Sixtieth Meeting of the North British Mathematical Physics Seminar

I will present recent developments in the study of integrable sigma models. After reviewing the definition of a coupled integrable model on an arbitrary number of copies of a Lie group, I will discuss its underlying geometry and its one-loop renormalisation group flow, leading to a conjecture for the renormalisation of a general class of integrable sigma models. I will then present the construction of new integrable sigma models defined on the quotient of copies of a Lie group by a well-chosen diagonal subgroup and their applications to \(T^{1,1}\) manifolds.

We will motivate and introduce the study of conformal defects in superconformal field theories (SCFTs). We will show how symmetries constrain the anomaly coefficients of BPS defects. In the case of N=(2,2) surface defects in four-dimensional N=2 SCFTs these anomaly coefficients can be computed by studying a protected subsector captured by a vertex operator algebra, or two-dimensional chiral algebra.

The Schur index of a 4d N=2 theory counts local operators in the cohomology of a particular supercharge Q. In superconformal theories, it was shown to coincide with the character of an associated VOA. I will discuss some current work with Wenjun Niu that goes in a (literally) orthogonal direction, explaining how local operators in Q-cohomology arise from a geometric description of the category of line operators in 4d N=2 theories (due to Kapustin and Cautis-Williams), and how this idea generalizes to operators at junctions of Wilson-'t Hooft lines, bound to surface defects, and in the presence of boundary conditions (half-indices). Mathematically, derived algebraic geometry plays a key role throughout.

I will briefly introduce the notion of para-Hermitian geometry and how to define a compatible generalised metric. This allows the construction of a sigma-model associated with these structures. Then I will discuss the reduction of such metric to the leaf space of a para-Hermitian manifold via a Lie algebroid gauging, thus obtaining a reduced sigma-model. I will also show how para-Hermitian morphisms play the role of generalised T-dualities and their relation with ordinary T-duality.