Papers by Raju Roychowdhury
Journal of Mathematical Physics, Aug 1, 2011
We reconsider the sub-leading quantum perturbative corrections to N = 2 cubic special Kähler geom... more We reconsider the sub-leading quantum perturbative corrections to N = 2 cubic special Kähler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations, dubbed Peccei-Quinn (PQ) transformations. Since PQ transformations do not belong to the d = 4 U-duality group G 4 , in symmetric cases they generally have a non-trivial action on the unique quartic invariant polynomial I 4 of the charge representation R of G 4. This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set I 4 to zero, or vice versa it can generate a non-vanishing I 4 : this corresponds to transitions between "large" and "small" charge orbits, which we classify in some detail within the "special coordinates" symplectic frame. Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of I 4 , with relevant application to black hole entropy.
International Journal of Modern Physics, Apr 10, 2010
We compute the effective black hole potential V BH of the most general N = 2,d = 4 (local) specia... more We compute the effective black hole potential V BH of the most general N = 2,d = 4 (local) special Kähler geometry with quantum perturbative corrections, consistent with axion-shift Peccei-Quinn symmetry and with cubic leading order behavior. We determine the charge configurations supporting axion-free attractors, and explain the differences among various configurations in relations to the presence of "flat" directions of V BH at its critical points. Furthermore, we elucidate the role of the sectional curvature at the non-supersymmetric critical points of V BH , and compute the Riemann tensor (and related quantities), as well as the so-called E-tensor. The latter expresses the non-symmetricity of the considered quantum perturbative special Kähler geometry.
arXiv (Cornell University), Jun 25, 2014
In the bottom-up approach of emergent gravity we attempt to find symplectic gauge fields emerging... more In the bottom-up approach of emergent gravity we attempt to find symplectic gauge fields emerging from Euclidean Schwarzschild instanton, which is studied as electromagnetism defined on the symplectic space $(M,\omega)$. Geometrical engineering with the emergent metric sets up the Seiberg Witten map between commutative and non-commutative gauge fields, preparing the ground for the evaluation of topological invariants in terms of the underlying gauge theory quantities.
Evolution Equations and Control Theory
We are concerned with the controllability of a general heterogeneous semilinear parabolic equatio... more We are concerned with the controllability of a general heterogeneous semilinear parabolic equation of the form \begin{document}$ y_t(x,t)=y_{xx}(x,t)+f(x,y(x,t)) $\end{document} in one space dimension using boundary static controls. More precisely, we provide conditions so that, using only static controls at the boundary, a certain target (a steady-state of the equation) is reached in infinite time. Applications of the main result are also presented, in particular, we provide several controllability results to heterogeneous equations of monostable and bistable types.
Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the ... more Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler Top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux-Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux-Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang-Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux-Halphen system and occurring in the study of Bianchi IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.
arXiv: High Energy Physics - Theory, 2016
The Darboux-Halphen system of equations have common or individual additive terms depending on the... more The Darboux-Halphen system of equations have common or individual additive terms depending on the matrices defining Yang-Mills gauge potential fields. Tod (Phys. Lett. A 190 (1994) 221-224), described a conserved quantity for the classical systems with no additive terms. We show that the conserved quantity apply even for the generalized cases with common additive terms. A theory has been presented, with an example, of how to formulate conserved quantities for equation with individual additive terms. We also briefly shed some light on the issues of surface motions of fluids in conncetion to Nahm`s equation and the self-duality and integrability of membrane dynamics.
This paper studies first the differential inequalities that make it possible to build a global th... more This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the differential inequalities describing pseudo-holomorphic functions can be used to define a one-real-dimensional manifold (by the vanishing of a function with nonzero gradient), which is here a 1-parameter family of plane curves. On studying the associated envelopes, such a parameter can be eliminated by solving two nonlinear partial differential equations. The classical differential geometry of curves can be therefore exploited to get a novel perspective on the equations describing the global theory of pseudo-holomorphic functions.
Many distinct problems give birth to Darboux-Halphen system of differential equations and here we... more Many distinct problems give birth to Darboux-Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in R 3. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss-Manin connection in disguise developed by one of the authors and finally in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.
In the bottom-up approach of emergent gravity we attempt to find out symplectic gauge fields emer... more In the bottom-up approach of emergent gravity we attempt to find out symplectic gauge fields emerging from Euclidean Schwarzschild metric which is a Ricci flat manifold but non-K\"ahler. The Euclidean Schwarzschild solution is studied extensively as electromagnetism defined on the symplectic space $(M,\omega)$. A detailed geometrical engineering with the emergent metric help us set up the Seiberg Witten map between commutative and non- commutative gauge fields and further paves the path for the evaluation of the topological invariants in terms of gauge theory quantities.
