Anomalies and Graded Coisotropic Branes

We compute the anomaly of the axial U(1) current in the A-model on a Calabi-Yau manifold, in the presence of coisotropic branes discovered by Kapustin and Orlov. Our results relate the anomaly-free condition to a recently proposed definition of graded coisotropic branes in Calabi-Yau manifolds. More specifically, we find that a coisotropic brane is anomaly-free if and only if it is gradable. We also comment on a different grading for coisotropic submanifolds introduced recently by Oh.Comment: AMS Tex, 11 page

Isotropic A-branes and the stability condition

The existence of a new kind of branes for the open topological A-model is argued by using the generalized complex geometry of Hitchin and the SYZ picture of mirror symmetry. Mirror symmetry suggests to consider a bi-vector in the normal direction of the brane and a new definition of generalized complex submanifold. Using this definition, it is shown that there exists generalized complex submanifolds which are isotropic in a symplectic manifold. For certain target space manifolds this leads to isotropic A-branes, which should be considered in addition to Lagrangian and coisotropic A-branes. The Fukaya category should be enlarged with such branes, which might have interesting consequences for the homological mirror symmetry of Kontsevich. The stability condition for isotropic A-branes is studied using the worldsheet approach.Comment: 19 page

Wilson-'t Hooft operators and the theta angle

We consider $(3+1)$-dimensional $SU(N)/\mathbb Z_N$ Yang-Mills theory on a space-time with a compact spatial direction, and prove the following result: Under a continuous increase of the theta angle $\theta\to\theta+2\pi$, a 't Hooft operator $T(\gamma)$ associated with a closed spatial curve $\gamma$ that winds around the compact direction undergoes a monodromy $T(\gamma) \to T^\prime(\gamma)$. The new 't Hooft operator $T^\prime(\gamma)$ transforms under large gauge transformations in the same way as the product $T(\gamma) W(\gamma)$, where $W(\gamma)$ is the Wilson operator associated with the curve $\gamma$ and the fundamental representation of SU(N).Comment: 7 page

Topological strings on noncommutative manifolds

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.Comment: 36 pages, AMS latex. v2: a reference to a related work has been added. v3: An error in the discussion of the Fourier-Mukai transform for twisted coherent sheaves has been fixed, resulting in several changes in Section 2. The rest of the paper is unaffected. v4: an incorrect statement concerning Lie algebroid cohomology has been fixe

Expectation values of chiral primary operators in holographic interface CFT

We consider the expectation values of chiral primary operators in the presence of the interface in the 4 dimensional N=4 super Yang-Mills theory. This interface is derived from D3-D5 system in type IIB string theory. These expectation values are computed classically in the gauge theory side. On the other hand, this interface is a holographic dual to type IIB string theory on AdS_5 x S^5 spacetime with a probe D5-brane. The expectation values are computed by GKPW prescription in the gravity side. We find non-trivial agreement of these two results: the gauge theory side and the gravity side.Comment: 17pages, no figur

Nonperturbative Tests of Three-Dimensional Dualities

We test several conjectural dualities between strongly coupled superconformal field theories in three dimensions by computing their exact partition functions on a three-sphere as a function of Fayet-Iliopoulos and mass parameters. The calculation is carried out using localization of the path integral and the matrix model previously derived for superconformal N = 2 gauge theories. We verify that the partition functions of quiver theories related by mirror symmetry agree provided the mass parameters and the Fayet-Iliopoulos parameters are exchanged, as predicted. We carry out a similar calculation for the mirror of N = 8 super-Yang-Mills theory and show that its partition function agrees with that of the ABJM theory at unit Chern-Simons level. This provides a nonperturbative test of the conjectural equivalence of the two theories in the conformal limit

Localized Tachyons and the Quantum McKay Correspondence

The condensation of closed string tachyons localized at the fixed point of a C^d/\Gamma orbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this paper we present some nontrivial evidence for this correspondence in the case C^2/Z_n by relating the intersection form of fractional branes to that of ``Higgs branch branes,'' the latter being branes which wrap nontrivial cycles in the resolved space.Comment: 25 pages; harvma

On Three-Dimensional Mirror Symmetry

Mirror Symmetry for a large class of three dimensional $\mathcal{N}=4$ supersymmetric gauge theories has a natural explanation in terms of M-theory compactified on a product of $\text{ALE}$ spaces. A pair of such mirror duals can be described as two different deformations of the eleven-dimensional supergravity background $\mathcal{M}=\mathbb{R}^{2,1} \times \text{ALE}_{1} \times \text{ALE}_{2}$, to which they flow in the deep IR. Using the $A-D-E$ classification of $\text{ALE}$ spaces, we present a neat way to catalogue dual quiver gauge theories that arise in this fashion. In addition to the well-known examples studied in \cite{Intriligator:1996ex}, \cite{deBoer:1996mp}, this procedure leads to new sets of dual theories. For a certain subset of dual theories which arise from the aforementioned M-theory background with an $A$-type $\text{ALE}_{1}$ and a $D$-type $\text{ALE}_2$, we verify the duality explicitly by a computation of partition functions of the theories on $S^3$, using localization techniques . We derive the relevant mirror map and discuss its agreement with predictions from the Type IIB brane construction for these theories.Comment: 50 pages, 12 figures; comments on the number of FI parameters adde

Localization and traces in open-closed topological Landau-Ginzburg models

We reconsider the issue of localization in open-closed B-twisted Landau-Ginzburg models with arbitrary Calabi-Yau target. Through careful analsysis of zero-mode reduction, we show that the closed model allows for a one-parameter family of localization pictures, which generalize the standard residue representation. The parameter $\lambda$ which indexes these pictures measures the area of worldsheets with $S^2$ topology, with the residue representation obtained in the limit of small area. In the boundary sector, we find a double family of such pictures, depending on parameters $\lambda$ and $\mu$ which measure the area and boundary length of worldsheets with disk topology. We show that setting $\mu=0$ and varying $\lambda$ interpolates between the localization picture of the B-model with a noncompact target space and a certain residue representation proposed recently. This gives a complete derivation of the boundary residue formula, starting from the explicit construction of the boundary coupling. We also show that the various localization pictures are related by a semigroup of homotopy equivalences.Comment: 36 page

Models for Modules

We recall the structure of the indecomposable sl(2) modules in the Bernstein-Gelfand-Gelfand category O. We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral discretization how a redefined action of the sl(2) algebra over the complex numbers can glue finite dimensional and infinite dimensional highest weight representations into indecomposable wholes. Furthermore, we discuss how projective cover representations arise in the tensor product of finite dimensional and Verma modules and give explicit tensor product decomposition rules. The tensor product spaces can be realized in terms of product path integrals. Finally, we discuss relations of our results to brane quantization and cohomological calculations in string theory.Comment: 18 pages, 6 figure