New Structures from Quantized Brane Moduli

Northwestern University At Chicago, Evanston IL

Investigators

Abstract

String theory has been a powerful thought laboratory both for physics and for mathematics. Phenomenological implications are being actively explored in the physical community, while lessons from the development of strings continually inform and improve our ability to apply physics to various problems such as the calculation of particle scattering or even the interaction of light with black holes. On the mathematical side, advances have continued apace for decades since the “second string revolution” incorporated extended states known as “branes.” These objects — which include black holes as a special case — have mathematical treatments that span a vast range of subjects from differential geometry and symplectic geometry to representation theory, sheaf theory and category theory. But once you have a brane, you can move it around and there are interesting families of branes. Since the overarching theory is quantum mechanical, points in these families behave like quantum particles, meaning the families get quantized. Algebraically, the quantization procedure has been explored by mathematicians in general, but quantization of brane moduli often has a richer, specific structure that can be revealed in examples. The projects in this grant explore these structures. Specifically, several projects will advance our understanding of quantization of brane moduli spaces. These include the moduli of augmentations of Legendrian knots, whose quantization can recover quantum groups in this setting. Brane moduli spaces have been quantized and connected to all-genus open Gromov-Witten invariants, and one project explores an upgrade of this from quantum tori to quantum character varieties through the skein algebras. Another project investigates the quantization of periods — which measure brane moduli — and connects the result to enumerative invariants. Other projects involve q-difference equations and constructible sheaves. All the while, PI Zaslow will continue to supervise graduate students, run seminars, and disseminate results at conferences, workshops and seminars. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.