$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

Ben‐Zvi, Dani; Nevins, Thomas

doi:10.1007/s00220-009-0925-y

Cited by 5 publications

(8 citation statements)

References 33 publications

(34 reference statements)

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“…Theorem E describes the exact size of the W -constraints of the tau functions associated to all points of the adelic Grassmannian. A geometric interpretation of the W -symmetries of the adelic Grassmannian was given by Ben-Zvi and Nevins in [7]. For all a ∈ C with a = 0 we have r i,a = i so that g a (W ) = 0.…”

Section: In This Casementioning

confidence: 99%

Reflective prolate-spheroidal operators and the KP/KdV equations

Casper

Grünbaum

Yakimov

et al. 2019

Proc. Natl. Acad. Sci. U.S.A.

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Beginning with the work of Landau, Pollak and Slepian in the 1960s on timeband limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation.We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian Gr ad gives rise to an integral operator TW , acting on L 2 (Γ) for a contour Γ ⊂ C, which reflects a differential operator with rational coefficients R(z, ∂z) in the sense that R(−z, −∂z) • TW = TW • R(w, ∂w) on a dense subset of L 2 (Γ). By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function ψW (x, z). The exact size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions ψW (x, −z) always reflect a differential operator. A 90 • rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels ψW (x, iz) admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.that is, the eigenfunctions of the integral operatorThis is a numerically extremely unstable problem. Landau, Pollak and Slepian [43,32] bypassed this issue based on the property thatcommutes with EE * . The differential operator R(z, ∂ z ) is the "radial part" of the Laplacian in prolate-spheroidal coordinates, and (the joint) eigenfunctions of it and the integral operator EE * are known as the prolate-spheroidal functions, which can be computed numerically in very stable ways. The commuting pair can be traced back to Bateman [6,

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Section: In This Casementioning

confidence: 99%

Reflective prolate-spheroidal operators and the KP/KdV equations

Casper

Grünbaum

Yakimov

et al. 2019

Proc. Natl. Acad. Sci. U.S.A.

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“…In [BN5], we study the factorization (or "vertex algebra space") structure [BD2] on the adelic Grassmannian. This structure is shown to encapsulate both (infinitesimally) the W 1+∞ -symmetry of the KP hierarchy and (globally) the Bäcklund transformations.…”

Section: Further Directionsmentioning

confidence: 99%

“…The directed system of ind-schemes itself is what is known as a factorization space [BD2]-see [BN5].…”

Section: The Adelic Grassmannianmentioning

confidence: 99%

$\mathcal D$-bundles and integrable hierarchies

Ben‐Zvi

Nevins

2011

J. Eur. Math. Soc.

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Abstract. We study the geometry of D-bundles-locally projective D-modules-on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of D-bundles; in particular, we prove that the local structure of D-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by D-bundles on cubic curves E, that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1. We develop a FourierMukai transform describing D-modules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle E over E. We then apply this transform to classify D-bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in E ). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.

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“…The irreducible quotient of M c by its maximal graded, proper D-submodule is a simple vertex algebra, and is often denoted by W 1+∞,c . These algebras have been studied extensively in both the physics and mathematics literature; see for example [1,2,3,7,8,14,22]. The above central extension is normalized so that M c is reducible if and only if c ∈ Z.…”

Section: Introductionmentioning

confidence: 99%

The Structure of the Kac–Wang–Yan Algebra

Linshaw

2015

Commun. Math. Phys.

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ABSTRACT. The Lie algebra D of regular differential operators on the circle has a universal central extensionD. The invariant subalgebraD + under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuumD + -module with central charge c ∈ C, and its irreducible quotient V c , possess vertex algebra structures, and V c has a nontrivial structure if and only if c ∈ 1 2 Z. We show that for each integer n > 0, V n/2 and V −n are W-algebras of types W(2, 4, . . . , 2n) and W(2, 4, . . . , 2n 2 + 4n), respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group O(n) and the symplectic group Sp(2n), respectively. Based on Sergeev's theorems on the invariant theory of Osp(1, 2n) we conjecture that V −n+1/2 is of type W(2, 4, . . . , 4n 2 + 8n + 2), and we prove this for n = 1. As an application, we show that invariant subalgebras of βγ-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

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$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

Cited by 5 publications

References 33 publications

Reflective prolate-spheroidal operators and the KP/KdV equations

Reflective prolate-spheroidal operators and the KP/KdV equations

$\mathcal D$-bundles and integrable hierarchies

The Structure of the Kac–Wang–Yan Algebra

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