Little strings, quasi-topological sigma model on loop group, and toroidal Lie algebras

Abstract

We study the ground states and left-excited states of the A k−1 N = (2, 0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP 1 with target space the based loop group of SU (k). The ground states, described by L 2 -cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.In loving memory of See-Hong Tan.

Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde Formula

“…Moreover, shrinking D allows us to identify the target of the A-model on Σ with the based loop group ΩG. Such an A-model is known to possess affine symmetry [7]. The corresponding A-model states form modules of an affine Lie algebra g aff , which span the space of g aff -modules on Σ that we denote by G mod (Σ).…”

“…This action can further be rewritten [7] as the sum of a Q-invariant term (in the perturbative regime), and a metric-independent one (topological) with mixed gauge coupling. Hence, Q-cohomology can also be defined on the topological sigma model on Σ.…”

“…Instanton Floer Homology of Σ × S 1 and Affine Algebra It was shown in [7] that the sigma model exhibits affine symmetry on the worldsheet Σ, generated by an affine Lie algebra g aff . In addition, A-model states form modules of g aff .…”