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Lectures on 𝐾-theoretic computations in enumerative geometry
... The principal objects of study in the modern theory are the vertex functions in the terminology of [44]. These are the generating functions Vertex(z, a) of counts of rational curves in X of all possible degrees. ...
... In particular, following this line of reasoning one can study the count of sections of X using the technique of degeneration of the base curve C, and this is precisely what is done in the K-theoretic setting in [44] when X is a Nakajima quiver variety. This leads one to define certain operators S σ acting in equivariant K-theory K C × q ×T (X) (or H • C × q ×T (X) in the cohomological setting, see [38]) which are referred to as shift operators and determine the difference equations in a-variables. ...
... In [44], the action of S σ for minuscule σ in the K-theoretic stable envelope basis is characterized via geometric R-matrices, and gives an explicit identification of the q-difference equation in the a-variables satisfied by Vertex(z, a) with the quantum Knizhnik-Zamolodchikov (qKZ) equation for a certain geometrically defined quantum group. ...
We introduce nonabelian analogs of shift operators in the enumerative theory of quasimaps. We apply them on the one hand to strengthen the emerging analogy between enumerative geometry and the geometric theory of automorphic forms, and on the other hand to obtain results about quantized Coulomb branch algebras. In particular, we find a short and direct proof that the equivariant convolution homology of the affine Grassmannian of is a quotient of a shifted Yangian.
... Our proof is an adaptation of the arguments in [Ok,Prop. 9 Proof. Let us fix a linear order of orbits such that ...
... Remark 5.6. The axiomatic characterization of motivic Chern classes in this section is motivated by the axiomatic characterization of certain K theoretic characteristic classes called K theoretic stable envelopes in [Ok,Section 9.1]. The similarity of the axiom systems imply certain coincidences that we explain now. ...
... Namely, the normalization and support axioms of both notions are the same. The Newton polytope axiom of [Ok,Section 9.1] requires that the small convex polytope remains inside the larger one even if shifted slightly towards the origin (at least for a specific choice of slope parameter [Ok,9.1.9], called antidominant alcove). ...
We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.
... The K-theoretic stable envelope maps for Nakajima varieties X are defined by Maulik and Okounkov in [MO2]. That definition is reproduced in [O,Section 9.1] and [OS, Section 2.1]. Here we recall their axiomatic definition of stable envelope classes (images of coordinate vectors under the stable envelope map) in the special case X = T * F λ acted upon by (C * ) n × C * . ...
... Definition. ( [MO2], [O,Section 9.1], [OS, Section 2.1]) An element Stab ∆ σ, I ∈ K T (T * F λ ) for an alcove ∆ ⊂ R N , σ ∈ S n , I ∈ I λ is called the stable envelope class, if it satisfies the following axioms: ...
... Remark. In fact, in [MO2,O,OS] the definition of stable envelope classes also depends on a choice of distinguishing half of the normal directions at each torus fixed point. For T * F λ there is a natural choice -distinguishing the cotangent directions -that we make throughout the paper. ...
In this paper we consider the cotangent bundles of partial flag varieties. We construct the K-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the K-theoretic stable envelopes and our elliptic stable envelopes. We show that the K-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the K-theoretic stable envelopes.
... For instance, the theory of canonical bases, the so called off-shell Bethe vectors [3,30,16] (which are the main object of investigation in the theory of the quantum integrable systems), weight functions for integral solutions of qKZ equations [33] are now understood as different incarnations of stable envelopes. Stable envelopes also appear as partition functions of integrable lattice models [13] and play important role in enumerative geometry [26]. The transition functions between elliptic stable envelopes corresponding to different chambers provide the so called quantum dynamical elliptic R-matrices. ...
... see Sections 7.1-7.2 of [5] for discussion of relations between elliptic, K-theoretic and cohomological versions Euler class. In this section we use this limit to obtain the explicit formulas for stable envelops in the equivariant K-theory of H. Recall that in the K-theory the stable envelope is a locally constant functions of a slope parameter [26,28,25]: ...
... They are described by Theorem 9. Let us note that the normalizations for K-theoretic stable envelopes used by different authors may differ from each other by a factor. In this section the Ktheoretic envelopes are normalized as in [26,28]. The stable envelopes defined in [25] differ by a line bundle factor. ...
We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the stable envelope in equivariant K-theory (with arbitrary slope) and equivariant cohomology.
