(NS5-brane, D5-brane, D3-brane) bound state, open D3-brane, open D5-brane limits, andSL(2,Z)duality

Abstract

We generalize the nonthreshold bound state in type IIB supergravity of the form (NS5-brane, D5-brane, D3-brane) constructed by the present authors [J. High Energy Phys. 02, 026 (2001)] to a nonzero asymptotic value of the axion $({\ensuremath{\chi}}_{0}).$ We identify the decoupling limits corresponding to both the open D3-brane theory and open D5-brane theory for this supergravity solution as expected. However, we do not find any noncommutative Yang-Mills theory (NCYM) limit for this solution in the presence of NS5-branes. We then study the $\mathrm{SL}(2,Z)$ duality symmetry of type IIB theory for both open D3-brane (OD3) limit and open D5-brane (OD5) limit. We find that for OD3 theory, a generic $\mathrm{SL}(2,Z)$ duality always gives another OD3 theory irrespective of the value of ${\ensuremath{\chi}}_{0}$ being rational or not. This indicates that OD3 theory is self-dual. But, under a special set of $\mathrm{SL}(2,Z)$ transformations for which ${\ensuremath{\chi}}_{0}$ is rational, OD3 theory goes over to a (5+1)-dimensional NCYM theory and these two theories in this case are related to each other by strong-weak duality symmetry. On the other hand, for OD5 theory, a generic $\mathrm{SL}(2,Z)$ duality gives another OD5 theory if ${\ensuremath{\chi}}_{0}$ is irrational, but when ${\ensuremath{\chi}}_{0}$ is rational it gives the little string theory limit indicating that OD5 theory is S dual to the type IIB little string theory.