We propose a triality relating the Double-Scaled SYK model, SL(2$\textit{,}$ ℂ) Chern-Simons theory on a disk with an irregular singularity at the center and the outcome of “real Schur quantization” applied to SU(2) Seiberg-Witten theory with Neumann boundary conditions. We give supporting evidence for our conjecture by establishing a precise match between a general class of correlators in all three systems.

Canada

Davide Gaiotto

United States

Herman Verlinde

SYK-Schur duality: double scaled SYK correlators from $$ \mathcal{N} $$ = 2 supersymmetric gauge theory

We discuss the role of formal deformation theory in quantum field theories and present various “higher operations” which control their deformations, (generalized) OPEs, and anomalies. Particular attention is paid to holomorphic-topological theories where we systematically describe and regularize the Feynman diagrams which compute these higher operations in free and perturbative scenarios, including examples with defects. We prove geometrically that the resulting higher operations satisfy expected “quadratic axioms,” which can be interpreted physically as a form of Wess-Zumino consistency condition for BRST symmetry. We discuss a higher-dimensional analogue of Kontsevich’s formality theorem, which proves the absence of perturbative corrections in TQFTs with two or more topological directions. We discuss at some length the relation of our work to the theory of factorization algebras and provide an introduction to the subject for physicists.

Justin Kulp

United Kingdom

Jingxiang Wu

Higher operations in perturbation theory

We provide a direct combinatorial proof of a Feynman graph identity which implies a wide generalization of a formality theorem by Kontsevich. For a Feynman graph Γ, we associate to each vertex a position $\textit{x}$ ∈ $\textit{ℝ}$ and to each edge $\textit{e}$ the combination $$ {s}_e={a}_e^{-\frac{1}{2}}\left({x}_e^{+}-{x}_e^{-}\right) $$ , where $$ {x}_e^{\pm } $$ are the positions of the two end vertices of $\textit{e}$, and $\textit{a}$ is a Schwinger parameter. The “topological propagator” $$ {P}_e={e}^{-{s}_e^2}{\textrm{d}s}_e $$ includes a part proportional to d$\textit{x}$ and a part proportional to d$\textit{a}$ . Integrating the product of all $\textit{P}$ over positions produces a differential form $\textit{α}$ in the variables $\textit{a}$ . We derive an explicit combinatorial formula for $\textit{α}$ , and we prove that $\textit{α}$ ∧ $\textit{α}$ = 0 for all graphs except for trees.

Paul-Hermann Balduf

Combinatorial proof of a non-renormalization theorem

We define and compute algebraically a “perturbative part” of protected sphere correlation functions in the M2 brane SCFTs. These correlation functions are expected to have a holographic description in terms of twisted, Ω-deformed M-theory. We uncover a hidden perturbative triality symmetry which supports this conjecture. We also discuss some variants of the setup, involving M2 branes at singularities and D3 branes with a transverse compact direction.

Jacob Abajian

Twisted M2 brane holography and sphere correlation functions

We propose a general strategy to build three-dimensional gauge theories with four supercharges which enjoy a supersymmetry enhancement in the IR. The resulting IR SCFTs admit topological twists with particularly nice properties, as well as boundary rational chiral algebras such that the associated Modular Tensor Categories are controlled by the topological twist. The theories arise from a twisted circle compactification of four-dimensional theories of Argyres-Douglas type. We develop a novel algorithm to compute or manipulate protected quantities associated to these theories, such as ellipsoid partition functions and superconformal indices and half-indices.

South Korea

Heeyeon Kim

3D TFTs from 4d

We employ the protected sphere correlation functions of three-dimensional Super Conformal Field Theories with eight supercharges in order to define a quantization of their Higgs and Coulomb branches of vacua as real phase spaces. We also employ hemisphere correlation functions to define a quantization of certain real loci in the Higgs and Coulomb branches. Localization formulae and dualities applied to these quantizations result in a body of predictions about unitary representations of certain algebras, which may perhaps be understood as an “analytic” form of the symplectic duality program. In particular, the protected correlation functions in the class of theories denoted as [ ] are naturally related to the theory of unitary representations of complex or real semi-simple Lie groups.

Sphere quantization of Higgs and Coulomb branches and Analytic Symplectic Duality

We derive and test a novel holographic duality in the B-model topological string theory. The duality relates the B-model on certain Calabi-Yau three-folds to two-dimensional chiral algebras defined as gauged systems. The duality conjecturally captures a topological sector of more familiar AdS /CFT holographic dualities.

Kevin Costello

Twisted holography

We explore the properties of Ω-deformed M-theory, with particular focus on the   <math> <msub> <mi>ℂ</mi> <msub> <mi>ϵ</mi> <mn>1</mn> </msub> </msub> </math>  $$ {\mathbb{C}}_{\epsilon_1} $$  ×   <math> <msub> <mi>ℂ</mi> <msub> <mi>ϵ</mi> <mn>2</mn> </msub> </msub> </math>  $$ {\mathbb{C}}_{\epsilon_2} $$  ×   <math> <msub> <mi>ℂ</mi> <msub> <mi>ϵ</mi> <mn>3</mn> </msub> </msub> </math>  $$ {\mathbb{C}}_{\epsilon_3} $$  background and coupling to Ω-deformed M2 and M5 brane world-volume theories.

Jihwan Oh

Aspects of Ω-deformed M-theory

We employ two-dimensional chiral algebra techniques to produce solutions of certain differential and integral equations which occur in the context of the Analytic Geometric Langlands Program. In particular, we build “Multiplication Kernels” K 3(x, x′, x′′) which intertwine the action of   <math> <mi>s</mi> <msub> <mi>l</mi> <mn>2</mn> </msub> </math>  $$ \mathfrak{s}{\mathfrak{l}}_2 $$  Gaudin Hamiltonians on three sets of variables.

Vertex algebra constructions for (analytic) geometric Langlands in genus zero

We study ’t Hooft lines in four-dimensional holomorphic-topological Chern-Simons theory. We relate them to Q-operators in the theory of integrable systems. We give a physical interpretation of the fundamental TQ and QQ relations satisfied by Q-operators and conventional transfer matrices.