A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

Xu, Hao

doi:10.1007/s00220-012-1531-y

Cited by 47 publications

(51 citation statements)

References 36 publications

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“…The generating functional Z[g] (see (52) for the definition) depends on the geometry of the manifold and the QH state. It encodes all universal properties of the QH states.…”

Section: Resultsmentioning

confidence: 99%

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Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

2015

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We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.

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“…The generating functional Z[g] (see (52) for the definition) depends on the geometry of the manifold and the QH state. It encodes all universal properties of the QH states.…”

Section: Resultsmentioning

confidence: 99%

“…where the first two terms are the same as in (2) for all FQHE states. The generating functional (52) for this state is presented in Appendix F. Here we present only the anomalous part of the functional (the generalization of the first line of (130))…”

Section: Other Fqhe Statesmentioning

confidence: 99%

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

2015

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“…The reader is also referred to [23] and [24] for a recursive formula for the coefficients a j 's and an alternative computation of a j for j ≤ 3 using Calabi's diastasis function (see also [40] for a graph-theoretic interpretation of this recursive formula).…”

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confidence: 99%

Two conjectures on Ricci-flat Kähler metrics

2018

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We propose two conjectures about Ricci-flat Kähler metrics:Conjecture 1: A Ricci-flat projectively induced metric is flat.Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the a n+1 coefficient of the TYZ expansion vanishes is flat.We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).2010 Mathematics Subject Classification. 53C55; 58C25; 58F06.

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“…Thanks to the Kähler condition dω = 0, the space of Weyl invariant polynomials on Kähler manifolds has a canonical basis represented by multi-digraphs. A closed formula has been discovered [51] for coefficients in the asymptotic expansion of the weighted Bergman kernel. We expect that a similar formula should exist for the heat kernel of the Laplacian operator on Kähler manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…As shown in [51,52], tensor calculus on Kähler manifolds could be naturally formulated in terms of graphs. In particular, the Weyl invariants [22] can be represented by multi-digraphs.…”

Section: Introductionmentioning

confidence: 99%

Bergman kernel and Kähler tensor calculus

2013

Pure and Applied Mathematics Quarterly

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Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman's asymptotic expansion building partly on Graham's work on CR invariants and Nakazawa's work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa's asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman's asymptotic expansion.

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A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

Cited by 47 publications

References 36 publications

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

Two conjectures on Ricci-flat Kähler metrics

Bergman kernel and Kähler tensor calculus

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