Tachyon condensation on the elliptic curve

Abstract

We use the framework of matrix factorizations to study topological B-type Dbranes on the cubic curve. Specifically, we elucidate how the brane RR charges are encoded in the matrix factors, by analyzing their structure in terms of sections of vector bundles in conjunction with equivariant R-symmetry. One particular advantage of matrix factorizations is that explicit moduli dependence is built in, thus giving us full control over the open-string moduli space. It allows one to study phenomena like discontinuous jumps of the cohomology over the moduli space, as well as formation of bound states at threshold. One interesting aspect is that certain gauge symmetries inherent to the matrix formulation lead to a non-trivial global structure of the moduli space. We also investigate topological tachyon condensation, which enables us to construct, in a systematic fashion, higher-dimensional matrix factorizations out of smaller ones; this amounts to obtaining branes with higher RR charges as composites of ones with minimal charges. As an application, we explicitly construct all rank two matrix factorizations.December, 2005 † On leave from the

D-brane monodromies from a matrix-factorization perspective

“…This matrix factorization does not depend on any open-string moduli, but it arises in the limit where the 3 × 3 factorization (4.4) becomes singular as one of the open-string parameters, α , approaches zero [17]. The U(1) R-symmetry representation (2.10) is given by 15) and the resulting three equivariant representations read 16) which label the branes, X a , in their equivariant orbit.…”

“…This means we need to add to the data of the brane, P , a Z d representation, R P , such that the matrix, Q P , fulfills the equivariance condition [11,33,17]:…”

“…Thus along the path the pair of branes, P and X, becomes unstable and an energetically favored bound state is formed via tachyon condensation. The matrix factorization, Q Con , of the condensate with the operator, Ψ (X,P ) , is easily realized as [44,17] …”

“…[16,17]. Here we briefly review the matrix factorizations of the 'long' and the 'short' branes, as we will study their monodromy transformations.…”

“…(2.15) the three equivariant 'short' branes, S a , distinguished by their Z 3 representations 13) JHEP02 (2007)006 with ω ≡ e 2πi 3 . Finally we introduce the exceptional 4 × 4-matrix factorization, which contains the pure D2-brane in its equivariant orbit [17] …”