Quantum Vacua of 2d Maximally Supersymmetric Yang-Mills Theory

We analyze the classical and quantum vacua of 2d $\mathcal{N}=(8,8)$ supersymmetric Yang-Mills theory with $SU(N)$ and $U(N)$ gauge group, describing the worldvolume interactions of $N$ parallel D1-branes with flat transverse directions $\mathbb{R}^8$. We claim that the IR limit of the $SU(N)$ theory in the superselection sector labeled $M \pmod{N}$ --- identified with the internal dynamics of $(M,N)$-string bound states of Type IIB string theory --- is described by the symmetric orbifold $\mathcal{N}=(8,8)$ sigma model into $(\mathbb{R}^8)^{D-1}/\mathbb{S}_D$ when $D=\gcd(M,N)>1$, and by a single massive vacuum when $D=1$, generalizing the conjectures of E. Witten and others. The full worldvolume theory of the D1-branes is the $U(N)$ theory with an additional $U(1)$ 2-form gauge field $B$ coming from the string theory Kalb-Ramond field. This $U(N)+B$ theory has generalized field configurations, labeled by the $\mathbb{Z}$-valued generalized electric flux and an independent $\mathbb{Z}_N$-valued 't Hooft flux. We argue that in the quantum mechanical theory, the $(M,N)$-string sector with $M$ units of electric flux has a $\mathbb{Z}_N$-valued discrete $\theta$ angle specified by $M \pmod{N}$ dual to the 't Hooft flux. Adding the brane center-of-mass degrees of freedom to the $SU(N)$ theory, we claim that the IR limit of the $U(N) + B$ theory in the sector with $M$ bound F-strings is described by the $\mathcal{N}=(8,8)$ sigma model into ${\rm Sym}^{D} ( \mathbb{R}^8)$. We provide strong evidence for these claims by computing an $\mathcal{N}=(8,8)$ analog of the elliptic genus of the UV gauge theories and of their conjectured IR limit sigma models, and showing they agree. Agreement is established by noting that the elliptic genera are modular-invariant Abelian (multi-periodic and meromorphic) functions, which turns out to be very restrictive.