Mathematical Physics

K-theoretical/Cohomological Hall algebras, associated with the stack of zero-dimensional sheaves on $\mathbb{C}^2$, play a prominent role in the proof, given by Schiffmann and Vasserot, of the AGT conjecture for

(pure) gauge theories on $\mathbb{C}^2$. In the present talk I will describe K-theoretic/Cohomological Hall algebras associated with stacks of torsion sheaves on A-type ALE spaces and their representations; the study of these algebras will provide eventually an approach to prove the AGT conjecture for (pure) gauge theories on A-type ALE spaces. (Work in progress with Olivier Schiffmann.)

We study the representation theory of truncated shifted Yangians. These algebras arise as quantizations of slices to Schubert varieties in the affine Grassmannian. We will describe the combinatorics of their highest weights, which is encoded in Nakajima's monomial crystal. We also prove Hikita's conjecture in this context.

I'll discuss the two topological twists of 3d N=4 theories, and explain how to understand them in the AKSZ/BV formalism, and how they relate to twists of 4d N =2 theories.  Symplectic duality then takes the form of an equivalence between 3d N=4 theories which interchanges the two topological twists.  I'll also introduce monopole operators and explain the role they play in symplectic duality.

After a quick review of the Higgs and Coulomb branches of 3d N=4 theories, I'll introduce some simple classes of boundary conditions and explain how they lead to (pairs of) modules for certain (pairs of) quantum algebras. I will focus on abelian theories, for which the relevant boundary conditions/modules can be described using the geometry of (pairs of) hyperplane arrangements. From this, the simplest examples of symplectic-dual modules will arise.

We consider the spectral action as an action functional for modified gravity on a spacetime that exhibits a fractal structure modeled on an Apollonian packing of 3-spheres (packed swiss cheese) or on a fractal arrangements of dodecahedral spaces. The contributions in the asymptotic expansion of the spectral action, that arise from the real poles of the zeta function, include the Einstein-Hilbert action with cosmological term and conformal and Gauss-Bonnet gravity terms. We show that these contributions are affected by the presence of fractality, which modifies the corresponding effective gravitational and cosmological constants, while an additional term appears in the action, which is entirely due to fractality. This term is further affected by a contribution of oscillatory terms coming from the poles of the zeta function that are off the real line, which are also a property specific to fractals. We show that the shape of the slow-roll potential obtained by scalar perturbations of the Dirac operators is also affected by the presence of fractality. The talk is based on joint work with Adam Ball.