Rukmini Dey

Geometric prequantization of the moduli space of the vortex equations on a Riemann surface

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact, Kahler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kahler) structures ΩΨ0 on the moduli space, parametrized by Ψ0, a section of a line bundle on the Riemann surface. Next, we show that corresponding to these, there is a family of prequantum line bundles PΨ0 on the moduli space whose curvature is proportional to the symplectic forms ΩΨ0.

The Weierstrass—Enneper representation using hodographic coordinates on a minimal surface

In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate ρ∈ D ⊂ ℂ and σ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.

Geometric quantization of the moduli space of the self-duality equations on a Riemann surface

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin showed that the moduli space M of solutions of the self-duality equations on a compact Riemann surface of genus g > 1 has a hyper-Kahler structure. In particular M is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which are missing in Hitchins paper. Next we apply Quillens determinant line bundle construction to show that M admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above.

Symplectic and hyperkähler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field

Abstract In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and we study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g ≥ 1, gives rise to a family of moduli spaces carrying a hyperkahler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkahler structure but has a natural symplectic structure. For the case of the torus, g = 1, we show that the full set of equations has a solution, different from the “vortex solutions”.

A variational proof for the existence of a conformal metric with preassigned negative Guassian curvature for compact Riemann surfaces of genus>1

Given a smooth functionK 1. We do so by minimizing an appropriate functional using elementary analysis. In particular forK a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genusg > 1.

Erratum: “Geometric prequantization of the moduli space of the vortex equations on a Riemann surface” [J. Math. Phys.47, 103501 (2006)]

In this erratum to a work done previously, we give an alternative description for the prequantization with respect to the forms ΩΨ0, where we do not need the 1-form θ which may not be globally defined. Next by modifying the Quillen metric of the usual determinant bundle suitably, we quantize the usual symplectic form Ω on the vortex moduli space. Next, we show that by modifying the Quillen metric, one can also interpolate between the forms Ω and ΩΨ0 and the corresponding prequantum line bundles are topologically equivalent. It is not clear whether they are holomorphically equivalent.

An analytical study on the multi-critical behaviour and related bifurcation phenomena for relativistic black hole accretion

We apply the theory of algebraic polynomials to analytically study the transonic properties of general relativistic hydrodynamic axisymmetric accretion onto non-rotating astrophysical black holes. For such accretion phenomena, the conserved specific energy of the flow, which turns out to be one of the two first integrals of motion in the system studied, can be expressed as a 8th degree polynomial of the critical point of the flow configuration. We then construct the corresponding Sturm’s chain algorithm to calculate the number of real roots lying within the astrophysically relevant domain of

Born–Infeld solitons, maximal surfaces, and Ramanujan’s identities

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