Heterotic/Type-I Duality in D<10 Dimensions, Threshold Corrections and D-Instantons
E. Kiritsis, N. A. Obers
TL;DR
The paper investigates heterotic/type-I duality in dimensions below ten by compactifying on tori and analyzing BPS-protected higher-derivative terms, notably $F^4$ and $R^4$, in the effective action. It shows that in the heterotic description these terms receive one-loop (and possibly higher-genus) corrections governed by the elliptic genus, while on the Type I side non-perturbative D1-instanton corrections reproduce these results, with the D1 determinant identified with the heterotic elliptic genus evaluated on the D1 world-volume modulus. A key finding is that the D1-instanton sector can be organized via Hecke operators, suggesting a matrix-model interpretation, and that the full threshold corrections can be expressed through generalized holomorphic prepotentials, even in eight dimensions with Wilson lines. The work extends to $D<8$, demonstrating that heterotic thresholds match the D1-instanton computations in these cases as well, and provides a framework to test dualities (including F-theory/heterotic) through explicit D1/D5/D1-brane instanton calculations. These results reveal a holomorphic or generalized-holomorphic structure underlying higher-derivative couplings in $N=4$ theories and point to intriguing links between elliptic genera, Jacobi forms, and D-brane matrix models in non-perturbative string dynamics. $
Abstract
We continue our study of heterotic/type-I duality in D<10 dimensions. We consider the heterotic and type-I theories compactified on tori to lower dimensions. We calculate the special (``BPS saturated'') F^4 and R^4 terms in the effective one-loop heterotic action. These terms are expected to be non-perturbatively exact for D>4. The heterotic result is compared with the associated type-I result. In D<9 dimensions, the type-I theory has instanton corrections due to D1 instantons. In D=8 we use heterotic-type I duality to give a simple prescription of the D-instanton calculation on the type I side. We allow arbitrary Wilson lines and show that the D1-instanton determinant is the affine character-valued elliptic genus evaluated at the induced complex structure of the D1-brane world-volume. The instanton result has an expansion in terms of Hecke operators that suggest an interpretation in terms of an SO(N) matrix model of the D1-brane. The total result can be written in terms of generalized prepotentials revealing an underlying holomorphic structure. In D<8 we calculate again the heterotic perturbative thresholds and show that they agree with the D1-instanton calculation using the rules derived in D=8.