R. Rimanyi, V. Schechtman, V. Tarasov, A. Varchenko
We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in \C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (\C^N)^{\otimes n}.
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We consider a version of the A_N Bethe equation of XXX type and introduce a reproduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl_{N+1} flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B_N and C_N. Populations of B_N and C_N type are isomorphic to the flag varieties of C_N and B_N types res…
Givental's recursion relations for the flag varieties $G/B$ are established.
We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct ex…
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…
As a generalization of our previous paper [GK], we formulate a residue formula and some simple behaviors of equivariant quantum cohomology applying to compute the quantum cohomology of partial flag manifolds $F_{k_1,\cdots , k_l} $with a try to give a rigorous definition of equivariant quantum cohomology.
This is an exposition of some recent developments related to the object in the title, particularly the combinatorial computation of the (genus 0) Gromov-Witten invariants of the flag manifold and the quadratic algebra approach. The notes are largely based on my joint papers with S.Gelfand, A.N.Kirillov, and A.Postnikov. This is by no means an exhaustive survey of the subject, but rather a casual introduction to its combinatorial aspects.
We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological-antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest-Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput-Manivel-Perrin and Golyshev-Manivel.
For the root system of type $A$ we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter system we construct an extension of the corresponding Nichols-Woronowicz algebra. In the case of finite crystallographic Coxeter systems we present a construction of extended Nichols-Woronowicz algebra mode…
We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ form a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\C)$ relying in particular on the positivity of the structure constants, which are enumerative Gromov--Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterso…
A 2-category was introduced in arXiv:0803.3652 [math.QA] that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N arepresentation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).
In the case of an almost simple algebraic group $G$ of type $G_2$ over a field of characteristic $p>0$ we study the cohomology modules of line bundles on the flag variety for $G$. Our main result is a complete determination of the vanishing behavior of such cohomology in the case where the line bundles in question are induced by characters from the lowest $p^2$-alcoves. When $U_q$ is the quantum group corresponding to $G$ whose parameter $q$ is a complex root of unity of order prime to 6 we give a complete (i.e. covering all characters) description of the vanishing behavior for the correspondi…
Using the Kontsevich's moduli space of stable maps, we define the equivariant quantum cohomology for generalized flag varieties and make a rigorous computation of quantum cohomology of flag varieties.
Bethe Algebra using Pure Spinors
提出so(2r)可积模型中Bethe代数的gl(r)-协变参数化，揭示其源于融合旗的几何本质，并据此计算了周期有理自旋链在不同秩r和Drinfeld多项式下的能谱。
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))…
Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $Λ_0, \ldots, Λ_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these oper…
In this paper we construct some quantum analogues of the global Cousin complex for the flag variety in positive characteristic. Just like in the positive characteristic case, we obtain some remarkable resolutions of the contragradient modules over the small quantum group. We use these resolutions to calculate semiinfinite cohomology of the small quantum group with coefficients in contragradient Weyl modules. It turns out that the described graded space of semiinfinite cohomology is naturally isomorphic to the space of distributions on the cotangent bundle of the flag variety with support in th…
Consider the infinite dimensional flag manifold $LK/T$ corresponding to the simple Lie group $K$ of rank $l$ and with maximal torus $T$. We show that, for $K$ of type $A$, $B$ or $C$, if we endow the space $H^*(LK/T)\otimes \bR[q_1,...,q_{l+1}]$ (where $q_1,...,q_{l+1}$ are multiplicative variables) with an $\bR[\{q_j\}]$-bilinear product satisfying some simple properties analogous to the quantum product on $QH^*(K/T)$, then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism t…
We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit "differential operator formula" for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations.
The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we give an algorithm for computing the multiplicative structure constants of this algebra (the 3-point genus zero Gromov-Witten invariants). The algorithm involves a Grobner basis calculation and the solution by quadrature of a system of differential equations. In particular we obtain quantum Schubert polynomials in a natural fashion.
In this paper we explain the parallelism in the classification of three different kinds of mathematical objects: (i) Classical r-matrices. (ii) Generalized cohomology theories that have Chern classes for complex vector bundles. (iii) 1-dimensional formal groups. The main point of the paper is a construction of the elliptic algebra associated to Belavin's classical elliptic r-matrix in terms of Equivariant elliptic cohomology of the Steinberg varieties associated to some partial flag manifolds.
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