Stacky dualities for the moduli of Higgs bundles

Derryberry, Richard

doi:10.1016/j.aim.2020.107152

“…Incidentally, the Liouville CFT has finally been defined mathematically from its path integral: see [938,939] and references therein. Other mathematical references include the study of 6j symbols of (the modular double of) U q (sl 2 ) [81], relations to the geometric Langlands correspondence or deformations thereof [754,755,[940][941][942][943], and more [75,205,574,944,945].…”

Section: Discussionmentioning

confidence: 99%

A slow review of the AGT correspondence

Floch¹

2020

Preprint

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Starting with a gentle approach to the Alday-Gaiotto-Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide survey of the literature on numerous extensions of the correspondence. This is the writeup of the lectures given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD to non-Lagrangian theories) obtained by twisted compactification of 6d N = (2, 0) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seiberg-Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of 4d N = 2 SU(2) superconformal quiver theories is equal to a Liouville CFT correlator of primary operators.Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres-Douglas (AD) theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilson-'t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.

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“…The last point of view fits nicely with the 3d/3d correspondence discussed in subsection 9.3, which involves G C Chern-Simons theory. Further works on the Hitchin system, opers, and Darboux coordinates on M include [50,343,356,366,415,488,[524][525][526][527][528][529][530][531][532][533][534][535] (some are reviewed in [536]); a different technique is based on spectral networks, which abelianize flat connections on C [537][538][539][540][541][542][543][544][545][546][547][548][549]; see also [49,[550][551][552].…”

Section: Line Operatorsmentioning

confidence: 99%

A slow review of the AGT correspondence

Floch¹

2022

J. Phys. A: Math. Theor.

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Starting with a gentle approach to the Alday–Gaiotto–Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School ‘YRISW 2020’ to appear in a special issue of J. Phys. A. Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD (quantum chromodynamics) to non-Lagrangian theories) obtained by twisted compactification of 6d N = ( 2 , 0 ) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seiberg–Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of a 4d N = 2 S U ( 2 ) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyres–Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilson–’t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries.

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“…In the vacuum b ∈ B \ ∆, the low-energy effective theory is a U (1) r gauge theory with electromagnetic charge lattice H 1 (F b , Z). The natural map Γ b → H 1 (F b , Z) is (possibly with punctures), the corresponding Seiberg-Witten complex integrable system is closely related to the Hitchin integrable system on the moduli space of G-Higgs bundles on C. Typically, the Seiberg-Witten integrable system is a self-dual fiberwise finite quotient of an Hitchin integrable system [22,34,37]. In particular, the base B is isomorphic to C r as a complex manifold.…”

Section: 1mentioning

confidence: 99%

“…Another reason to expect this correspondence is the work of Gaiotto-Moore-Neitzke [34,37] in which the hyperkähler metric on the Seiberg-Witten integrable system M is reconstructed from "quantum corrections" determined by the BPS spectrum. As M is self-mirror [22], compatibility between the SYZ mirror construction and the Gaiotto-Moore-Neitzke description of the hyperkähler metric requires a matching between counts of BPS states and counts of J ζ -holomorphic curves. Conjecture 4.4 provides a categorification of this correspondence and is formulated in a way leading to a natural physics derivation presented in the next section.…”

Section: 1mentioning

confidence: 99%

Holomorphic Floer theory and Donaldson-Thomas invariants

Bousseau¹

2022

Preprint

0

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We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a N = 2 4-dimensional quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system.

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Stacky dualities for the moduli of Higgs bundles

Cited by 4 publications

References 56 publications

A slow review of the AGT correspondence

A slow review of the AGT correspondence

A slow review of the AGT correspondence

Holomorphic Floer theory and Donaldson-Thomas invariants

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