Equivariant Cohomology Research Articles

The syzygy order of big polygon spaces

Big polygon spaces are compact orientable manifolds with a torus action whose equivariant cohomology can be torsion-free or reflexive without being free as a module over $H^*(BT)$. We determine the exact syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in $H^2(BT)$ adapted to the isotropy groups occurring in a given $T$-space.

Correction to: An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Correction to: An algebraic model for rational naïve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

The freeness theorem for equivariant cohomology of Rep(C2)-complexes

Let C2 be the cyclic group of order two. We show that the RO(C2)-graded Bredon cohomology of a finite Rep(C2)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor F2_. This paper corrects some errors in Kronholm's proof of this freeness theorem. It also extends the freeness result to finite type complexes, those with finitely many cells of each fixed-set dimension. We give a counterexample showing the theorem does not hold for locally finite complexes.

Equivariant one-parameter deformations of Lie triple systems

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie triple systems.

Equivariant cohomology for differentiable stacks

We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott’s spectral sequence which converges to the cohomology of the classifying space of a Lie group.

GLSMs for exotic Grassmannians

In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.

Bisymplectic Grassmannians of planes

The bisymplectic Grassmannian $${{\,\mathrm{I_2Gr}\,}}(k,V)$$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when $$k=2$$ ; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of $${{\,\mathrm{I_2Gr}\,}}(2, {\mathbb {C}}^6)$$ , which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology.

Topological correlators and surface defects from equivariant cohomology

We find a one-dimensional protected subsector of mathcal{N} = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general mathcal{N} = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

Cohomological localization for transverse Lie algebra actions on Riemannian foliations

We prove localization and integration formulas for the equivariant basic cohomology of Riemannian foliations. As a corollary we obtain a Duistermaat–Heckman theorem for transversely symplectic foliations.

EQUIVARIANT DERIVED CATEGORIES FOR TOROIDAL GROUP IMBEDDINGS

Let X denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on X is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded Ext. Only special cases of this conjecture have been proven so far. The purpose of this paper is to provide a proof of this conjecture for all projective toroidal imbeddings of complex reductive groups. In fact, we show that the methods used by Lunts for a proof in the case of toric varieties can be extended with suitable modifications to handle the toroidal imbedding case. Since every equivariant imbedding of a complex reductive group is dominated by a toroidal imbedding, the class of varieties for which our proof applies is quite large.We also show that, in general, there exist a countable number of obstructions for this conjecture to be true and that half of these vanish when the odd dimensional equivariant intersection cohomology sheaves on the orbit closures vanish. This last vanishing condition had been proven to be true in many cases of spherical varieties by Michel Brion and the author in prior work.

A closed formula for the evaluation of foams

We give a purely combinatorial formula for evaluating closed, decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral, equivariant version of colored Khovanov–Rozansky link homology categorifying the \mathfrak {sl}_N link polynomial. We also provide connections to the equivariant cohomology rings of partial flag varieties.

BPS Wilson loops and quiver varieties

Three dimensional supersymmetric field theories have large moduli spaces of circular Wilson loops preserving a fixed set of supercharges. We simplify previous constructions of such Wilson loops and amend and clarify their classification. For a generic quiver gauge theory we identify the moduli space as a quotient of for some m by an appropriate symmetry group. These spaces are quiver varieties associated to a cover of the original quiver or a subquiver thereof. This moduli space is generically singular and at the singularities there are large degeneracies of operators which seem different, but whose expectation values and correlation functions with all other gauge invariant operators are identical. The formulation presented here, where the Wilson loops are on S 3 or squashed also allows to directly implement a localization procedure on these observables, which previously required an indirect cohomological equivalence argument.

