Ruan Cohomology Research Articles

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

Chen–Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

Let \((X,\,D)\) be an m-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\), fix full flag and concentrated weight system \(\alpha \). Let \(P \mathcal {M}_{\xi }\) denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant \(\xi \) over X with weight system \(\alpha \), where r is a prime number and \(\xi \) is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on \(P \mathcal {M}_{\xi }\) of the group of r-torsion points in \(\mathrm{Pic}^0(X)\).

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Abstract We show how a novel construction of the sheaf of Cherednik algebras $\mathscr {H}_{1, c, X, G}$ on a quotient orbifold $Y:=X/G$ in author’s prior work leads to results for $\mathscr {H}_{1, c, X, G}$, which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of $\mathscr {H}_{1, c, X, G}$, which generalizes the standard Engeli–Felder’s trace density construction for the sheaf of differential operators $\mathscr {D}_X$. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb {C}_{Y}\llbracket \hbar \rrbracket $-modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras $\mathscr {H}_{1, \hbar , X, G}$. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold $Y$ with values in the ring of formal power series $\mathbb {C}\llbracket \hbar \rrbracket $. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathscr {H}_{1, 0, X, G}$ has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of $Y$. Finally, we prove an algebraic index theorem.

McKay correspondence in quasi-SL quasitoric orbifolds

We show McKay correspondence for Betti numbers of Chen–Ruan cohomology of omnioriented quasi-\({ SL}\) quasitoric orbifolds. This generalizes a correspondence for \({ SL}\) projective toric orbifolds due to Batyrev and Dias (Topology 35(4) (1996) 901–929) to a setting that does not require a complex or even an almost complex structure. In previous works with Ganguli and Poddar (Osaka J. Math. 50(2) (2013) 397–415; 50(4) (2013) 977–1005), we have proved the correspondence in dimensions four and six. Here we deal with the general case.

Ruan cohomologies of the compactifications of resolved orbifold conifolds

In this paper, we study the Ruan cohomologies of $X^s$ and $X^{sf}$, the natural compactifications of $V^s$ and $V^{sf}$, where $V^s$ and $V^{sf}$ are the two small resolutions of \[ V=\{(x,y,z,w)\mid xy-zw=0\}/\mu _r(1,-1,0,0),\quad r>1, \] the finite group quotient of the singular conifold. There is an additive isomorphism between the Chen-Ruan cohomologies $\phi :H^*_{CR}(X^s)\to H^*_{CR}(X^{sf})$. We study the three-point orbifold Gromov-Witten invariants of the exceptional curves $\Gamma ^s$ on $X^s$ and $\Gamma ^{sf}$ on $X^{sf}$ and show that the corresponding Ruan cohomology ring structures on the Chen-Ruan cohomologies of $X^s$ and $X^{sf}$, defined by these three-point functions, are isomorphic to each other under the map $\phi $ and the identification $[\Gamma ^s]\leftrightarrow -[\Gamma ^{sf}]$.

Ruan’s conjecture on singular symplectic flops of mixed type

In this paper, we study the global singular symplectic flops related to the following affine hypersurface with cyclic quotient singularities, $$V_{r,b} = \{ (x,y,z,t) \in \mathbb{C}^4 |xy - z^{2r} + t^2 = 0\} /\mu _r (a, - a,b,0),r \geqslant 2,$$ where b = 1 appears in Mori’s minimal model program and b ≠ 1 is a new class of singularities in symplectic birational geometry. We prove that two symplectic 3-orbifolds which are singular flops to each other have isomorphic Ruan cohomology rings. The proof is based on the symplectic cutting argument and virtual localization technique.

Chen–Ruan Cohomology of \mathcal{M}_{1,n} and \overline{\mathcal{M}}_{1,n}

In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a subring of the Chen--Ruan cohomology and of the stringy Chow ring.

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

We study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman and Weitsman’s proof of the GKM theorem [TW] in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S ⊂ T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen–Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen–Ruan cohomology in terms of orbifold fixed point data.

The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen–Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen–Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen–Ruan cohomology ring as an algebra on the ordinary cohomology.

CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

Chen–Ruan cohomology of some moduli spaces of parabolic vector bundles

Let ( X , D ) be an ℓ-pointed compact Riemann surface of genus at least two. For each point x ∈ D , fix parabolic weights ( α 1 ( x ) , α 2 ( x ) ) such that ∑ x ∈ D ( α 1 ( x ) − α 2 ( x ) ) < 1 / 2 . Fix a holomorphic line bundle ξ over X of degree one. Let P M ξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights { ( α 1 ( x ) , α 2 ( x ) ) } x ∈ D . The group of order two line bundles over X acts on P M ξ by the rule E ∗ ⊗ L ↦ E ∗ ⊗ L . We compute the Chen–Ruan cohomology ring of the corresponding orbifold.

Singular symplectic flops and Ruan cohomology

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient W r = { ( x , y , z , t ) ∣ x y − z 2 r + t 2 = 0 } / μ r ( a , − a , 1 , 0 ) , r ≥ 1 , which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let X and Y be two symplectic orbifolds connected by such a flop. We study orbifold Gromov–Witten invariants of exceptional classes on X and Y and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.

The Chen–Ruan Cohomology of Some Moduli Spaces

Let X be a compact connected Riemann surface of genus at least two. We compute the Chen-Ruan cohomology ring of the moduli space of stable -bundles over X of the nontrivial second Stiefel-Whitney class.

CHEN–RUAN COHOMOLOGY OF ADE SINGULARITIES

We study Ruan's cohomological crepant resolution conjecture [41] for orbifolds with transversal ADE singularities. In the An-case, we compute both the Chen–Ruan cohomology ring [Formula: see text] and the quantum corrected cohomology ring H*(Z)(q1,…,qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between [Formula: see text] and H*(Z)(-1) in the A1-case, verifying Ruan's conjecture. In the An-case, the family H*(Z)(q1,…,qn) is not defined for q1 = ⋯ = qn = -1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conjecture 1.9). Finally, we prove Conjecture 1.9 in the A2-case by constructing an explicit isomorphism.

The Chen–Ruan Cohomology of Weighted Projective Spaces

AbstractIn this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space

Frobenius action on ℓ-adic Chen–Ruan cohomology

We extend the definition of the Chen-Ruan cohomology ring to smooth, proper, tame, Deligne-Mumford stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product.

Mellin–Barnes integrals as Fourier–Mukai transforms

We study the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky and its relationship with the toric Deligne–Mumford (DM) stacks recently studied by Borisov, Chen and Smith. We construct series solutions with values in a combinatorial version of the Chen–Ruan (orbifold) cohomology and in the K-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the K-theory action of the Fourier–Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation transformations of Mellin–Barnes type.

Mukai flop and Ruan cohomology

Suppose that two compact symplectic manifolds X,X′ are connected by a sequence of simple Mukai flops. In this paper, we construct a ring isomorphism between cohomology rings of X and X′. Using the localization technique, we prove that the quantum corrected products on X,X′ are the ordinary intersection products. Furthermore, X,X′ have isomorphic Ruan cohomology, i.e. we verify the cohomological minimal model conjecture proposed by Ruan for the pair (X,X′).