Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups

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

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Equivariant $K$ -theory of affine flag manifolds and affine Grothendieck polynomials

Duke Mathematical Journal ◽

10.1215/00127094-2009-032 ◽

2009 ◽

Vol 148 (3) ◽

pp. 501-538 ◽

Cited By ~ 10

Author(s):

Masaki Kashiwara ◽

Mark Shimozono

Keyword(s):

Flag Manifolds ◽

K Theory ◽

Affine Flag ◽

Grothendieck Polynomials

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Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture

Topics in Singularity Theory - American Mathematical Society Translations: Series 2 ◽

10.1090/trans2/180/07 ◽

1997 ◽

pp. 103-115 ◽

Cited By ~ 29

Author(s):

Alexander Givental

Keyword(s):

Stationary Phase ◽

Flag Manifolds ◽

Toda Lattices

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Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

International Mathematics Research Notices ◽

10.1093/imrn/rnab049 ◽

2021 ◽

Author(s):

Leonardo C Mihalcea ◽

Hiroshi Naruse ◽

Changjian Su

Keyword(s):

Divided Difference ◽

Quantum Cohomology ◽

Flag Manifold ◽

Leibniz Rule ◽

Flag Manifolds ◽

Difference Operators ◽

Left Multiplication ◽

Schubert Classes ◽

Left And Right ◽

K Theory

Abstract We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

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Finite difference quantum Toda lattice via equivariant K-theory

Transformation Groups ◽

10.1007/s00031-005-0402-4 ◽

2005 ◽

Vol 10 (3-4) ◽

pp. 363-386 ◽

Cited By ~ 16

Author(s):

Alexander Braverman ◽

Michael Finkelberg

Keyword(s):

Finite Difference ◽

Toda Lattice ◽

K Theory ◽

Quantum Toda Lattice

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QUANTUM GROUPS, SQUEEZING, BLOCH, AND THETA FUNCTIONS

Modern Physics Letters B ◽

10.1142/s0217984993001363 ◽

1993 ◽

Vol 07 (20) ◽

pp. 1321-1329 ◽

Cited By ~ 5

Author(s):

E. CELEGHINI ◽

S. DE MARTINO ◽

S. DE SIENA ◽

G. VITIELLO ◽

M. RASETTI

Keyword(s):

Finite Difference ◽

Quantum Groups ◽

Theta Functions ◽

Heisenberg Algebra ◽

Quantum Systems ◽

Difference Operators ◽

Algebra Structure ◽

Lattice Quantum ◽

Bargmann Representation

We realize the deformation of the Weyl–Heisenberg algebra in terms of finite difference operators within the Fock–Bargmann representation. This allows us to incorporate in a unified q-algebra structure, the notions of squeezing and lattice quantum systems resorting to the properties of theta functions.

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