arXiv subjects
Publications and source records attributed to Koushik Ray.
We construct a three-dimensional locally scale invariant version of the free Schroedinger field theory. It is shown to provide an effective theory of superfluid turbulence in Helium IV.
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.
An algebraic geometry method is used to calculate the moments of the electron density of states as a function of the energy for lattices in the tight binding approximation. Interpreting the moments as the Mellin transform of the density allows writing down a formula for the density as an inverse Mellin transform. The method is illustrated by working out the density function for the two-dimensional square and honeycomb lattices.
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 SO(1,1) and SO(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.
We study the moduli space of scalars in the BLG theory with and without a constant background four-form field. The classical vacuum moduli space is sixteen-dimensional in the absence of the four-form field. In its presence, however, the moduli space of BPS configurations may be reduced in dimension. We exemplify this with a BPS configuration having $SO(1,2)$ world-volume symmetry and $SO(4) \times SO(4)$ R-symmetry in the presence of a four-form field, by constructing an explicit solution.
We use a generalized Jordan algebra of the second kind to study the recently proposed BLG theory of multiple M2-branes. We find the restriction imposed on the ternary product from its consistency with the BLG theory.
A prescription for obtaining supergravity solutions for planar (p,q)-string networks is presented, based on earlier results. It shows that networks may be looked upon as tropical curves emerging as the spine of the amoeba of a holomorphic curve in M-theory. The Kaehler potential of supergravity is identified with the corresponding Ronkin function. Implications of this identification in counting dyons is discussed.
Various topological properties of D-branes in the type--IIA theory are captured by the topologically twisted B-model, treating D-branes as objects in the bounded derived category of coherent sheaves on the compact part of the target space. The set of basic D-branes wrapped on the homology cycles of the compact space are taken to reside in the heart of t-structures of the derived category of coherent sheaves on the space at any point in the Kähler moduli space. The stability data entails specifying a t-structure along with a grade for sorting the branes. Considering an example of a degenerate Calabi-Yau space, obtained via geometric engineering, that retains but a projective curve as the sole non-compact part, we identify the regions in the Kähler moduli space of the curve that pertain to the different t-structures of the bounded derived category of coherent sheaves on the curve corresponding to the different phases of the topological branes.
Problems of stabilizing moduli of the type--IIB string theory on toroidal orientifolds $\T^6/\Z_2$, in presence of worldvolume fluxes on various D-branes, are considered. For $Z_2$ actions, introducing either O9 or O3 planes, we rule out the possibility of moduli stabilization in a wide class of models with $\mathcal{N}=1$ 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 $Z_2$ orientifolds of $T^6$ 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.
We study fractional D-branes in the Type-IIA theory on a non-compact orientifold of the orbifold C^3/Z_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 Z_2.
We study stabilization of moduli in the type--IIB superstring theory on the six-dimensional toroidal orientifold $\T^6/Ω\cdot(-1)^{F_L}\cdot\Z_2$. We consider background space-filling D9-branes wrapped on the orientifold along with non-Abelian fluxes on its world-volume and demonstrate with two examples that this can stabilize all the complex structure moduli and some of the Kähler moduli.
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 $\cp{1}$ in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Pi-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 Pi-stability through the central charges of the corresponding objects.
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 on the non-compact orbifold $\C^3/\Z_5$ are studied. First, the boundary state description of the fractional branes are obtained. The open-string Witten index calculated using these states reproduces the adjacency matrix of the quiver of $\Z_5$. Then, using the toric crepant resolution of the orbifold $\C^3/\Z_5$ and invoking the local mirror principle, B-type branes wrapped on the holomorphic cycles of the resolution are studied. The boundary states corresponding to the five fractional branes are identified as bound states of BPS D-branes wrapping the 0-, 2- and 4-cycles in the exceptional divisor of the resolution of $\C^3/\Z_5$.
We study D1-branes on the fourfold $\C^4/(\Z_2\times\Z_2\times\Z_2)$, in the presence of discrete torsion. Discrete torsion is incorporated in the gauge theory of the D1-branes by considering a projective representation of the finite group $\Z_2\times\Z_2\times\Z_2$. The corresponding orbifold is then deformed by perturbing the F-flatness condition of the gauge theory. The moduli space of the resulting gauge theory retains a stable singularity of codimension three.
We find, from the toric description of the moduli space of D3-branes on non-compact six-dimensional singularities $\C^3/\Z_3$ and $\C^3/\Z_5$ in the blown-down limit, the four-dimensional bases on which these singular spaces are complex cones, and prove the existence of Kähler-Einstein metrics on these four-dimensional bases. This shows, in particular, that one can use the horizons obtained from these base spaces by a U(1)-foliation as compact parts of the target space for Type-IIB string theory with $\ads{5}$ in the context of the AdS-CFT correspondence.
In this note we study the resolution of conifold singularity by D-branes by considering compactification of D-branes on $\C^3/(\Z_2\times\Z_2)$. The resulting vacuum moduli space of D-branes is a toric variety which turns out to be a resolved conifold, that is a nodal variety in $\C^4$. This has the implication that all the corresponding phases of Type--II string theory are geometrical and are accessible to the D-branes, since they are related by flops.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction $g = 2$. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order.