... In this Type IIB frame, we will show that the various components of the Miura transformation are equal to a certain half-index of the 3d theory on the D3 brane. 9 For us, this half-index is a count of BPS states defined in the UV, for any Lagrangian 3d N = 2 theory [28][29][30][31]: one replaces R 2 εc with a finite disk/hemisphere D 2 , and imposes a specific set of 1/2-BPS N = (0, 2) boundary conditions at finite distance on S 1 (R) × S 1 D 2 = T 2 , which will flow to a superconformal point in the IR. 10 In geometry, the half-index we compute is a generating function of quasimaps CP 1 /{0} → X of all degrees in equivariant quantum K-theory, where X is the Higgs branch of the 3d theory [32][33][34][35]; by excising the origin of CP 1 , we are allowing for line operator insertions supported on {0} × S 1 to contribute to the quasimap count (see for instance [36,37]); these are the 1-dimensional degrees of freedom supported at a totally transverse NS5-D3 brane intersection. In section 4, we find: ...
... Because boundary conditions have to be specified for the fields "at infinity" of R 2 ε 3 , holomorphic blocks are intrinsically defined in the IR. 29 In Mathematics, this index is called vertex function, or generating function of quasimaps CP 1 → X of all degrees in equivariant quantum K-theory, where X is the Higgs branch of the 3d theory [32][33][34][35]. Quasimaps are solutions to the vortex equations of the 3d theory, subjected to certain regularity conditions on the gauge and matter fields, and K-theory enters here instead of cohomology because we are studying a 3-dimensional theory compactified on a circle instead of a purely 2-dimensional one. ...
... Instead, we chose to interpret (4.8) from a 3-dimensional point of view. 35 In this spirit, we will count the 1d N = 4 multiplets of the D3-NS5 intersection as arising from a coupled 3d/1d half-index, by which we mean the half-index of a 3d bulk theory supported on R 2 ε 3 × S 1 (R), coupled to a codimension-2 line defect along S 1 (R). ...
We propose that Miura operators are R-matrices of certain infinite-dimensional quantum algebras. We test our proposal by realizing Miura operators of q-deformed W- and Y-algebras in terms of R-matrices of the quantum toroidal algebra of . Physically, the representations of this toroidal algebra arise from the algebra of local operators on M2-branes and M5-branes, in M-theory subject to an -background. We associate an R-matrix to each M2-M5 brane crossing, by studying its description as a gauge-invariant intersection of a topological line defect and a holomorphic surface defect in 5-dimensional non-commutative Chern-Simons theory. The Miura transformation is engineered using multiple M2-M5 intersections, relying crucially on the properties of the underlying R-matrices. We thereby identify each R-matrix with a Miura operator. In a dual Type IIB frame, the components of the Miura transformation are shown to coincide with the half-index of a 3d supersymmetric gauge theory on a Hanany-Witten system of D3-NS5 branes. As a further application, we demonstrate that qq-characters can be algebraically constructed from the Miura transformation.
... The K-theoretic capped vertex functions were defined in [6] as partition functions of relative quasimaps to Nakajima varieties. In this paper we consider the capped vertex functions for a variety X = Hilb n (C 2 ) given by Hilbert scheme of n-points in the plane C 2 . ...
... Content of the Paper. In [6], Okounkov introduced the capped vertex function with descendents as the following partition function: Definition 1.1. (Section 7 of [6]) The capped vertex with descendent τ for a Nakajima quiver variety X is the following generating function: ...
... In [6], Okounkov introduced the capped vertex function with descendents as the following partition function: Definition 1.1. (Section 7 of [6]) The capped vertex with descendent τ for a Nakajima quiver variety X is the following generating function: ...
We obtain explicit formulas for capped descendent vertex functions of for descendents given by chern classes of tautological bundles. The expression is the result of twisting a well known generating function for normalized Macdonald polynomials. This gives an explicit description of the rational function the capped vertex is an expansion of. Along the way we prove various limit expressions of the capping operator and the bare vertex function.
... Part of their conjecture is an action of the spectrum-generating algebra of a quantum integrable system, Yangian for example, on the cohomology (or generalized cohomologies) of the Higgs branch of the corresponding gauge theory. The correspondence and its string-theory origin, when g is a bosonic Lie algebra, have been extensively studied [31,[33][34][35][36][37][38][39][40][41][42][43][44][45][46][47], generalized to higher dimensions [32,35,[48][49][50][51][52][53], and the mathematical formulation has been established [54][55][56][57][58][59]. For g being a Lie superalgebra, the correspondence and its string-theory origin were studied recently [60][61][62][63] but despite progress [30,64], a full mathematical treatment was lacking. ...
