Exact Description of D-branes via Tachyon Condensation

TL;DR

This paper provides an exact description of D-branes through tachyon condensation, demonstrating how fluctuations reproduce D-brane fields and actions, and establishing algebraic and geometric equivalences including the Atiyah-Singer index theorem.

Contribution

It offers a unified approach to describe D-branes via boundary states and string field theory, proving descent/ascent relations and connecting geometric and algebraic frameworks.

Findings

Reproduces D-brane fields and actions exactly from fluctuations.

Establishes equivalence between geometric and algebraic D-brane descriptions.

Derives the Atiyah-Singer index theorem from boundary state analysis.

Abstract

We examine the fluctuations around a Dp-brane solution in an unstable D-brane system using boundary states and also boundary string field theory. We show that the fluctuations correctly reproduce the fields on the Dp-brane. Plugging these into the action of the unstable D-brane system, we recover not only the tension and RR charge, but also full effective action of the Dp-brane exactly. Our method works for general unstable D-brane systems and provides a simple proof of D-brane descent/ascent relations under the tachyon condensation. In the lowest dimensional unstable D-brane system, called K-matrix theory, D-branes are described in terms of operator algebra. We show the equivalence of the geometric and algebraic descriptions of a D-brane world-volume manifold using the equivalence between path integral and operator formulation of the boundary quantum mechanics. As a corollary, the Atiyah-Singer index theorem is naturally obtained by looking at the coupling to RR-fields. We also generalize the argument to type I string theory.