Quantum cohomology of the infinite-dimensional generalized flag manifolds

Mare, Augustin-Liviu

doi:10.1016/s0001-8708(03)00229-9

Cited by 4 publications

(27 citation statements)

References 12 publications

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“…Remark 1.2. Theorem 1.1 was used in [10] and [11] in connection with the quantum cohomology ring of affine flag manifolds.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On the complete integrability of the periodic quantum Toda lattice

Mare

2017

Forum Mathematicum

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Abstract. We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and E 6 by Goodman and Wallach at the beginning of the 1980's. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.

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“…Remark 1.2. Theorem 1.1 was used in [10] and [11] in connection with the quantum cohomology ring of affine flag manifolds.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On the complete integrability of the periodic quantum Toda lattice

Mare

2017

Forum Mathematicum

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“…An influential result of Givental and Kim (for

G = {SL}_{n + 1} false(double-struckC false)

) and Kim (for all

G

) states that the ideal of relations in the quantum cohomology ring

{prefixQH}^{*} false(G / B false)

of the generalized flag manifold

G / B

is generated by the integrals of motion of the Toda lattice associated to the dual root system of

G

. Soon after that, Guest and Otofuji (for

G = {SL}_{n + 1} false(double-struckC false)

) and Mare (for

G

of Lie types

A_{n}, B_{n}, C_{n}

) assumed that there exists a (still undefined) quantum cohomology algebra

{prefixQH}^{*} false({scriptF ℓ}_{G} false)

for the affine flag manifold

{scriptF ℓ}_{G}

, which satisfies the analogues of certain natural properties enjoyed by quantum cohomology, such as associativity, commutativity, the divisor axiom, etc. The list of conjectured properties includes the fact that the quantum multiplication of Schubert divisor classes satisfies a quadratic relation determined by the Hamiltonian of the dual periodic Toda lattice.…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of this paper is to rigorously define quantum products on the rings

{prefixH}_{aff}^{*} false(G / B false)

and

{prefixH}^{#} false({scriptF ℓ}_{G} false)

, for

G

of all Lie types, which will satisfy the analogues of the properties predicted in . We will then identify the ideal of relations in the quantum ring

{prefixQH}_{aff}^{*} false(G / B false)

determined by

{prefixH}_{aff}^{*} false(G / B false)

with the conserved quantities of the periodic Toda lattice associated to the dual of the extended Dynkin diagram for

G

(these diagrams correspond to twisted affine Lie algebras).…”

Section: Introductionmentioning

confidence: 99%

“…Since

G / T

and

G / T

are affine bundles over the corresponding flag manifolds

{scriptF ℓ}_{G}

and

G / B

, this induces a map

{normale}_{1}^{*} : {prefixH}^{*} false(G / B false) \to {prefixH}^{*} false({scriptF ℓ}_{G} false)

. We analyze this map in § 4 and prove that it is injective and that

{normale}_{1}^{*} false(σ_{i} false) = ε_{i} - m_{i} ε_{0}, false(1 ⩽ i ⩽ n false), where θ^{\lor} = m_{1} α_{1}^{\lor} + \dots + m_{n} α_{n}^{\lor} .

In the topological category this map was studied by Mare and it played a key role in Peterson's ‘quantum=affine’ phenomenon proved by Lam and Shimozono . Using this map define

{prefixH}_{aff}^{*} false(G / B false) : = {normale}_{1}^{*} false(H^{*} (G / B) false)

, which is a graded subalgebra of

{prefixH}^{*} false({scriptF ℓ}_{G} false)

isomorphic to

{prefixH}^{*} false(G / B false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Let

\nabla_{\partial / \partial z_{i}}

be the derivation corresponding to the vector field

\partial / \partial z_{i}

, where

z_{i}

is the coordinate on

{prefixH}^{2} false(G / B false)

corresponding to the Schubert class

σ_{i}

. The flatness of the Dubrovin connection

\nabla^{ℏ}

together with an argument of Mare imply that the system of quantum differential equations

