arXiv subjects

Claudia Scheimbauer

Publications and source records attributed to Claudia Scheimbauer.

9 recordsLinked to original sources

arXiv2026

On dualizability and invertibility in the higher Morita category

We prove a conjecture of Lurie characterizing $(n+1)$-dualizability in higher Morita categories of $\mathbb{E}_n$-algebras in terms of dualizability over certain factorization homologies. A key ingredient is a higher Morita category based on the recently developed framework of pointless factorization algebras of Karlsson and the first author. We also verify an invertibility conjecture of Brochier--Jordan--Safranov--Snyder as an immediate corollary of our main result. Moreover, we prove a relative version of the dualizability conjecture, yielding a new criterion for relative/twisted field theories. We give some examples, including Dirichlet and Neumann relative theories.

math.CT↗

arXiv2024

Simons Lectures on Categorical Symmetries

Global Categorical Symmetries are a powerful new tool for analyzing quantum field theories. This volume compiles lecture notes from the 2022 and 2023 summer schools on Global Categorical Symmetries, held at the Perimeter Institute for Theoretical Physics and at the Swiss Map Research Station in Les Diableret. Specifically, this volume collects the lectures: * An introduction to symmetries in quantum field theory, Kantaro Ohmori * Introduction to anomalies in quantum field theory, Clay C\'ordova * Symmetry Categories 101, Michele Del Zotto * Applied Cobordism Hypothesis, David Jordan * Finite symmetry in QFT, Daniel S. Freed These volumes are devoted to interested newcomers: we only assume (basic) knowledge of quantum field theory (QFT) and some relevant maths. We try to give appropriate references for non-standard materials that are not covered. Our aim in this first volume is to illustrate some of the main questions and ideas together with some of the methods and the techniques necessary to begin exploring global categorical symmetries of QFTs.

math-ph↗

arXiv2024

Dagger $n$-categories

Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.

math.CT↗

arXiv2023

Relative field theories via relative dualizability

We investigate relative versions of dualizability designed for relative versions of topological field theories (TFTs), also called twisted TFTs, or quiche TFTs in the context of symmetries. In even dimensions we show an equivalence between lax and oplax fully extended framed relative topological field theories valued in an $(\infty , N)$-category in terms of adjunctibility. Motivated by this, we systematically investigate higher adjunctibility conditions and their implications for relative TFTs. Summarizing we arrive at the conclusion that oplax relative TFTs is the notion of choice. Finally, for fun we explore a tree version of adjunctibility and compute the number of equivalence classes thereof.

math.CT↗

arXiv2022

The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory

We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the AKSZ construction, first implemented in this context by Pantev-Toën-Vaquié-Vezzosi, and show that it induces a family of symmetric monoidal functors from oriented stacks to symplectic stacks. Finally, we construct forgetful functors from the unoriented bordism $(\infty,n)$-category to cospans of spaces, and from the oriented bordism $(\infty,n)$-category to cospans of spaces equipped with an orientation; the latter combines with the AKSZ functors by viewing spaces as constant stacks, giving the desired field theories.

math.CT↗

arXiv2018

Duals and adjoints in higher Morita categories

We study duals for objects and adjoints for $k$-morphisms in $\operatorname{Alg}_n(\mathcal{S})$, an $(\infty,n+N)$-category that models a higher Morita category for $E_n$ algebra objects in a symmetric monoidal $(\infty,N)$-category $\mathcal{S}$. Our model of $\operatorname{Alg}(\mathcal{S})$ uses the geometrically convenient framework of factorization algebras. The main result is that $\operatorname{Alg}_n(\mathcal{S})$ is fully $n$-dualizable, verifying a conjecture of Lurie. Moreover, we unpack the consequences for a natural class of fully extended topological field theories and explore $(n+1)$-dualizability.

math.CT↗

arXiv2018

A note on the $(\infty,n)$-category of cobordisms

In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\operatorname{Cob}$ and the cobordism bicategory $n\operatorname{Cob}^{ext}$ from it.

math.AT↗

arXiv2016

(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong $(\infty, n)$-functors. We construct a double $(\infty,n)$-category built out of the target $(\infty, n)$-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty, n)$-category $\mathcal{C}$ to an $(\infty, n+d)$-category using the higher morphisms in $\mathcal{C}$.

math.CT↗

arXiv2014

Lectures on mathematical aspects of (twisted) supersymmetric gauge theories

Supersymmetric gauge theories have played a central role in applications of quantum field theory to mathematics. Topologically twisted supersymmetric gauge theories often admit a rigorous mathematical description: for example, the Donaldson invariants of a 4-manifold can be interpreted as the correlation functions of a topologically twisted N=2 gauge theory. The aim of these lectures is to describe a mathematical formulation of partially-twisted supersymmetric gauge theories (in perturbation theory). These partially twisted theories are intermediate in complexity between the physical theory and the topologically twisted theories. Moreover, we will sketch how the operators of such a theory form a two complex dimensional analog of a vertex algebra. Finally, we will consider a deformation of the N=1 theory and discuss its relation to the Yangian, as explained in arXiv:1308.0370 and arXiv:1303.2632.

math.QA↗