Algebras of "symmetries" in QFT and applications to solitons (1)
2025-02-18 16:15 - 18:15 (America/New_York) | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard CMSA and Math
Title: Algebras of “symmetries” in QFT and applications to solitons (1)
https://arxiv.org/abs/2403.08883

CMSA Q&A Seminar: Phillip Matchett Wood
2025-02-19 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Phillip Matchett Wood, Harvard University
Topic: Info session on the CMSA/Mathematics Summer REU Program (Research Experience for Undergraduates)

Toward constructing a large-scale quantum computer based on TQFT
2025-02-24 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Liyuan Chen ( Harvard)
Title: Toward constructing a large-scale quantum computer based on TQFT
Abstract: Topological quantum computation, motivated by topological quantum field theory (TQFT), offers a promising path toward fault-tolerant universal quantum computation. However, the practical realization of such a system remains challenging due to the difficulty of finding suitable topological materials. In this work, we provide a comprehensive blueprint for constructing a fault-tolerant universal quantum computer based on the quantum double model $mathcal{D}(S_3)$, a specific non-Abelian topological order. We implement logical computations using quantum circuits on qubits and qutrits, including a single non-Clifford gate, compatible with near-term quantum devices. This work bridges the gap between abstract mathematical frameworks and noise-resilient quantum computation on near-term devices. Our proposal offers a promising path to realize a non-Abelian anyon-based large scale quantum computer.

Algebras of "symmetries" in QFT and applications to solitons (2)
2025-02-25 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard CMSA and Math
Title: Algebras of “symmetries” in QFT and applications to solitons (2)
https://arxiv.org/abs/2403.08883

Datasets for Math: From AIMO Competitions to Math Copilots for Research
2025-02-26 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Simon Frieder, Oxford
Title: Datasets for Math: From AIMO Competitions to Math Copilots for Research
Abstract: This talk begins with a brief exposition of the AI Mathematical Olympiad (AIMO) on Kaggle, now in its second iteration, outlining datasets and models available to contestants. Taking a broader perspective, I then examine 1) the overarching issues the current datasets suffer from—such as binary evaluation or constrained sets of use cases— and 2) the trajectory they set for competition-style mathematical problem-solving, which is different from mathematical research practice. I argue for a fundamental shift in dataset structure and composition, both for training and evaluation, and introduce the idea of mapping mathematical workflows to data, a key example underscoring the need for this shift. I touch upon new thinking LLMs and their role in redefining LLM math evaluation, highlighting their implications for dataset design. Finally, I propose general improvements to the current state of mathematical datasets, including mathematical adaptations of dataset documentation (e.g., datasheets).

The Combinatorics of the Amplituhedron – Tiles, Tilings, and Cluster Algebras
2025-02-28 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Matteo Parisi
Title: The Combinatorics of the Amplituhedron – Tiles, Tilings, and Cluster Algebras
Abstract: The amplituhedron is the image of the positive Grassmannian—the region of the Grassmannian where all Plücker coordinates are nonnegative—under a totally positive linear map. It is a far-reaching generalization of cyclic polytopes and hyperplane arrangements, and the positive Grassmannian itself. The “volume” of the amplituhedron encodes probabilities of particle interactions in the quantum field theory N=4 super Yang-Mills, and calculating this volume involves decomposing (or tiling) the amplituhedron into smaller pieces (or tiles) and summing their volumes. This talk will delve into the rich combinatorics of these tiles and tilings, presenting recent results on some of the central conjectures in this area, including the magic number, BCFW tiling, and cluster adjacency conjectures.

Quantum Field Theory and Physical Mathematics
2025-03-03 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Kai Xu, Harvard
Title: Finite Landscape of 6d N=(1,0) Supergravity
Abstract: We present a bottom-up argument showing that the number of massless fields in six-dimensional quantum gravitational theories with eight supercharges is uniformly bounded. Specifically, we show that the number of tensor multiplets is bounded by T≤193, and the rank of the gauge group is restricted to r(V)≤480. Given that F-theory compactifications on elliptic CY 3-folds are a subset, this provides a bound on the Hodge numbers of elliptic CY 3-folds: h1,1(CY3)≤491, h1,1(Base)≤194 which are saturated by special elliptic CY 3-folds. This establishes that our bounds are sharp and also provides further evidence for the string lamppost principle.