ℏ \partial / \partial z_{i} false(s false) = σ_{i} ★_{aff} s ⟺ \nabla_{\partial / \partial z_{i}} false(s false) = 0; i \in false{1, \dots, n false}

has non‐trivial solutions

s

in the ring of formal power series

C false[[q_{0}, \dots, q_{n}] false] false[z_{1}, \dots, z_{n} false] false[ℏ^{- 1} false]

where

q_{i} = e^{z_{i}}

for

1 ⩽ i ⩽ n

and where there is a relation

q^{c} = q_{0} q_{1}^{m_{1}} \dots q_{n}^{m_{n}} = 1

. See Remark for an interpretation of this relation.…”

Section: Introductionmentioning

confidence: 99%

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An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

Mare

Mihalcea

2017

Proc. London Math. Soc.

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Consider the generalized flag manifold G/B and the corresponding affine flag manifold F G. In this paper we use curve neighborhoods for Schubert varieties in F G to construct certain affine Gromov-Witten invariants of F G, and to obtain a family of 'affine quantum Chevalley' operators Λ0, . . . , Λn indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H * (F G) with coefficients in Z[q0, . . . , qn]. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for G = SLn(C). The first quantum ring is a deformation of the subalgebra of H * (F G) generated by divisors. The second ring, denoted QH * aff (G/B), deforms the ordinary quantum cohomology ring QH * (G/B) by adding an affine quantum parameter q0. We prove that QH * aff (G/B) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of QH * aff (G/B) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of G.Contents Schubert classes ε 0 , . . . , ε n . The main result of this paper, proved in § 7, is the following.Theorem 1.1. The affine quantum Chevalley operators Λ i satisfy the following properties.(a) The operators Λ i commute up to the imaginary coroot, that is, for any w ∈ W aff and any 0 i, j n we haveThe modified operators Λ i − m i Λ 0 commute (without any additional constraint), that is, for any w ∈ W aff and any 1 i, j n we have

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Instanton counting via affine Lie algebras. I. Equivariant 𝐽-functions of (affine) flag manifolds and Whittaker vectors

Braverman¹

2004

CRM Proceedings and Lecture Notes

222

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Let g be a simple complex Lie algebra, G -the corresponding simply connected group; let also g aff be the corresponding untwisted affine Lie algebra. For a parabolic subgroup P ⊂ G we introduce a generating function Z aff G,P which roughly speaking counts framed G-bundles on P 2 endowed with a P -structure on the horizontal line (the formal definition uses the corresponding Uhlenbeck type compactifications studied in [3]). In the case P = G the function Z aff G,P coincides with Nekrasov's partition function introduced in [23] and studied thoroughly in [24] and [22] for G = SL(n). In the "opposite case" when P is a Borel subgroup of G we show that Z aff G,P is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebraǧ aff -the Langlands dual Lie algebra of g aff . This clarifies somewhat the connection between certain asymptotic of Z aff G,P (studied in loc. cit. for P = G) and the classical affine Toda system. We also explain why the above result gives rise to a calculation of (not yet rigorously defined) equivariant quantum cohomology ring of the affine flag manifold associated with G. In particular, we reprove the results of [13] and [18] about quantum cohomology of ordinary flag manifolds using methods which are totally different from loc. cit.We shall show in a subsequent publication how this allows one to connect certain asymptotic of the function Z aff G,P with the Seiberg-Witten prepotential (cf.[2], thus proving the main conjecture of [23] for an arbitrary gauge group G (for G = SL(n) it has been proved in [24] and [22] by other methods.

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Quantum cohomology of the infinite-dimensional generalized flag manifolds

Cited by 4 publications

References 12 publications

On the complete integrability of the periodic quantum Toda lattice

On the complete integrability of the periodic quantum Toda lattice

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

Instanton counting via affine Lie algebras. I. Equivariant 𝐽-functions of (affine) flag manifolds and Whittaker vectors

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