The Refined BPS Index from Stable Pair Invariants

Choi, Jinwon; Katz, Sheldon; Klemm, Albrecht

doi:10.1007/s00220-014-1978-0

Cited by 88 publications

(195 citation statements)

References 64 publications

(218 reference statements)

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“…It is "refined" partly in the sense that it is the virtual motive of M ∞ X (d, χ), which specializes to the PandharipandeThomas invariant [24]. A product formula for these refined PandharipandeThomas invariants is conjectured in [3], which is consistent with B-model calculation in physics. It is expected that the lower degree correction terms produced from the product formula are in correspondence with the wallcrossing terms in our paper.…”

Section: 2supporting

confidence: 62%

“…It is expected that the lower degree correction terms produced from the product formula are in correspondence with the wallcrossing terms in our paper. Details can be found in [3].…”

Section: 2mentioning

confidence: 99%

“…In [3], new definition of the refined PandharipandeThomas invariant is proposed via an extension of the Bia lynicki-Birula decomposition to singular moduli spaces. It is "refined" partly in the sense that it is the virtual motive of M ∞ X (d, χ), which specializes to the PandharipandeThomas invariant [24].…”

Section: 2mentioning

confidence: 99%

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Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Choi

Chung

2016

J. Math. Soc. Japan

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Abstract. We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.1. Introduction 1.1. Motivation and Results. In moduli theory, for a given quasi-projective moduli space M 0 , various compactifications stem from the different view points for the moduli points of M 0 . After we obtain various compactified moduli spaces of M 0 , it is quite natural to ask the geometric relationship among them. Sometimes, this question is answered by birational morphisms between them, which enables us to obtain some geometric information (for example, the cohomology groups) of one space from that of the other [28,5].In this paper, we study the moduli space of semistable sheaves of dimension one on smooth projective surfaces [27], which recently gains interests in both mathematics and physics. This is an example of compactifications of the relative Jacobian variety, where we regard its general point as a sheafon a smooth curve C with pole along points p i of general position. In general, the moduli space of semistable sheaves is hard to study due to the lack of geometry of its boundary points. However, if n is equal to the genus of C, the sheaf F has a unique section up to scalar. So, we may alternatively consider the general point as a sheaf with a section, which in turn leads to another compactification, so called the moduli space of α-semistable pairs (more generally, the coherent systems [18]). When α is large, it can be shown that the moduli spaces of α-stable pairs are nothing but the relative 2010 Mathematics Subject Classification. 14D20.

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Section: 2supporting

confidence: 62%

“…It is expected that the lower degree correction terms produced from the product formula are in correspondence with the wallcrossing terms in our paper. Details can be found in [3].…”

Section: 2mentioning

confidence: 99%

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Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Choi

Chung

2016

J. Math. Soc. Japan

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“…The refined topological string free energy is a function of the Kähler moduli and of two parameters 1,2 , which "refine" the topological string coupling constant. We also introduce (see for example [36,37])…”

Section: Jhep05(2016)133mentioning

confidence: 99%

“…indicate the BPS generating function (2.37). This function can be also derived by many different methods [36,37,39,40], but for our purposes it is more convenient to use the (refined) topological vertex [42,47,48], or, equivalently, Nekrasov's five dimensional instanton partition function [16,46]. We will spell out some details of such a calculation in the case N = 3, in the next section.…”

Section: Jhep05(2016)133mentioning

confidence: 99%

Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N = 3. Our proposal can be generalized to arbitrary toric Calabi-Yau manifolds and might solve the corresponding quantum integrable system of Goncharov and Kenyon.

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Exact Results on $${\mathcal N}=2$$ Supersymmetric Gauge Theories

Teschner

2015

New Dualities of Supersymmetric Gauge Theories

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The following is meant to give an overview over our special volume. The first three sections 1-3 are intended to give a general overview over the physical motivations behind this direction of research, and some of the developments that initiated this project. These sections are written for a broad audience of readers with interest in quantum field theory, assuming only very basic knowledge of supersymmetric gauge theories and string theory. This will be followed in Section 4 by a brief overview over the different chapters collected in this volume, while Section 5 indicates some related developments that we were unfortunately not able to cover here.Due to the large number of relevant papers the author felt forced to adopt a very restrictive citation policy. With the exception of very few original papers only review papers will be cited in Sections 1 and 2. More references are given in later sections, but it still seems impossible to list all papers on the subjects mentioned there. The author apologises for any omission that results from this policy. A citation of the form [V:x] refers to article number x in this volume.

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The Refined BPS Index from Stable Pair Invariants

Cited by 88 publications

References 64 publications

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Exact quantization conditions for the relativistic Toda lattice

Exact Results on $${\mathcal N}=2$$ Supersymmetric Gauge Theories

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