The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Chongying Dong - One of the best experts on this subject based on the ideXlab platform.
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determinants for integral forms in lattice type Vertex Operator algebras
Journal of Algebra, 2020Co-Authors: Chongying Dong, Robert L. GriessAbstract:Abstract We prove a determinant formula for the standard integral form of a lattice Vertex Operator algebra.
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trace functions of the parafermion Vertex Operator algebras
Advances in Mathematics, 2019Co-Authors: Chongying Dong, Victor G Kac, Li RenAbstract:Abstract The trace functions for the Parafermion Vertex Operator algebra associated to any finite dimensional simple Lie algebra g and any positive integer k are studied and an explicit modular transformation formula of the trace functions is obtained.
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retracted article modular framed Vertex Operator algebras
European Journal of Mathematics, 2018Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed Vertex Operator algebras over any algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed Vertex Operator algebras is established. For a code Vertex Operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed Vertex Operator algebra over C is constructed. As a result, one can obtain a modular framed Vertex Operator algebra from any framed Vertex Operator algebra over C.
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Modular framed Vertex Operator algebras
arXiv: Quantum Algebra, 2017Co-Authors: Chongying Dong, Ching Hung Lam, Li RenAbstract:Framed Vertex Operator algebras over any algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed Vertex Operator algebras is established. For a code Vertex Operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[\frac{1}{2}]-form for any framed Vertex Operator algebra over C is constructed. As a result, one can obtain a modular framed Vertex Operator algebra from any framed Vertex Operator algebra over C.
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representations of the parafermion Vertex Operator algebras
Advances in Mathematics, 2017Co-Authors: Chongying Dong, Li RenAbstract:Abstract The rationality of the parafermion Vertex Operator algebra K ( g , k ) associated to any finite dimensional simple Lie algebra g and any nonnegative integer k is established and the irreducible modules are determined.
Geoffrey Mason - One of the best experts on this subject based on the ideXlab platform.
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free bosonic Vertex Operator algebras on genus two riemann surfaces i
Communications in Mathematical Physics, 2010Co-Authors: Geoffrey Mason, Michael P TuiteAbstract:We define the partition and n-point functions for a Vertex Operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice Vertex Operator algebras and a continuous orbifolding of the rank two fermion Vertex Operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.
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torus n point functions for mathbb r graded Vertex Operator superalgebras and continuous fermion orbifolds
Communications in Mathematical Physics, 2008Co-Authors: Geoffrey Mason, Michael P Tuite, Alexander ZuevskyAbstract:We consider genus one n-point functions for a Vertex Operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion Vertex Operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.
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the genus two partition function for free bosonic and lattice Vertex Operator algebras
arXiv: Quantum Algebra, 2007Co-Authors: Geoffrey Mason, Michael P TuiteAbstract:We define the n-point function for a Vertex Operator algebra on a genus two Riemann surface in two separate sewing schemes where either two tori are sewn together or a handle is sewn to one torus. We explicitly obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and lattice Vertex Operator algebras in both sewing schemes. We prove that the partition functions are holomorphic in the sewing parameters on given suitable domains and describe their modular properties. Finally, we show that the partition functions cannot be equal in the neighborhood of a two-tori degeneration point where they can be explicitly compared.
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twisted representations of Vertex Operator algebras
Mathematische Annalen, 2006Co-Authors: Chongying Dong, Geoffrey MasonAbstract:This paper gives an analogue of Ag(V) theory for a Vertex Operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.
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rational Vertex Operator algebras and the effective central charge
International Mathematics Research Notices, 2004Co-Authors: Chongying Dong, Geoffrey MasonAbstract:We establish that the Lie algebra of weight 1 states in a (strongly) rational Vertex Operator algebra is reductive, and that its Lie rank 1 is bounded above by the effective central charge c~. We show that lattice Vertex Operator algebras may be characterized by the equalities c~=l=c, and in particular holomorphic lattice theories may be characterized among all holomorphic Vertex Operator algebras by the equality l = c.
Thomas Creutzig - One of the best experts on this subject based on the ideXlab platform.
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A quasi-Hopf algebra for the triplet Vertex Operator algebra
Communications in Contemporary Mathematics, 2019Co-Authors: Thomas Creutzig, Azat Gainutdinov, Ingo RunkelAbstract:We give a new factorisable ribbon quasi-Hopf algebra U , whose underlying algebra is that of the restricted quantum group for sℓ(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet Vertex Operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet Vertex Operator algebra M(p), and our construction is parallel to extending M(p) to W(p). We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra CZ to a quasi-Hopf algebra for CZ_{2p}, which corresponds to passing from the Heisenberg Vertex Operator algebra to a lattice extension.
