[Paper Review] D-branes and K-theory in 2D topological field theory

Gregory W. Moore, Graeme Segal|ArXiv.org|Sep 5, 2006

Black Holes and Theoretical Physics49 references95 citations

TL;DR

This paper establishes that in 2D topological field theories with semisimple closed string algebras, D-branes are classified as (G-twisted) vector bundles on spacetime due to sewing constraints—primitive worldsheet locality—providing a foundational link between D-branes and K-theory. The results extend to G-equivariant theories, showing D-branes correspond to B-twisted G-vector bundles, with the category of boundary conditions equivalent to that of orbifold theories up to tensoring with G-line bundles.

ABSTRACT

This expository paper describes sewing conditions in two-dimensional open/closed topological field theory. We include a description of the G-equivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints -- the most primitive form of worldsheet locality -- already imply that D-branes are (G-twisted) vector bundles on spacetime. We comment on extensions to cochain-valued theories and various applications. Finally, we give uniform proofs of all relevant sewing theorems using Morse theory.

Motivation & Objective

• To clarify the mathematical classification of D-branes in 2D topological field theories using sewing consistency conditions.
• To establish a direct link between D-branes and K-theory in the simplest possible setting—semisimple 2D TFTs.
• To generalize the classification to G-equivariant topological field theories, relevant for orbifold D-brane physics.
• To show that the category of boundary conditions in semisimple theories is equivalent to the category of finite-dimensional vector bundles on spacetime.
• To provide a uniform proof of sewing theorems using Morse theory, unifying the treatment of open and closed string amplitudes.

Proposed method

• Use of sewing conditions as the fundamental worldsheet locality constraints to derive the structure of boundary conditions.
• Application of Morse theory to give uniform, geometric proofs of all sewing theorems, treating critical points of Morse functions as elementary cobordisms.
• Adaptation of Frobenius algebra techniques to the equivariant case, incorporating group actions and G-bundles on worldsheets.
• Construction of maps between vector spaces of open strings using holonomy data and group actions (ρg), with dual bases and trace maps.
• Introduction of B-fields and G-invariant dilaton fields to describe G-equivariant TFTs, with the closed string algebra encoding spacetime data.
• Use of A∞-category formalism to generalize results beyond the semisimple case, suggesting cochain-complex-valued theories as the natural framework for non-semisimple extensions.

Experimental results

Research questions

• RQ1 What is the complete set of D-branes allowed in a 2D topological field theory with a semisimple closed string algebra?
• RQ2 How do sewing constraints—worldsheet locality—force the D-brane category to be equivalent to vector bundles on spacetime?
• RQ3 What is the role of the G-action in classifying D-branes in equivariant topological field theories?
• RQ4 How does the presence of a B-field affect the classification of D-branes in G-equivariant TFTs?
• RQ5 Can the category of boundary conditions in a 2D TFT be reconstructed from the closed string algebra, and if so, under what conditions?

Key findings

• In semisimple 2D TFTs, the category of D-branes is equivalent to the category of finite-dimensional complex vector bundles on spacetime, with the equivalence defined up to tensoring with a fixed line bundle.
• The choice of a maximal category of D-branes corresponds to a choice of square root of the dilaton field θx at each point x of the finite spacetime X.
• In the G-equivariant case, D-branes are classified as B-twisted G-vector bundles on X, with the classification invariant under tensoring with G-line bundles.
• The category of D-branes in the G-equivariant theory is equivalent to that of the corresponding orbifold theory, but the full equivariant structure is lost in the orbifold limit.
• The sewing theorems—essential for consistency of the theory—are uniformly proven using Morse theory, with critical points of a Morse function encoding the elementary cobordisms.
• The results suggest that K-theory classification of D-branes is not merely a consequence of anomaly cancellation, but arises more fundamentally from worldsheet locality and sewing constraints.