Worldsheet (anti)instanton bound states in type II on T2

Anirban Basu

doi:10.1007/jhep04(2020)168

Abstract

The 1/8 BPS D6ℛ4 coupling in type II string theory compactified on T2 receives contributions from worldsheet instantons and anti-instantons wrapping the T2, up to genus three in string perturbation theory. These involve contributions separately from bound states of instantons and anti-instantons, which are qualitatively similar to such contributions to the 1/2 and 1/4 BPS couplings. At genus two, the D6ℛ4 coupling also receives contributions from instanton/anti-instanton bound states unlike the 1/2 and 1/4 BPS couplings, which is a consequence of a T-duality invariant eigenvalue equation a term in the coupling satisfies. We solve this eigenvalue equation to obtain the complete structure of the worldsheet (anti)instanton contributions. In the type IIB theory, strong weak coupling duality leads to certain contributions involving bound states of D string (anti)instantons wrapping the T2.

Highlights

The 1/8 BPS D6R4 coupling in type II string theory compactified on T 2 receives contributions from worldsheet instantons and anti-instantons wrapping the T 2, up to genus three in string perturbation theory
The D6R4 coupling receives contributions from instanton/anti-instanton bound states unlike the 1/2 and 1/4 BPS couplings, which is a consequence of a T-duality invariant eigenvalue equation a term in the coupling satisfies
In the type IIB theory, strong weak coupling duality leads to certain contributions involving bound states of D stringinstantons wrapping the T 2

Summary

The mode carrying no NS charge

First let us consider the mode F0(T2) in (2.1) which carries no NS charge. Using (1.9) and the large T2 expansion of E1 given in (A.3), we have that. Using (B.2), we see that Qn(T2) is an infinite series in powers of T2 in the large T2 expansion. Since this contribution is weighted by e−4πnT2, it follows that it arises from the bound state of worldsheet instantons/anti-instantons carrying equal and opposite NS charge n. Using (B.3) we see that the singular terms in the small. For small T2, the singularity in F0(T2) is only logarithmic, and is weaker than the bound in (2.2). When expanded for large T2, we see that F0(T2) has terms that are power behaved and logarithmic in T2, as well as terms that are exponentially suppressed in T2. To F0(T2) from the instanton/anti-instanton sector with weight e−4πnT2

The modes carrying NS charge

The solution for In(T2)

The solution for In1,n2(T2) Like before, we express

S-duality and an elementary consequence for D string instanton contributions

B The exponential integral function