Nonabelian Localization Research Articles

A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.

Volume formula for N-fold reduced products

Let G be a semisimple compact connected Lie group. An N-fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N-fold reduced product of G. Suzuki and Takakura gave a volume formula for the N-fold reduced product of \( \mathbf {SU}(3) \) in [25] by using geometric quantization and the Riemann–Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of \( \mathbf {SU}(3) \).

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

Quantum K-Theory of Grassmannians and Non-Abelian Localization

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

Virtual equivariant Grothendieck-Riemann-Roch formula

For a G -scheme X with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of B. Fantechi and L. Göttsche [Geom. Topol. 14, No. 1, 83–115 (2010; Zbl 1194.14017)] to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over \mathbb{C} with proper actions.

Residue formulas for push-forwards in equivariant cohomology: a symplectic approach

The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.

Adams operations on the virtual K-theory of ℙ(1,n)

We analyze the structure of the virtual (orbifold) [Formula: see text]-theory ring of the complex orbifold [Formula: see text] and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin–Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant [Formula: see text]-theory and Riemann–Roch for quotients, Adv. Math. 198(2) (2005) 547–582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual [Formula: see text]-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual [Formula: see text]-theory ring of [Formula: see text] to the ordinary [Formula: see text]-theory ring of a crepant resolution of the cotangent bundle of [Formula: see text] which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of [Formula: see text] which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin–Jarvis–Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial [Formula: see text]-theory, Ann. K-Theory (2016)], who proved the latter for [Formula: see text].

Witten Non Abelian Localization for Equivariant K-Theory, and the [𝑄,𝑅]=0 Theorem

The purpose of the present paper is two-fold. First, we obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, we deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, we use this general approach to reprove the [Q,R] = 0 theorem of Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general spin^c Dirac operators (see the preprint arXiv:1411.7772).

Localization in abelian Chern-Simons theory

Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected, and abelian. The abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. The partition function is then formally computed using the technique of non-abelian localization. This study leads to a natural identification of the abelian Reidemeister-Ray-Singer torsion as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections for the class of Sasakian three-manifolds. The torsion part of the abelian Chern-Simons partition function is computed explicitly in terms of Seifert data for a given Sasakian three-manifold.

Non-Abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on a Seifert manifold

We derive non-Abelian localization formulae for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold M, which is the three-dimensional space of a circle bundle over a two-dimensional Riemann surface \Sigma, by using the cohomological approach introduced by Kallen. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on S^1x\Sigma. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the localization.

Equivariant Volumes of Non-Compact Quotients and Instanton Counting

Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kahler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $\R^{4}$ and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.

Nonabelian localization for gauge theory on the fuzzy sphere

We apply nonabelian equivariant localization techniques to Yang-Mills theory on the fuzzy sphere to write the partition function entirely as a sum over local contributions from critical points of the action. The contributions of the classical saddle-points are evaluated explicitly, and the partition function of ordinary Yang-Mills theory on the sphere is recovered in the commutative limit.

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

We study the relations between two-dimensional Yang-Mills theory on the torus, topological string theory on a Calabi-Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern-Simons theory on torus bundles. The chiral partition function of the Yang-Mills gauge theory in the large N limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang-Mills theory as an efficient tool for the computation of Gromov-Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar-Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured nonperturbative relation between two-dimensional Yang-Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson-Thomas theory of the Calabi-Yau background. We match the instanton representation of Yang-Mills theory on the torus with the nonabelian localization of Chern-Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of BPS black holes in the local Calabi-Yau background.

Localization for Yang-Mills Theory on the Fuzzy Sphere

We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.

New polytope decompositions and Euler–Maclaurin formulas for simple integral polytopes

We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence–Varchenko and Brianchon–Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler–Maclaurin formulas.

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303–368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.

The Kirwan map, equivariant Kirwan maps, and their kernels

Consider a Hamiltonian action of a compact Lie group K on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κ K . We start with the case when K is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [ R. F. Goldin , An effective algorithm for the cohomology ring of symplectic reductions, Geom. Func. Anal. 12 (2002), 567–583]) corresponding to a generic circle S ⊂ T , and show how to recover from this the kernel of κ T , as described by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [ L. C. Jeffrey , The residue formula and the Tolman-Weitsman theorem, J. reine angew. Math. 562 (2003), 51–58], [ L. C. Jeffrey and A.-L. Mare , The kernel of the equivariant Kirwan map and the residue formula, Quart. J. Math. Oxford 54 (2004), 435–444].) For a compact nonabelian Lie group K we will use the ‘‘non-abelian localization formula’’ of [ L. C. Jeffrey and F. C. Kirwan , Localization for nonabelian group actions, Topology 34 (1995), 291–327] and [ L. C. Jeffrey and F. C. Kirwan , Localization and the quantization conjecture, Topology 36 (1995), 647–693] to establish relationships—some of them obtained by Tolman and Weitsman in [ S. Tolman and J. Weitsman , The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]—between Ker( κ K ) and Ker( κ T ), where T ⊂ K is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of μ T .

Nonabelian localization in equivariant [formula omitted]-theory and Riemann–Roch for quotients

We prove a localization formula in equivariant algebraic K-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to nondiagonalizable groups the localization formulas of Nielsen [Diagonalizably linearized coherent sheaves, Bull. Soc. Math. France 102 (1974) 85–97] and Thomason [Une formule de Lefschetz en K-théorie équivariante algébrique, Duke Math. J. 68(3) (1992) 447–462]. As an application we give a Riemann–Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of Toen [Théorèmes de Riemann–Roch pour les champs de Deligne-Mumford, K-Theory 18(1) (1999) 33–76] and the authors [Riemann–Roch for quotients and Todd classes of simplicial toric varieties, Comm. Algebra 31 (2003) 3735–3752].

Non-Abelian localization for Chern-Simons theory

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

On path integral localization and the Laplacian

We introduce a new localization principle, which is a generalized canonical transformation. It unifies BRST localization, the non-Abelian localization principle and a special case of the conformal Duistermaat–Heckman integration formula of Paniak, Semenoff, and Szabo. The heat kernel on compact Lie groups is localized in two ways. First, using a non-Abelian generalization of the derivative expansion localization of Palo and Niemi and second, using the BRST localization principle and a configuration space path integral. In addition, we present some new formulas on homogeneous spaces, which might be useful in a possible localization of Selberg’s trace formula on locally homogeneous spaces.