Arrangements of hyperplanes and Lie algebra homology

Schechtman, Vadim; Varchenko, Alexander

doi:10.1007/bf01243909

Cited by 321 publications

(380 citation statements)

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“…where a label the set of couplings and the operators H a on the right hand side are κ-independent, show up in mathematical physics on several occasions (Knizhnik-Zamolodchikov connection [61,105], t-part of tt * -connection [19], Gauss-Manin connection for exponential periods [66,68], λ-connection associated to the solution of the WDVV equations [45,62], and more recently, e.g. [15]).…”

Section: Fractional Instantons and Quantum Differential Equationsmentioning

confidence: 99%

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

Nekrasov

2016

J. High Energ. Phys.

213

466

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We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the qq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger-type relations. These relations imply that the supersymmetric partition functions in the presence of Ω-deformation and defects obey the Ward identities of two dimensional conformal field theory and its q-deformations. The details will be discussed in the companion papers.

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Section: Fractional Instantons and Quantum Differential Equationsmentioning

confidence: 99%

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

Nekrasov

2016

J. High Energ. Phys.

213

466

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“…There is a neat relationship between the wave vector satisfying the OSBAE (1.6) and the vector-valued solutions of the KZ equation (1.4): the general vector valued solution of the KZ equation for an arbitrary simple Lie algebra was found by Schechtman and Varchenko [12]. It can be represented as a multiple contour integral…”

Section: Introductionmentioning

confidence: 96%

TheA⁽²⁾₂Gaudin model and its associated Knizhnik–Zamolodchikov equation

Kurak

Lima-Santos

2004

J. Phys. A: Math. Gen.

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The semiclassical limit of the algebraic Bethe Ansatz for the Izergin-Korepin 19-vertex model is used to solve the theory of Gaudin models associated with the twisted A (2) 2 R-matrix. We find the spectra and eigenvectors of the N − 1 independents Gaudin Hamiltonians . We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik-Zamolodchikov equations.

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“…We refer the reader to [1], [14], [15], [16], [21], [25] for related works with the twisted cohomology and homology.…”

Section: Twisted Cohomology and Homologymentioning

confidence: 99%

“…a system of rationally holonomic differential equations [3], [21].The purpose of the present paper is to study the (ir)reducibility conditions of the Gauss-Manin system associated with the integral above. Here the (ir)reducibility of a system means the (ir)reducibility of its monodromy representation.…”

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Reductibility and irreducibility of the Gauss-Manin system associated with a Selberg type integral

Mimachi

1993

Nagoya Mathematical Journal

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Dedicated to professor Joji Kajiwara on his 60th birthdyi.e. a system of rationally holonomic differential equations [3], [21].The purpose of the present paper is to study the (ir)reducibility conditions of the Gauss-Manin system associated with the integral above. Here the (ir)reducibility of a system means the (ir)reducibility of its monodromy representation.Throughout our arguments, we adopt a framework of the theory of twisted rational de Rham cohomology and twisted homology, which will be briefly reviewed in Section 1. Let the many-valued n-ίorm Φdt γ .. .dt n be the integrand of (0.1). A basis of the twisted n-th de Rham cohomology group attached to Φ naturally induces a system of matrix valued differential equations of the first order, which is called the Gauss-Manin system. Our first result is its explicit expression (Proposition 2.1). Luckily enough, our system happens to have logarithmic poles with residues of constant matrices, which makes our discussion much simpler. A study about (ir)reducibility is made in Section 3. Our second result is the irreducibility condition when v = 0. In this case, it is shown that if λj (1 < j < m) <£ Z and Σjli λj £7A then our system is irreducible (Theorem 3.1). This suggests thatReceived May 7, 1993. our Gauss-Manin system is generically irreducible, though we could not succeed in finding (ir)reducibility conditions in general setting. If a given system of differential equations has a subsystem, then the monodromy representation of the original one is reducible, of course. It seems an interesting problem to find a subsystem of our system. We find four reducible cases, when m -3 (Theorem 3.3). We also prove that the Lie algebra generated by the residue matrices is isomorphic to the general linear Lie algebra Qi(n + 1 C) (Proposition 3.4). The author believes that this Lie algebra should shed light on the irreducibility conditions in the future.When m is arbitrary, we obtain a reducibility condition and a corresponding subsystem in Theorem 3.5: We find that if 0 < £ < n -1 and 2λ } ,+ v£ = 0 (j = 1,... ,rn -2) then our system is reducible. By virtue of the formula in Theorem 3.5 in the case £ -1, we can derive a system of the differential equations of the second order which is satisfied by the integral (0.1) (Theorem 4.1). This system is related with the spherical functions of BC type defined in [11].We finally give a comment that the integral (0. is generated by the symmetrization of the logarithmic forms(2) If we suppose Π \λ m + -w k) ^ 0, in addition to the assumption (Mon),then a basis of H (12* (*/)), V ω ) n is given by the symmetrization ofThis theorem was first proved by Aomoto [3]. His proof depends on the arguments for the generalized Pochhammer differential equations. On the other hand, Esnault-Schechtman-Viehweg gave another proof of (1) Let S ω be the local system on X defined by the monodromy of Φ and S ω the dual local system of S ω . In this paper we consider only the (S n -) symmetric (co)homology, i.e., the (twisted) cycles Γ in the integral (0.1) ar...

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Arrangements of hyperplanes and Lie algebra homology

Cited by 321 publications

References 12 publications

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

TheA⁽²⁾₂Gaudin model and its associated Knizhnik–Zamolodchikov equation

Reductibility and irreducibility of the Gauss-Manin system associated with a Selberg type integral

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Arrangements of hyperplanes and Lie algebra homology

Cited by 321 publications

References 12 publications

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

TheA(2)2Gaudin model and its associated Knizhnik–Zamolodchikov equation

Reductibility and irreducibility of the Gauss-Manin system associated with a Selberg type integral

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TheA⁽²⁾₂Gaudin model and its associated Knizhnik–Zamolodchikov equation