Vertex Operators Research Articles

Bosonic Ghostbusting: The Bosonic Ghost Vertex Algebra Admits a Logarithmic Module Category with Rigid Fusion

The rank 1 bosonic ghost vertex algebra, also known as the beta gamma ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor C_2-cofinite. We identify a module category, denoted category mathscr {F}, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category mathscr {F} has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as C_2-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors’ knowledge this is the first example of a proof of rigidity for a logarithmic non-C_2-cofinite vertex algebra.

On irreducible characters of the Iwahori-Hecke algebra in type A

In this paper, we use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type A. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. We also formulate a determinant type Murnaghan-Nakayama formula and give another proof of the combinatorial Murnaghan-Nakayama rule. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Leduc-Ram formula.

Scattering amplitudes of Kaluza-Klein strings and extended massive double-copy

We study the scattering amplitudes of massive Kaluza-Klein (KK) states of open and closed bosonic strings under toroidal compactification. We analyze the structure of vertex operators for the KK strings and derive an extended massive KLT-like relation which connects the N-point KK closed-string amplitude to the products of two KK open-string amplitudes at tree level. Taking the low energy field-theory limit of vanishing Regge slope, we derive double-copy construction formula of the N-point massive KK graviton amplitude from the sum of proper products of the corresponding KK gauge boson amplitudes. Then, using the string-based massive double-copy formula, we derive the exact tree-level four-point KK gauge boson amplitudes and KK graviton amplitudes, which fully agree with those given by the KK field-theory calculations. With these, we give an explicit prescription on constructing the exact four-point KK graviton amplitudes from the sum of proper products of the corresponding color-ordered KK gauge boson amplitudes. We further analyze the string-based double-copy construction of five-point and six-point scattering amplitudes of massive KK gauge bosons and KK gravitons.

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

AbstractWe continue our study of effective field theory via homotopy transfer of ‐algebras, and apply it to tree‐level non‐Wilsonian effective actions of the kind discussed by Sen in which the modes integrated out are comparable in mass to the modes that are kept. We focus on the construction of effective actions for string states at fixed levels and in particular on the construction of weakly constrained double field theory. With these examples in mind, we discuss closed string theory on toroidal backgrounds and resolve some subtle issues involving vertex operators, including the proper form of cocycle factors and of the reflector state. This resolves outstanding issues concerning the construction of covariant closed string field theory on toroidal backgrounds. The weakly constrained double field theory is formally obtained from closed string field theory on a toroidal background by integrating out all but the ‘doubly massless’ states and homotopy transfer then gives a prescription for determining the theory's vertices and symmetries. We also discuss consistent truncation in the context of homotopy transfer.

Euclidean quantum field formulation of p-adic open string amplitudes

We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.

On the vertex operator representation of Lie algebras of matrices

The polynomial ring Br:=Q[e1,…,er] in r indeterminates is a representation of the Lie algebra of all the endomorphism of Q[X] vanishing at Xj for all but finitely many j. We determine the structural formal power series of the Br-representation of gl∞(Q). This is a formal power series in r+2 indeterminates encoding the images of all the basis elements of Br under the action of the generating function of elementary endomorphisms of Q[X]. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes the generating functions for the images of specified basis elements. For sake of completeness, in the last section we construct the structural formal power series of the B=B∞-representation of gl∞(Q). It consists in the evaluation of a bosonic vertex operator against the generating function of the standard Schur basis of B. It provides an alternative description of the bosonic representation of gl∞, due to Date, Jimbo, Kashiwara and Miwa, which does not explicitly involve exponentials of differential operators.

Local Energy Bounds and Strong Locality in Chiral CFT

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.

Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠Crev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X′⊠X is a commutative algebra, where X′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi⁎.These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.

The Conway-Miyamoto correspondences for the Fischer 3-transposition groups

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups F i 23 \mathrm {Fi}_{23} and F i 22 \mathrm {Fi}_{22} and c = 25 / 28 c=25/28 and c = 11 / 12 c=11/12 Virasoro vectors of subalgebras of the moonshine vertex operator algebra.

QFT entanglement entropy, 2D fermion and gauge fields

Entanglement and the Rényi entropies for Dirac fermions on 2 dimensional torus in the presence of chemical potential, current source, and topological Wilson loop are unified in a single framework by exhausting all the ingredients of the electromagnetic vertex operators of Zn orbifold conformal field theory. We employ different normalizations for different topological sectors to organize various topological phase transitions in the context of entanglement entropy. Pictorial representations for the topological transitions are given for the n=2 Rényi entropy.Our analytic computations reveal numerous novelties and provide resolutions for existing issues. We have settled to provide non-singular entanglement entropies that are also continuous across the topological sectors. Surprisingly, in infinite space, these entropies become exact and depend only on the Wilson loop. On a circle, we resolve to find the entropies subtly depend on the chemical potential at zero temperature, which is useful for probing the ground state energy levels of quantum systems.

Urod algebras and Translation of W-algebras

Abstract In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak {g}$ , we have the vertex algebra isomorphism $H_{DS,f}^{0}(V\otimes L)\cong H_{DS,f}^{0}(V)\otimes L$ , where in the left-hand-side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } L$ . The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima–Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb {CP}^{2}$ of an arbitrary rank.

