[Paper Review] Homotopy field theory in dimension 2 and group-algebras

Vladimir Turaev|ArXiv.org|Oct 4, 1999

Algebraic structures and combinatorial modelsMathematics8 references77 citations

TL;DR

This paper introduces homotopy quantum field theories (HQFTs) in dimension 2, generalizing topological quantum field theories by incorporating maps into a space $X = K(\pi,1)$. It establishes a complete classification of $ (1+1) $-dimensional HQFTs via crossed $\pi$-algebras, showing that semi-cohomological HQFTs over characteristic 0 fields are classified by semisimple crossed group-algebras and satisfy a Verlinde-type formula.

ABSTRACT

We apply the idea of a topological quantum field theory (TQFT) to maps from manifolds into topological spaces. This leads to a notion of a (d+1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed d-dimensional manifolds and (d+1)-dimensional cobordisms endowed with homotopy classes of maps into a given space. For a group $π$, we introduce cohomological HQFT's with target $K(π,1)$ derived from cohomology classes of $π$ and its subgroups of finite index. The main body of the paper is concerned with (1+1)-dimensional HQFT's. We classify them in terms of so called crossed group-algebras. In particular, the cohomological (1+1)-dimensional HQFT's over a field of characteristic 0 are classified by simple crossed group-algebras. We introduce two state sum models for (1+1)-dimensional HQFT's and prove that the resulting HQFT's are direct sums of rescaled cohomological HQFT's. We also discuss a version of the Verlinde formula in this setting.

Motivation & Objective

• To generalize topological quantum field theories (TQFTs) by incorporating homotopy classes of maps into a space $X$, leading to the notion of $(d+1)$-dimensional homotopy quantum field theory (HQFT).
• To classify $ (1+1) $-dimensional HQFTs with target $X = K(\pi,1)$ using algebraic structures, specifically crossed $\pi$-algebras.
• To construct state sum models for $ (1+1) $-dimensional HQFTs using biangular and non-degenerate $\pi$-algebras, proving their homotopy invariance and relation to cohomological HQFTs.
• To establish a Verlinde-type formula for semi-cohomological $ (1+1) $-dimensional HQFTs and show that over algebraically closed fields of characteristic 0, all such HQFTs are semi-cohomological.

Proposed method

• Introduces the concept of a $(d+1)$-dimensional HQFT as a TQFT enriched with homotopy classes of maps from manifolds and cobordisms into a fixed space $X$, with $X = K(\pi,1)$ being central to the $ (1+1) $-dimensional case.
• Defines a $\pi$-algebra as an associative algebra $L$ graded by a group $\pi$ with $L_\alpha L_\beta \subset L_{\alpha\beta}$, and introduces the subclass of crossed $\pi$-algebras satisfying additional compatibility conditions with group action and inner product.
• Constructs two state sum models for $ (1+1) $-dimensional HQFTs: one based on biangular $\pi$-algebras and another using non-degenerate $\pi$-algebras, where the latter requires summing over all $\pi$-systems in a fixed homotopy class to ensure homotopy invariance.
• Uses the partition function of a $\pi$-system (defined via the $\pi$-algebra structure) to define a state sum invariant, proving that the resulting assignment yields a well-defined HQFT.
• Applies the theory of Frobenius algebras and group cohomology to construct cohomological HQFTs from 2-cocycles in $H^2(\pi, K^*)$, and shows that these correspond to crossed $\pi$-algebras.
• Proves that over a field of characteristic 0, every $ (1+1) $-dimensional HQFT arising from a non-degenerate $\pi$-algebra is semi-cohomological, and hence satisfies a Verlinde-type formula.

Experimental results

Research questions

• RQ1 How can topological quantum field theories be generalized to incorporate maps into a topological space $X$, leading to a notion of homotopy quantum field theory (HQFT)?
• RQ2 What algebraic structures classify $ (1+1) $-dimensional HQFTs with target $X = K(\pi,1)$, and how do they relate to group cohomology and Frobenius algebras?
• RQ3 Can state sum models be constructed for $ (1+1) $-dimensional HQFTs using $\pi$-algebras, and do they yield homotopy-invariant invariants?
• RQ4 Under what conditions does a $ (1+1) $-dimensional HQFT satisfy a Verlinde-type formula, and how is this related to the semisimplicity of the underlying crossed $\pi$-algebra?
• RQ5 Are all $ (1+1) $-dimensional HQFTs over an algebraically closed field of characteristic 0 semi-cohomological, and what does this imply for their classification?

Key findings

• There is a bijective correspondence between isomorphism classes of $ (1+1) $-dimensional HQFTs with target $X = K(\pi,1)$ and crossed $\pi$-algebras, establishing a complete algebraic classification.
• Semi-cohomological $ (1+1) $-dimensional HQFTs over a field of characteristic 0 are classified by semisimple crossed $\pi$-algebras, and such HQFTs satisfy a Verlinde-type formula.
• The state sum model based on a biangular $\pi$-algebra yields a homotopy-invariant partition function, defining a well-behaved $ (1+1) $-dimensional HQFT.
• For non-degenerate $\pi$-algebras, the partition function is not generally homotopy-invariant, but summing over all $\pi$-systems in a fixed homotopy class restores invariance and yields an HQFT.
• Over an algebraically closed field of characteristic 0, all $ (1+1) $-dimensional HQFTs arising from non-degenerate $\pi$-algebras are semi-cohomological, implying they satisfy the Verlinde formula.
• The construction of crossed $\pi$-algebras from group cohomology classes in $H^2(\pi, K^*)$ yields cohomological HQFTs, and when $\pi = 1$, this recovers the classical classification of $ (1+1) $-dimensional TQFTs by commutative Frobenius algebras.