Balanced Metrics and Noncommutative Kähler Geometry

Lukić, Sergio

doi:10.3842/sigma.2010.069

Cited by 5 publications

(2 citation statements)

References 26 publications

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“…In other words, if we think of M as parametrizing embeddings, we pull back the Fubini-Study metric on U ∨ r . Conversely, we saw in (3.28) that we had the assignment Incidentally, there is a sense in which balanced metrics may be regarded as quantized versions of hermitian-Einstein metrics, with = 1/m [53,54]. It is currently not completely clear to us what the significance of this is in the context of phenomenological string compactifications (see [55,56] for a possible interpretation in a slightly different setting), but it would surely be interesting if balanced metrics have some physical significance beyond serving as approximations of hermitian-Einstein metrics.…”

Section: Numerical Approach With Balanced Metricsmentioning

confidence: 97%

Gluing branes — I

Donagi

Wijnholt

2013

J. High Energ. Phys.

109

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We consider several aspects of holomorphic brane configurations. We recently showed that an important part of the defining data of such a configuration is the gluing morphism, which specifies how the constituents of a configuration are glued together, but is usually assumed to be vanishing. Here we explain the rules for computing spectra and interactions for configurations with non-vanishing gluing VEVs. We further give a detailed discussion of the D-terms for Higgs bundles, spectral covers and ALE fibrations. We highlight a stability criterion that applies to degenerate configurations of the spectral data, and address an apparent discrepancy between the field theory and ALE descriptions. This allows us to show that one gets walls of marginal stability in F -theory even though they are absent in the 11d supergravity description. We also propose a numerical approach for approximating the hermitian-Einstein metric of the Higgs bundle using balanced metrics.

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Section: Numerical Approach With Balanced Metricsmentioning

confidence: 97%

Gluing branes — I

Donagi

Wijnholt

2013

J. High Energ. Phys.

109

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“…In this context the Bergman kernel is the "reproducing kernel" studied in [20], see [17] for a review. For recent work on applications of the Bergman kernel to quantization of Kähler manifolds see [21,22]. Another recent paper discussing the topic is [23].…”

Section: Introductionmentioning

confidence: 99%

Bergman Kernel from Path Integral

Douglas

Klevtsov

2009

Commun. Math. Phys.

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We rederive the expansion of the Bergman kernel on Kahler manifolds developed by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation theory, and generalize it to supersymmetric quantum mechanics. One physics interpretation of this result is as an expansion of the projector of wave functions on the lowest Landau level, in the special case that the magnetic field is proportional to the Kahler form. This is relevant for the quantum Hall effect in curved space, and for its higher dimensional generalizations. Other applications include the theory of coherent states, the study of balanced metrics, noncommutative field theory, and a conjecture on metrics in black hole backgrounds. We give a short overview of these various topics. From a conceptual point of view, this expansion is noteworthy as it is a geometric expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time expansion for the heat kernel, but in this case describing the long time limit, without depending on supersymmetry.Comment: 27 page

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Balanced metrics on some Hartogs type domains over bounded symmetric domains

Feng

2014

Ann Glob Anal Geom

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The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain k j=1 Ω j B d 0 (µ) is defined as the Hartogs type domain constructed over the product k j=1 Ω j of irreducible bounded symmetric domains Ω j (1 ≤ j ≤ k), with the fiber over each point (z 1 , · · · , z k ) ∈ k j=1 Ω j being a ball in C d 0 of the radius k j=1 N Ω j (z j , z j ) µ j 2 of the product of positive powers of their generic norms. Any such domain k j=1 Ω j B d 0 (µ) (k ≥ 2) is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric αg(µ) (α > 0) on the domain k j=1 Ω j B d 0 (µ) to be a balanced metric, where g(µ)is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.

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Balanced Metrics and Noncommutative Kähler Geometry

Cited by 5 publications

References 26 publications

Gluing branes — I

Gluing branes — I

Bergman Kernel from Path Integral

Balanced metrics on some Hartogs type domains over bounded symmetric domains

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