Matroid Theory and Chern-Simons

TL;DR

This paper explores how matroid theory can serve as a mathematical framework for duality symmetries in quantum field theories and M-theory, linking topological invariants with physical models.

Contribution

It introduces a novel application of matroid theory to duality symmetries in M-theory and quantum gravity, connecting combinatorial structures with physical theories.

Findings

Matroid theory provides a natural framework for duality symmetries.

The Thistlethwaite theorem links graph polynomials to knot invariants in this context.

Connections between the Fano matroid and D=11 supergravity are discussed.

Abstract

It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang-Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern-Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D=11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in Ashtekar formalism.