Twisted K-theory in $g>1$ from D-branes

TL;DR

This paper explores the use of noncommutative geometry and twisted K-theory to analyze D-branes wrapped on higher genus Riemann surfaces in type IIB string theory, providing explicit constructions and extending known spectra.

Contribution

It introduces a novel framework for studying D-branes on genus g>1 surfaces using noncommutative C*-algebras and constructs projective modules with constant-curvature connections, extending the g=1 BPS spectrum.

Findings

Explicit construction of noncommutative C*-algebras for g>1 surfaces.

Development of projective modules with constant-curvature connections.

Extension of the BPS spectrum to higher genus surfaces.

Abstract

We study the wrapping of N type IIB Dp-branes on a compact Riemann surface $\Sigma$ in genus $g>1$ by means of the Sen-Witten construction, as a superposition of N' type IIB Dp'-brane/antibrane pairs, with $p'>p$. A background Neveu-Schwarz field B deforms the commutative $C^{\star}$-algebra of functions on $\Sigma$ to a noncommutative $C^{\star}$-algebra. Our construction provides an explicit example of the $N'\to\infty$ limit advocated by Bouwknegt-Mathai and Witten in order to deal with twisted K-theory. We provide the necessary elements to formulate M(atrix) theory on this new $C^{\star}$-algebra, by explicitly constructing a family of projective $C^{\star}$-modules admitting constant-curvature connections. This allows us to define the $g>1$ analogue of the BPS spectrum of states in $g=1$, by means of Donaldson's formulation of the Narasimhan-Seshadri theorem.