Koushik Ray

Seiberg duality as derived equivalence for some quiver gauge theories

We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derived equivalence between the categories of modules of representation of the path algebras of the quivers. Starting from the projective modules of one quiver we construct tilting complexes whose endomorphism algebra yields the path algebra of the dual quiver. Finally, we present a general scheme for obtaining Seiberg dual quiver theories by constructing quivers whose path algebras are derived equivalent. We also discuss some combinatorial relations between this approach and some of the other approaches which has been used to study such dualities.

Fractional branes in non-compact type IIA orientifolds

We study fractional D-branes in the Type-IIA theory on a non-compact orientifold of the orbifold 3/3 in the boundary state formalism. We find that the fractional D0-branes of the orbifold theory become unstable due to the presence of a tachyon, while there is a stable D-instanton whose tachyon gets projected out. We propose that the D-instanton is obtained after tachyon condensation. We evidence this by calculating the Whitehead group of the Abelian category of objects corresponding to the boundary states as being isomorphic to 2.

Strings, junctions and stability

Identification of string junction states of pure SU(2) Seiberg-Witten theory as B-branes wrapped on a Calabi-Yau manifold in the geometric engineering limit is discussed. The wrapped branes are known to correspond to objects in the bounded derived category of coherent sheaves on the projective line 1 in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Π-stability. The spiral strings in the weak coupling region are interpreted as certain projective resolutions of the invertible sheaves. We discuss transitions between the spiral strings and junctions using the grade introduced for Π-stability through the central charges of the corresponding objects.

Some BPS configurations of the BLG theory

Abstract We obtain BPS configurations of the BLG theory and its variant including mass terms for scalars and fermions in addition to a background field with different world-volume and R -symmetries. Three cases are considered, with world-volume symmetries S O ( 1 , 1 ) and S O ( 2 ) and preserving different amounts of supersymmetry. In the former case we obtain a singular configuration preserving N = ( 3 , 3 ) supersymmetry and an one-quarter BPS configuration corresponding to intersecting M2-M5-M5-branes. In the latter instance the BPS equations are reduced to those in the self-dual Chern–Simons theory with two complex scalars. In want of an exact solution, we find a topological vortex solution numerically in this case. Other solutions are given by combinations of domain walls.

Forbidden territories in the string landscape

Problems of stabilizing moduli of the type-IIB string theory on toroidal orientifolds T6/Z2, in presence of worldvolume fluxes on various D-branes, are considered. For Z2 actions, introducing either O9 or O3 planes, we rule out the possibility of moduli stabilization in a wide class of models with = 1 supersymmetry, characterized by the type of fluxes turned on along D-brane worldvolume. Our results, in particular, imply that Abelian worldvolume fluxes can not by themselves stabilize closed string moduli, in a consistent supersymmtric model, for above orientifold compactifications. We also discuss other Z2 orientifolds of T6 and show that certain other brane wrappings are also ruled out by similar consistency requirements. In specific setups we consider examples with D9-branes wrapping on a complex three-torus with its world-volume fluxes taken to be semi-homogeneous bundles and D7-branes wrapping holomorphic four-cycles of the complex three-torus carrying world-volume fluxes.

Bulk reconstruction in AdS and Gel'fand-Graev-Radon Transform

A bstractThe bulk reconstruction formula for a Euclidean anti-de Sitter space is directly related to the inverse of the Gel’fand-Graev-Radon transform. Correlation functions of a conformal scalar field theory in the boundary are thereby related to correlation functions of a self-interacting scalar field theory in the bulk at different loop orders.

Determination of many-electron basis functions for a quantum Hall ground state using Schur polynomials

A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the 5/2 quantum Hall state with a few particles.

Beta-gamma system, pure spinors and Hilbert series of arc spaces

A bstractAlgorithms are presented for calculating the partition function of constrained beta-gamma systems in terms of the generating functions of the individual fields of the theory, the latter obtained as the Hilbert series of the arc space of the algebraic variety defined by the constraint. Examples of a beta-gamma system on a complex surface with an A1 singularity and pure spinors are worked out and compared with existing results.

Partition function of beta-gamma system on orbifolds

A bstractPartition function of beta-gamma systems on the orbifolds C2/ZN and C3/ZM × ZN are obtained as the invariant part of that on the respective affine spaces, by lifting the geometric action of the orbifold group to the fields. Interpreting the sum over roots of unity as an elementary contour integration, the partition function evaluates to a generalized Molien series counting invariant monomials composed of basic operators of the theory at each mass level.

BPS state counting on singular varieties

We define new partition functions for theories with targets on toric singularities via products of old partition functions on crepant resolutions. We compute explicit examples and show that the new partition functions turn out to be homogeneous on MacMahon factors.