Higher Algebra and Mathematical Physics
The Euler characteristic of a compact complex manifold M is a classical cohomological invariant. Depending on the viewpoint, it is most natural to interpret it as an index of an elliptic diﬀerential operator on M, or as a supersymmetric index in superconformal ﬁeld theories “on M”. Reﬁning the Euler characteristic but keeping with both index theoretic interpretations, one arrives at the notion of complex elliptic genera.
Topological ﬁeld theories in the sense of Atiyah–Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A ﬁeld theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant.
We describe joint work with L. Hollands on the geometry of the moduli space of flat connections over a Riemann surface. On the one hand, we generalize and compute certain "complexified Fenchel-Nielsen" coordinates for SL(2)-connections to higher rank using the spectral network "abelianization" approach of Gaiotto-Moore-Neitzke.
In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G.
According to the principle of locality in physics, events taking place at diﬀerent locations should behave independently of each other, a feature expected to be reﬂected in the measurements. We propose an algebraic locality framework to keep track of the independence, where sets are equipped with a binary symmetric relation we call a locality relation on the set, this giving rise to a locality set category. In this algebraic locality setup, we implement a multivariate regularisation, which gives rise to multivariate meromorphic functions.
I will describe joint work with Sachin Gautam where we give a deﬁnition of the category of ﬁnite-dimensional representations of an elliptic quantum group which is intrinsic, uniform for all Lie types, and valid for numerical values of the deformation and elliptic parameters. We also classify simple objects in this category in terms of elliptic Drinfeld polynomials. This classiﬁcation is new even for sl(2), as is our deﬁnition outside of type A.
Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of categoriﬁed homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss a step towards this goal: a categoriﬁcation of the classical Dold–Kan correspondence
In this talk I will present some upcoming work on Bridgeland stability conditions on partially wrapped Fukaya categories of topological surfaces. The main result is a proof that the stability conditions defined by Haiden, Katzarkov and Kontsevich using quadratic differentials cover the entire stability space. This proof uses a definition of the new concept of relative stability conditions, which is a relative version of Bridgeland's definition, with functorial behavior analogous to compactly supported cohomology.
There are various ways to define factorization algebras: one can define a factorization algebra that lives over the open subsets of some fixed manifold; or, alternatively, one can define a factorization algebra on the site of all manifolds of a given dimension (possibly with a specified geometric structure). In this talk, I will outline a comparison between G-equivariant factorization algebras on a fixed model space M to factorization algebras on the site of all manifolds equipped with a (M, G)-structure, given by an atlas with charts in M and transition maps given by elements of G.
This talk is based on joint work with Yuri Manin. The idea of a “geometry over the ﬁeld with one element F1” arises in connection with the study of properties of zeta functions of varieties deﬁned over Z. Several diﬀerent versions of F1 geometry (geometry below Spec(Z)) have been proposed over the years (by Tits, Manin, Deninger, Kapranov–Smirnov, etc.) including the use of homotopy theoretic methods and “brave new algebra” of ring spectra (To¨en–Vaqui´e).