... A similar kind of Gauss decomposition of R-matrices has been discussed by Khoroshkin and Tolstoy in the algebraic setting [68,69]. Since the paper of Maulik and Okounkov came out, the construction of stable envelopes has been extended to K-theory [55] and elliptic cohomology [57], giving rise to trigonometric and elliptic solutions to the Yang-Baxter equations respectively. ...
... However, the mathematical formulation of the BGC shows that more basic objects to study are stable envelopes through which R-matrix can be constructed easily. It would be very interesting to provide a 4d Chern-Simons perspective of the mathematical works [54][55][56][57] and specially the construction of various stable envelopes. The string-theory realization of the Bethe-Gauge Correspondence in which 4d Chern-Simons theory plays a crucial role should be a very useful guide in providing such a perspective (see [44,62,96] for some related works in this direction). ...
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 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 spin chain with fundamental representations using the corresponding 3d SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on for an interval I and an elliptic curve compute the stable envelopes, and in turn the geometric R-matrix, of the 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 the results of Rim\'anyi and Rozansky.
... A key concept of this paper, the "stable envelope", is due to A. Okounkov and his co-authors [28,36,1]. It is that rare notion in mathematics that not only organizes and generalizes several earlier mathematical objects (in this case, from enumerative geometry and representation theory), but also opens up a plethora of new connections, found in places like geometry, quantum integrable systems, KZ-type PDE's, difference equations, quantum groups, and quantum cohomology. ...
... All the maps are obvious, and the diagrams are Cartesian (although we don't need this property for the second one). By Proposition 5.13, we have (36) Stab X C (f ) = (p 13 ) * ∆ * Stab X C (F )(z)) Stab F C/C (f )(z ±γ(f ) ) . Recall that although the map p 13 is not proper in general, its restriction to the support of ∆ * Stab X C (f ) Stab X C/C (f )(z ±γ(f ) ) is proper, so the pushforward in elliptic cohomology is well defined. ...
... In the first step, we applied equation (36). In the second one, we used the compatibility of pushforward and pullback induced by the first Cartesian diagram above. ...
In this paper we study the elliptic characteristic classes known as ''stable envelopes'', which were introduced by M. Aganagic and A. Okounkov. We prove that for a rich class of holomorphic symplectic varieties\unicode{x2013}called Cherkis bow varieties\unicode{x2013}their elliptic stable envelopes exhibit a duality property inspired by mirror symmetry in d=3, quantum field theories. A crucial step of our proof involves the process of ''resolving'' large charge branes into multiple smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover various new features in the geometry of bow varieties.
... Matrix model, defined by the W -representation [26], like Z := eŴ · 1 = < 1 > Its superintegrability property [27] S λ ∼ S λ Sets (rays) of commuting Hamiltonians, made from iteratedÊ andF [28][29][30][31] W 1+∞ -algebras and VOAs [32,33] Stable envelopes [34] . . . ...
... In this context a family of toric Calabi-Yau 3-folds represented by generalized conifolds associated with affine Yangians Y( gl m|n ) is interesting since it presents quiver varieties beyond the ones of the Nakajima type [46], and a simple resolved conifoldalgebra Y( gl 1|1 ) -delivers the simplest example of a non-Nakajima quiver. This quiver variety has no canonical symplectic structure pairing present in the Nakajima quiver varieties [34], and, therefore, the standard enumerative geometry methods should be improved to capture the construction of stable envelopes for these varieties as well. We believe that our construction of super-Schur polynomials and, especially, a similarity between languages of symmetric polynomials in this note and works [47,48] A Semi-Fock representations of Y( gl 1|1 ) from D-brane dynamics ...
... Mathematically literature [34] defines of a stable envelope as a unique function on fixed points λ satisfying the following rules: ...
We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian . This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables . The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.
... In Section 3 we study the Vertex function introduced in [28]. The main outcome of this section is that the Vertex V is computed by the topologically twisted index on S 1 × ε CP 1 (in the sense of 3d A-twist). ...
