An introduction to affine Grassmannians and
                    the geometric Satake equivalaence

“…Recall that the affine Grassmannian Gr G is stratified by dominant coweights ω ∨ , and there is an with respect to the standard partial order on dominant coweights (again, this is explained in [Zhu17]). We can then define…”

Section: There Is a Canonical Inclusionmentioning

confidence: 99%

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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“…The E 2 -algebra structure present on ΩG(C) is encoded by a more elaborate object, the Beilinson-Drinfeld Grassmannian. A good general reference for both of these objects is [Zhu16], whose presentation we will more or less follow below.…”

Section: The E 2 Schubert Filtrationmentioning

confidence: 99%

Multiplicative structure in the stable splitting of ΩSLn(C)

Hahn

Yuan

2019

Advances in Mathematics

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The space of based loops in SLn(C), also known as the affine Grassmannian of SLn(C), admits an E 2 or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell-Richter splitting is coherently multiplicative, but not E 2 . Nonetheless, we show that the splitting becomes E 2 after base-change to complex cobordism. Our proof of the A∞ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson-Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.

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An introduction to affine Grassmannians and the geometric Satake equivalaence

Abstract: We introduce various affine Grassmannians, study their geometric properties, and give some applications. We also discuss the geometric Satake equivalence. These are the expanded lecture notes for a mini-course in 2015 PCMI summer school.

Cited by 79 publications

References 49 publications

Geometric Satake, categorical traces, and arithmetic of Shimura varieties

Geometric Satake, categorical traces, and arithmetic of Shimura varieties

Multiplicative Hitchin systems and supersymmetric gauge theory

Multiplicative structure in the stable splitting of ΩSLn(C)

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