The Abelian/Nonabelian correspondence and Frobenius manifolds

By Ionut Ciocan-Fontanine, Bumsig Kim, C. Sabbah. 2006

80 citations recorded by the source corpus.

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ebc6b95d891e6b29140717eabee04c3d63556cfb

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Two-Sphere Partition Functions and Gromov–Witten Invariants (2012, methodology): “[73], an appropriate version of the quantum hyperplane theorem [74, 75] (also known as the quantum restriction formula [17]) should allow one to calculate the genus zero Gromov–Witten invariants of any Calabi–Yau threefold that arises as the zero locus of a vector bundle over a geometric quotient V/”
Quantum periods for 3-dimensional Fano manifolds (2013, methodology): “Instead, our preferred tools are those for which we have a good structural understanding on the Gromov–Witten side: Givental’s mirror theorem [21], the Quantum Lefschetz theorem of Coates–Givental [15], and the Abelian/non-Abelian correspondence of Bertram–Ciocan-Fontanine–Kim–Sabbah [8].”
Higher genus quasimap wall-crossing for semi-positive targets (2013, methodology): “ IONUT¸ CIOCAN-FONTANINE AND BUMSIG KIM of Theorem 1.3.4 as well. Note that explicit closed formulas for the small I-functions of these targets are easily obtained from the results in [BCK1], [BCK2], [CKS]. For example, the T-equivariant small Ifor the local Grass(r,n) is Ism= 1 + X d>0 qd ndY−1 ”
Wall-crossing in genus zero quasimap theory and mirror maps (2013, background): “ formulas are due to Givental, see [G2]; • for ﬂag varieties of classical types A,B,Cand D, as well as zero loci of regular sections of homogeneous vector bundles on such ﬂag varieties, see [BCK] and [CKS]; • the I-function of a local target E//G over a base W//G can be written explicitly in terms o”
A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, background): “In relation to [22], consider the T -equivariant embedding X ⊂ P(V ), where V = H(X,OX(D)).”
Defects and quantum Seiberg-Witten geometry (2014, background): “Thus using the results of [64] one can derive quantum J-function for the cotangent bundle to the flag variety in the framework of [65] (see [63] for details).”