... The main object that we study in this paper is the index counting holomorphic quasi-maps [28,[30][31][32] from CP 1 into the Higgs or Coulomb branch of vacua. It is known under various names in the literature: the vortex partition function [33][34][35][36], the holomorphic block [37], the homological block [38] (in a narrower context), the half-index [23], or the Vertex function [28]. ...
... The main object that we study in this paper is the index counting holomorphic quasi-maps [28,[30][31][32] from CP 1 into the Higgs or Coulomb branch of vacua. It is known under various names in the literature: the vortex partition function [33][34][35][36], the holomorphic block [37], the homological block [38] (in a narrower context), the half-index [23], or the Vertex function [28]. ...
The supersymmetric cigar (half-)index or cigar partition function of 3d gauge theories contains a wealth of information. Physically, it captures the spectrum of BPS states, the non-perturbative corrections to various partition functions, the effective twisted superpotential and the data of supersymmetric vacua. Mathematically, it defines the K-theoretic Vertex counting vortices/quasimaps, and connects to quantum K-theory, as well as elliptic cohomology and stable envelopes. We explore these topics from the physics standpoint, systematically developing the foundations and explaining various mathematical properties using the quantum field theory machinery.
... (1. 3) where (X, Y ) are hypermultiplet scalars and (σ, φ) are vector multiplet scalars. In the Ctwist, the moduli space of solutions to the above equations has an algebraic description as the moduli space of quasimaps Σ → M H to the Higgs branch [25] (in the H or A twist, one recovers the twisted quasimaps of [26]). The twisted index on Σ × S 1 computes the equivariant Euler characteristic of this moduli space. ...
... We then turn to one of the main conceptual results of this work, which is to explicitly relate the above partition function to the enumerative geometry of based quasimaps QM α from P 1 → M H [25]. This moduli space splits into components labelled by the degree k of the G bundle over P 1 , which we denote QM k α . ...
... (we refer the reader to [25] for an excellent review of enumerative geometry and quasimaps), which is an equivariant count of based quasimaps to the Higgs branch. We show that they are related to the hemisphere path integral up to perturbative contributions, for which we give a precise geometrical interpretation. ...
We study the partition functions of topologically twisted 3d gauge theories on a hemisphere spacetime with boundary . We show that the partition function may be localised to either the Higgs branch or the Coulomb branch where the contributions to the path integral are vortex or monopole configurations respectively. Turning to supersymmetry, we consider partition functions for exceptional Dirichlet boundary conditions that yield a complete set of `IR holomorphic blocks'. We demonstrate that these correspond to vertex functions: equivariant Euler characteristics of quasimap moduli spaces. In this context, we explore the geometric interpretation of both the Higgs and Coulomb branch localisation schemes in terms of the enumerative geometry of quasimaps and discuss the action of mirror symmetry.
... For a quiver variety X, stable envelope is a map from the equivariant cohomology of the torus A-fixed point set X A to the equivariant cohomology of X. It was extended to equivariant Ktheory [43] and equivariant elliptic cohomology [1]. In terms of stable envelopes Maulik and Okoukov constructed rational R matrices geometrically and obtained a geometric realization of the Yangian Y Q associated with a quiver Q [35]. ...
... In terms of stable envelopes Maulik and Okoukov constructed rational R matrices geometrically and obtained a geometric realization of the Yangian Y Q associated with a quiver Q [35]. Such geometric construction of R matrices was extended to the trigonometric [43] and the elliptic [1] cases. The stable envelopes also proved to be useful in solving integrable systems [2,35,49]. ...
We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with on the Gelfand-Tsetlin basis of the tensor product of the n-vector representations. The result is described in a combinatorial way by using the partitions of [1,n]. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology of the cotangent bundle X of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on .
... The notion of stable envelopes was initiated in [35,41]. They provide a good basis of equivariant cohomology and K-theory of Nakajima quiver varieties X [37,38] and has nice applications to enumerative geometry, geometric representation theory and quantum integrable systems. ...
... In addition, for the same case X = T * f l, it was shown that the vertex operators yield a K-theoretic vertex functions [33,41,42] as a vacuum expectation value of a composition of the vertex operators [30]. The vertex function is a generating function of the counting of quasi maps from X to P 1 . ...
