Introduction to lie algebras and representation theory

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. We prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for Y_Q/ equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of Y_Q. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, we give an extended example of the general theory for moduli spaces of sheaves on C^2, framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization/ the corresponding R-matrix is closely related to the reflection operator in Liouville field theory. We show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, we obtain an action of the W-algebra W(gl(r)) on the equivariant cohomology of rank $r$ moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

http://inspirehep.net/record/1201562/files/arXiv%3A1211.1287.pdf

Quantum Groups and Quantum Cohomology

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toen and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toen in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (n−d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T∗X), for E an elliptic curve and T∗X is the total space of the cotangent bundle of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).

/pdf/shifted-symplectic-structures-55h4ws750t.pdf

Shifted Symplectic Structures

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.

https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n3-p01-s.pdf

From real affine geometry to complex geometry

/pdf/logarithmic-deformations-of-normal-crossing-varieties-and-255zbavefo.pdf

Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties

By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially a few examples of symplectic manifolds are discovered, it seems an important task to seek new symplectic manifolds by deforming symplectic varieties. In this paper we shall study a projective symplectic variety from a view point of deformation theory. If X has a resolution π : X → X such that (X, πω) is a symplectic manifold, we say that X has a symplectic resolution. Our first results are concerned with a birational contraction map of a symplectic manifold.

Deformation theory of singular symplectic n-folds

This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. 
We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.

https://www.degruyter.com/view/j/crll.2001.2001.issue-539/crll.2001.070/crll.2001.070.pdf

Extension of 2-forms and symplectic varieties

Yoshinori Namikawa I,*, J.H.M. Steenbrink 2 I Department of Mathematics, Sophia University, Kioi-Cho, Tokyo 102, Japan 2 Mathematical Institute, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Oblatum 10-X-1994 & 13-VI-1995 Introduction Friedman [Fr] has studied the relationship between local and global deforma- tions of a threefold Z with isolated hypersurface singularities which admits small resolutions. One of his main results is as follows. Let Z be a Moishezon threefold with only ordinary double points { Pl ..... p. }. Assume that the canon- ical line bundle

Global smoothing of Calabi-Yau threefolds.

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introduced a Poisson scheme and studied a Poisson deformation of it. A Poisson deformation is the deformation of the pair of a scheme itself and a Poisson structure on it. When X is a convex symplectic variety, X admits a natural Poisson structure induced from a symplectic 2-form ω; hence one can consider its Poisson deformations. Then they are controlled by the Poisson cohomology. The Poisson cohomology has been extensively studied by Fresse [Fr 1], [Fr 2]. In some good cases, it can be described by well-known topological data (Corollary 10). The first application of the Poisson deformation theory is the following two results:

/pdf/flops-and-poisson-deformations-of-symplectic-varieties-2mzhqef1fo.pdf

Yoshinori Namikawa

Papers

Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties

Deformation theory of singular symplectic n-folds

Extension of 2-forms and symplectic varieties

Global smoothing of Calabi-Yau threefolds.

Flops and Poisson deformations of symplectic varieties