Vertex operator algebras and operads

Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal to $0$, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (two-dimensional) ``complex'' geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) ``real'' geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level.Comment: 16 pages. Only the definitions of "partial operad" and of "rescaling group" have been improve

The tensor structure on the representation category of the Wp\mathcal{W}_p triplet algebra

We study the braided monoidal structure that the fusion product induces on the abelian category $\mathcal{W}_p$-mod, the category of representations of the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of $\mathcal{W}_p$-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of $\mathcal{W}_p$-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective $\mathcal{W}_p$-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.Comment: 58 pages; edit: added references and revisions according to referee reports. Version to appear on J. Phys.

Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad).Comment: 42 page

Phase behavior of the Lattice Restricted Primitive Model with nearest-neighbor exclusion

The global phase behavior of the lattice restricted primitive model with nearest neighbor exclusion has been studied by grand canonical Monte Carlo simulations. The phase diagram is dominated by a fluid (or charge-disordered solid) to charge-ordered solid transition that terminates at the maximum density, $\rho^*_{max}=\sqrt2$ and reduced temperature $T^*\approx0.29$. At that point, there is a first-order phase transition between two phases of the same density, one charge-ordered and the other charge-disordered. The liquid-vapor transition for the model is metastable, lying entirely within the fluid-solid phase envelope.Comment: 6 pages, color. submitted to J. Chem. Phy

Progress in Time-Dependent Density-Functional Theory

The classic density-functional theory (DFT) formalism introduced by Hohenberg, Kohn, and Sham in the mid-1960s, is based upon the idea that the complicated N-electron wavefunction can be replaced with the mathematically simpler 1-electron charge density in electronic struc- ture calculations of the ground stationary state. As such, ordinary DFT is neither able to treat time-dependent (TD) problems nor describe excited electronic states. In 1984, Runge and Gross proved a theorem making TD-DFT formally exact. Information about electronic excited states may be obtained from this theory through the linear response (LR) theory formalism. Begin- ning in the mid-1990s, LR-TD-DFT became increasingly popular for calculating absorption and other spectra of medium- and large-sized molecules. Its ease of use and relatively good accuracy has now brought LR-TD-DFT to the forefront for this type of application. As the number and the diversity of applications of TD-DFT has grown, so too has grown our understanding of the strengths and weaknesses of the approximate functionals commonly used for TD-DFT. The objective of this article is to continue where a previous review of TD-DFT in this series [Annu. Rev. Phys. Chem. 55: 427 (2004)] left off and highlight some of the problems and solutions from the point of view of applied physical chemistry. Since doubly-excited states have a particularly important role to play in bond dissociation and formation in both thermal and photochemistry, particular emphasis will be placed upon the problem of going beyond or around the TD-DFT adiabatic approximation which limits TD-DFT calculations to nominally singly-excited states. Posted with permission from the Annual Review of Physical Chemistry, Volume 63 \c{opyright} 2012 by Annual Reviews, http://www.annualreviews.org

Meromorphic open-string vertex algebras

A notion of meromorphic open-string vertex algebra is introduced. A meromorphic open-string vertex algebra is an open-string vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions for vertex operators. The vertex operator map for a meromorphic open-string vertex algebra satisfies rationality and associativity but in general does not satisfy the Jacobi identity, commutativity, the commutator formula, the skew-symmetry or even the associator formula. Given a vector space \mathfrak{h}, we construct a meromorphic open-string vertex algebra structure on the tensor algebra of the negative part of the affinization of \mathfrak{h} such that the vertex algebra struture on the symmetric algebra of the negative part of the Heisenberg algebra associated to \mathfrak{h} is a quotient of this meromorphic open-string vertex algebra. We also introduce the notion of left module for a meromorphic open-string vertex algebra and construct left modules for the meromorphic open-string vertex algebra above.Comment: 43 pape

Singular vectors of the WA2WA_2 algebra

The null vectors of an arbitrary highest weight representation of the $WA_2$ algebra are constructed. Using an extension of the enveloping algebra by allowing complex powers of one of the generators, analysed by Kent for the Virasoro theory, we generate all the singular vectors indicated by the Kac determinant. We prove that the singular vectors with given weights are unique up to normalisation and consider the case when $W_0$ is not diagonalisable among the singular vectors.Comment: 10 pages, LaTeX with 3 figures in LaTe

Phase diagrams in the lattice RPM model: from order-disorder to gas-liquid phase transition

The phase behavior of the lattice restricted primitive model (RPM) for ionic systems with additional short-range nearest neighbor (nn) repulsive interactions has been studied by grand canonical Monte Carlo simulations. We obtain a rich phase behavior as the nn strength is varied. In particular, the phase diagram is very similar to the continuum RPM model for high nn strength. Specifically, we have found both gas-liquid phase separation, with associated Ising critical point, and first-order liquid-solid transition. We discuss how the line of continuous order-disorder transitions present for the low nn strength changes into the continuum-space behavior as one increases the nn strength and compare our findings with recent theoretical results by Ciach and Stell [Phys. Rev. Lett. {\bf 91}, 060601 (2003)].Comment: 7 pages, 10 figure

Difference L operators and a Casorati determinant solution to the T-system for twisted quantum affine algebras

We propose factorized difference operators L(u) associated with the twisted quantum affine algebras U_{q}(A^{(2)}_{2n}),U_{q}(A^{(2)}_{2n-1}), U_{q}(D^{(2)}_{n+1}),U_{q}(D^{(3)}_{4}). These operators are shown to be annihilated by a screening operator. Based on a basis of the solutions of the difference equation L(u)w(u)=0, we also construct a Casorati determinant solution to the T-system for U_{q}(A^{(2)}_{2n}),U_{q}(A^{(2)}_{2n-1}).Comment: 15 page

Structural and Linear Elastic Properties of DNA Hydrogels by Coarse-Grained Simulation

© 2019 American Chemical Society. We introduce a coarse-grained numerical model that represents a generic DNA hydrogel consisting of Y-shaped building blocks. Each building block comprises three double-stranded DNA arms with single-stranded DNA sticky ends, mimicked by chains of beads and patchy particles, respectively, to allow for an accurate representation of both the basic geometry of the building blocks and the interactions between complementary units. We demonstrate that our coarse-grained model reproduces the correct melting behavior between the complementary ends of the Y-shapes, and their self-assembly into a percolating network. Structural analysis of this network reveals three-dimensional features consistent with a uniform distribution of inter-building-block dihedral angles. When applying an oscillatory shear strain to the percolating system, we show that the system exhibits a linear elastic response when fully connected. We finally discuss to what extent the system's elastic modulus may be controlled by simple changes to the building block complementarity. Our model offers a computationally tractable approach to predicting the structural and mechanical properties of DNA hydrogels made of different types of building blocks