We propose new vertex operators, both the type I and the type II dual, of the elliptic quantum toroidal algebra U_{t_1,t_2,p}(gl_{1,tor}) by combining representations of U_{t_1,t_2,p}(gl_{1,tor}) and the notions of the elliptic stable envelopes for the instanton moduli space M(n,r). The vertex operators reproduce the shuffle product formula of the elliptic stable envelopes by their composition. We also show that the vacuum expectation value of a composition of the vertex operators gives a correct formula of the K-theoretic vertex function for M(n,r). We then derive exchange relations among the vertex operators and construct a L-operator satisfying the RLL=LLR^* relation with R and R^* being elliptic dynamical instanton R-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of L, we define a comultiplication \Delta in terms of it. It turns out that the new vertex operators are intertwining operators of the U_{t_1,t_2,p}(gl_{1,tor})-modules w.r.t \Delta.
... The generating function of D0-D6 partition functions with varying number of D0-branes can be interpreted as the instanton partition function of the (6 + 1)-dimensional noncommutative U(n) super Yang-Mills theory on S 1 t × C 3 [9,10], and can be computed exactly [11]. Mathematically, this instanton partition function computes rank n K-theoretic Donaldson-Thomas invariants of C 3 [12][13][14]. Upon lifting to M-theory, the D0-branes become Kaluza-Klein (KK) modes of the graviton, and the D6-branes become KK monopoles, which can be described geometrically by an n-centered multi-Taub-NUT space TN n [15,16]. ...
... Based on the standard lore of Witten index, Z X only receives contributions from the ground states of SQM (X), which are constant modes along S 1 t . Furthermore, if Z X gets contributions only from supergravity fields, including the eleven-dimensional graviton g µν , a Majorana gravitino Ψ µ , and a three-form potential A µνρ , then a simple expression for Z X was derived by Nekrasov and Okounkov [13,14], ...
The duality between type IIA superstring theory and M-theory enables us to lift bound states of D0-branes and n parallel D6-branes to M-theory compactified on an n-centered multi-Taub-NUT space . As a consequence, the rank n K-theoretic Donaldson-Thomas invariants of are connected with the index of M-theory on . In this paper, we extend the correspondence by considering intersecting D6-branes. In the presence of a suitable Neveu-Schwarz B-field, the system preserves two supercharges. This system is T-dual to the configuration of tetrahedron instantons which we introduced in \cite{Pomoni:2021hkn}. We conjecture a closed-form expression for the K-theoretic tetrahedron instanton partition function, which is the generating function of the D0-D6-brane partition functions. We find that the tetrahedron instanton partition function coincides with the partition function of the magnificent four model for special values of the parameters, leading us to conjecture that our system of intersecting D6-branes can be obtained from the annihilation of D8-branes and anti-D8-branes. Remarkably, the K-theoretic tetrahedron instanton partition function allows an interpretation in terms of the index of M-theory on a noncompact Calabi-Yau fivefold which is related to a superposition of Kaluza-Klein monopoles. The dimensional reduction of the system allows us to express the cohomological tetrahedron instanton partition function in terms of the MacMahon function, generalizing the correspondence between Gromov-Witten invariants and Donaldson-Thomas invariants for Calabi-Yau threefolds.
... Recall the coefficients T μ,ν α,β and (T −1 ) μ,ν α,β from (21). The connection between f α,β (u) and Xμ,ν(u) is ...
... Substituting the above formula into the second equation of (143) and using (21) to express (T −1 ) ...
We consider the problem of the R-matrix of the quantum toroidal algebra [Formula: see text] in the Fock representation. Using the connection between the R-matrix R( u) ( u being the spectral parameter) and the theory of Macdonald operators, we obtain explicit formulas for R( u) in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator, which completely describe the functional dependence of R( u) on the spectral parameter u. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with R( u) and thus gives another approach to the study of the poles of the R-matrix as a function of u.
... Physically, it has most recently attracted attention in the context of five-dimensional gauge theories, where the finite difference parameter q is the exponentiated radius of the fifth dimension [48] in the spirit of Kaluza and Klein. Mathematically, q-numbers appear in enumerative geometry of symplectic resolutions as K-theory weights associated with fixed points of equivariant torus action [49]. From the perspective of integrability theory, q-numbers correspond to trigonometric integrable models, which occupy an intermediate level of complexity between rational (corresponding to usual numbers) and elliptic (corresponding to elliptic numbers, [50]) models. ...
We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarthmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential.