Large value estimates in number theory and computer science
2025-03-03 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Larry Guth, MIT
Title: Large value estimates in number theory and computer science
Abstract: A large value estimate for a matrix M is a simple type of estimate in quantitative linear algebra. Estimates of this type appear in many parts of math, both pure and applied. One example is the large value problem for Dirichlet polynomials from analytic number theory, which is related to estimates about the zeroes of the Riemann zeta function. We will also give some examples from computer science. Many large value problems are difficult. On the pure math side, the sharp conjecture about large values of Dirichlet polynomials has been open for a long time and is out of reach of current methods. On the computer science side, we don’t know any efficient algorithm to approximately solve the large value problem for a given matrix M. Many experts think that such an algorithm does not exist. In this talk we will survey how large value estimates come up, the known methods for working on them, and some of the obstacles to fully understanding them.

CMSA Q&A Seminar: Puskar Mondal
2025-03-05 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Puskar Mondal, Harvard CMSA
Topic: What is the positive energy theorem?

Machine Learning G2 Geometry
2025-03-05 14:00 - 15:00 | Hybrid
New Technologies in Mathematics Seminar
Speaker: Elli Heyes, Imperial College
Title: Machine Learning G2 Geometry
Abstract: Compact Ricci-flat Calabi-Yau and holonomy G2 manifolds appear in string and M-theory respectively as descriptions of the extra spatial dimensions that arise in the theories. Since 2017 machine-learning techniques have been applied extensively to study Calabi-Yau manifolds but until 2024 no similar work had been carried out on holonomy G2 manifolds. In this talk, I will firstly show how topological properties of these manifolds can be learnt using neural networks. I will then discuss how one could try to numerically learn metrics on compact holonomy G2 manifolds using machine-learning and why these approximations would be useful in M-theory.

A Tetrahedral Approach to Calabi-Yau Geometry
2025-03-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Charles Doran, CMSA
Title: A Tetrahedral Approach to Calabi-Yau Geometry
Abstract: We will open with a quick introduction to the what and why of Calabi-Yau geometry. Following this, we will consider the problem of deforming tetrahedra while preserving the areas of their faces, following our noses to discover a beautiful path to elliptic curves, K3 surfaces, and beyond. Time permitting, we will also discuss motivations and applications across physics. The talk should be broadly accessible.

Comments on Non-Invertible Symmetries in K3 CFTs and the Conway Moonshine Module
2025-03-10 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Sarah Harrison, Northeastern
Title: Comments on Non-Invertible Symmetries in K3 CFTs and the Conway Moonshine Module
Abstract: There is an established connection between discrete symmetry groups of K3 non-linear sigma models and a distinguished N=1 chiral SCFT called the Conway moonshine module. More specifically, all symmetry groups of K3 NLSMs preserving the N=4 superconformal algebra can be obtained as subgroups of “Conway zero”, the group of symmetries of the Conway module, and their explicit action on the BPS spectrum can (almost always) be obtained via traces in the Conway module. A natural question is whether this relation extends to fusion category symmetry of these theories. I will discuss positive evidence in this direction, by exploring examples of non-invertible topological defect lines in K3 NLSMs and the Conway module. This is based on work in progress with R. Angius, S. Giaccari, and R. Volpato.

Introduction to the probabilistic approach to Louville theory
2025-03-11 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Leon Liu, Harvard
Title: Introduction to the probabilistic approach to Louville theory
Abstract: I will give an introduction to the probabilistic approach to Louville theory, following Hairer’s notes.

CMSA Q&A Seminar: Dan Freed
2025-03-12 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Dan Freed, Harvard University
Topic: What are spectra (in homotopy theory)?

Discovery in Mathematics with Automated Conjecturing
2025-03-12 14:00 - 15:00 | Hybrid – G10
New Technologies in Mathematics Seminar
Speaker: Randy Davila, RelationalAI and Rice University
Title: Discovery in Mathematics with Automated Conjecturing
Abstract: Automated conjecturing is a form of artificial intelligence that applies heuristic-driven methods to mathematical discovery. Since the late 1980s, systems such as Fajtlowicz’s Graffiti, DeLaViña’s Graffiti.pc, and TxGraffiti have collectively contributed to over 130 publications in mathematical journals. In this talk, we outline the evolution of automated conjecturing, focusing on TxGraffiti, a program that employs linear optimization methods and several distinct heuristics to generate mathematically meaningful conjectures. We will then introduce GraphMind, a dueling framework where the Optimist proposes conjectures while the Pessimist seeks counterexamples, fostering a feedback loop that strengthens automated reasoning. Finally, we will present GraffitiAI, a Python package that extends automated conjecturing across various mathematical domains.
Bio: Randy R. Davila is a Lecturer in the Department of Computational Applied Mathematics & Operations Research at Rice University and a Library Engineer at RelationalAI, specializing in relational knowledge graph systems for intelligent data management. He earned his PhD in Mathematics from the University of Johannesburg in 2019, with research focused on graph theory and combinatorial optimization. His work explores artificial intelligence in mathematical conjecture generation, graph theory, and neural network applications to combinatorial problems. As the creator of TxGraffiti, he has developed AI-driven systems that have contributed to numerous mathematical publications. His recent projects include GraphMind, a dueling agent-based framework that pairs conjecture generation with counterexample discovery, and GraffitiAI, a Python package for automated conjecturing across mathematical disciplines.

