We study the duality group of $\widehat{A}_{n-1}$ quiver gauge theories, primarily using their M5-brane construction. For $\mathcal{N}=2$ supersymmetry, this duality group was first noted by Witten to be the mapping class group of a torus with $n$ punctures. We find that it is a certain quotient of this group that acts faithfully on gauge couplings. This quotient group contains the affine Weyl group of $\widehat{A}_{n-1}$, $\mathbb{Z}_n$ and $SL(2,\mathbb{Z})$. In fact there are $n$ non-commuting $SL(2,\mathbb{Z})$ subgroups, related to each other by conjugation using the $\mathbb{Z}_n$. When supersymmetry is broken to $\mathcal{N}=1$ by masses for the adjoint chiral superfields, a renormalization group (RG) flow ensues which is believed to terminate at a conformal field theory (CFT) in the infrared. We find the explicit action of this duality group for small values of the adjoint masses, paying special attention to when the sum of the masses is non-zero. In the $\mathcal{N}=1$ CFT, Seiberg duality acts non-trivially on both gauge couplings and superpotential couplings and we interpret this duality as inherited from the $\mathcal{N}=2$ parent theory. We conjecture the action of $S$-duality in the CFT based on our results for small mass deformations. We also consider non-conformal deformations of these $\mathcal{N}=1$ theories. The cascading RG flows that ensue are a one-parameter generalization of those found by Klebanov and Strassler and by Cachazo et al. The universality exhibited by these flows is shown to be a simple consequence of paths generated by the action of the affine Weyl group.

We show how a Kähler spacetime foam in four dimensional conformal (super)gravity may be mapped to twistor spaces carrying the D1 brane charge of the B model topological string theory. The spacetime foam is obtained by blowing up an arbitrary number of points in $\mathbb{C}^2$ and can be interpreted as a sum over gravitational instantons. Some twistor spaces for blowups of $\mathbb{C}^2$ are known explicitly. In these cases, we write down a meromorphic volume form and suggest a relation to a holomorphic superform on a corresponding super Calabi-Yau manifold.

The purpose of this paper is to investigate the possibility of a physical 12-dimensional $F$-theory. We study the question of geometric interaction terms in the $F$-theory Lagrangians. We also introduce a new supergravity multiplet in dimension (9, 3) which is based on a particle with 3-dimensional time-like world volume. A construction of signature (9, 3) $F$-theory is given using dualities analogous to those considered by Hull, and possible matches of $F$-theory's low energy fields with the (9, 3)-supergravity field content are given. Finally, preliminary suggestions are made regarding a possible phenomenological compactification of $F$-theory from dimension (9, 3) to (3, 1).

We construct a gauge-fixed action for topological membranes on $G_2$-manifolds such that its bosonic part is the standard membrane theory in a particular gauge. We prove that the path integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$, the theory naturally reduces to the standard A-model on Calabi--Yau manifold and to a membrane theory localized on special Lagrangian submanifolds. We discuss some properties of topological membrane theory on $G_2$-manifolds. We also generalize our construction to topological $p$-branes on special manifolds by exploring a relation between vector cross product structures and TFTs.

When a spacetime takes Bondi radiating metric and is vacuum and asymptotically flat at spatial infinity which ensures the positive mass theorem, we prove that the standard ADM energy--momentum is the past limit of the Bondi energy--momentum. We also derive a formula relating the ADM energy--momentum of any asymptotically flat spacelike hypersurface to the Bondi energy--momentum of any null hypersurface. The formula indicates that the Bondi mass is always less than the ADM total energy if the system has news. The assumed asymptotic flatness precludes gravitational radiation. We therefore study further the relation between the ADM total energy and the Bondi mass when gravitational radiation emits. We find that, in this case, the ADM total energy is no longer the past limit of the Bondi mass. They differ by certain quantity relating to the news of the system.