... P. Aluffi writes "Segre classes provide a systematic framework for enumerative geometry computation; but this is of relatively little utility, as Segre classes are in general hard to compute" [A]. More recent applications of CSM classes (under the name of "stable envelope classes") are in [MO,Ok,RTV2] and references therein. ...
The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas---including residue generating sequences---for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (2-step) positivity properties for some relevant representations.
... Recalling that stable pairs on C 2 × C are the same thing as quasi-maps from C to Hilb d (C 2 ) (c.f. [31,Exercise 4.3.22]), this corresponds to the fact that the quantum cohomology of Hilb d (C 2 ) is generated by divisors. ...
We give an explicit formula for the descendent stable pair invariants of all (absolute) local curves in terms of certain power series called Bethe roots, which also appear in the physics/representation theory literature. We derive new explicit descriptions for the Bethe roots which are of independent interest. From this we derive rationality, functional equation and a characterization of poles for the full descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. We also sketch how our methods give a new approach to the spectrum of quantum multiplication on .
... See for example [24]. A key to this is the identification of the weight functions in the representation theory of quantum groups [13,24,31,32] with the stable envelopes in the geometry of quiver varieties [1,26,30]. In fact, it has been shown for the case of the elliptic quantum group U q,p ( gl N ) that the Gelfand-Tsetlin bases are obtained by transforming the standard bases with the change of basis matrix given by the elliptic weight functions [24] (see also [31] for the affine quantum group case). ...
We study the level-0 representations of the elliptic quantum group . We give a classification theorem of the finite-dimensional irreducible representations of in terms of the theta function analogue of the Drinfeld polynomial for the quantum affine algebra . We also construct the Gelfand-Tsetlin bases for the level-0 -modules following the work by Nazarov-Tarasov for the Yangian -modules. This is a construction in terms of the Drinfeld generators. For the case of tensor product of the vector representations, we give another construction of the Gelfand-Tsetlin bases in terms of the L-operators and make a connection between the two constructions. We also compare them with those obtained by the first author by using the -action realized by the elliptic dynamical R-matrix on the standard bases. As a byproduct, we obtain an explicit formula for the partition functions of the corresponding 2-dimensional square lattice model in terms of the elliptic weight functions of type .
... The notion of stable envelopes was initiated in [57,65]. They form a good basis of equivariant cohomology and K-theory of Nakajima quiver varieties [61,62] and has nice applications to enumerative geometry, geometric representation theory and quantum integrable systems. ...
We expose the elliptic quantum groups in the Drinfeld realization associated with both the affine Lie algebra \g and the toroidal algebra \g_tor. There the level-0 and level \not=0 representations appear in a unified way so that one can define the vertex operators as intertwining operators of them. The vertex operators are key for many applications such as a derivation of the elliptic weight functions, integral solutions of the (elliptic) q-KZ equation and a formulation of algebraic analysis of the elliptic solvable lattice models. Identifying the elliptic weight functions with the elliptic stable envelopes we make a correspondence between the level-0 representation of the elliptic quantum group and the equivariant elliptic cohomology. We also emphasize a characterization of the elliptic quantum groups as q-deformations of the W-algebras.
... The corresponding theory for threefolds (not necessarily CY) has been studied in great generality by the Okounkov school [Oko15], building on prior milestones [Nak94;NS15]. From there, it is clear that the object of our study can be looked at from various angles (with some subtle differences): GLSM partition functions on the disk and D-brane central charges [HR13], holomorphic blocks [BDP14], and Givental's I-function [Giv96]. ...
We present formulas for the K-theoretic Pandharipande-Thomas vertex of fourfolds, for the case of one non-trivial leg. They are obtained from computations in a three-dimensional supersymmetric gauge theory, where we identify the field content and boundary conditions that correspond to the vertex with tautological insertions.
... For bosonic simply laced Lie algebras, the contemporary understanding of the theory is due to Maulik and Okounkov [69], followed by [8,9,[75][76][77][78] and building on work of Nekrasov and Shatashvili [70]. It is based on quantum geometry of certain holomor-phic symplectic varieties that arise as moduli spaces of three-dimensional quiver gauge theories with N = 4 supersymmetry. ...