Freedman CMSA Seminar: Michael Freedman (CMSA) & Elia Portnoy (MIT)
2025-03-12 15:00 - 17:00 | Hybrid – G10
Freedman CMSA Seminar
Speaker: Michael Freedman, Harvard CMSA (3:00–4:00 pm ET)
Title: How many links can you fit in a box?
Abstract: I’ll discuss a “made up” problem on the interface of topology and packing, which may well be classified as “recreational math”. Here is the first question suppose you have a unit box, how many unlinked (split) copies of the Hopf link (c_1,i,c_2,i) and be embedded so that for each copy the two components c_1,i and c_2,i maintain a distance of at least some fixed epsilon >0. Is this number even finite?

Speaker: Elia Portnoy, MIT (4:00–5:00 pm ET)
Title: An explicit packing of links in a box and some progress in quantitative embeddings
Abstract: Following Freedman’s talk, I’ll begin by showing how to pack a large number of links in a box with certain geometric and topological constraints (joint with Fedya Manin). If time permits, I’ll also discuss some progress and open questions for the following quantitative embedding problem: given a simplicial complex X, what is the smallest size of a map from X to R^n so that the preimage of each unit ball intersects a small constant number of simplices?

The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
2025-03-24 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Ka Ho Wong (Yale)
Title: The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
Abstract: In the early 2010s, Andersen and Kashaev defined a TQFT based on quantum Teichmuller theory. In particular, they define a partition function for every ordered ideal triangulation of hyperbolic knot complement in $mathbb{S}^3$ equipped with an angle structure. The Andersen-Kashaev volume conjecture suggests that the partition function can be expressed in terms of a Jones function of the knot which, in its semi-classical limit, decay exponentially with decay rate the hyperbolic volume of the knot complement. In this talk, we will introduce a purely combinatorial condition on triangulations which, together with the geometricity of the triangulations, imply the Andersen-Kashaev volume conjecture and its generalization. This talk is based on the joint work with Fathi Ben Aribi.

The Toda Lattice as a Soliton Gas
2025-03-24 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Amol Aggarwal, Columbia University
Title: The Toda Lattice as a Soliton Gas
Abstract: A basic tenet of integrable systems is that, under sufficiently irregular initial data, they can be thought of as dense collections of many solitons, or “soliton gases.” In this talk we focus on the Toda lattice, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system, under certain random initial data, can be interpreted through solitons, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.

AlphaProof: when reinforcement learning meets formal mathematics
2025-03-27 10:00 - 11:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Thomas Hubert (Google DeepMind)
Title: AlphaProof: when reinforcement learning meets formal mathematics
Abstract: Galileo, the renowned Italian astronomer, physicist, and mathematician, famously described mathematics as the language of the universe. Progress since only confirmed his intuition as the world we live in can be described with extreme precision with just a few mathematical equations.
In the last 70 years, the rise of computers has also enriched our understanding of and revolutionized the world we live in. Mathematics tremendously benefited from this digital revolution as well: while Gauss had to compute primes by hand, computers and computation are now routinely used in research mathematics and contribute to grand problems like the Birch and Swinnerton-Dyer conjecture, one of the Millennium Prize Problems.
Today, computers are entering a new age, one in which computation can be transformed into reasoning. In this talk, I would like to discuss two such developments that will undoubtedly have an integral role to play in the future of mathematics: the concurrent rise of formal mathematics and of machine intelligence.

