K-theory in cutoff version of Vacuum String Field Theory

TL;DR

This paper applies K-theory to the cutoff regularized version of Vacuum String Field Theory, showing that nonperturbative solutions like the sliver and butterfly states can be mathematically described as Schwartz functions, aligning with classical D-brane classifications.

Contribution

It introduces a K-theoretic classification of regularized VSFT solutions, connecting them to the topological K-theory of D-branes, and provides a rigorous mathematical framework for these states.

Findings

Sliver and butterfly states are Schwartz functions with a well-defined *-product.

K-theory classification of regularized solutions matches classical D-brane K-theory.

Regularization resolves zero-norm issues of VSFT solutions.

Abstract

Solutions of the Vacuum String Field Theory (VSFT) equation of motion involving matter part are given by projectors, and they represent nonperturbative solutions (e.g. the sliver) interpreted as D25-branes (or lower dimensional branes), but they are not mathematically well defined as they have zero norm. In this work we will use a regularization procedure based on the cutoff version of Moyal String Field Theory (MSFT), a particular version of VSFT, and we will see that both the sliver and the butterfly states, in this regime, have a good mathematical description. In particular they are exponential functions belonging to $\Sc(\RR^{2Nd})$, the space of Schwartzian functions equipped with the *-product. Then we prove that if we classify those regularized solutions with K-theory group built out of the C*-algebra $\bar{\Sc}(\RR^{2Nd})$ we find exactly the same result obtained considering a K-theoretic classification of D25-branes in usual string theory, using the topological K-theory of vector bundles over the D25-brane worldvolume. We then comment on the meaning of this result and possible physical implications.