Mathematical Physics

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture…

Topologically ordered phases of matter have long been regarded as lying beyond the Landau–Ginzburg paradigm of symmetry breaking. Our recent studies of anyon condensation, however, suggest that such phases may, in fact, be brought back within the Landau–Ginzburg framework. In this talk, I will…

Consider a finite modular tensor category A. In [DGGPR] the authors exhibit a 3-dimensional topological field theory Z_A: Bord(A) -> Vect, which, in the case where A is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT Z_Ch(A)…

Recent work of Bonifacio-Hinterbichler, Bonifacio, and Kravchuk-Mazáč-Pal introduced an analogy at the level of representation theory between conformal field theories and hyperbolic surfaces. In the spectral theory of hyperbolic surfaces, there are analogs of scaling dimensions of local operators…

In one dimensional quantum mechanics, the all-orders WKB method leads to `quantum periods' which are formal power series in \hbar whose coefficients are certain period integrals. These periods, which limelight in supersymmetric/string theories, have rich structure and can be computed in a number of…

3d N=4 gauge theories admit topological A- and B-twists; boundary conditions for these twisted theories are expected to form 2-categories. The 3d mirror symmetry duality should identify the A-twist of such a theory with a B-twist of a dual theory. This duality can be used to understand the A-side 2…

A celebrated 1997 theorem of Kontsevich shows that every Poisson manifold (classical phase space) can be "quantized" to produce a noncommutative algebra (the corresponding quantum observables). He gives an explicit Feynman-style series expansion for the quantum product, but as with many perturbative…

Recently, Bozec–Calaque–Scherotzke have articulated a noncommutative version of the AKSZ construction, which associates to a smooth Calabi–Yau category, a fully extended TQFT valued in a category of iterated Calabi–Yau cospans. In this talk, I will study a class of examples of such theories that…

Surfaceology is a novel framework to think about scattering amplitudes in a wide variety of gauge theories. Rather than breaking them up into Feynman diagrams, amplitudes are presented as integrals over the moduli spaces of Riemann surfaces and its tropicalization. Over the last few years, this new…

We formulate the holomorphic twists of the 6d N=(0,1) and (0,2) abelian tensor multiplets as moduli spaces in derived geometry, using Deligne cohomology as a key tool. This description allows one to mimic the Beilinson-Drinfeld construction of lattice chiral algebras to quantize these 6d theories…

The moduli space of quasimaps gives a partial compactification of maps from an algebraic curve to a variety. In physics, the cohomology of this moduli space can be viewed as the state space of a (twisted) supersymmetric gauge theory. It is shown by Bullimore-Dimofte-Gaiotto-Hilburn-Kim that when the…

In a joint work with Yalong Cao, Andrei Okounkov, and Zijun Zhou, we introduce stable envelopes in critical cohomology and K-theory for symmetric quiver varieties with potentials and related geometries. Critical stable envelopes are compatible with dimensional reductions, specializations, Hall…