The Competition Complexity of Dynamic Pricing
2025-03-28 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Tomer Ezra
Title: The Competition Complexity of Dynamic Pricing
Abstract: One of the most fundamental questions in mechanism design is the tradeoff between simplicity and optimality. A canonical example of this tradeoff is competition complexity in auctions, which quantifies how many additional bidders are needed for a simple mechanism to (approximately) match the revenue of the optimal mechanism.
In this talk, we analyze the competition complexity of dynamic pricing in the setting of selling a single item. We establish tight asymptotic guarantees for various scenarios, including when bidder values are i.i.d., independent, or correlated. Our results characterize the performance of different classes of dynamic pricing algorithms and provide insights into their effectiveness under varying market conditions.

Homotopical Methods for Free and Interacting Fermionic SPTs
2025-03-31 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Cameron Krulewski, MIT
Title: Homotopical Methods for Free and Interacting Fermionic SPTs
Abstract: We develop and compute homotopical “free-to-interacting maps” to compare classifications of fermionic symmetry-protected topological phases (SPTs), determining when such phases are stable under interactions or, alternatively, interaction-enabled. Generalizing work of Freed-Hopkins, we construct maps from K-theory to Anderson-dual spin bordism in two new situations: for weak phases, which are SPTs protected by discrete translation symmetry, and for the “Bott spiral” studied by Queiroz-Khalaf-Stern, who observed that the tenfold way classification of free theories breaks down to a large 2-torsion interacting classification. Along the way, we employ T-duality, generalizations of the spin orientation of KO-theory, and the Adams spectral sequence.
This talk is based on joint work with Omar Antolín Camarena, Arun Debray, Natalia Pacheco-Tallaj, Daniel Sheinbaum, and Luuk Stehouwer.

Introduction to the probabilistic approach to Louville theory III
2025-04-01 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Bowen Yang, CMSA
Title: Introduction to the probabilistic approach to Louville theory III
Abstract: I will continue with the construction of Liouville CFT from probabilistic methods, following a review by Guillarmou, Kupiainen, and Rhodes (arXiv:2403.12780). The talk will emphasize technical aspects of Gaussian free fields (GFF) and contrast them with Brownian motion. The references is arXiv:2004.04720

Learning Dynamical Transport without Data
2025-04-02 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Michael Albergo (Harvard)
Title: Learning Dynamical Transport without Data
Abstract: Algorithms based on dynamical transport of measure, such as score-based diffusion models, have resulted in great progress in the field of generative modeling. However, these algorithms rely on access to an abundance of data from the target distribution. A complementary problem to this is learning to generate samples from a target distribution when only given query access to the unnormalized log-likelihood or energy function associated to it, with myriad application in statistical physics, chemistry, and Bayesian inference. I will present an algorithm based on dynamical transport to sample from a target distribution in this context, which can be seen as an augmentation of annealed importance sampling and sequential Monte Carlo. Time permitting, I will also discuss how to generalize these ideas to dynamics of discrete distributions. This is joint work with Eric Vanden-Eijnden, Peter Holderrieth, and Tommi Jaakkola.

Learning diffusion models in high-dimensions
2025-04-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Hugo Cui
Title: Learning diffusion models in high-dimensions
Abstract: We consider the problem of learning a generative model parametrized by a two-layer auto-encoder, and trained with online stochastic gradient descent, to sample from a high-dimensional data distribution with an underlying low-dimensional structure. We provide a tight asymptotic characterization of low-dimensional projections of the resulting generated density, and evidence how mode(l) collapse can arise. On the other hand, we discuss how in a case where the architectural bias is suited to the target density, these simple models can efficiently learn to sample from a binary Gaussian mixture target distribution.

Quantum Field Theory and Physical Mathematics
2025-04-07 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Matthew Yu (Oxford)
Title: A new tangential structure for type IIA string theory
Abstract: The Green-Schwarz anomaly cancellation condition says that the target space of heterotic string theory must come with a string structure for the theory to be consistent. In this talk we discuss a new tangential structure called string^h, first introduced by Devalapurkar, as a spin^c analogue of string. Approximating the correct tangential structure of string theory has many implications, primarily is the Swampland program via the cobordism conjecture of Mcnamara and Vafa. We will show that the spectrum of string^h has the notable property that it orients tmf_1(n), just like how the spectrum of string orients tmf, by the work of Ando-Hopkins-Rezk. Finally we will show that the anomaly condition of the partition function of M-theory, studied by Diaconescu-Moore-Witten is implied by a string^h structure on the target space of type IIA, in parallel to the Green-Schwarz anomaly for heterotic string theory, and discuss applications for anomaly cancellation.

