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 $\boldsymbol{z}$ with poles in $\boldsymbol{a}$ to a basis of solutions holomorphic in $\boldsymbol{a}$ with poles in $\boldsymbol{z}$. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this way recover the vertex functions for the same variety under an exchange of the parameters $\boldsymbol{a} \leftrightarrow \boldsymbol{z}$. As a corollary of this, we deduce the expected 3d mirror relationship for the elliptic stable envelope.
We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl_n partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.
Let $P$ be a parabolic subgroup in $SL_n(\mathbb C)$. We show that there is a $SL_n(\mathbb C)$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}(\mathbb C)}$) which is a natural compactification of the cotangent bundle to $SL_n(\mathbb C)/P$. As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.
Following the program of algebraic Frobenius splitting begun by Kumar and Littelmann, we use representation-theoretic techniques to construct a Frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic. We also show that this splitting is the same as one of the splittings constructed by Kumar, Lauritzen, and Thomsen.
We show that there is a ${SL_n}$-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}}$) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety ${SL_n/B}$.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_λ;\C) of the cotangent bundle of a partial flag variety F_λparametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=λ_i, as the Yangian Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_λ;\C) as the algebra of functions on fibers of a discr…
Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same card…
Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle $T^*G/P$. Restricting this identification to the conormal variety $N^*X(w)$ of a Schubert divisor $X(w)$ in $G/P$, we show that there is a compactification of $N^*X(w)$ as an affine Schubert variety. It follows that $N^*X(w)$ is normal, Cohen-Macaulay, and Frobenius split.
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所及数学系）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Department of Mathematics, Virginia Tech（弗吉尼亚理工大学数学系）；The Chinese University of Hong Kong（香港中文大学）
We show that there is an affine Schubert variety in the infinite dimensional partial Flag variety (associated to the two- step parabolic subgroup of the Kac-Moody group {\hat SL(n)}, corresponding to omitting α_0,α_d) which is a natural compactification of the cotangent bundle to the Grassmann variety.
A theorem of the first author states that the cotangent bundle of the type $A$ Grassmannian variety can be embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety. We extend this result to cotangent bundles of cominuscule generalized Grassmannians of arbitrary Lie type.
We consider the cotangent bundle $T^*F_λ$ of a $GL_n$ partial flag variety, $λ=(λ_1,...,λ_N)$, $|λ|=\sum_iλ_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_λ)$. We introduce K-theoretic stable envelope maps $\Stab_σ: \oplus_{|λ|=n} K_T((T^*F_λ)^T)\to\oplus_{|λ|=n}K_T(T^*F_λ)$, where $σ\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|λ|=n}K_T(T^*F_λ)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_λ))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_λ))…
We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the $K$-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solu…
We obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.
We consider the cotangent bundle T^*F_λof a GL_n partial flag variety, λ= (λ_1,...,λ_N), |λ|=\sum_iλ_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*_T(T^*F_λ). In [MO], a Yangian module structure was introduced on \oplus_{|λ|=n} H^*_T(T^*F_λ). We identify this Yangian module structure with the Yangian module structure introduced in [GRTV]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*F_λ), described in [MO], with the action of the dynamical Hamiltonians from [TV2, MTV1, GRTV]. To construct these identifications we provide a formula for the stable …
In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.
We introduce a superpotential for partial flag varieties of type $A$. This is a map $W: Y^\circ \to \mathbb{C}$, where $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster chart for $Y$, our superpotential agrees with earlier mirrors constructed by Eguchi--Hori--Xiong and Batyrev--Ciocan-Fontanine--Kim--van Straten. Our main tool is quantum Sch…
In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ngô-Yun, and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries we obtain combinatorial identities for counts of rational curve…
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of $F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ..., n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a gene…
Given a flag variety $Fl(n;r_1, \dots , r_ρ)$, there is natural ring morphism from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions $λ$, the image of $s_λ$ under the ring homomorphism is a Schubert class which is described by partitioning $λ$ into a quantum hook (or $q$-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpote…
Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of Kostant-Kumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable …
Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anti-canonical hypersurfaces in G/B. Recently Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev's in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of deg…
We construct pseudotoric structures (à la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror.
In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral $\int_{X}τ^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ for a flag manifold $X=\Fl_{r_1, ..., r_I}({\Bbb C}^n)$ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar{\cal M}_{0,0}(\CP^1\times X)$ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot({\cal E}^n)$. In this article, the detail of this localization computation on $\HQuot({\cal E}^n)$ is carried out. The necessary ingredients in the computation,…
We show that (quantum) D-modules of Givental type on affine flag varieties are transformed to D-modules of Beilinson-Drinfeld type on $n$-reduced Sato Grassmannians via affine Miura transformation.
In this paper we prove a formula relating the equivariant Euler characteristic of $K$-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.
We describe \,$q$-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^*F_λ$ of a partial flag variety \,$F_λ$\,. These \,$q$-hypergeometric solutions manifest a Landau-Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for \,$T^*F_λ$ \,says that the leading term of the asymptotics of the \,$q$-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of $T^*F…
Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of the flag variety itself as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.
We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters …
We show that normalized quantum K-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars-Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik-Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions.
Let G be a semisimple complex algebraic group with Borel subgroup B and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of G/P. Ranee Brylinski discovered a connection between cohomology of G-equivariant line bundles on T*(G/B) and the so-called Brylinski-Kostant filtration, which describes the action of principal sl_2 triples on G-representations. In this paper we generalize these results to a larger class of sl_2 triples. Along the way we also obtain generalizations of results due to Broer on cohomology of G-equivariant bundles on T*(G/P) for various parabolics P.
An algebro-geometric setting for the study of the Painlevé VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.
We explain how A. Givental's mirror symmetric family to the type A flag variety and its proposed generalization to partial flag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y in SL_n/B. We then use this theory to describe the totally nonnegative part of Y.
We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.
We prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli space of strongly parabolic Higgs bundles of rank two or three, with full flags. Although the main theorem is proved only for rank at most three, most of the results are proved for any prime rank.
In this paper we investigate the idea of a tropical critical point of the superpotential for the full flag variety of type A. Recall that associated to an irreducible representation of G=SLn(C) are various polytopes whose integral points parameterize a basis for the representation, e.g. the Gelfand-Zetlin polytope. Such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function, and in fact, the function involved coincides with the superpotential from the Landau-Ginzburg model for G/B coming from mirror symmetry. In mirror symmetry a special role is p…
We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d $\mathcal{N}=(2,2)$ theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is …
Let G be a simple simply connected complex algebraic group. We give a Lie theoretic construction of a conjectural mirror family associated to a general flag variety G/P, and show that it recovers the Peterson variety presentation for the T-equivariant quantum cohomology rings qH*_T(G/P)_(q) with quantum parameters inverted. For SL_n/B we relate our construction to the mirror family defined by Givental and its T-equivariant analogue due to Joe and Kim.
We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ 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…
