Noncommutative and Ordinary Super Yang-Mills on (D$(p-2)$, D$p$) Bound States

TL;DR

This paper explores the thermodynamics and dualities of noncommutative super Yang-Mills theories realized via D-brane bound states with B fields, revealing equivalences with ordinary theories in lower dimensions and matching probe brane free energies.

Contribution

It demonstrates the thermodynamic equivalence between noncommutative and ordinary super Yang-Mills theories across different dimensions, and establishes relations via probe methods.

Findings

Thermodynamics of D$p$-branes with B field matches that of D$(p-2)$-branes with smeared coordinates.

Noncommutative SYM with gauge group U(N_p) in (p+1) dimensions is equivalent to ordinary SYM with U(∞) in (p-1) dimensions.

Probe brane free energies with B field coincide with those without B field.

Abstract

We study properties of (D$(p-2)$, D$p$) nonthreshold bound states ($2 \le p \le 6 $) in the dual gravity description. These bound states can be viewed as D$p$-branes with a nonzero NS $B$ field of rank two. We find that in the decoupling limit, the thermodynamics of the $N_p$ coincident D$p$-branes with $B$ field is the same not only as that of $N_p$ coincident D$p$-branes without $B$ field, but also as that of the $N_{p-2}$ coincident D$(p-2)$-branes with two smeared coordinates and no $B$ field, for $N_{p-2}/N_p= \tilde{V}_2/[(2\pi)^2 \tilde{b}]$ with $\tilde{V}_2$ being the area of the two smeared directions and $\tilde{b}$ a noncommutativity parameter. We also obtain the same relation from the thermodynamics and dynamics by probe methods. This suggests that the noncommutative super Yang-Mills with gauge group $U(N_p)$ in ($p+1$) dimensions is equivalent to an ordinary one with gauge group $U(\infty)$ in ($p-1$) dimensions in the limit $\tilde{V}_2 \to \infty$. We also find that the free energy of a D$p$-brane probe with $B$ field in the background of D$p$-branes with $B$ field coincides with that of a D$p$-brane probe in the background of D$p$-branes without $B$ field.