3-d Mirror Symmetry
2025-04-07 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ben Webster, University of Waterloo & Perimeter Institute
Title: 3-d Mirror Symmetry
Abstract: I’ll give an introduction (or update, for those who’ve been introduced) to 3d mirror symmetry from the perspective of a mathematician.

CMSA Q&A Seminar: Eric Maskin
2025-04-09 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Eric Maskin, Harvard University
Topic: The Mathematics of Voting

Can Transformers Do Enumerative Geometry?
2025-04-09 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Baran Hashemi, Technical University of Munich
Title: Can Transformers Do Enumerative Geometry?
Abstract: How can Transformers model and learn enumerative geometry? What is a systematic procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a Neural Enumerative Reasoning model for computation of ψ-class intersection numbers on the moduli space of curves. By reformulating the problem as a continuous optimization task, we compute intersection numbers across a wide value range from 10e-45 to 10e45. To capture the recursive nature inherent in these intersection numbers, we propose the Dynamic Range Activator (DRA), a new activation function that enhances the Transformer’s ability to model recursive patterns and handle severe heteroscedasticity. Given precision requirements for computing the intersections, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window adaptive to the partitions of equivalent number of marked points. Beyond simply computing intersection numbers, we explore the enumerative “world-model” of Transformers. Our interpretability analysis reveals that the network is implicitly modeling the Virasoro constraints in a purely data-driven manner. Moreover, through abductive hypothesis testing, probing, and causal inference, we uncover evidence of an emergent internal representation of the large-genus asymptotic of ψ-class intersection numbers. This opens up new possibilities in inferring asymptotic closed-form expressions directly from limited amount of data.
This talk is based on https://openreview.net/pdf?id=4X9RpKH4Ls.

Homological Invariants in Translation-Invariant Pauli Stabilizer Codes
2025-04-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Bowen Yang
Title: Homological Invariants in Translation-Invariant Pauli Stabilizer Codes
Abstract: Pauli stabilizer codes serve as foundational models in quantum error correction and the study of exotic quantum phases. In this talk, we explore the application of homological methods to translationally invariant Pauli stabilizer codes with qudits of varying dimensions. We introduce a series of invariants, termed charge modules, and delve into their properties and physical interpretations. A key focus is on codes whose charge modules exhibit zero Krull dimension, a condition indicative of the mobility of excitations. Notably, we demonstrate that this condition is universally met in two-dimensional codes with a unique ground state in infinite volume, extending prior findings beyond the realm of uniform, prime qudit dimensions. For systems where all excitations are mobile, we establish the existence of p-dimensional excitations and associated (D−p−1)-form symmetries corresponding to each element of the p-th charge module. Additionally, we define a braiding pairing between charge modules in complementary degrees.

The Extended Vertex Algebra of 4d N = 2 SCFTs and their Higher Products
2025-04-14 15:00 - 16:00 | Hybrid – G10
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Mitch Weaver, KAIST
Title: The Extended Vertex Algebra of 4d N = 2 SCFTs and their Higher Products
Abstract: Every 4d N=2 superconformal field theory contains a BPS protected sub-algebra of local operators that has the structure of a vertex operator algebra (VOA). This VOA is identified by passing to the cohomology of a nilpotent supercharge, T, whose local operator cohomology is represented by twist-translated Schur operators with support in a Euclidean two-plane. When working in 4d Minkowski space, this cohomology admits a web of three extended operators (called descent operators) that are constructed from each Schur operator in the VOA, have worldvolume support in the Lorentzian two-plane that is transverse to the Euclidean plane supporting the VOA, and behave as point-like insertions in the plane of the VOA, i.e. as new chiral operators. The combined result is the extended vertex algebra (EVA): a universal extension of the VOA that canonically has the structure of a quasi-VOA, i.e. a vertex algebra (VA) with no conformal vector but which still possesses a representation of sl(2). After reviewing the VOA of Schur operators, I will explain the origin of the descent operators and present the OPEs for a subsector of the EVA in the free hyper SCFT.
Time permitting, I will also describe the construction and basic properties of a set of higher products that are associated to each descent operator. Such products function as higher dimensional versions of 2d chiral algebra λ-brackets, i.e. positive mode operators: they are defined on the EVA and map to the operators appearing in the singular terms of OPEs involving descent operators. Their existence offers a route toward sl(2) symmetry enhancement of the EVA and suggests the latter has structural properties that are common to the higher dimensional chiral algebras describing the minimal twists of 3d N = 2 and 4d N = 1 SQFTs. This talk is based on [2211.04410] and forthcoming work.
