Kernel functions, analytic torsion, and moduli spaces

Fay, John D.

doi:10.1090/memo/0464

Cited by 117 publications

(256 citation statements)

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“…We will show the construction of the Laughlin wave function on Riemann surfaces and indicate some interesting relations with recent results appeared in the mathematical literature [3].…”

Section: Introductionmentioning

confidence: 53%

“…This metric is proportional to the Bergman reproducing Kernel and it is also proportional to the curvature of the Arakelov metric [3]. Its most interesting feature (for our study here) is that it is proportional to the Chern form of the θ-function line bundle [9], implying the covariant derivatives match the transformation properties of the θ-function.…”

Section: Introductionmentioning

confidence: 86%

“…The transformations of the prime form and half -differential under the Fuchsian transformations can be found in Ref. [3]. ν(B, γ) corresponds to the boundary condition parameters and the characteristics a and b in Eq.…”

Section: The Wave Functions Of the Landau Levelsmentioning

confidence: 99%

“…The formula for the wave function would be in this case a generalization of Eq. (27) [3]. For those characteristics there exists also a Landau level, generically with multiplicity 1, for those s < 1/2 for which (1 − 2s)(g − 1) is a positive integer.…”

Section: The Wave Functions Of the Landau Levelsmentioning

confidence: 99%

“…Although not directly accessible to experiments, the problem of the physics on the Riemann surface happens to have deep relations with modern investigations on some interesting problems, like the occurrence of chaos in the surface with a negative curvature [2], and recent developments in the theory of Riemann surfaces, for example, the moduli of the surface and the vector bundles defined on the moduli [3].…”

Section: Introductionmentioning

confidence: 99%

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Quantum mechanics and quantum Hall effect on Reimann surfaces

Iengo

1994

Nuclear Physics B

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The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the θ-bundle, and the wave functions of the Landau levels in the case of the the Poincaré metric. The degeneracy of the the Landau levels is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin wave function on Riemann surfaces and discuss the mathematical structure hidden in the Laughlin wave function. Moreover the degeneracy of the Laughlin states is also discussed.

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“…We will show the construction of the Laughlin wave function on Riemann surfaces and indicate some interesting relations with recent results appeared in the mathematical literature [3].…”

Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 86%

Section: The Wave Functions Of the Landau Levelsmentioning

confidence: 99%

Section: The Wave Functions Of the Landau Levelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum mechanics and quantum Hall effect on Reimann surfaces

Iengo

1994

Nuclear Physics B

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Indefinite Hardy Spaces on Finite Bordered Riemann Surfaces

Alpay¹,

Vinnikov²

2000

Journal of Functional Analysis

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It is well known that one may study Hardy spaces with the domain a finite bordered Riemann surface rather than the unit disk. However, for domains with more than one boundary component it is natural to consider, besides the usual positive definite inner product on H 2 , indefinite inner products obtained by picking up different signs (or in the vector-valued case, different signature matrices) when integrating over different components. In this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a Kre@$ n space. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. As is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point is to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface.

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Matrix Riemann-Hilbert Problems Related to Branched Coverings of ℂℙ1

Korotkin

2003

Factorization and Integrable Systems

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In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with an arbitrary quasi-permutation monodromy group. The solution is given in terms of Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe divisor of zeros of the tau-function (so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to determinant of Laplacian operator on the Riemann surface.

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Kernel functions, analytic torsion, and moduli spaces

Cited by 117 publications

References 0 publications

Quantum mechanics and quantum Hall effect on Reimann surfaces

Quantum mechanics and quantum Hall effect on Reimann surfaces

Indefinite Hardy Spaces on Finite Bordered Riemann Surfaces

Matrix Riemann-Hilbert Problems Related to Branched Coverings of ℂℙ1

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