The cobordism hypothesis says, in its basic form, that a TQFT (on plain manifolds) are determined from its value on a point. One can consider more interesting TQFTs by allowing defects (singularities along a stratification) on manifolds. In this talk, I will describe a formal construction of such cobordism categories from the plain ones and how the cobordism hypothesis with defects follows from the plain ones as a formal consequence of (pre)stability of symmetric monoidal (infinity, n)-categories.
Factorization algebras, introduced by Costello and Gwilliam, encode the structure of observables in perturbative quantum field theory, capturing concepts like the operator product and correlation functions. Beyond this, their local structures encompass notions such as associative and $A_\infty$-algebras, vertex algebras, bimodules, and $E_n$-algebras. Examples of the latter are (possibly braided) tensor categories and n-fold loop spaces.In this talk, I will provide an overview of the surprisingly recently well-studied theory of locally constant factorization algebras. Building on this foundation, I will introduce a broader class of constructible factorization algebras, defined on manifolds with corners or, more generally, stratified manifolds. Examples are given by factorization homology. I will report on recent progress regarding the additivity of factorization algebras, including the additivity of the generalized Swiss cheese operad- a result inspired by Dunn’s theorem for the little disks operad. This is based on joint work with Victor Carmona. In our approach, we use the reformulation of factorization algebras within the framework of $\infty$-operads, which offers powerful tools to study their behavior over product spaces.
Given a Fano or a general type variety, one can study the position of the roots of its Hilbert polynomial H(z) with respect to the anticanonical class. By intersection theory, its coefficients are simple expressions in the Chern numbers. The well-known Routh-Hurwitz stability criterion says that for H(z) to have all roots in a given vertical strip, or on a line, a system of polynomial inequalities in Chern numbers must be satisfied. Work by Manivel and others establishes vertical strip properties for certain classes of Fano varieties.
Now, the generating function of the sequence P(n) of values of any polynomial is rational. Its numerator is another polynomial, called the transform of P. Motivated by the above considerations I will discuss some polynomials arising from dual lattice polytopes, their transforms, and their role in mirror symmetry.
Calabi-Yau compactifications of superstring theory yield a surprisingly varied set of theories, ranging from cosmological flux vacua to supersymmetric black holes. It turns out that in many cases the physical properties of these theories are related to intricate arithmetic properties of the compactification manifold. In this talk, I discuss how techniques from number theory and arithmetic geometry can be used to find some of these solutions and compute many of their relevant physical quantities, concentrating mostly on the relation between modularity of Calabi-Yau manifolds and supersymmetric Minkowski flux vacua. I introduce briefly the zeta function of a manifold, discuss how it is computed and how it can be used to find these vacua. In special cases, the zeta function is related to (elliptic) modular forms, and conjectures in number theory relate the modular forms to physical quantities, such as the vacuum expectation value of the background fields. I present a simple method that can be used to construct many families of supersymmetric flux vacua for which we have explicitly tested these conjectures. The talk is based on several works together with Candelas, de la Ossa, McGovern, Jockers, and Kotlewski.
One of the best ways to study a theory of particles (or strings) is to collide the components and measure the probability of different scenarios. The mathematical objects describing these probabilities are scattering amplitudes. In this talk, we use the genus expansion of string theory and study the amplitude of one-loop closed strings. They are integrals over the moduli space of the punctured torus. We see how "Modular Graph Forms" (MGFs) arise from the perturbative solution of the integrals over the configuration space of punctures. The main focus of the talk will be the iterated integral representation of MGFs, their number-theoretical properties, and their integral over the moduli space of the torus.
Orbifold data are defects in topological quantum field theories which can be gauged to obtain a new TQFT. Examples include gaugings of (higher) group actions, state sum models, and more generally gaugings of "non-invertible symmetries". The defining conditions of orbifold data encode invariance unter the choice of defect network used in the gauging process, which has been rigorously developed in arbitrary dimensions. The talk gives an introduction to the orbifold construction and illustrates it with examples in dimensions 2,3, and 4.
In the classical Batalin–Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology. In this talk, we will explore generalisations of (non-)commutative Batalin-Vilkovisky algebras for differential operators of arbitrary order, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. This is joint work with V. Dotsenko and S. Shadrin.
