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Geometric quantization and the metric dependence of the self-dual field theory
High Energy Physics - Theory · Submitted 25 Nov 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold.
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The abstract, sentence by sentence
We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. | contribution 0.84 | 0.90 | 0.05 | asserted 0.70 |
| s1 | Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. | method 0.95 | 0.95 | 0.08 | asserted 1.00 |
| s2 | This connection governs metric dependence of the partition function of the self-dual field. | interpretation 0.54 | 0.78 | 0.23 | asserted 1.00 |
| s3 | We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. | result 1.00 | 0.93 | 0.66 | asserted 0.99 |
| s4 | We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. | result 0.53 | 0.93 | 0.07 | asserted 1.00 |
| s5 | As an application, we show that the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model. | result 0.98 | 0.93 | 0.10 | asserted 1.00 |
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The paper
Original abstract
We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.
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References
How the literature uses it
12 arXiv papers cite this in 35 sentences
Papers citing it · 29 sentences without a confident label
Compares with it 1 paper
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This agrees with twice the expression derived in [ref] for the curvature of the anomaly bundle of a single self-dual field, using geometric quantization techniques.
The anomaly line bundle of the self-dual field theory · The partition function as a section of a line bundle · 0.99
Cites it as support 1 paper
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Moreover, $𝒟$ is isomorphic to $(𝒦)^-1$ [ref] .
Global anomalies and chiral p-forms · Line bundles over the space of metrics modulo diffeomorphisms · 0.97
Cites it as background 4 papers
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The information summarized above about this one-form will be sufficient for our purpose, but the interested reader can find details in [ref] .
The global anomaly of the self-dual field in general backgrounds · The connection on the anomaly bundle · 0.85
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[See Section 2 of [ref] for a pedagogical introduction to differential cocycles, as well as Section 3.1 of [ref] for details about how to see the spin Chern-Simons field as a differential cocycle.] We fix a flux background on $M$, which is an integral class $a$ in $H^2ℓ+2(M,Z)$, to be understood as the class of the field strength of $Č$.
The global gravitational anomaly of the self-dual field theory · The combined gauge-gravitational anomaly · 0.96
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Much better understanding of the dynamics of self-dual fields [ref] [ref] [ref] and their partition functions [ref] [ref] [ref] [ref] [ref] [ref] [ref] .
On global anomalies in type IIB string theory · Introduction · 0.84
What it cites · 20 sentences without a confident label
Compares with 2 papers
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Using our result about the self-dual field, we compute the norm of the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold, and show that it coincides with the square root of the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) torsion [ref] .
Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes · Introduction and summary · 0.85
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Recalling that $h = -i(τ- τ̅)$, this reproduces the expression obtained in [ref] , equation (6.35), for the norm of the partition function.
Holographic Action for the Self-Dual Field · Normalization factor · 0.97
Cites as support 1 paper
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This subtle point, that we missed in the first version of this paper, is nicely confirmed by the path integral construction of the partition function of the self-dual field theory [ref] .
The anomaly line bundle of the self-dual field theory · Local gravitational anomaly · 0.96
Cites as background 6 papers
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We refer the reader to the two papers [ref] for a detailed exposition of these ideas.
Holographic Action for the Self-Dual Field · Symplectic reduction on the space of $2ℓ+ 1$ forms · 0.87
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We refer the reader to the two papers [ref] for a detailed exposition of these ideas.
Five-Brane Effective Action In M-Theory · Symplectic reduction on the space of $2ℓ+ 1$ forms · 0.87
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Note as well that the quantization of the intermediate Jacobian of a Calabi-Yau threefold also appears naturally in relation to the B-model [ref] .
Topological wave functions and heat equations · The determinant of the (2,0) supermultiplet on a Calabi-Yau threefold[The work pertaining to this section was started up · 0.91
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