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higgs and coulomb branches from Vertex Operator algebras
Journal of High Energy Physics, 2019Co-Authors: Kevin Costello, Thomas Creutzig, Davide GaiottoAbstract:We formulate a conjectural relation between the category of line defects in topologically twisted 3d $$ \mathcal{N} $$ = 4 supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local Operators for the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.
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self dual Vertex Operator superalgebras and superconformal field theory
Journal of Physics A, 2018Co-Authors: Thomas Creutzig, John F R Duncan, Wolfgang RiedlerAbstract:Recent work has related the equivariant elliptic genera of sigma models with K3 surface target to a Vertex Operator superalgebra that realizes moonshine for Conway's group. Motivated by this we consider conditions under which a self-dual Vertex Operator superalgebra may be identified with the bulk Hilbert space of a superconformal field theory. After presenting a classification result for self-dual Vertex Operator superalgebras with central charge up to 12 we describe several examples of close relationships with bulk superconformal field theories, including those arising from sigma models for tori and K3 surfaces.
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tensor categories for Vertex Operator superalgebra extensions
arXiv: Quantum Algebra, 2017Co-Authors: Thomas Creutzig, Shashank Kanade, Robert McraeAbstract:Let $V$ be a Vertex Operator algebra with a category $\mathcal{C}$ of (generalized) modules that has Vertex tensor category structure, and thus braided tensor category structure, and let $A$ be a Vertex Operator (super)algebra extension of $V$. We employ tensor categories to study untwisted (also called local) $A$-modules in $\mathcal{C}$, using results of Huang-Kirillov-Lepowsky showing that $A$ is a (super)algebra object in $\mathcal{C}$ and that generalized $A$-modules in $\mathcal{C}$ correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a $\mathcal{C}$-algebra and (under suitable conditions) of generalized $A$-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended Vertex Operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of $V$-modules to $A$-modules is a Vertex tensor functor. We give two applications. First, we derive Verlinde formulae for regular Vertex Operator superalgebras and regular $(1/2)\mathbb{Z}$-graded Vertex Operator algebras by realizing them as (super)algebra objects in the Vertex tensor categories of their even and $\mathbb{Z}$-graded components, respectively. Second, we analyze parafermionic cosets $C=\mathrm{Com}(V_L,V)$ where $L$ is a positive definite even lattice and $V$ is regular. If the category of either $V$-modules or $C$-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.
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on regularised quantum dimensions of the singlet Vertex Operator algebra and false theta functions
International Mathematics Research Notices, 2016Co-Authors: Thomas Creutzig, Antun Milas, Simon WoodAbstract:We study a family of non-C2-cofinite Vertex Operator algebras, called the singlet Vertex Operator algebras, and connect several important concepts in the theory of Vertex Operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet Vertex Operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ϵϵ. We then apply modular transformation properties of regularisedpartial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ϵϵ. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ϵϵ, while for sufficiently negative real part of ϵϵ its surjective image gives the fusion ring of a rational Vertex Operator algebra. The category of modules of this rational Vertex Operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this Vertex Operator algebra can be also detected from vector-valued quantum modular forms formed by distinguished atypical characters.
Yi-zhi Huang - One of the best experts on this subject based on the ideXlab platform.
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lower bounded and grading restricted twisted modules for affine Vertex Operator algebras
Journal of Pure and Applied Algebra, 2021Co-Authors: Yi-zhi HuangAbstract:Abstract We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine Vertex (Operator) algebras. We prove that these universal twisted modules for affine Vertex (Operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.
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braided tensor categories and extensions of Vertex Operator algebras
Communications in Mathematical Physics, 2015Co-Authors: Yi-zhi Huang, Alexander Kirillov, James LepowskyAbstract:Let V be a Vertex Operator algebra satisfying suitable conditions such that in particular its module category has a natural Vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.
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generalized twisted modules associated to general automorphisms of a Vertex Operator algebra
Communications in Mathematical Physics, 2010Co-Authors: Yi-zhi HuangAbstract:We introduce a notion of a strongly \({\mathbb{C}^{\times}}\)-graded, or equivalently, \({\mathbb{C}/\mathbb{Z}}\)-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a Vertex Operator algebra. We also introduce a notion of a strongly \({\mathbb{C}}\)-graded generalized g-twisted V-module if V admits an additional \({\mathbb{C}}\)-grading compatible with g. Let \({V=\coprod_{n\in \mathbb{Z}}V_{(n)}}\) be a Vertex Operator algebra such that \({V_{(0)}=\mathbb{C}\mathbf{1}}\) and V(n) = 0 for n < 0 and let u be an element of V of weight 1 such that L(1)u = 0. Then the exponential of \({2\pi \sqrt{-1}\; {\rm Res}_{x} Y(u, x)}\) is an automorphism gu of V. In this case, a strongly \({\mathbb{C}}\)-graded generalized gu-twisted V-module is constructed from a strongly \({\mathbb{C}}\)-graded generalized V-module with a compatible action of gu by modifying the Vertex Operator map for the generalized V-module using the exponential of the negative-power part of the Vertex Operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening Operators on certain Vertex Operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group \({\mathbb{C}/\mathbb{Z}}\) or \({\mathbb{C}}\) and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted Vertex Operators in general involve the logarithm of the formal variable.