An algebraic model for rational na\xefve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Equipping a non-equivariant topological text {E}_infty -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take G=SO(2) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an text {E}_infty -ring spectrum. Moreover, the category of modules over that text {E}_infty -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

Unramified affine Springer fibers and isospectral Hilbert schemes

For any connected reductive group G over \({{\mathbb {C}}}\), we revisit Goresky–Kottwitz–MacPherson’s description of the torus equivariant Borel–Moore homology of affine Springer fibers \({\mathrm {Sp}}_{\gamma }\subset {{\,\mathrm{Gr}\,}}_G\), where \(\gamma =zt^d\) and z is a regular semisimple element in the Lie algebra of G. In the case \(G=GL_n\), we relate the equivariant cohomology of \({\mathrm {Sp}}_\gamma \) to Haiman’s work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of (n, dn)-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve \(\{x^n=y^{dn}\}\).

3d mathcal{N} = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians

We study a correspondence between 3d mathcal{N} = 2 topologically twisted Chern-Simons-matter theories on S1× Σg and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a parameter β associated with an algebraic Bethe ansatz introduced by Gorbounov-Korff. They showed that the Frobenius algebra with β = −1 is isomorphic to the (equivariant) small quantum K -ring of the Grassmannian, and the Frobenius algebra with β = 0 is isomorphic to the equivariant small quantum cohomology of the Grassmannian. We apply supersymmetric localization formulas to the correlation functions of supersymmetric Wilson loops in the Chern-Simons-matter theory and show that the algebra of Wilson loops is isomorphic to the Frobenius algebra with β = −1. This allows us to identify the algebra of Wilson loops with the quantum K - ring of the Grassmannian. We also show that correlation functions of Wilson loops on S1× Σg satisfy the axiom of 2d TQFT. For β = 0, we show the correspondence between an A-twisted GLSM, the Frobenius algebra for β = 0, and the quantum cohomology of the Grassmannian. We also discuss deformations of Verlinde algebras, omega-deformations, and the K -theoretic I -functions of Grassmannians with level structures.

Equivariant K-Theory and Refined Vafa\u2013Witten Invariants

In Maulik and Thomas (in preparation) the Vafa–Witten theory of complex projective surfaces is lifted to oriented {mathbb {C}}^*-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in t^{1/2} invariant under t^{1/2}leftrightarrow t^{-1/2} which specialise to numerical Vafa–Witten invariants at t=1. On the “instanton branch” the invariants give the virtual chi ^{}_{-t}-genus refinement of Göttsche–Kool, extended to allow for strictly semistable sheaves. Applying modularity to their calculations gives predictions for the contribution of the “monopole branch”. We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Göttsche–Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon–Northcott complexes, and show they calculate refined Vafa–Witten invariants. Using this Laarakker (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385) proves universality results for the invariants.

Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group

We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.

The equivariant cohomology of complexity one spaces

Complexity one spaces are an important class of examples in symplectic geometry. They are less restrictive than toric symplectic manifolds. Delzant has established that toric symplectic manifolds are completely determined by their moment polytope. Danilov proved that the ordinary and equivariant cohomology rings are dictated by the combinatorics of this polytope. These results are not true for complexity one spaces. In this paper, we describe the equivariant cohomology for a Hamiltonian S^1\circlearrowright M^4 . We then assemble the equivariant cohomology of a complexity one space from the equivariant cohomology of the 2- and 4 -dimensional pieces, as a subring of the equivariant cohomology of its fixed points. We also show how to compute equivariant characteristic classes in dimension four.

Cohomology rings of the real and oriented partial flag manifolds

We show, in contrast to the cases of complex and quaternionic partial flag manifolds, the rational cohomology rings of the real and oriented partial flag manifolds are generated not only by the characteristic classes of their canonical vector bundles, but also by exterior algebras arising from the rational cohomology rings of real Stiefel manifolds. We also discuss the rational and the mod-2 equivariant cohomology rings of these partial flag manifolds for certain canonical actions of tori and 2-tori.

Manifolds of Isospectral Matrices and Hessenberg Varieties

Abstract We consider the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology rings of $X_h$ are described using GKM theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and regular semisimple Hessenberg varieties well known in algebraic geometry. Both spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families, which can be generalized to other submanifolds of the flag variety.