Mathematical Physics

The theory of resurgence connects perturbative and non-perturbative physics. Focusing on certain one-dimensional quantum mechanical systems with degenerate harmonic minima, I will explain how the resurgent trans-series expansions for the low lying energy eigenvalues follow from the exact quantization condition via the uniform WKB approach. In the opposite spectral region (with high lying eigenvalues), in contrast to the divergent asymptotic expansions expressed as trans-series, the relevant expansions are convergent. However, due to the poles in the expansion coefficients, they contain non-perturbative contributions which can be identified with complex instantons. I will demonstrate that in each spectral region there are striking relation between perturbative and non-perturbative expansions even though the nature of these expansions are very different. Notably, the quantum mechanical examples that I will discuss encode the vacua of certain supersymmetric gauge theories in their spectra.

The first lecture will be devoted to the review of the classical theory of the Witten Laplacian, the second -- to the concepts of resurgent analysis. The third -- to applications of the resurgent analysis to the Witten Laplacian. Time permitting, we will touch upon some foundational questions of resurgent analysis.

These lectures will focus on the geometry of ambitwistor string theories. These are infinite tension analogues of conventional strings and provide the theory that leads to the remarkable formulae for tree amplitudes that have been developed by Cachazo, He and Yuan based on the scattering equations. Although the bosonic ambitwistor string action is expressed in space-time, it will be seen that its target is classically `ambitwistor space', the space of complexified null geodesics in the complexification of a space-time. The lectures will review Ambitwistor constructions from the 70's and 80's that extend the Penrose-Ward twistor constructions for self-dual Yang-Mills and gravitational fields in four dimensions to arbiitrary fields in general dimension. LeBrun showed that the conformal geometry of a space-time is encoded into the complex structure of ambitwistor space. The linearized version encodes linear fields on space-time into sheaf cohomology classes on ambitwistor space. In the case of momentum eigenstates, these give the `scattering equations' that underly the CHY formulae and the ambitwistor string can be used to compute amplitudes via these formulae. If there is time, the lectures will discuss how different matter theories can be obtained, different geometric realizations of ambitwistor space lead to different formulae, the relationship between the asymptotic symmetries of space-time and Weinberg's soft theorems concerning the behaviour of amplitudes when momenta become small, and/or extensions of the ideas to loop amplitudes.