## Daniel Friedan (26/08/16)

Math Colloquium

### Speaker: Daniel Friedan, Rutgers University and University of Iceland

Title: Quasi Riemann Surfaces

Location: TG-227 (Tæknigarður, 2nd floor)

Time: Friday, August 26 at 13:20.

### Abstract:

This will be a talk about some speculative mathematics (analysis) with

possible applications in quantum field theory. I will leave any mention

of quantum field theory to the end. I will try to define everything

from scratch, but it probably will help to have already seen the basics

of manifolds, differential forms, and Riemann surfaces.

The talk is taken from my recent paper

“Quantum field theories of extended objects”, arXiv:1605.03279 [hep-th]

which is a mixture of speculative quantum field theory and speculative

mathematics. In the talk, the speculative mathematics will be presented

on its own, without the motivations from quantum field theory.

Below is the abstract from a note I am presently writing to try to

interest mathematicians in looking at this structure:

I will describe an attempt to define a mathematical structure – “quasi

Riemann surface” – in which the points of a Riemann surface are replaced

by the integral 0-currents. The same structure is found in the spaces

of integral relative (n-1)-cycles in a conformal manifold M of dimension

d=2n >= 4. The quasi Riemann surfaces associated to M form a natural

fiber bundle Q(M) -> B(M) over the space of integral lines in the space

of integral (n-2)-currents in M.

A goal is to re-express analysis in Riemann surfaces as analysis in the

corresponding quasi Riemann surfaces, so that it will apply also to the

spaces of integral relative (n-1)-cycles in conformal manifolds M. I

hope to use this structure to construct a new class of quantum field

theories in space-time manifolds of dimension d=2n >= 4 from the known

quantum field theories in two-dimensional manifolds.

I offer some speculations on the classification of quasi Riemann

surfaces, leading to pictures of universal homogeneous bundles in which

all the Q(M) -> B(M) are naturally embedded. The strongest form of this

speculation would equate every quasi Riemann surface to a quasi Riemann

surface associated to some two-dimensional manifold. Such isomorphisms

could be used to transport two-dimensional quantum field theories

directly to higher dimensional manifolds M.

I am seeking mathematical advice — is this structure known? is it

feasible? — and trying to raise interest in making the structure

mathematically solid. Aside from possible applications to quantum field

theory, the structure might be useful in the study of manifolds.