Heterotic versus Type I

TL;DR

This work probes the heterotic $Spin(32)/Z_2$ and Type I duality by analyzing higher-derivative terms, notably $F^4$ and $R^4$, in toroidally compactified effective actions. It shows that D-brane contributions are constrained: D-strings do not run in loops while their wrapped instantons contribute at orbifold fixed points, with the instanton sum expressible as a sum of elliptic genera of matrix models. A key technical device is the unfolding trick, which relates heterotic one-loop integrals to Type I open-string loops and reveals a Hecke-operator structure governing multi-instanton sectors, interpreted via a symmetric orbifold matrix-model moduli space. The framework yields a coherent, non-perturbative bridge between heterotic and Type I descriptions and points toward extending to D5-branes to illuminate Seiberg–Witten theory through D-brane instantons.

Abstract

I compare the calculations of special $F^4$ and $R^4$ terms in the (toroidally-compactified) heterotic and type-I effective actions. Besides checking duality, this elucidates the quantitative rules of D-brane calculus. I explain in particular (a) why D-branes do not run in loops, and (b) how their instanton contributions arise from orbifold fixed points of their moduli space. The instanton sum has a simple representation as a sum of the elliptic genera of matrix models.

Figures (1)

• Figure 1: The three distinct ways in which the (shaded) heterotic world sheet can wrap twice around the target-space torus.