arXiv subjects
Publications and source records attributed to Maxim Zabzine.
We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space.
We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\times S^1$ quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. In particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the $CY_3$, we find that the amplitudes of our model are related to Joyce invariants.
The geometry of the target space of an N=(2,2) supersymmetry sigma-model carries a generalized Kahler structure. There always exists a real function, the generalized Kahler potential K, that encodes all the relevant local differential geometry data: the metric, the B-field, etc. Generically this data is given by nonlinear functions of the second derivatives of K. We show that, at least locally, the nonlinearity on any generalized Kahler manifold can be explained as arising from a quotient of a space without this nonlinearity.
In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class $[α]$. The same condition was considered before by Crainic and Zhu but in a different context. In the case when $[α]$ is in the image of the sharp map we reproduce the Vaisman's condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show, with a different procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end we discuss the case of a generic coisotropic D-brane.
We give a physical derivation of generalized Kahler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri regarding the equivalence between generalized Kahler geometry and the bi-hermitean geometry of Gates-Hull-Rocek. When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.
We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kahler potential for any generalized Kahler manifold; this potential is the superspace Lagrangian.
We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.
We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically 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.
In this note we clarify the relation between extended world-sheet supersymmetry and generalized complex structure. The analysis is based on the phase space description of a wide class of sigma models. We point out the natural isomorphism between the group of orthogonal automorphisms of the Courant bracket and the group of local canonical transformations of the cotangent bundle of the loop space. Indeed this fact explains the natural relation between the world-sheet and the geometry of T+T^*. We discuss D-branes in this perspective.
In this note we generalize a result by Alekseev and Strobl for the case of $p$-branes. We show that there is a relation between anomalous free current algebras and "isotropic" involutive subbundles of $T\oplus \wedge^p T^*$ with the Vinogradov bracket, that is a generalization of the Courant bracket. As an application of this construction we go through some interesting examples: topological strings on symplectic manifolds, topological membrane on $G_2$-manifolds and topological 3-brane on $Spin(7)$ manifolds. We show that these peculiar topological theories are related to the physical (i.e., Nambu-Goto) brane theories in a specific way. These topological brane theories are proposed as microscopic description of topological M/F-theories.
Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N=(2,2) nonlinear sigma-models. The most direct relation is obtained at the N=(1,1) level when the sigma model is formulated with an additional auxiliary spinorial field. We revive a formulation in terms of N=(2,2) semi-(anti)chiral multiplets where such auxiliary fields are naturally present. The underlying generalized complex structures are shown to commute (unlike the corresponding ordinary complex structures) and describe a Generalized Kahler geometry. The metric, B-field and generalized complex structures are all determined in terms of a potential K.
We derive the most general local boundary conditions necessary for T-duality to be compatible with superconformal invariance of the two-dimensional N=1 supersymmetric nonlinear sigma model with boundaries. To this end, we construct a consistent gauge invariant parent action by gauging a U(1) isometry, with and without boundary interactions. We investigate the behaviour of the boundary conditions under T-duality, and interpret the results in terms of D-branes.
We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two-dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in context of deformation quantization.
We give a world-sheet description of D-brane in terms of gluing conditions on T+T^*. Using the notion of generalized Kahler geometry we show that A- and B-types D-branes for the general N=(2,2) supersymmetric sigma model (including a non-trivial NS-flux) correspond to the (twisted) generalized complex submanifolds with respect to the different (twisted) generalized complex structures however.
We discuss the notion of tensionless limit in quantum bosonic string theory, especially in flat Minkowski space, noncompact group manifolds (e.g., SL(2,R)) and coset manifolds (e.g., AdS). We show that in curved space typically there exists a critical value of the tension which is related to the critical value of the level of the corresponding affine algebra. We argue that at the critical level the sring theory becomes tensionless and that there exists a huge new symmetry of the theory. We dicuss the appearence of the higher spin massless states at the critical level.
We consider the N=1 supersymmetric two-dimensional non-linear sigma model with boundaries and nonzero B-field. By analysing the appropriate currents we describe the full set of boundary conditions compatible with N=1 superconformal symmetry. Using this result the problem of finding a correct action is discussed. We interpret the supersymmetric boundary conditions as a maximal integral submanifold of the target space manifold, and speculate about a new geometrical structure, the deformation of an almost product structure.
We study the Poisson sigma model which can be viewed as a topological string theory. Mainly we concentrate our attention on the Poisson sigma model over a group manifold G with a Poisson-Lie structure. In this case the flat connection conditions arise naturally. The boundary conditions (D-branes) are studied in this model. It turns out that the D-branes are labelled by the coisotropic subgroups of G. We give a description of the moduli space of classical solutions over Riemann surfaces both without and with boundaries. Finally we comment briefly on the duality properties of the model.
We review the most general, local, superconformal boundary conditions for the two-dimensional N=1 and N=2 non-linear sigma models, and analyse them for the N=1 and N=2 supersymmetric WZW models. We find that the gluing map between the left and right affine currents is generalised in a very specific way as compared to the constant Lie algebra automorphisms that are known.