Holomorphic Symplectic Structure Research Articles

Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the $R$-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the elliptic stable envelopes, and in turn the geometric elliptic $R$-matrix, of the anisotropic $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the $R$-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.

On the symplectic structure over the moduli space of projective structures on a surface

The moduli space of projective structures on a compact oriented surface Σ has a holomorphic symplectic structure, which is constructed by pulling back, using the monodromy map, the Atiyah–Bott–Goldman symplectic form on the character variety Hom(π1(Σ),PSL(2,C))//PSL(2,C). We produce another construction of this symplectic form. This construction shows that the symplectic form on the moduli space is actually algebraic. Note that the monodromy map is only holomorphic and not algebraic, so the first construction does not give algebraicity of the pulled back form.

Symplectic realizations of holomorphic Poisson manifolds

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, \pi)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subseteq T^*X$.

Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

Complex Lagrangians in a hyperKähler manifold and the relative Albanese

AbstractLet M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let ω̄ : 𝒜̂ → M be the relative Albanese over M. We prove that 𝒜̂ has a natural holomorphic symplectic structure. The projection ω̄ defines a completely integrable structure on the symplectic manifold 𝒜̂. In particular, the fibers of ω̄ are complex Lagrangians with respect to the symplectic form on 𝒜̂. We also prove analogous results for the relative Picard over M.

Vector bundles on Fano threefolds and K3 surfaces

Let X be a Fano threefold, and let $$S\subset X$$ be a K3 surface. Any moduli space $${\mathscr {M}}_{S}$$ of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form a Lagrangian subvariety of $${\mathscr {M}}_{S}$$ . We illustrate this with a number of concrete examples.

Secondary Products in Supersymmetric Field Theory

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 {mathcal {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 Omega -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 {mathcal {N}}=4 super-Yang–Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

The hyperkähler metric on the almost-Fuchsian moduli space

Donaldson [11] constructed a hyperkähler moduli space \mathcal{M} associated to a closed oriented surface \Sigma with genus (\Sigma) \geq 2 . This embeds naturally into the cotangent bundle T^*\mathcal T(\Sigma) of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to \Sigma . The latter is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in (-1,1) . Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bundle of Teichmüller space, the hyperkähler structure on \mathcal M can be viewed as the Feix–Kaledin hyperkähler extension of the Weil–Petersson metric. The almost-Fuchsian moduli space embeds into the SL (2,\mathbb C) -representation variety of \Sigma and the hyperkähler structure on \mathcal M extends the Goldman holomorphic symplectic structure. Here, the natural complex structure corresponds to the second complex structure in the first picture. Moreover, the area of the minimal surface in an almost-Fuchsian manifold provides a kähler potential for the hyperkähler metric. The various identifications are obtained using the work of Uhlenbeck [33] on germs of hyperbolic 3-manifolds, an explicit map from \mathcal M to \mathcal T(\Sigma)\times \overline{\mathcal T(\Sigma)} found by Hodge [20], the simultaneous uniformization theorem of Bers [2], and the theory of Higgs bundles introduced by Hitchin [18].

Moduli Spaces of Framed G–Higgs Bundles and Symplectic Geometry

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

From hypercomplex to holomorphic symplectic structures

The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex structures and holomorphic symplectic 2-forms on manifolds respectively. Basic properties of such structures are established.

Non-squeezing property for holomorphic symplectic structures

Non-squeezing property for holomorphic symplectic structures

Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact, simple hyperkähler manifold with b2⩾7 admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkähler manifolds are never Kobayashi hyperbolic.

HOLOMORPHIC SYMPLECTIC AND POISSON STRUCTURES ON THE HOLOMORPHIC COTANGENT BUNDLE OF A COMPLEX LIE GROUP AND OF A HOLOMORPHIC PRINCIPAL BUNDLE

In this paper we introduce holomorphic symplectic and Poisson structures on the holomorphic cotangent bundle of a complex Lie group and of a holomorphic principal bundle.

BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN TOPOLOGICAL FIELD THEORIES

We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi–Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here, [Formula: see text] is the initial nondeformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We discuss the identification of the harmonic structure (HT•(X);HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

Toward a classification of real compact solvmanifolds with real symplectic structures

In this paper, we deal with the problem of classifying compact complex solvmanifolds with holomorphic symplectic structures and obtain some structure results which make the classification possible. In particular, we reduce the classification to the nilpotent case with the same dimension, which we call the nilpotent reduction. The same method also works for the real compact solvmanifolds with real symplectic structures. The real six-dimensional case was treated completely. This is one of the major steps to obtain further examples of compact holomorphic symplectic manifolds. For example, the Kodaira–Thurston surface is NOT a complex homogeneous manifold with a transitive Lie group action which keeps the complex structure invariant, but a real solvmanifold with a complex structure.

Real projective structures on a real curve

Given a compact connected Riemann surface X equipped with an antiholomorphic involution τ, we consider the projective structures on X satisfying a compatibility condition with respect to τ. For a projective structure P on X, there are holomorphic connections and holomorphic differential operators on X that are constructed using P. When the projective structure P is compatible with τ, the relationships between τ and the holomorphic connections, or the differential operators, associated to P are investigated. The moduli space of projective structures on a compact oriented C∞ surface of genus g≥2 has a natural holomorphic symplectic structure. It is known that this holomorphic symplectic manifold is isomorphic to the holomorphic symplectic manifold defined by the total space of the holomorphic cotangent bundle of the Teichmüller space Tg equipped with the Liouville symplectic form. We show that there is an isomorphism between these two holomorphic symplectic manifolds that is compatible with τ.

BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS

The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A,Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D -branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A -model, and a Calabi-Yau manifold with boundaries in the B -model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra, [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)nQ+d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D -brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A,Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS

The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A, Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D-branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D-brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A, Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

ON MODULI SPACE OF HIGGS Gp(2n, ℂ)-BUNDLES OVER A RIEMANN SURFACE

Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].

Modular Galois covers associated to symplectic resolutions of singularities

Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Namikawa's work. The modular Galois group G is the product of Weyl groups of finite type, indexed by such irreducible components B. Each Weyl group factor W_B is that of a Dynkin diagram, obtained as a quotient of the Dynkin diagram of the singularity-type of B, by a group of Dynkin diagram automorphisms. Finally we consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities. We prove that the morphism f from Def(X) to Def(Y) is a Galois cover of its image. This explains the analogy between the above results and related work of Nakajima, on quiver varieties, and of Szendroi on enhanced gauge symmetries for Calabi-Yau threefolds.