Daniel Friedan (26/08/16)

Sigurður Örn Stefánsson, ágúst 22, 2016

Málstofa í stærðfræði

Fyrirlesari: Daniel Friedan, Rutgers University and University of Iceland
Titill: Quasi Riemann Surfaces

Staðsetning: TG-227 (Tæknigarður, 2. hæð)
Tími: Föstudagur 26. ágúst kl. 13:20.


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.