Geometric Langlands twists of $N=4$ gauge theory from derived algebraic geometry

Elliott, Chris; Yoo, Philsang

doi:10.4310/atmp.2018.v22.n3.a3

Cited by 20 publications

(52 citation statements)

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“…This is the shifted cotangent space whose base is the moduli space of G-bundles on C × Σ with a flat connection on Σ and a Higgs field on C (note that this twisted theory is defined where C and Σ are any curves, not necessarily Calabi-Yau). This agrees with the calculation performed in [EY18]. In particular, if we set Σ = C the result is…”

Section: 2supporting

confidence: 91%

“…Instead the moduli space will appear as the moduli space of solutions to the equations of motion in a twisted 5d N = 2 supersymmetric gauge theory, compactified on a circle. This story will be directly analogous to the occurence of the ordinary moduli stack of Higgs bundles in a holomorphic twist of 4d N = 4 theory (see [Cos13a,EY18] for a discussion of this story); we'll recover that example in the limit where the radius of the circle shrinks to zero.…”

Section: Twisted Gauge Theorymentioning

confidence: 87%

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Multiplicative Hitchin systems and supersymmetric gauge theory

Elliott

Pestun

2019

Sel. Math. New Ser.

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Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where C = CP 1 with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. We also discuss quantization of these moduli spaces, yielding the modules for the Yangian Y (g) discovered by Gerasimov, Kharchev, Lebedev and Oblezin.Date: July 12, 2019. 1 Usually Higgs bundles on a curve are defined to be sections of the coadjoint bundle twisted by the canonical bundle. There isn't an obvious replacement for this twist in the multiplicative context, but we'll mostly be interested in the case where C is Calabi-Yau, and therefore this twist is trivial. 1 arXiv:1812.05516v4 [math.AG] 11 Jul 2019 G (CP 1 , D, ω ∨ ) of framed ε-connections on CP 1 . 1.2. Moduli Space of Monopoles. If we focus on the example where a real three-dimensional Riemannian manifold M = C × S 1 splits as the product of a compact Riemann surface and a circle, for the rational case C = C the moduli space of monopoles on M was studied by Cherkis and Kapustin [CK98, CK01, CK02] from the perspective of the Coulomb branch of vacua of 4d N = 2 supersymmetric gauge theory. For general C the moduli space of monopoles on M = C × S 1 was studied by Charbonneau-Hurtubise [CH10], for G R = U(n), and by Smith [Smi16] for general G.These moduli spaces were also studied where C = C with more general boundary conditions than the framing at infinity which we consider. This example has also been studied in the mathematics literature by Foscolo [Fos16, Fos13] and by Mochizuki [Moc17b] -note that considering weaker boundary conditions at infinity means they need to use far more sophisticated analysis in order to work with hyperkähler structures on their moduli spaces than we'll consider in this paper.The connection between periodic monopoles and multiplicative Higgs bundles is provided by the following theorem.Theorem 1.2 (Charbonneau-Hurtubise, Smith). There is an analytic isomorphismbetween the moduli space of (polystable) multiplicative G-Higgs bundles on a compact curve C with singularities at points z 1 , . . . , z k and residues ω ∨ z 1 , . . . , ω ∨ z k and the moduli space of periodic monopoles on C × S 1 with Dirac singularities at (z 1 , 0), . . . , (z k , 0) with charges ω ∨ z 1 , . . . , ω ∨ z kThe morphism H, discussed first by Cherkis and Kapustin in [CK01] and later in [CH10], [Smi16], [NP12] defines the value of the multiplicative Higgs field at z ∈ C to be equal to the holonomy of the monopole connection (complexified by the scalar field) along the fiber circl...

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Section: 2supporting

confidence: 91%

Section: Twisted Gauge Theorymentioning

confidence: 87%

Multiplicative Hitchin systems and supersymmetric gauge theory

Elliott

Pestun

2019

Sel. Math. New Ser.

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“…Holomorphic twists were studied in the factorization algebra formalism of [33] in e.g. [93,78,94]. They form an important bridge between full, physical SUSY QFTs and topologically twisted ones, and they admit higher products closely analogous to those of TQFTs, whose general structure was briefly outlined in [35].…”

Section: Further Directionsmentioning

confidence: 99%

“…Naively, this seems to imply that the equivalence between A G and B G ∨ is non-canonical in the topological theory, but this turns out not to be the case. In the algebraic description of the B-model B G ∨ described in [113,94,114], the local operators naturally appear as invariant polynomials of a complex coadjoint scalar,…”

Section: Local Operatorsmentioning

confidence: 99%

Secondary Products in Supersymmetric Field Theory

Beem

Ben‐Zvi

Bullimore

et al. 2020

Ann. Henri Poincaré

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The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky-Witten twist of three-dimensional N = 4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an Ω-background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky-Witten theories and in four-dimensional N = 4 super Yang-Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

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“…Other physical realizations of derived geometry in different contexts have also appeared in e.g. [75][76][77].…”

Section: Derived Schemesmentioning

confidence: 99%

Categorical Equivalence and the Renormalization Group

Sharpe

2019

Fortschritte der Physik

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In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and ‘decomposition,’ in which two‐dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D‐branes, antibranes, and tachyons. In the third part of this article, we review how Landau–Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs.

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Geometric Langlands twists of $N=4$ gauge theory from derived algebraic geometry

Cited by 20 publications

References 40 publications

Multiplicative Hitchin systems and supersymmetric gauge theory

Multiplicative Hitchin systems and supersymmetric gauge theory

Secondary Products in Supersymmetric Field Theory

Categorical Equivalence and the Renormalization Group

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