M5-branes from gauge theories on the 5-sphere

Abstract

We use the 5-sphere partition functions of supersymmetric Yang-Mills theories to explore the (2, 0) superconformal theory on S 5 × S 1 . The 5d theories can be regarded as Scherk-Schwarz reductions of the 6d theory along the circle. In a special limit, the perturbative partition function takes the form of the Chern-Simons partition function on S 3 . With a simple non-perturbative completion, it becomes a 6d index which captures the degeneracy of a sector of BPS states as well as the index version of the vacuum Casimir energy. The Casimir energy exhibits the N 3 scaling at large N . The large N index for U (N ) gauge group also completely agrees with the supergravity index on AdS 7 × S 4 .

Gauge theories with 16 supersymmetries on spheres

“…The perturbative results are described in terms of matrix models for which we have briefly discussed their large Nbehaviour. Unfortunately, the localization locus for both S 6 and S 7 appears difficult to analyze explicitly. In particular it seems that extended instanton solutions are not ruled out.…”

“…Moreover we conjecture the form of the full answer based on the factorization properties of the perturbative answer. In section 6 we present the detailed calculation for S 6 and following an analogy with S 4 we conjecture the full answer. In section 7 we briefly discuss the properties of 6D and 7D matrix models and argue for N 2 -behaviour at large N .…”

“…In [4,5] it was shown how to localize N = 1 in five dimensions with a vector multiplet and hypermultiplets in arbitrary representations. In [6] it was also shown how to extend this to N = 2, the maximal case for d = 5. Going down a dimension, there are an impressive number of results for three dimensional Chern-Simons theories that are derived using localization, starting with the work in [7] and to more general theories in [8].…”

“…In the case of d = 5, (2.18) is equivalent to the N = 2 Lagrangian in [6], after a Euclidean rotation of φ 0 . If d ≤ 5, we can split up Ψ into parts that are eigenstates of Γ ≡ Γ 6789 , with the even eigenstates, ψ = +Γψ making up the fermions in the vector multiplet and the odd eigenstates, χ = −Γχ making up those in an adjoint hypermultiplet.…”

Little strings and T-duality

“…Finally, the self-dual string elliptic genera in 6d SCFTs turn out to be related to other interesting observables, such as the superconformal indices [33][34][35][36][37][38]. It will be interesting to see if the elliptic genera for little strings also find similar interesting applications.…”