The moduli dependence of D4-branes on a Calabi–Yau manifold is studied using attractor flow trees, in the large volume limit of the Kähler cone. One of the moduli-dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS partition function is therefore convergent. From non-primitive wall crossing is deduced that the S-duality invariant partition function must be a generating function of the rational invariants $\bar{\Omega} (\Gamma) = \Sigma_{m|\Gamma} =\frac {\Omega(\Gamma/m)}{m^2}$ instead of the integer invariants $\Omega(\Gamma)$.

Given two two-dimensional conformal field theories, a domain wall — or defect line—between them is called invertible if there is another defect with which it fuses to the identity defect. A defect is called topological if it is transparent to the stress tensor. A conformal isomorphism between the two CFTs is a linear isomorphism between their state spaces which preserves the stress tensor and is compatible with the operator product expansion. We show that for rational CFTs there is a one-to-one correspondence between invertible topological defects and conformal isomorphisms if both preserve the rational symmetry. This correspondence is compatible with composition.

We continue our program initiated in "Integration over the u-plane in Donaldson theory with surface operators," J. High Energy Phys. 05 (2011), to consider supersymmetric surface operators in a topologically twisted ${\cal N}=2$ pure ${\it SU}(2)$ gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsáth and Szabó and Taubes, will be furnished. In particular, we will show that Taubes’ groundbreaking and difficult result — that the ordinary SW invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold $X$ — nonetheless lends itself to a simple and concrete physical derivation in the presence of “ordinary” surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and “ramified” SW invariants of $X$, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the “ramified” Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from “ramified” instantons by Kronheimer and Mrowka; and (ii) monopole Floer homology and SW theory on symplectic four-manifolds by Kutluhan–Taubes. Supersymmetry, as well as other physical concepts such as $R$-invariance, electric–magnetic duality, spontaneous gauge symmetry breaking and localization onto supersymmetric configurations in topologically twisted quantum field theories, play a pivotal role in our story.

We study a supersymmetry-preserving solution-generating method in heterotic supergravity. In particular, we use this method to construct one-parameter non-Kähler deformations of Calabi–Yau manifolds with a $U(1)$ isometry, in which the complex structure remains invariant. We explain how to obtain corresponding solutions to heterotic string theory, up to first order in $\alpha'$, by a modified form of the standard embedding. In the course of the paper we also show that Abelian heterotic supergravity embeds into type II supergravity, and note that the solution-generating method in this context is related to a dipole-type deformation when there is a field theory dual.

We show stable causality of the Pomeransky–Senkov black rings.

In this paper, we use trivial defects to define global taffy-like operations on string worldsheets, which preserve the field theory. We fold open and closed strings on a space $X$ into open strings on products of multiple copies of $X$, and perform checks that the “taffy-folded” worldsheets have the same massless spectra and other properties as the original worldsheets. Such folding tricks are a standard method in the defects community; the novelty of this paper lies in deriving mathematical identities to check that e.g., massless spectra are invariant in topological field theories. We discuss the case of the B model extensively, and also derive the same identities for string topology, where they become statements of homotopy invariance. We outline analogous results in the A model, B-twisted Landau–Ginzburg models, and physical strings. We also discuss the understanding of the closed string states as the Hochschild homology of the open string algebra, and outline possible applications to elliptic genera.