Lagrangian Subvarieties of Hyperspherical Varieties Related to $$G_2$$

Abstract.We show that the image of the moduli space of stable bundles on an Enriques surface by the pull back map is a Lagrangian subvariety in the moduli space of stable bundles, which is a symplectic variety, on the covering K3 surface. We also describe singularities and some other features of it.

On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties

We consider the moduli space $\mathcal{M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. This is a hyperk\"ahler manifold homeomorphic to the moduli space $\mathcal{R}(G)$ of representations of the fundamental group of $X$ in $G$. In this paper we study finite order automorphisms of $\mathcal{M}(G)$ obtained by combining the action of an element of order $n$ in $H^1(X,Z)\rtimes \mbox{Out}(G)$, where $Z$ is the centre of $G$ and $\mbox{Out}(G)$ is the group of outer automorphisms of $G$, with the multiplication of the Higgs field by an $n$th-root of unity, and describe the subvarieties of fixed points. We give special attention to the case of involutions, defined by the action of an element of order $2$ in $H^1(X,Z)\rtimes\mbox{Out}(G)$ combined with the multiplication of the Higgs field by $\pm 1$. In this situation, the subvarieties of fixed points are hyperk\"ahler submanifolds of $\mathcal{M}(G)$ in the (+1)-case, corresponding to the moduli space of representations of the fundamental group in certain reductive complex subgroups of $G$ defined by holomorphic involutions of $G$; while in the (-1)-case they are Lagrangian subvarieties corresponding to the moduli space of representations of the fundamental group of $X$ in real forms of $G$ and certain extensions of these. We illustrate the general theory with the description of involutions for $G=\mbox{SL}(n,\mathbb{C})$ and involutions and order three automorphism defined by triality for $G=\mbox{Spin}(8,\mathbb{C})$.

Let BunG be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T∗BunG. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.

Quantization of restricted Lagrangian subvarieties in positive characteristic

We prove that every irreducible component of every fibre of Lagrangian fibrations on holomorphic symplectic manifolds is a Lagrangian subvariety. Espe- cially, Lagrangian fibrations are equidimensional.

Geometric construction of crystal bases for quantum generalized Kac–Moody algebras

We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$ and a symplectic flat connection $\nabla$ such that the metric $g$ is locally the Hessian of a function. We prove that any Lagrangian subvariety $Z\subset M$ which intersects smooth fibers of $\pi$ and smoothly projects to $\pi(Z)$ is a toric fibration over its image $\pi(Z)$ in $B$, and this image is also special K\"ahler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.

This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid and the elliptic solitons. We then describe some systems in which the spectral curve is replaced by various higher dimensional analogues: a spectral cover of an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is that it is integrable analytically, but not algebraically: the Liouville tori (on which the system is linearized) are the intermediate Jacobians of a family of Calabi-Yau manifolds. Most of the results concerning these three types of non-curve-based systems are quite recent. Some of them, as well as the compatibility between spectral systems and the KP hierarchy, are new, while other parts of the story are scattered through several recent preprints. As best we could, we tried to maintain the survey style of this article, starting with some basic notions in the field and building gradually to the recent developments.

Mukai [M1] showed that there is a nondegenerate symplectic structures on the moduli space of stable vector bundles on a K3 surface. Later Tyurin [T2] studied (generalized) symplectic structures on the moduli space of stable vector bundles on any smooth regular surface X with P9 > 0. In the work of Tyurin, a symplectic structure means a nonzero regular two form on the moduli spaces, in particular it may degenerate. In this paper, we define a Lagrangian subvariety of the moduli space to be a subvariety on the Zariski tangent space (at any point) of which the given symplectic two form is identically zero. Note that we do not impose any restriction on the dimension of a Lagrangian subvariety. This is because symplectic structures considered here may degenerate. The purpose of this paper is to use Bril l-Noether theory for curves to construct explicit ly a family of Lagrangian subvarieties of the moduli space of stable vector bundles on a regular surface with pg > 0 . Le t . P/~ be a generically smooth and irreducible component of the moduli space of Gieseker-stable bundles of rank r + 1 with respect to a fixed polarization D on a regular algebraic surface X with p~ > 0. By the boundedness of ..Jf/~ (see [Ma]), after possibly twisted by the same negative line bundle . ~ on X, we can assume that for any point [E] C ,//J~, (i) E * is generated by global sections. (ii) hl(E *) = hZ(E *) : hi(E) : O. (iii) h l (de t E *) = hZ(det E *) = 0. For any point [E] c .//Z. Choose a (r + 1) dimensional subspace V C H~ and consider an evaluation map e v : V | P x --~ E* . For a general V, we can make e v degenerate exactly along a smooth curve C C X and coker e v is a line bundle

We generalize Voisin's theorem on deformations of pairs of a symplectic manifold and a Lagrangian submanifold to the case of Lagrangian normal crossing subvarieties. Partial results are obtained for arbitrary Lagrangian subvarieties. We apply our results to the study of singular bers of Lagrangian brations.

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space ℒ K (M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of ℒ K (M) as a Lagrangian subvariety in the symplectic moduli space $$ {\mathcal{X}}_K^{\mathrm{un}}\left(\partial M\right) $$ of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of ℒ K (M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒ K (M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒ K (M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d $$ \mathcal{N}=2 $$ superconformal field theories T K [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T K [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N f = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T K [M] grow cubically in K.

On the variety of triangles for a hyper-Kähler fourfold constructed by Debarre and Voisin