The cohomology of the free loop spaces of $SU(n+1)/T^n$

TL;DR

This paper develops new methods to compute the cohomology of free loop spaces of complex flag manifolds, revealing rich combinatorial structures and providing explicit calculations and computational tools.

Contribution

It introduces a novel integral Gr"obner basis procedure for computing cohomology of free loop spaces of homogeneous spaces, with explicit examples and a Python library.

Findings

Computed the integral cohomology of the free loop space of $SU(4)/T^3$

Revealed new combinatorial structures in the cohomology algebra

Developed a general computational procedure applicable to similar spectral sequences

Abstract

We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(\Lambda(SU(n+1)/T^{n});\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gr\"obner basis procedure for computation. This procedure is applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. The power of this procedure is illustrated by the explicit calculation of $H^*( \Lambda(SU(4)/T^3);\mathbb{Z})$. We also provide a python library with examples of all procedures used in the paper.