted to lie in the Cartan subalgebra. Costello-Witten-Yamazaki gives physical interpretation to these R-matrices and the classification results in terms of the 4 dimensional Chern-Simons theory. Following their ideas, I will explain how to construct new classical and quantum dynamical R-matrices over higher genus curves, based on Kapustin’s twist of 4d N=2 gauge theories. These R-matrices are smooth (rather than holomorphic) functions valued in the cotangent Lie algebra of a simple Lie algebra, and their space of dynamical parameters can be taken to be in some open neighborhood of the moduli space of G-bundles. I will compare this construction and traditional method of obtaining spectral dynamical R-matrices, and show that it provides clarity to the traditional method. This is based on joint work to appear with R. Abedin and F. Moosavian.  

Takahiro Nishinaka

Chern-Simons matter theories associated with the R-twisted 3d reduction of Argyres-Douglas theories 

It has recently been pointed out that some 3d N=2 Chern-Simons (CS) matter theories can flow to the same IR fixed points as the R-twisted circle reduction of 4d N=2 superconformal field theories (SCFTs) called Argyres-Douglas (AD) theories. This particularly means that, when an appropriate monopole superpotential is turned on, the N=2 supersymmetry of the CS matter theories is enhanced to N=4 in the infrared. Furthermore, it reproduces the vertex operator algebra (VOA) associated with the parent 4d N=2 SCFT as the VOA associated with the 3d N=4 IR fixed point defined on a half-space. In this talk, I will discuss our recent work on a small but non-trivial generalization of this story. 

Pietro Longhi

BPS states in coupled 3d-5d systems, quantum curves and skein-valued uplifts

In this talk I will discuss aspects of BPS states in 3d-5d systems. When a 3d N=2 QFT is coupled to an ambient 5d N=1 QFT, their respective BPS sectors become part of a broader spectrum of 3d-5d BPS states. In 5d systems arising from M-theory CY3 compactifications, 3d N=2 defect QFTs can be introduced by considering M5 branes supported on Lagrangian submanifolds. In this setting both 3d and 5d BPS states have well-understood geometric duals corresponding respectively to open and closed topological string sectors. An interpretation of the broader 3d-5d spectrum can be provided on the mirror geometry through the framework of exponential networks. 

Networks compute 3d-5d states, and through their wall-crossing they capture the spectra of both 5d and 3d BPS states at genus zero. The extension to higher genus invariants leads to a quantization of the mirror geometry in which 3d-5d states play the role of Stokes data. Recently, a skein-valued generalization of quantum curves for stacked M5 branes has been found to exist for several geometric configurations. If time permits, I will discuss the skein-valued trace map—an extension of spectral networks establishing a map between skein modules associated with UV and IR Coulomb phases of the ambient QFT. 

Fabrizio Del Monte

BPS State Counting on Local Calabi-Yau Threefolds

In this talk I will report new results on BPS counting for local Calabi-Yau threefolds that permit the complete solution of the BPS spectral problem, i.e. the computation of all semistable states and their refined BPS invariants (motivic DT) for infinite classes of cases, at the same time challenging some widespread beliefs regarding M-theoretic geometric engineering. 

First, I will present a theorem on how to induce stability conditions on (resolved) orbifolds of local CY3s and its implications for the spectrum, as wel as relations with the BPS Riemann-Hilbert problem and WKB for q-difference equations. As an application, I will give a closed formula for the spectrum of stable BPS states and for the Kontsevich–Soibelman wall-crossing invariant for the local Calabi–Yau threefolds Y^{(N,0)} for any N, geometrically engineering 5d SU(N) super Yang-Mills.

I will conclude by applying the same techniques to orbifold of nontoric CY3s, and describe some entirely novel aspects of the resulting physical theories and their geometric origin. These include a surprising infinite family of rank-1 theories that evades all known classifications. 

Philsang Yoo

A Geometric Model for Tensor-Product BPS Quivers

BPS quivers are one of the most fundamental invariants of four-dimensional N=2 supersymmetric field theory. In this talk, I will discuss the BPS quivers of a class of such theories, originating from Argyres-Douglas theories associated with pairs of ADE diagrams, and their vast generalizations. The goal is to explain how these BPS quivers arise from Fukaya categories of certain Calabi–Yau manifolds. This is based on joint work in progress with Sangjin Lee. 

Eric Sharpe

Generalized symmetries and gauge theory multiverses: an introduction to decomposition

In this talk, I will outline higher-form symmetries in quantum field theories, focusing on local d-dimensional theories with (d-1)-form symmetries. Such theories are equivalent to superpositions (disjoint unions) of quantum field theories, and frequently exhibit consistent instanton truncations, naively violating Weinberg's old postulate.

This equivalence is known as `decomposition,' with the constituent subtheories (analogous to but distinct from superselection sectors) known as universes. I will outline some basic examples and applications, including to consistent anomaly restriction, as well as recent results involving continuous families of universes. 

Yuto Moriwaki

Hilbert-Space Refinements of Factorization Algebras in Conformal Field Theory

Factorization algebras provide a natural framework for describing local observables in quantum field theory, yet it is not clear in general how a physical Hilbert space should emerge from such a structure. In this talk, we discuss this question through the prefactorization algebra associated with the conformal Laplacian, a model of free conformal field theory.