Smash product construction of modular lattice vertex algebras

<abstract><p>Motivated by a work of Li, we study nonlocal vertex algebras and their smash products over fields of positive characteristic. Through smash products, modular vertex algebras associated with positive definite even lattices are reconstructed. This gives a different construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess in lattice vertex operator algebras over a field of characteristic zero.</p></abstract>

Argyres–Douglas theories, modularity of minimal models and refined Chern–Simons

The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.

Celestial OPEs and w1+\u221e algebra from worldsheet in string theory

Celestial operator product expansions (OPEs) arise from the collinear limit of scattering amplitudes and play a vital role in celestial holography. In this paper, we derive the celestial OPEs of massless fields in string theory from the worldsheet. By studying the worldsheet OPEs of vertex operators in worldsheet CFT and further examining their behaviors in the collinear limit, we find that new vertex operators for the massless fields in string theory are generated and become dominant in the collinear limit. Mellin transforming to the conformal basis yields exactly the celestial OPEs in celestial CFT. We also derive the celestial OPEs from the collinear factorization of string amplitudes and the results derived in these two different methods are in perfect agreement with each other. Our final formulae of celestial OPEs are applicable to general dimensions, corresponding to Einstein-Yang-Mills theory supplemented by some possible higher derivative interactions. Specializing to 4D, we reproduce all the celestial OPEs for gluon and graviton in the literature. We consider various string theories, including the open and closed bosonic string, as well as the closed superstring theory with mathcal{N} = 1 and mathcal{N} = 2 worldsheet supersymmetry. In the case of mathcal{N} = 2 string, we also derive all the overline{mathrm{SL}left(2,mathbb{R}right)} descendant contributions in the celestial OPE; the soft sector of such OPE just yields the w1+∞ algebra after rewriting in terms of chiral modes. Our stringy derivation of celestial OPEs thus initiates the first step towards the microscopic realization of celestial CFT dual to string theory in flat spacetime.

ПОВЫШЕНИЕ ЭФФЕКТИВНОСТИ ПЕРВИЧНОЙ ОБРАБОТКИ МОЛОКА НА ФЕРМАХ

Technological processes of primary processing and storage of livestock products require large volumes of low-potential thermal energy. One of the effective ways to preserve the high quality of milk, reduce the energy consumption of primary processing and storage of livestock products is the use of alternative energy sources. The author has analyzed systems for primary processing and storage of milk based on the use of natural cold, vacuum and thermoelectric modules. The article shows the advantages of such systems as compared with traditional ones and gives the classification of equipment using natural cold. It is noted that when using natural cold, the cost of electricity for cooling milk is reduced to 10…12 kWh/t, and when using a hybrid vacuum-evaporation plant – up to 15…18 kWh/t. The article presents a technological scheme of a thermoelectric installation for milk cooling and pasteurization and recuperative heating of washing water on farms with milking robots. Based on the performed analysis the author makes recommendations on the use of alternative energy sources for milk pre-cooling on farms with milking robots. It is established that the use of alternative energy sources provides for environmentally safe, energy-saving and reliable primary processing and storage of livestock products in the places of its production, and contributes to a significant reduction in losses and preservation of high quality. Studies have proved the effectiveness of using combined cooling and storage systems for livestock products using natural cold in combination with thermoelectric, vacuum, and ground cooling.

Categories of weight modules for unrolled restricted quantum groups at roots of unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules. Bibliography: 44 titles.

Categories of weight modules for unrolled restricted quantum groups at roots of unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules.

On the Heisenberg algebra associated with the rational R-matrix

We associate a deformation of the Heisenberg algebra with the suitably normalized Yang R-matrix, and we investigate its properties. Moreover, we construct new examples of quantum vertex algebras, which possess the same representation theory as the aforementioned deformed Heisenberg algebra.

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

Critical non-Abelian vortices and holography for little string theory

It has been shown that non-Abelian vortex string supported in four dimensional (4D) ${\mathcal N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge group and $N_f = 4$ quark flavors becomes a critical superstring. This string propagates in the ten dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau non-compact threefold, namely, the conifold. The spectrum of closed string states of the associated string theory was obtained using the equivalence between the critical string on the conifold and the non-critical string on the semi-infinite cigar described by SL($2, \mathbb{R}$)/U(1) Wess-Zumino-Novikov-Witten model. This spectrum was identified with the spectrum of hadrons in 4D ${\mathcal N}=2$ SQCD. In order to describe effective interactions of these 4D hadrons in this paper we study correlation functions of normalizable vertex operators localized near the tip of the SL($2, \mathbb{R}$)/U(1) cigar. We also compare our solitonic string approach to the gauge-string duality to the AdS/CFT-type holography for little string theories (LSTs). The latter relates off mass-shell correlation functions on the field theory side to correlation functions of non-normalizable vertex operators on the cigar. We show that in most channels holographic approach works in our theory because normalizable and non-normalizable vertex operators with the same conformal dimension are related due to the reflection from the tip of the cigar. However, we find that holography does not work for lightest hadrons with given baryonic charge.