Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Direct Sum Decompositions and Indecomposable TQFT's

Stephen Sawin

TL;DR

This paper develops a complete decomposition framework for TQFTs over algebraically closed fields, showing unitary theories decompose into direct sums of theories with one-dimensional sphere state spaces and providing a full classification of indecomposable 2D TQFTs. The central tool is the Frobenius-algebra structure on Z(S^{d-1}), with TQFT decomposition corresponding to a direct-sum decomposition of this algebra; indecomposables in 2D are classified into simple $S_\lambda$ and nilpotent $N_{A,\mu}$ types. In two dimensions, each Frobenius algebra determines a unique TQFT, enabling explicit descriptions of indecomposables such as $Z_\lambda$ and $Z_{A,\mu}$, where the latter can annihilate higher-genus cobordisms. The results reveal both a semisimple-like decomposition in the unitary case and intrinsic obstructions to reconstruction from closed-manifold data in the nonunitary setting, and point toward state-sum and categorical formulations of these theories.

Abstract

The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT's in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one-dimensional, and indecomposable two-dimensional theories are classified.

Direct Sum Decompositions and Indecomposable TQFT's

TL;DR

This paper develops a complete decomposition framework for TQFTs over algebraically closed fields, showing unitary theories decompose into direct sums of theories with one-dimensional sphere state spaces and providing a full classification of indecomposable 2D TQFTs. The central tool is the Frobenius-algebra structure on Z(S^{d-1}), with TQFT decomposition corresponding to a direct-sum decomposition of this algebra; indecomposables in 2D are classified into simple and nilpotent types. In two dimensions, each Frobenius algebra determines a unique TQFT, enabling explicit descriptions of indecomposables such as and , where the latter can annihilate higher-genus cobordisms. The results reveal both a semisimple-like decomposition in the unitary case and intrinsic obstructions to reconstruction from closed-manifold data in the nonunitary setting, and point toward state-sum and categorical formulations of these theories.

Abstract

The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT's in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one-dimensional, and indecomposable two-dimensional theories are classified.

Paper Structure

This paper contains 5 sections, 6 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Axiomatic TQFT's and Direct Sums
  3. Frobenius Algebras and Decomposition of TQFT's
  4. TQFT's in Two Dimensions
  5. Remarks

Key Result

Proposition 1

If $\cal{Z}$ is a $d$-dimensional TQFT, then $\cal{Z}$ gives $\cal{Z}(S)$ the structure of a commutative Frobenius algebra, and an action of this algebra on $\cal{Z}(\Sigma)$ for each connected $(d-1)$-manifold $\Sigma$. If $\cal{Z}$ is unitary, $\cal{Z}(S)$ is a $C^*$-Frobenius algebra and the acti

Figures (5)

  • Figure 1: The structure of a Frobenius algebra
  • Figure 2: The axioms of a Frobenius algebra
  • Figure 3: Removing two balls is the same as removing one
  • Figure 4: The action of $p_i$ intertwines composition
  • Figure 5: Moves of Cerf theory

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Proposition 2
  • Corollary 2
  • Theorem 2