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generalized twisted modules associated to general automorphisms of a Vertex Operator algebra
arXiv: Quantum Algebra, 2009Co-Authors: Yi-zhi HuangAbstract:We introduce a notion of strongly C^{\times}-graded, or equivalently, C/Z-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a Vertex Operator algebra. We also introduce a notion of strongly C-graded generalized g-twisted V-module if V admits an additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a Vertex Operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let u be an element of V of weight 1 such that L(1)u=0. Then the exponential of 2\pi \sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this case, a strongly C-graded generalized g_{u}-twisted V-module is constructed from a strongly C-graded generalized V-module with a compatible action of g_{u} by modifying the Vertex Operator map for the generalized V-module using the exponential of the negative-power part of the Vertex Operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening Operators on certain Vertex Operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group C/Z or C and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted Vertex Operators in general involve the logarithm of the formal variable.
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two dimensional conformal geometry and Vertex Operator algebras
1997Co-Authors: Yi-zhi HuangAbstract:The focus of this volume is to formulate and prove one main theorem, the equivalance between the algebraic and geometric formulations of the notion of Vertex Operator algebra. The author introduces a geomatric notion of Vertex Operator algebra in terms of complex powers of the determinant line bundles over certain moduli spaces (parameter spaces) of spheres (genus-zero Riemann surfaces) with punctures and local analytic co-ordinates, and seeks to prove that this notion is precisely equivalent to the algebraic notion of Vertex Operator algebra. In particular, a detailed algebraic and analytic study of the sewing operation in the moduli space is presented.
Xingjun Lin - One of the best experts on this subject based on the ideXlab platform.
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on the unitary structures of Vertex Operator superalgebras
Journal of Algebra, 2017Co-Authors: Xingjun LinAbstract:Abstract In this paper, the notion of unitary Vertex Operator superalgebra is introduced. It is proved that Vertex Operator superalgebras associated to unitary highest weight representations for the Neveu–Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary Vertex Operator superalgebras. The unitary structures are then used to study the structures of Vertex Operator superalgebras, it is proved that any unitary Vertex Operator superalgebra is a direct sum of strong CFT type unitary simple Vertex Operator superalgebras. The classification of unitary Vertex Operator superalgebras generated by subspaces with conformal weights less than or equal to 1 is also considered.
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modularity of n point functions and theta functions for Vertex Operator superalgebras
Journal of Algebra, 2015Co-Authors: Chongying Dong, Xingjun LinAbstract:Abstract The modularity of the n -point trace functions and theta functions associated with Vertex Operator superalgebras are established.
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on the unitary structures of Vertex Operator superalgebras
arXiv: Quantum Algebra, 2015Co-Authors: Xingjun LinAbstract:In this paper, the notion of unitary Vertex Operator superalgebra is introduced. It is proved that the Vertex Operator superalgebras associated to the unitary highest weight representations for the Neveu-Schwarz Lie superalgebra, Heisenberg superalgebras and to positive definite integral lattices are unitary Vertex Operator superalgebras. The unitary structures are then used to study the structures of Vertex Operator superalgebras, it is proved that any unitary Vertex Operator superalgebra is a direct sum of strong CFT type unitary simple Vertex Operator superalgebras. The classification of unitary Vertex Operator superalgebras generated by the subspaces with conformal weights less than or equal to $1$ is also considered.
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unitary Vertex Operator algebras
Journal of Algebra, 2014Co-Authors: Chongying Dong, Xingjun LinAbstract:Abstract Unitary Vertex Operator algebras are introduced and studied. It is proved that most well-known rational Vertex Operator algebras are unitary. The classification of unitary Vertex Operator algebras with central charge c ⩽ 1 is also discussed.
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unitary Vertex Operator algebras
arXiv: Quantum Algebra, 2013Co-Authors: Chongying Dong, Xingjun LinAbstract:Unitary Vertex Operator algebras are introduced and studied. It is proved that most well-known rational Vertex Operator algebras are unitary. The classification of unitary Vertex Operator algebras with central charge c less than or equal to 1 is also discussed.
