The heterotic string, the tangent bundle and derived categories

Abstract

We consider the compactification of the E 8 ×E 8 heterotic string on a K3 surface with "the spin connection embedded in the gauge group" and the dual picture in the type IIA string (or F-theory) on a Calabi-Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 point-like instantons. X is necessarily singular, and we see that this singularity allows the Ramond-Ramond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the point-like instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of T-duality to the heterotic string on a K3 surface. e-print archive: http://xxx.lanl.gov/abs/hep-th/9806094

Small instanton transitions in heterotic M-theory

“…Equivalently, V can be constructed directly from C and N using the Fourier-Mukai transformation, as discussed in [15,21,22]. Both of these constructions work in reverse, yielding the spectral data (C, N ) up to the overall factor of K B given the vector bundle V .…”

D-branes, categories and N=1 supersymmetry

“…At the end of the discussion we will regard the simultaneous translation of all objects by E → E [1] combined with reversing all the K theory classes as a gauge symmetry, but until we get there, we will regard these as distinct objects.…”

“…Actually this is only a choice of notation: for example we could define D ≡ A [1] and find that the morphism f appeared to have degree one. If one consistently identifies * There is a technical point which all the references emphasize, namely that this construction is noncanonical: given a specific chain map f , one is making further choices in writing an explicit complex representing C f .…”

Stable sheaves on elliptic fibrations

“…The spectral cover. In this section we give a construction of the spectral cover similar to the one described in [12], [14] (sections 4.3 and 5.1) and [4].…”

“…Recently there has been a growing interest in the moduli spaces of stable vector bundles on elliptic fibrations. Aside from their mathematical importance, these moduli spaces provide a geometric background to the study of some recent developments in string theory, notably in connection with the conjectural duality between F-theory and heterotic string theory ( [13], [14], [4], [10]). …”