Related papers: A-branes and Noncommutative Geometry
We propose a new class of non-factorising D-branes in the product group GxG where the fluxes and metrics on the two factors do not necessarily coincide. They generalise the maximally symmetric permutation branes which are known to exist…
An important subclass of D-branes on a Calabi-Yau manifold, X, are in 1-1 correspondence with objects in D(X), the derived category of coherent sheaves on X. We study the action of the monodromies in Kaehler moduli space on these D-branes.…
We construct a noncommutative geometry with generalised `tangent bundle' from Fell bundle $C^*$-categories ($E$) beginning by replacing pair groupoid objects (points) with objects in $E$. This provides a categorification of a certain class…
We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen-Wallach pp-wave supported by a constant Neveu-Schwarz three-form flux. We find a class of flat noncommutative euclidean D3-branes analogous to branes…
We prove that the complex of proper non-trivial non-degenerate subspaces of a finite-dimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same…
We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror…
We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of…
We study marginal and relevant supersymmetric deformations of the N=4 super-Yang-Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The…
We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these…
A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov -…
In light of recent attempts to extend the Cieliebak-Mohnke approach for constructing Gromov-Witten invariants to positive genera, we compare the absolute and relative Gromov-Witten invariants of compact symplectic manifolds when the…
We describe the formulation of Fukaya categories of symplectic manifolds with $B$-fields. In addition, we give a formula for how the $A_\infty$ structure maps change as we deform an object by a Lagrangian isotopy.
We consider non-compact branes in topological string theories on a class of Calabi-Yau spaces including the resolved conifold and its mirror. We compute the amplitudes of the insertion of non-compact Lagrangian branes in the A-model on the…
This paper proposes a new notion of smoothness of algebras, termed differential smoothness, that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincar\'e duality…
Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…
We study local mirror symmetry on non-compact Calabi-Yau manifolds (conifold type of singularities) in the presence of D3 brane probes. Using an intermediate brane setup of NS 5-branes `probed' by D4 resp. D5 branes, we can explicitly…
I provide two solutions to the problem of categorifying quantum link invariants, which work uniformly for all gauge groups and originate in geometry and string theory. The first is based on a category of equivariant B-type branes on ${\cal…
This article is devoted to a world sheet analysis of A-type D-branes in N=(2,2) supersymmetric non-linear sigma models. In addition to the familiar Lagrangian submanifolds with flat connection we reproduce the rank one A-branes of Kapustin…
Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $\delta(C)\in…
Noncommutative versions of theories with a gauge freedom define (when they exist) consistent deformations of their commutative counterparts. General aspects of Seiberg-Witten maps are discussed from this point of view. In particular, the…