The N=1* Theories on R^{1+2} X S^1 with Twisted Boundary Conditions

TL;DR

The paper studies ${\rm N}=1^{*}$ theories compactified on ${\bf R}^{1+2}\times S^1$ with twisted boundary conditions, classifying the twists by twisted affine Lie algebras ${\frak G}^{(L)}$ and showing that the low-energy dynamics are governed by a holomorphic superpotential linked to elliptic Calogero-Moser models. By combining monopole-instanton sums, Weyl symmetry, 3D reductions, and a crucial ${\rm SL}(2,\mathbb{Z})$ modular constraint (supported by M-theory realizations), the authors derive exact twisted superpotentials. They further demonstrate universality of the glueball sector via a dual transformation, showing the Veneziano–Yankielowicz glueball potential is twist-independent to several orders and that twist-induced differences reside only in the irreducible ${\cal W}({\boldsymbol{X}})$, not in ${\cal W}(S)$. The results establish a deep link between nonperturbative dynamics in twisted compactifications and integrable models, with implications for dualities and for generalizations to other dimensions and representations.

Abstract

We explore the N=1* theories compactified on a circle with twisted boundary conditions. The gauge algebra of these theories are the so-called twisted affine Lie algebra. We propose the exact superpotentials by guessing the sum of all monopole-instanton contributions and also by requiring SL(2,Z) modular properties. The latter is inherited from the N=4 theory, which will be justified in the M theory setting. Interestingly all twisted theories possess full SL(2,Z) invariance, even though none of them are simply-laced. We further notice that these superpotentials are associated with certain integrable models widely known as elliptic Calogero-Moser models. Finally, we argue that the glueball superpotential must be independent of the compactification radius, and thus of the twisting, and confirm this by expanding it in terms of glueball superfield in weak coupling expansion.

Figures (3)

• Figure 1: The four diagrams show M theory realization of four types of $O3$ planes compactified on a circle. After a T-duality, which splits a single $O3$ to a pair of $O2$'s, we lift the configurations to M theory by adding a $x^{11}$ circle. Each $O2$ is composed of two $OM2$'s, separated along $x^{11}$, of which there are two types, $OM2^+$ and $OM2^-$. These are denoted by red $+$'s and red $-$'s on the torus. The four dimensional limit corresponds to shrinking the torus while maintaining its complex modulus fixed.
• Figure 2: M theory realization of $O2$ planes that gives the twisted theories with classical groups upon T-duality. Again each $O2$ is composed of two $OM2$'s, separated along $x^{11}$, of which there are two types, $OM2^+$ and $OM2^-$, denoted by red $+$'s and red $-$'s on the torus. Together with four possibilities of Figure 1, these exhaust all possible such combination of $OM2$'s up to overall translations of the torus.
• Figure 3: Dynkin Diagrams for Affine Algebra with comarks