Invariant of the hypergeometric group associated to the quantum cohomology of the projective space

Tanabé, Susumu

doi:10.1016/j.bulsci.2004.05.001

Cited by 9 publications

(9 citation statements)

References 13 publications

(20 reference statements)

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“…In particular, following observations of Cecotti and Vafa [10] and Zaslow [29], Dubrovin conjectured [15] that the derived category of a Fano variety Y has a full exceptional collection (E 0 , E 1 , · · · , E n−1 ) if and only if the quantum cohomology of Y is generically semisimple, and that in this case the Stokes matrix S ij of the corresponding Frobenius manifold should coincide with the Gram matrix χ(E i , E j ) for the Euler form of D(Y ). This statement has been verified for projective spaces [21,28].…”

Section: Introductionmentioning

confidence: 55%

Stability Conditions on a Non-Compact Calabi-Yau Threefold

Bridgeland

2006

Commun. Math. Phys.

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Section: Introductionmentioning

confidence: 55%

Stability Conditions on a Non-Compact Calabi-Yau Threefold

Bridgeland

2006

Commun. Math. Phys.

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“…The Gamma conjectures for projective spaces also follow from the computations in [44,45,49]. Corollary 5.0.2 (Guzzetti [35], Tanabé [60]). Dubrovin's conjecture holds for P = P N −1 .…”

Section: Gamma Conjectures For Projective Spacesmentioning

confidence: 89%

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

Galkin¹,

Golyshev²,

Iritani³

2016

Duke Math. J.

184

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Abstract. We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class AF to a Fano manifold F . We say that F satisfies Gamma Conjecture I if AF equals the Gamma class ΓF . When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by ΓF Ch(Ei) for an exceptional collection {Ei} in the derived category of coherent sheaves D b coh (F ). Gamma Conjecture II refines part (3) of Dubrovin's conjecture [18]. We prove Gamma Conjectures for projective spaces and Grassmannians.

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“…This is used by Dubrovin (see [10,Lemma 5.4,p. 97]), D. Guzzetti, H. Iritani, S. Tanabe, K. Ueda et al in computations of the Stokes matrix for quantum differential equations, see [8,9,10,15,16,25,26,27,28].…”

Section: Example: Unramif Ied Casesmentioning

confidence: 99%

The Stokes Phenomenon and Some Applications

Put¹

2015

SIGMA

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Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations

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Invariant of the hypergeometric group associated to the quantum cohomology of the projective space

Abstract: We present a simple method to calculate the Stokes matrix for the quantum cohomology of the projective spaces CP k−1 in terms of certain hypergeometric group. We present also an algebraic variety whose fibre integrals are solutions to the given hypergeometric equation.

Cited by 9 publications

References 13 publications

Stability Conditions on a Non-Compact Calabi-Yau Threefold

Stability Conditions on a Non-Compact Calabi-Yau Threefold

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

The Stokes Phenomenon and Some Applications

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