Virasoro Correlation Functions for Vertex Operator Algebras

Hurley, Donny; Tuite, Michael P.

doi:10.1142/s0129167x12501066

Cited by 8 publications

(12 citation statements)

References 19 publications

(17 reference statements)

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“…where i 1 = 3, j = 5, in the second cycle. Noting that these graphs are essentially the same as what is called 'Virasoro graphs' in [HT12]. For every such directed graph we can get a corresponding element in DR(T ) in the obvious way.…”

Section: Diagrams Derangements and Some Necessary Notationsmentioning

confidence: 99%

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On correlation functions of vertex operator algebras associated to Jordan algebras

Zhao¹

2016

Journal of Algebra

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Given a Z ≥0 graded vertex operator algebra (VOA) V = ∞ i=0 V i with dim(V 0 ) = 1, then V 1 has a structure of Lie algebra, with the operation given by [a, b] = a(0)b, ∀a, b ∈ V 1 . The affine vertex algebra V r (g) of level r provides such an example, with V 1 = g as a Lie algebra.When dim(V 0 ) = 1, dim(V 1 ) = 0, then V 2 has a structure of commutative (but not necessarily associative) algebra, with the operation aGriess algebra of V . In [HL96] and [HL99], Lam constructed vertex algebras whose Griess algebras are simple Jordan algebras of type A, B, C. In [AM09], Ashihara and Miyamoto constructed for a Jordan algebra J of Type B, a family of VOAs V J ,r parameterized by a complex number r, such that (V J ,r ) 0 = C1, (V J ,r ) 1 = {0}, and (V J ,r ) 2 ∼ = J as Jordan algebra, and Lam's example for type B Jordan algebra is a quotient of V J ,1 . The main result of this paper is a formula about the genus zero correlation function of generating fields in V J ,r .A. Albert classified finite dimensional simple Jordan algebras over an algebraic closed field F with char(F) = 0 [Alb47]. We have a brief review of simple Jordan algebra of type B, and we only consider the case F = C. Let (h, (·, ·)) be a finite dimensional vector space with a non-degenerate symmetric bilinear form (·, ·) and dim(h) = d. Then h ⊗ h has an associative algebra structure:which induces a Jordan algebra structure on h ⊗ h:Let J be the Jordan subalgebra of h ⊗ h consists of symmetric tensors:Then J is essentially the Jordan algebra of symmetric d × d matrices over C, which is called simple Jordan algebra of type B according to Jacobson's notation[JJ49].

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Section: Diagrams Derangements and Some Necessary Notationsmentioning

confidence: 99%

“…where i 1 = 3, j = 5, in the second cycle. Noting that these graphs are essentially the same as what is called 'Virasoro graphs' in [HT12].…”

Section: Introductionmentioning

confidence: 99%

On correlation functions of vertex operator algebras associated to Jordan algebras

Zhao¹

2016

Journal of Algebra

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“…M (h, x 1 ; h, x 2 ; h, y 1 ; h, y 2 ) − dx 1 dx 2 (x 1 − x 2 ) 2 ω(y 1 , y 2 )Z This differential equation is similar in form to the genus one case [30] q∂ q + 2 r=1…”

Section: A Differential Equation For the Normalised Bidifferentialmentioning

confidence: 91%

Genus Two Zhu Theory for Vertex Operator Algebras

Gilroy,

Tuite

2015

Preprint

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We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two n-point correlation function in terms of (n − 1)-point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half plane for holomorphic 1-differentials, the normalised bidifferential of the second kind, the projective connection and the Heisenberg partition function. We prove that the holomorphic mapping from the sewing parameter domain to the Siegel upper half plane is injective but not surjective. We also demonstrate that genus two differential equations arising from Virasoro singular vectors have holomorphic coefficients.

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“…Note, in particular, that the one-point Virasoro vector function for the Heisenberg vertex operator algebra is given by [7] ( ) ( ) ( ) ( ) ( )…”

Section: Torus Two-point Functionmentioning

confidence: 99%