Academic Year 2013/2014

In this talk I will survey recent progress in heterotic string compactifications and two-dimensional theories with (0,2) supersymmetry. For the first half of the talk, I will give an overview of `quantum sheaf cohomology,' a heterotic analogue or generalization of quantum cohomology. For the second half of the talk, I will discuss various new and old dualities in two dimensions. Specifically, I will first review progress towards (0,2) mirror symmetry, a generalization of ordinary mirror symmetry. I'll then turn to gauge dualities in two dimensions, and how easy bits of math can be used to derive such dualities. Finally, I'll discuss decomposition for two-dimensional nonabelian gauge theories with center-invariant matter, which is the observation that such theories can decompose into a disjoint union of closely related theories.

The generally covariant approach to quantum field theory is reviewed for the simple example of the Klein-Gordon field under a slightly different perspective. The basic idea is to assign to each globally hyperbolic spacetime a suitable space of linear functionals to test solutions of the field equation. The choice of functionals is made in such a way that the evaluation on solutions defines a non-degenerate pairing. It turns out that this perspective reproduces the result of the usual approach. In a second moment, a quantization prescription is applied. In the second part of the talk, we focus the attention on linear gauge theories in the context of general local covariance. We illustrate how, for a simple model, namely Maxwell $k$-forms, the perspective presented in the first part (adapted to gauge theories) provides a definite answear for the choice of functionals to be used to test field configurations. An explicit characterization of this space of functionals can be obtained by means of standard Poincare' duality for de Rham cohomology and its analogue for causally restricted supports.

This is intended as a quick introduction to noncommutative geometry from the perspective of analysis. Starting from the Weyl law, we will discuss applications and drawback of these techniques by means of examples.

Given a compact Lie group and a principal bundle over a closed Riemannian manifold, the quotient space of connections, modulo the action of the group of gauge transformations, has fundamental significance for algebraic geometry, low-dimensional topology, the classification of smooth four-dimensional manifolds, and high-energy physics. The quotient space of connections is equipped with the Yang-Mills energy functional and Atiyah and Bott (1983) had proposed that its gradient flow with respect to the natural Riemannian metric on the quotient space should prove to be an important tool for understanding the topology of the quotient space via an infinite-dimensional Morse theory. The critical points of the energy functional are gauge-equivalence classes of Yang-Mills connections. However, thus far, smooth solutions to the Yang-Mills gradient flow have only been known to exist for all time and converge to critical points, as time tends to infinity, in relatively few cases, including (1) when the base manifold has dimension two or three (Rade, 1991 and 1992, in dimension two and three; G. Daskalopoulos, 1989 and 1992, in dimension two), (2) when the base manifold is a complex algebraic surface (Donaldson, 1985), and (3) in the presence of rotational symmetry in dimension four (Schlatter, Struwe, and Tahvildar-Zadeh, 1998). Global existence of solutions with up to finitely many point singularities (caused by the ``bubbling'' phenomenon) at up to finitely many times was proved independently by Struwe (1994) and Kozono, Maeda, and Naito (1995). However, the question of global existence and convergence of smooth solutions over general compact, Riemannian, four-dimensional base manifolds has thus far remained unresolved. In this talk we shall describe our proof of the following result: Given a compact Lie group and a smooth initial connection on a principal bundle over a compact, Riemannian, four-dimensional manifold, there is a unique, smooth solution to the Yang-Mills gradient flow which exists for all time and converges to a smooth Yang-Mills connection on the given principal bundle as time tends to infinity.

Monopole dynamics provides plentiful information about both gravitational instantons and about quantum field theories in various dimensions. Doubly periodic monopoles, which are the subject of this talk, are of particular interest, as their moduli spaces are expected to be the spaces of vacua of five-dimensional quantum theories compactified on a two-torus. At the same time, when four-dimensional, these moduli spaces are self-dual gravitational instantons of lowest volume growth. In this talk we identify all doubly periodic monopoles with four moduli. For a general case we commute the asymptotic metric on the moduli space. Surprisingly, the answer is best given in terms of crystals, secondary polytopes, and amoebas.

The study of consistent truncations of gravitational theories is an old subject going back to work of Jordan and Pauli. It is well known that reductions on group manifolds are consistent, but there is also a mysterious set of truncations on very particular coset manifolds with non-trivial form- field fluxes. We show how generalised geometry gives a unified description of such reductions. In particular we demonstrate that all round-sphere S^d geometries admit "generalised parallelisations", giving a simple way to understand the remarkable consistent truncations on S7, S5 and S4.

Deformed Virasoro algebra introduced in the middle of 90's resembles the well-known deformation of Jack to Macdonald polynomials. More precisely, singular vectors in Verma modules over the Virasoro algebra are Jack polynomials while singular vectors for the deformed Virasoro are described by Macdonald polynomials. Such a natural deformation of the Virasoro algebra appeared to be useful in many applications to off-critical statistical models and massive integrable field theories. In the talk, I present my recent results on a surprising realization of this algebra in finite and critical quantum spin-chains when the deformation parameters are at roots of unity. More formally, it turns out that the deformed Virasoro at N-th root of unity has representations realized by the Temperley-Lieb algebras with N generators. Such finite-dimensional representations are of the cyclic type (similar to those for quantum groups at roots of unity) and were never observed before.