Taub-NUT as Bertrand spacetime with magnetic fields
Based on symmetries Taub-NUT shares with Bertrand spacetime, we cast it as the latter with magnet... more Based on symmetries Taub-NUT shares with Bertrand spacetime, we cast it as the latter with magnetic fields. Its nature as a Bianchi-IX gravitational instanton and other related geometrical properties are reviewed. We provide an easy derivation and comparison between the spatial Killing-Yano tensors deduced from first-integrals and the corresponding hyperk¨ahler structures and finally verify the existence of a graded Lie-algebra structure via SchoutenNijenhuis brackets
In the bottom-up approach of emergent gravity we attempt to find out symplectic gauge fields emer... more In the bottom-up approach of emergent gravity we attempt to find out symplectic gauge fields emerging from Euclidean Schwarzschild metric which is a Ricci flat manifold but non-Kahler. The Euclidean Schwarzschild solution is studied extensively as a electromagnetism defined on the symplectic space (M, {\omega}). A detailed geometrical engineering with the emergent metric help us set up the Seiberg Witten map between commutative and non- commutative gauge fields and further paves the path for the evaluation of the topological invariants in terms of gauge theory quantities.
Emergent gravity is based on the Darboux theorem or the Moser lemma in symplectic geometry statin... more Emergent gravity is based on the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as U(1) gauge theory is defined on a spacetime with symplectic structure. In this approach, the spacetime geometry is defined by U(1) gauge fields on noncommutative (NC) spacetime. Accordingly the topology of spacetime is determined by the topology of NC U(1) gauge fields. We show that the topology change of spacetime is ample in emergent gravity and the subsequent resolution of spacetime singularity is possible in NC spacetime. Therefore the emergent gravity approach provides a well-defined mechanism for the topology change of spacetime which does not suffer any spacetime singularity in sharp contrast to general relativity.
In the Einestein-dilaton theory with a Liouville potential parameterized by $\eta$, we find a Sch... more In the Einestein-dilaton theory with a Liouville potential parameterized by $\eta$, we find a Schwarzschild-type black hole solution. This black hole solution, whose asymptotic geometry is described by the warped metric, is thermodynamically stable only for $0 \le \eta < 2$. Applying the gauge/gravity duality, we find that the dual gauge theory represents a non-conformal thermal system with the equation of state depending on $\eta$. After turning on the bulk vector fluctuations with and without a dilaton coupling, we calculate the charge diffusion constant, which indicates that the life time of the quasi normal mode decreases with $\eta$. Interestingly, the vector fluctuation with the dilaton coupling shows that the DC conductivity increases with temperature, a feature commonly found in electrolytes.
In this paper we examine a small but detailed test of the emergent gravity picture with explicit ... more In this paper we examine a small but detailed test of the emergent gravity picture with explicit solutions in gravity and gauge theory. We first derive symplectic U(1) gauge fields starting from the Eguchi-Hanson metric in four-dimensional Euclidean gravity. The result precisely reproduces the U(1) gauge fields of the Nekrasov-Schwarz instanton previously derived from the top-down approach. In order to clarify the role of noncommutative spacetime, we take the Braden-Nekrasov U(1) instanton defined in ordinary commutative spacetime and derive a corresponding gravitational metric. We show that the K\"ahler manifold determined by the Braden-Nekrasov instanton exhibits a spacetime singularity while the Nekrasov-Schwarz instanton gives rise to a regular geometry-the Eguchi-Hanson space. This result implies that the noncommutativity of spacetime plays an important role for the resolution of spacetime singularities in general relativity. We also discuss how the topological invariants associated with noncommutative U(1) instantons are related to those of emergent four-dimensional Riemannian manifolds according to the emergent gravity picture.
Emergent gravity is aimed at constructing a Riemannian geometry from U(1) gauge fields on a nonco... more Emergent gravity is aimed at constructing a Riemannian geometry from U(1) gauge fields on a noncommutative spacetime. But this construction can be inverted to find corresponding U(1) gauge fields on a (generalized) Poisson manifold given a Riemannian metric (M, g). We examine this bottom-up approach with the LeBrun metric which is the most general scalar-flat Kahler metric with a U(1) isometry and contains the Gibbons-Hawking metric, the real heaven as well as the multi-blown up Burns metric which is a scalar-flat Kahler metric on C^2 with n points blown up. The bottom-up approach clarifies some important issues in emergent gravity.
Arxiv preprint arXiv: …, Jan 1, 2010
We study some problems arising from the introduction of a complex scalar field in cosmology, mode... more We study some problems arising from the introduction of a complex scalar field in cosmology, modelling its possible behaviors in both the inflationary and dark energy stages of the universe.
Arxiv preprint arXiv:0805.1182, Jan 1, 2008
Electrodynamics in curved space-time can be studied in the Eastwood-Singer gauge, which has the a... more Electrodynamics in curved space-time can be studied in the Eastwood-Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations.
Arxiv preprint arXiv:0902.2098, Jan 1, 2009
This paper studies spinor two-point functions for spin-1/2 and spin-3/2 fields in maximally symme... more This paper studies spinor two-point functions for spin-1/2 and spin-3/2 fields in maximally symmetric spaces such as de Sitter(dS) spacetime, by using intrinsic geometric objects. The Feynman, positive-and negative-frequency Green functions are then obtained for these cases, from which we eventually display the supercommutator and the Peierls bracket under such a setting in twocomponent-spinor language.
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Papers by Raju Roychowdhury