Supersymmeric Gauge Theory on Curved 7‐Branes

Abstract

We construct novel 7d supersymmetric gauge theories which include a Chern‐Simons‐like term on curved spaces. In order to so, we examine the supersymmetry constraints for E7‐branes in type IIA* theory, rather than making use of an off‐shell supergravity. We find two classes of solutions to the constraints, expressed in terms of a G2‐structure with non‐vanishing intrinsic torsion. The supersymmetric gauge theories are then obtained by coupling flat gauge theory to such supersymmetric backgrounds. Various examples are given, including round and squashed S7 as well as S3×M4 with M4 hyperkähler.

7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

“…We remark that although supersymmetry can be taken offshell also for proper G 2 -manifolds (see e.g. [7]), the cohomological complex would be of a different nature compared to the Sasaki-Einstein case above since there is only one solution to (1.1). We hope to return to the proper G 2 case in future work.…”

“…To do this the susy must be realised off-shell. For 7D SYM, the supersymmetry can be taken off-shell for any of the cases listed above, see [6,7], and in [5,6] the cohomological complex was written down for the manifolds admitting at least two Killing spinors (Sasaki-Einstein, 3-Sasakian and S 7 ). With this complex, the equivariant localisation, applied formally to the path integral, reduces the latter to computing the one-loop approximation round the instanton background.…”

“…A non-degenerate matrix Q is called admissible if each of its (k + 1) × k minors can be completed into an SL(k + 1, Z)-matrix. 7 If Q is non-degenerate and admissible, then it gives rise to a free action and C(X) = µ −1 (0)/K is the cone over a smooth 3-Sasakian manifold X.…”

“…Note that the matrix we call Q is the transpose of the matrix called Ω in[17] 7. A (k + 1) × k matrix can be completed into an SL(k + 1, Z)-matrix iff the determinants of its k × k minors have gcd 1.…”

On supersymmetric AdS3 solutions of Type II

“…Let us now give several examples of solutions to the Bianchi identities, which have been reduced to the constraint (5.12). First, we consider M 7 = S 3 × M 4 , with standard G 2 -structure, trivial warp factor A = 0, and where M 4 is any hyper-Kähler manifold [17]. This recovers the near-horizon limit of D1-and D5-branes, with N = (4, 4) supersymmetry [18].…”