Combinatoric topological string theories and group theory algorithms

This is the writeup of an expository talk. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call higher twisted sectors in the theory of orbifolds.

We construct an extended oriented $(2+\epsilon)$-dimensional topological field theory, the character field theory $X_G$ attached to a affine algebraic group in characteristic zero, which calculates the homology of character varieties of surfaces. It is a model for a dimensional reduction of Kapustin-Witten theory ($N=4$ $d=4$ super-Yang-Mills in the GL twist), and a universal version of the unipotent character field theory introduced in arXiv:0904.1247. Boundary conditions in...

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions (HWGs) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and als...

We study interacting features of $S_N$ Orbifold CFTs. Concentrating on characters (associated with $S_N$ Orbifold primaries) we first formulate a novel procedure for evaluating them through $GL(\infty)_+$ tracing. The result is a polynomial formula which we show gives results equivalent to those found by Bantay. From this we deduce a hierarchy of commuting Hamiltonians featuring locality in the induced space, and nonlinear string-type interactions.

We study the cohomology of $G$-representation varieties and $G$-character stacks by means of a topological quantum field theory (TQFT). This TQFT is constructed as the composite of a so-called field theory and the 6-functor formalism of sheaves on topological stacks. We apply this framework to compute the cohomology of various $G$-representation varieties and $G$-character stacks of closed surfaces for $G = \text{SU}(2), \text{SO}(3)$ and $\text{U}(2)$. This work can be seen ...

This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and examples which are included to give the reader a sense of the context. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.

In this note, based on a conference talk, we show how a 3 dimensional topological field theory leads to an algebraic gadget roughly equivalent to a quantum group. This is an expository version of some material in hep-th/9212115 (where we also carry out computations for a specific finite example). We also explain how to incorporate the central extensions usually explained via ``framings'', and we show how to recover invariants of framed tangles. This paper is written using AMS...

The character table of the symmetric group $S_n$, of permutations of $n$ objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity, which has a geometrical interpretation in combinatorial topological field theories, relating the column sum of normalised central characters of $S_n$ to a sum of structure constants of multiplication in the centre of the group algebra of $S_...

For a tuple of square complex-valued $N\times N$ matrices $A_1,\dots,A_n$ the determinant of their linear combination $x_1A_1+\cdots +x_nA_n$, which is called \textit{a pencil}, is a homogeneous polynomial of degree $N$ in $\C[x_1,...x_n]$. Zero-set of this polynomial is an algebraic set in the projective space $\C\Po^{n-1}$. This set is called the determinantal hypersurface or determinantal manifold of the tuple $(A_1,...,A_n)$. It was shown in Cuckovic, Stessin, Tchernev (2...

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a ...