Equivariant mirrors and the Virasoro conjecture for flag manifolds

By Dosang Joe, Bumsig Kim. 2002

47 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2544f95464850c381470dd56734f459e2f4b2aa5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A mirror symmetric construction of qHT∗(G/P)(q) (2005, methodology): “…to Givental [17] and Eguchi, Hori and Xiong [11] mirror symmetry should have an extension to Fano manifolds X , where it means in essence a ‘mirror side’ representation of the quantum cohomology D-module, or quantum differential equations, of X by complex oscillatory integrals.”
Symplectic geometry of Frobenius structures (2003, methodology): “• In non-equivariant Gromov–Witten theory, the formula (14) 〈D〉 = Ŝ Ψ R̂ e 〈D pt 〉 is known to be correct for complex projective spaces and other Fano toric manifolds [24] and for complete flag manifolds [30] (which in particular implies that the Virasoro conjecture holds true for these manifolds).”
Convergence of quantum cohomology by quantum Lefschetz (2005, background): “The R-conjecture has been proved for complete flag [20], Grassmannian [2] and partial flag varieties [3].”
Toric degenerations of Gelfand-Cetlin systems and potential functions (2008, methodology): “Now we recall the quantum Toda lattice and its relation with the J-function of the full flag manifold following Givental and Kim [12] (see also Kim [18] and Joe and Kim [17]).”
The spaces of Laurent polynomials, Gromov-Witten theory of ℙ1-orbifolds and integrable hierarchies (2008, background): “This conjecture is proven for toric manifolds (see [18] for toric Fano case and [22] for general toric case), Flag manifolds [24], and Grassmannians [5].”
Baxter Operator and Archimedean Hecke Algebra (2007, methodology): “These integral operators are closely related with the recursion operators in the Givental integral representation of Whittaker functions (see [Gi], [ JK ] for gll+1 and [GLO3] for other classical Lie algebras).”