There is a generalization of Heegaard-Floer theory from to other Lie (super)algebras . The corresponding category of A-branes is solvable explicitly and categorifies quantum link invariants. The theory was discovered in \cite{A1,A2}, using homological mirror symmetry. It has novel features, including equivariance and, if , coefficients in categories. In this paper, we describe the theory and how it is solved in detail in the two simplest cases: the theory itself, categorifying the Alexander polynomial, and the theory, categorifying the Jones polynomial. Our approach to solving the theory is new, even in the familiar case.
... 2/ theory in six dimensions, localizes on the defects, yielding quiver gauge theories whose Higgs branches are Nakajima quiver varieties. Using powerful methods of enumerative equivariant K-theory [42], the authors of [1] were able to express these deformed conformal blocks in terms of solutions of the qKZ equations. ...
We introduce the notions of (G,q) -opers and Miura (G,q) -opers, where G is a simply connected simple complex Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q) -opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a q DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ( q -differential equations). If \mathfrak{g} is simply laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra U_q \widehat{\mathfrak{g}} . However, if \mathfrak{g} is non-simply-laced, then these equations correspond to a different integrable model, associated to U_q {}^L\widehat{\mathfrak{g}} where ^L\widehat {\mathfrak{g}} is the Langlands dual (twisted) affine algebra. A key element in this q DE/IM correspondence is the QQ -system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category \mathcal{O} of the relevant quantum affine algebra.
... Denote by : × → the multiplication and by : × → the action. An equivariant structure on a sheaf F is an isomorphism : * → * of sheaves on × such that the cocycle condition holds on × × : * 23 •(id × ) * = ( ×id ) * , where 23 is the projection to the second and third factor. ...
We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.
We study enumerative aspects of moduli space of sheaves on smooth quasi-projective varieties of dimension three and four. Our focus is on moduli spaces of rank one sheaves and complexes, including Donaldson-Thomas theory and Pandharipande-Thomas’s stable pairs. Unlike the moduli space of sheaves on curves, the moduli space of sheaves on higher dimensional varieties can have arbitrary singularities, making the study of its geometry difficult. Nevertheless, there remains a method for constructing virtual cycles that enables to study of enumerative aspects.
In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as , where there is canonical instanton action suppression, and annihilates a particle in the vacuum, whereas creates a particle in the vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection i.e. by a quantum R-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the R-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. Operators , might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object -- a ``tunneling algebra''. The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this note we revise few successful cases of the aforementioned correspondence: quantum algebras and affine Yangians . For affine Yangians we demonstrate explicitly how instantons ``perform'' equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.
In this paper we prove the rationality of the capped vertex function with descendents for arbitrary Nakajima quiver varieties with generic stability conditions. We generalise the proof given by Smirnov to the general case, which requires to use techniques of tautological classes rather than the fixed-point basis. This result confirms that the "monodromy" of the capped vertex function is trivial, which gives a strong constraint for the monodromy of the capping operators. We will also provide a GIT wall-crossing formula for the capped vertex function in terms of the quantum difference operators and fusion operators.
Let G be a linear semisimple algebraic group and B its Borel subgroup. Let be the maximal torus. We study the inductive construction of Bott–Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory , the right and left Demazure–Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on with respect to the Kirwan map. We show that the left and right twisted Demazure–Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.
In this paper we prove the isomorphism of the positive half of the quantum toroidal algebra and the positive half of the Maulik-Okounkov quantum affine algebra of affine type A via the monodromy representation for the Dubrovin connection. The main tool is on the proof of the fact that the degeneration limit of the algebraic quantum difference equation is the same as that of the Okounkov-Smirnov geometric quantum difference equation.
We consider the quantum difference equation of the Hilbert scheme of points in C2. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande in [27]. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra . In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in equivariant K-theory and elliptic cohomology. Using the second approach we identify the monodromy matrices for the differential equation with the K-theoretic R-matrices of cyclic quiver varieties, which appear as subvarieties in the 3D-mirror Hilbert scheme. Most of the results in the paper are illustrated by explicit examples for cases n=2n=2 and n=3n=3 in the Appendix.