Motivated by reflection positivity, we observe that sections over unit disks, with suitable boundary conditions, naturally embed into a Hilbert Fock space. However, the structure maps of naive factorization algebra are generally unbounded in the Hilbert-space norm. We show that, after imposing natural geometric conditions on conformal embeddings and disk configurations, this structure admits a bounded refinement valued in ind-Hilbert spaces. We also discuss connections with factorization homology, sphere partition functions, and the Heisenberg vertex operator algebra. 

Mauricio Romo

Mirror symmetry for Calabi-Yau singular double covers and their pre-quotients

I will discuss mirror symmetry of singular CY double covers, originally proposed by Hosono-Lee-Lian-Yau (HLLY) from the point of view of gauged linear sigma models. I will argue that a physics realization of noncommutative (nc) resolutions can be used to characterize the mirror of such singular double covers and make computation of B-brane central charges and other properties, very explicit. I will then present a generalization of  the construction of nc resolutions to more general covers than HLLY and a conjecture relating their A-periods to certain smooth CICY varieties. 

Meng-Chwan Tan

Topological-Holomorphic N=4 Gauge Theory: From Langlands Duality of Holomorphic Invariants to Mirror Symmetry of Quasi-topological Strings

In this talk, we will consider a topological-holomorphic twist of N=4 supersymmetric gauge theory on a four-manifold of the form  M4 = Sigma_1 x  Sigma_2, and elucidate the mathematical implications of its physics. In particular,  we will study the cohomology of different linear combinations of its resulting scalar supercharges under S-duality, and exploit its topological-holomorphic nature,  in order to (i) derive novel topological and holomorphic invariants of M4 and their Langlands duals; (ii) relate these 4d invariants and their Langlands duals to the mirror symmetry of Higgs bundles and that of quasi-topological strings described by the sheaf of chiral differential operators; and (iii) obtain a fundamental understanding from 4d gauge theory, why chiral differential operators are purely perturbative objects whence mathematicians have not been able to construct their quantum cohomology.    

Uhi Rinn Suh

SUSY W-algebras and the N=2 W infinity algebra 

SUSY W-algebras arise as Hamiltonian reductions of vertex superalgebras and are naturally endowed with a super-conformal vector. In particular, the principal SUSY W-algebras of type A are known to be isomorphic to the corresponding ordinary W-superalgebras. Nevertheless, they provide a useful framework for revealing the underlying N=2 structure and for understanding these algebras as supersymmetric analogues of the W_N-algebras.

In this talk, I will introduce the mathematical construction of SUSY W-algebras and the N=2 W infinity algebra, which can be viewed as the supersymmetric analogue of the W infinity algebra. 

Cyril Closset

Three-dimensional TQFTs from Argyres-Douglas SCFTs via the 3d/3d correspondence

I will discuss new work on the dimensional reduction of 4d N=2 SCFTs of Argyres-Douglas type to 3d N=4 with a U(1)_r twist. Upon further 3d topological twisting, one thus constructs 3d TQFTs that admit the VOA of the Beem et al. VOA/SCFT correspondence and their (Galois) cousins at its holomorphic boundary. We construct these 3d SQFT and TQFTs as 3d N=2 abelian Chern-Simons-matter theories of Dimofte--Gaiotto--Gukov type using the 3d/3d correspondence on a `punctured' lens space. We give very fine checks of our proposal using supersymmetric partition functions on Seifert manifolds. A particular care with the phase of the partition function allows us to keep track of the central charge of the (potential) boundary VOAs. [Based on work with Adam Keyes and Sungjoon Kim.] 

Dongmin Gang

Refined 3D Index

After introducing the 3D index based on a Dehn surgery representation and ideal triangulation of a 3-manifold, I will present a refined version of this index. This refinement naturally arises from an accidental symmetry in the 3D gauge theory associated with the 3-manifold. 

Sunghyuk Park

Flow loops and quantum groups

BPS q-series, also known as ẑ-invariants, are physically predicted 3-manifold invariants that unify various quantum invariants; for knot complements, they recover all colored Jones polynomials, as well as ADO invariants. In this talk, I will explain how, for fibered knot complements, these series — usually constructed using Verma modules for quantum groups — can instead be obtained by counting Morse flow loops, and how this bridge arises from topological string theory.  

Daniel Bryan

Wall-crossing, unlinking and Yangian algebras in 4d and 3d N=2 theories 

In this talk I will explain a general relation between the BPS spectrum of 4d N=2 theories encoded by the BPS quiver and that of 3d N=2 theories associated to the doubled symmetric quiver. 

This is derived by formulating the Kontsevich-Soibelman wall-crossing formula as a duality between two 3d theories. These are described by the doubled BPS quiver in the minimal chamber and a new infinite symmetric quiver on the other side of the wall.  For special 4d framed states the formulation simplifies into a step by step unlinking process which realises modules of the quiver Yangian. 

This talk is based on joined work with Wei Li and Piotr Sułkowski.