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Mixed anomalies of chiral algebras compactified to smooth quasi-projective surfaces
High Energy Physics - Theory · Submitted 14 Dec 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The obstruction classes of gerbes of them coincide with the second Chern characters obtained by the Riemann-Roch theorem and in particular vanishes for $1$ point blowup, which means that one of the gravitational anomalies vanishes for a non-Calabi-Yau manifold compactification.
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The abstract, sentence by sentence
Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected to play a central role in the convergence of divergence in mathematical physics of superstring theory for quantization of gauge theory and gravity. Naively, this algebra plays an important role in a holomorphic conformal field theory with a non-negative integer graded conformal dimension, whose target space does not necessarily have the vanishing first Chern class. This algebra has two definitions until now: one is that by Malikov-Schechtman-Vaintrob by gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop spaces. I will use the new definition of Nekrasov by simplifying Malikov-Schechtman-Vaintrob in order to compute the obstruction classes of gerbes of chiral differential operators. In this paper, I will examine the two independent Ans$\"{a}$tze (or working hypotheses) of Witten's $\mathcal{N}=(0,2)$ heterotic strings and Nekrasov's generalized complex geometry, after Hitchin and Gualtieri, are consistent in the case of $\mathbb{CP}^2$, which has $3$ affine patches and is expected to have the "first Pontryagin anomaly". I also scrutinized the physical meanings of $2$ dimensional toric Fano manifolds, or rather toric del Pezzo surfaces, obtained by blowing up the non-colinear $1, 2, 3$ points of $\mathbb{CP}^2$. The obstruction classes of gerbes of them coincide with the second Chern characters obtained by the Riemann-Roch theorem and in particular vanishes for $1$ point blowup, which means that one of the gravitational anomalies vanishes for a non-Calabi-Yau manifold compactification. The future direction towards the geometric Langlands program is also discussed in the last section.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected to play a central role in the convergence of divergence in mathematical physics of superstring theory for quantization of gauge theory and gravity. | background 1.00 | 0.03 | 0.02 | hedged possibility 0.49 |
| s1 | Naively, this algebra plays an important role in a holomorphic conformal field theory with a non-negative integer graded conformal dimension, whose target space does not necessarily have the vanishing first Chern class. | background 1.00 | 0.06 | 0.03 | asserted 0.79 |
| s2 | This algebra has two definitions until now: one is that by Malikov-Schechtman-Vaintrob by gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop spaces. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s3 | I will use the new definition of Nekrasov by simplifying Malikov-Schechtman-Vaintrob in order to compute the obstruction classes of gerbes of chiral differential operators. | method 0.91 | 0.71 | 0.06 | asserted 0.97 |
| s4 | In this paper, I will examine the two independent Ans$\"{a}$tze (or working hypotheses) of Witten's $\mathcal{N}=(0,2)$ heterotic strings and Nekrasov's generalized complex geometry, after Hitchin and Gualtieri, are consistent in the case of $\mathbb{CP}^2$, which has $3$ affine patches and is expected to have the "first Pontryagin anomaly". | contribution 0.98 | 0.63 | 0.25 | hedged possibility 0.67 |
| s5 | I also scrutinized the physical meanings of $2$ dimensional toric Fano manifolds, or rather toric del Pezzo surfaces, obtained by blowing up the non-colinear $1, 2, 3$ points of $\mathbb{CP}^2$. | contribution 0.99 | 0.84 | 0.05 | asserted 0.87 |
| s6 | The obstruction classes of gerbes of them coincide with the second Chern characters obtained by the Riemann-Roch theorem and in particular vanishes for $1$ point blowup, which means that one of the gravitational anomalies vanishes for a non-Calabi-Yau manifold compactification. | result 0.91 | 0.87 | 0.58 | asserted 1.00 |
| s7 | The future direction towards the geometric Langlands program is also discussed in the last section. | future work 1.00 | 0.47 | 0.03 | asserted 0.87 |
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The paper
Original abstract
Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected to play a central role in the convergence of divergence in mathematical physics of superstring theory for quantization of gauge theory and gravity. Naively, this algebra plays an important role in a holomorphic conformal field theory with a non-negative integer graded conformal dimension, whose target space does not necessarily have the vanishing first Chern class. This algebra has two definitions until now: one is that by Malikov-Schechtman-Vaintrob by gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop spaces. I will use the new definition of Nekrasov by simplifying Malikov-Schechtman-Vaintrob in order to compute the obstruction classes of gerbes of chiral differential operators. In this paper, I will examine the two independent Ans$\"{a}$tze (or working hypotheses) of Witten's $\mathcal{N}=(0,2)$ heterotic strings and Nekrasov's generalized complex geometry, after Hitchin and Gualtieri, are consistent in the case of $\mathbb{CP}^2$, which has $3$ affine patches and is expected to have the "first Pontryagin anomaly". I also scrutinized the physical meanings of $2$ dimensional toric Fano manifolds, or rather toric del Pezzo surfaces, obtained by blowing up the non-colinear $1, 2, 3$ points of $\mathbb{CP}^2$. The obstruction classes of gerbes of them coincide with the second Chern characters obtained by the Riemann-Roch theorem and in particular vanishes for $1$ point blowup, which means that one of the gravitational anomalies vanishes for a non-Calabi-Yau manifold compactification. The future direction towards the geometric Langlands program is also discussed in the last section.
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What it cites · 3 sentences without a confident label
Uses the method or tool of 2 papers
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Let us introduce an ansatz of Nekrasov [ref] for a coordinate change $γ^i$ to $γ̃^a$ such that the OPE of the $βγ$-system is unchanged after the coordinate transformation.
Lectures on curved beta-gamma systems, pure spinors, and anomalies · The target space coordinate transformations · 0.99
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In the following subsections, as [ref] did, we will use the notation $v^[i], w^[i], V^[i], W^[i]$, instead of $γ, β$.
Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects · $U_0 →U_1$ · 0.96
Cites as background 1 paper
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During the preparation of this paper, Kapustin-Witten [ref] submitted a paper to the preprint, which is related to this paper.
Electric-Magnetic Duality And The Geometric Langlands Program · Anomaly 2-forms for 3 points blowups of $ℂℙ^2$ · 0.88
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