We describe the Stokes phenomenon, its main significance, and its emergence in various landscapes. We explain how to use Stokes phenomena to enrich the classical dynamics of complex dynamical systems, defining some wild dynamics. In the same line, we describe how, adding Stokes multipliers to the classical monodromy, it is possible to classify a lot of complex dynamical systems (linear or not), up to “gauge transformations” (using some cohomological invariants), and to get generalizations of the Riemann-Hilbert correspondence, with a lot of applications. In the linear case, we describe the relations between Stokes phenomena and differential Galois theory and some important consequences. We end with a short description of many other incarnations of Stokes phenomenon (singular perturbations, resurgence, difference and q-difference equations, theoretical physics …), with some insights in the several variables cases. All along our article, we insist on the historical roots and on some simple geometric ideas. The interpretation of the divergence of some power series as expressing a form of branching for an analytic function is a leitmotiv and a red thread all along the text.
In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, . This integral representation can be used to compute the limit of the vertex function, where denotes the equivariant parameter of a torus acting on X by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the A-series of the Grassmannian with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.
We discuss the q q qq -systems, the functional form of the Bethe ansatz equations for the twisted Gaudin model from a new geometric point of view. We use a concept of G G -Wronskians, which are certain meromorphic sections of principal G G -bundles on the projective line. In this context, the q q qq -system, similar to its difference analog, is realized as the relation between generalized minors of the G G -Wronskian. We explain the link between G G -Wronskians and twisted G G -oper connections, which are the traditional source for the q q qq -systems.
We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters.
The goal of this paper is to better understand the quasimap vertex functions of type A Nakajima quiver varieties. To that end, we construct an explicit embedding of any type A quiver variety into a type A quiver variety with all framings at the rightmost vertex of the quiver. Then, we consider quasimap counts, showing that the map induced by this embedding on equivariant K-theory preserves vertex functions.
This paper describes a relation between the elliptic stable envelopes of a hypertoric variety XX and a distinguished K-theory class on the product of the loop hypertoric space L~X and its symplectic dual PX!. This class intertwines the K-theoretic stable envelopes in a certain limit. Our results are suggestive of a possible categorification of elliptic stable envelopes.
We prove that the Hilbert scheme of k points on C 2 ( Hilb k [ C 2 ] ) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the C ħ × -action. First, we find a two-parameter family X k , l of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb k [ C 2 ] is obtained via direct limit l ⟶ ∞ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted ħ -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank- N sheaves on P 2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.
In this article we use the philosophy in \cite{OS22} to construct the quantum difference equation of affine type A quiver varieties in terms of the quantum toroidal algebra . In the construction, and we define the set of wall for each quiver varieties by the action of the universal R-matrix, which is shown to be almost equivalent to that of the K-theoretic stable envelope on each interval in . We also give the examples of the instanton moduli space M(n,r) and the Hilbert scheme to show the explicit form of the quantum difference operator.
This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.
In order to extend the geometrization of Yangian R-matrices from Lie algebras to superalgebras , we introduce new quiver-related varieties which are associated with representations of . In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre transforms. We present superalgebra versions of stable envelopes for the new quiver-like varieties that generalize the cotangent bundle of a Grassmannian. We define superalgebra generalizations of the Tarasov–Varchenko weight functions, and show that they represent the super stable envelopes. Both super stable envelopes and super weight functions transform according to Yangian -matrices of with M+N=2.
We introduce a couple of technical tools that will be needed in the construction of the perfect obstruction theory on the Hilbert scheme of points on a 3-fold.
This thesis studies the algebro-geometric aspects of supersymmetric abelian gauge theories in three dimensions. The supersymmetric vacua are demonstrated to exhibit a window phenomenon in Chern-Simons levels, which is analogous to the window phenomenon in quantum K-theory with level structures. This correspondence between three-dimensional gauge theories and quantum K-theory is investigated from the perspectives of semi-classical vacua, twisted chiral rings, and twisted indices. In particular, the twisted index admits an algebro-geometric interpretation as the supersymmetric index of an effective quantum mechanics. Via supersymmetric localisation, the contributions from both topological and vortex saddle points are shown to agree with the Jeffrey-Kirwan contour integral formula. The algebro-geometric construction of Chern-Simons contributions to the twisted index from determinant line bundles provides a natural connection with quantum K-theory.
We propose a variation of the classical Hilbert scheme of points, the double nested Hilbert scheme of points , which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss K -theoretic refinements à la Nekrasov–Okounkov.
Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of q-difference equations holomorphic in variables with poles in variables to a basis of solutions holomorphic in with poles in . The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that the functions obtained by applying the elliptic stable envelope to the vertex functions of the cotangent bundle of the full flag variety are precisely the vertex functions for the same variety under an exchange of the parameters . As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope.
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