## Titles and Abstracts

**Dan Berwick-Evans:****Lecture Notes**

**Title**: How do field theories detect the torsion in topological modular forms? **Abstract**: Since the 80s there have been hints of a deep connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner’s conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2-dimensional supersymmetric field theories. Properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2-dimensional field theories that realize certain torsion classes in TMF. In particular, this leads to a description of the generator of \(\pi_3(TMF)=\mathbb Z/24\) in terms of the supersymmetric sigma model with target \(S^3\).

**Ivan Contreras: Lecture Notes**,**Annotated Lecture Notes**

**Title:** Frobenius objects in the category of spans and the symplectic category.**Abstract:** It is well known that Frobenius algebras are in correspondence with 2-dimensional TQFT. In this talk, we introduce Frobenius objects in any monoidal category and in particular, in the category where objects are sets and morphisms are spans of sets. We prove the existence of a simplicial set that encodes the data of the Frobenius structure in this category. This serves as a (simplicial) toy model of the Wehrheim-Woodward construction for the symplectic category.

This is part of a program that intends to describe, in terms of category theory, the relationship between symplectic groupoids and topological field theory, via the Poisson sigma model. Based on joint work with Rajan Mehta and Molly Keller (arXiv:2106.14743), and ongoing work with Rajan Mehta and Walker Stern.

**Chris Elliott**:**Lecture Notes**

**Title**: Supersymmetry and Pure Spinors**Abstract**: While many examples of supersymmetric field theories in various dimensions are known, their construction can sometimes feel a little ad-hoc. The introduction of the BV formalism already clarifies things quite a bit, as one realizes that it is often cleaner to construct an action of a supersymmetry algebra on the BV fields homotopically (i.e. to construct an \(L_\infty\) action) rather than strictly; constructing a quasi-isomorphic strict model leads to the “auxiliary fields” one sees in the literature. I will discuss a new perspective on a procedure in physics known as the “pure spinor superfield formalism” that gives a systematic way of constructing supersymmetric field theories in the BV formalism starting from algebraic data. For us, the input datum is a sheaf of Lie algebras on the “derived space of pure spinors”. This perspective can be viewed as an equivalence of dg-categories, meaning in particular that all supersymmetric theories arise in this way. This is based on joint work with Fabian Hahner and Ingmar Saberi.

**Owen Gwilliam:****Lecture Notes**

**Title:** Framed \(E_n\) algebras from topological quantum field theories**Abstract:** Elliott and Safronov showed that a topological AKSZ theory on n-dimensional manifolds provides an algebra over the little n-disks operad, using the BV formalism and factorization algebras. Building upon that work, Elliott and I have characterized the obstruction complex for lifting that \(E_n\) algebra to a framed \(E_n\) algebra, showing how this kind of framing anomaly relates to Pontryagin classes. I will review their work and ours, offering as well some applications to the Kapustin-Witten theories in 4-dimensions.

**Si Li**:**Lecture Notes**

**Title**: Elliptic chiral homology and chiral index**Abstract**: We present an effective quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].

**Andrey Losev**:**Lecture Notes**

**Title:** Tau-theory**Abstract:** David Gross and Edward Witten said, “Everyone in string theory is convinced…that spacetime is doomed. But we don’t know what it’s replaced by.” Nathan Seiberg said, “I am almost certain that space and time are illusions. These are primitive notions that will be replaced by something more sophisticated.”Tau-theory (Tau stands for Tensor Algebra Universe) is an attempt to introduce such replacement. In the contrast with the expectations of abovementioned scientists the replacement is not that sophisticated, it is rather simple: the space of solutions to A-infinity equations in tensor algebra. Being space of solutions it forms what I call a tau-landscape.Most of points of tau-landscape are not interesting for physics like most of exoplanets are not interesting for biology. However, we may look at commutative associative (super)algebras, concentrating on those whose spectrum is smooth. These are like planets suitable for life but with no life yet.Things become more interesting if we study tau-landscape in the neighborhood of such points. The first order deformations of A-infinity algebra are Hochschild cohomology of the A-infinity algebra, and for smooth scheme they are polyvector fields on these scheme (due to Hochschild-Kostant-Rosenberg theorem) that would form fields of the emerging QFT. The arising world would be made out of “fluctuations of the multiplication table”.Having spacetime and fields we would need equations of motion of these fields. We do not need to invent them since they naturally come from the obstruction for the first order deformations to be deformations of A-infinity algebra to the second order. And this obstruction is given by Schouten bracket on polyvectors. Thus we have universal equations, this is like to find life on a planet. Biologists would be satisfied at that point (like mathematical physicists that like to play with model theories) but it would be much interesting to find intelligent life. For physicists this would mean to find something like a gauge theory with matter fields.I will show that equations of motion in tau-theory are exactly equations of motion in B-type topological gravity. By a KK-like construction we can get get holomorphic CS theory.Moreover, using pure spinors one can get along this way equations of motion of N=1 d=10 SYM.I will also discuss problems in obtaining supergravity.

**Eugene Rabinovich:****Lecture Notes**

**Title**: Classical Bulk-Boundary Correspondences via Factorization Algebras**Abstract**: A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin-Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a ”sufficiently nice” such factorization algebra on a manifold \(N\), one may associate to it a factorization algebra on \(N\times\mathbb R_{\ge 0}\). The aim of the talk is to explain the sense in which the latter factorization algebra “knows all the classical data” of the former. This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.

**Pavel Safronov:****Lecture Notes**

**Title**: Whitehead torsion in string topology**Abstract**: String topology operations are interesting operations, such as the string product and the string coproduct, on the homology of the free loop space of a manifold. The string product appears as a natural pair-of-pants product in a 2d TQFT and is known to be homotopy invariant. I will explain that the string coproduct is an operation in a “compactified” 2d TQFT. It turns out that the compactified 2d TQFT is not homotopy-invariant and depends on the simple homotopy type of the manifold. In particular, under homotopy equivalences the string coproduct changes by a term involving the Whitehead torsion. This is a report on work in progress joint with Florian Naef.

**Konstantin Wernli**:**Lecture Notes**

**Title**: On the globalization of the perturbative Chern-Simons partition function**Abstract**: Using the BV formalism one can define the perturbative Chern-Simons partition function for any reference flat connection \(A_0\).The family of partition functions defined in this way gives rise to a volume form defined on the smooth part of the moduli space of flat connections. This talk is based on ongoing joint work with P. Mnev.

**Brian Williams**:**Lecture Notes**

**Title**: A holomorphic approach to fivebranes. **Abstract**: The six-dimensional superconformal field theory associated to an ADE Lie algebra is a key player in various mathematical subjects inspired by QFT including the AGT correspondence and Geometric Langlands. Nevertheless, its explicit description as a QFT has remained elusive. Motived by (twisted) holography (following Costello, Gaiotto, Li, and Paquette) I propose a holomorphic model describing the minimal twist of the six-dimensional theory when the rank of the Lie algebra is small. We will also exhibit a large N analysis and compare partition functions to expectations in the physics literature. This talk reports on joint work with Surya Raghavendran.