[Paper Review] Twisted K-Theory and TQFT

Ulrich Bunke, Ingo Schröder|ArXiv.org|Apr 22, 2005

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper provides a stack-theoretic, geometric proof of the twisted K-theory of a compact Lie group G acting on itself by conjugation, using smooth stacks and formal properties of K-theory. It establishes that the complexified twisted K-theory is isomorphic to the quotient of the complexified representation ring by an ideal, and constructs a 1+1-dimensional TQFT structure with explicit formulas for the product, identity, and co-form via geometric correspondences and K-orientations.

ABSTRACT

The goal of the present paper is the calculation of the equivariant twisted K-theory of a compact Lie group which acts on itself by conjugations, and elements of a TQFT-structure on the twisted K-groups. These results are originally due to D.S.Freed, M.J.Hopkins and C.Teleman. In this paper we redo their calculations in the framework of topological and differentiable stacks. We also show how moduli spaces of flat connections on surfaces give rise to trivializations of twists.

Motivation & Objective

• To provide a geometric, stack-based proof of the twisted G-equivariant K-theory of a compact Lie group G under conjugation, avoiding analytic methods.
• To construct a 1+1-dimensional topological quantum field theory (TQFT) structure on the twisted K-theory using moduli stacks of flat G-connections.
• To establish explicit formulas for the TQFT operations—product, identity, and co-form—using geometric correspondences and K-orientations.
• To show that the complexified twisted K-theory is isomorphic to a quotient of the complexified representation ring R(G)ℂ/I, using a detection map to finite group representations.
• To clarify the role of trivializations and orientations in TQFT constructions, particularly for boundary evaluation maps and glueing of surfaces.

Proposed method

• The paper uses smooth stacks and local quotient stacks to formalize geometric constructions, assuming the existence of a K-theory functor with standard functorial properties.
• It constructs elements of twisted K-theory via a composition R!∘Φ, where Φ is a geometric pullback and R! is a pushforward along a correspondence.
• A key detection tool is the embedding Θ: twisted K-theory → representation ring of a finite group, inspired by C. Teleman, enabling element detection via restriction.
• The TQFT structure is built using moduli stacks of flat G-connections on surfaces, with boundary evaluation maps and natural trivializations of twists via determinant line bundles on restricted Grassmannians.
• K-orientations of boundary evaluation maps are constructed using Lie algebra orientations, and compatibility under glueing is verified using the Leray-Serre spectral sequence and surjectivity of pullback maps.
• The product is defined via a correspondence of stacks, and its associativity and unitality are proven using a distinguished K-orientation and isomorphism of twists induced by glueing.

Experimental results

Research questions

• RQ1 How can the twisted K-theory of a compact Lie group G under conjugation be computed purely geometrically using smooth stacks and formal K-theory properties?
• RQ2 What is the precise algebraic structure of the twisted K-theory as a module over the representation ring, and how does it relate to the complexified group ring?
• RQ3 How can a 1+1-dimensional TQFT be naturally realized on the twisted K-theory using moduli spaces of flat connections and their boundary maps?
• RQ4 What role do K-orientations and trivializations of twists play in ensuring functoriality and compatibility under surface glueing?
• RQ5 Can the TQFT product, identity, and co-form be explicitly computed in terms of geometric correspondences and stack-theoretic constructions?

Key findings

• The complexified twisted K-theory of G is isomorphic to the quotient R(G)ℂ/I, where I is the ideal generated by the relations of the representation ring.
• The TQFT product on the twisted K-theory is commutative and associative, with identity element E constructed from the disk correspondence.
• The identity element E corresponds to the inverse of the central charge section c, i.e., E ≅ c⁻¹, under the isomorphism with the line bundle V.
• The product structure is explicitly computed via the composition (qa)!∘r*∘qi*, where qa is the evaluation map on the cylinder, and the isomorphism r is induced by glueing.
• The construction ensures that the TQFT operations are compatible with glueing, provided K-orientations and twist isomorphisms are chosen consistently.
• The method provides a framework generalizable to other twisted cohomology theories beyond K-theory, relying only on geometric stacks and formal K-theory properties.