Vertex Operators Research Articles

Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.

Vertex algebras and TKK algebras

In this paper, we associate the TKK algebra Gˆ(J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ, the category of restricted Gˆ(J)-modules of level ℓ is canonically isomorphic to the category of σ-twisted VCgˆ(ℓ,0)-modules, where VCgˆ(ℓ,0) is a vertex algebra arising from the toroidal Lie algebra of type C2 and σ is an isomorphism of VCgˆ(ℓ,0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ, the integrable restricted Gˆ(J)-modules of level ℓ are exactly the σ-twisted modules for the quotient vertex algebra LCgˆ(ℓ,0) of VCgˆ(ℓ,0). Finally, we classify the irreducible 12N-graded σ-twisted LCgˆ(ℓ,0)-modules.

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions “counting” finite-dimensional representations of A-type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are “classically free” as vertex algebras.We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L1(so5), L1(so8) and L1(g2). We derive a new character formula for the principal subspace of L1(so5), proving that it is classically free, and present evidence that the principal subspaces of L1(so8) and of L1(g2) are also classically free.

Tensor category [formula omitted] via minimal affine W-algebras at the non-admissible level [formula omitted

We prove that the Kazhdan-Lusztig category of slˆm at level k, KLk(slm), is a semi-simple, rigid braided tensor category for all even m≥4, and k=−m+12. Moreover, all modules in KLk(slm) are simple-currents and they appear in the decomposition of conformal embeddings glm↪slm+1 at level k=−m+12. For this we inductively identify minimal affine W-algebra Wk−1(slm+2,θ) as simple current extension of Lk(slm)⊗H⊗M, where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c=−2. The proof uses previously obtained results for the tensor categories of singlet algebra. We also classify all irreducible ordinary modules for Wk−1(slm+2,θ). The semi-simple part of the category of Wk−1(slm+2,θ)–modules comes from KLk−1(slm+2), using quantum Hamiltonian reduction, but this W-algebra also contains indecomposable ordinary modules.

Classification of irreducible admissible modules for 𝒮-local vertex operator algebra V #H

For an [Formula: see text]-local vertex operator algebra [Formula: see text], where [Formula: see text] is a simple vertex operator algebra, and [Formula: see text] is the group algebra of a finite subgroup of [Formula: see text], Frobenius reciprocity is investigated. We give an explicit construction and classification of admissible [Formula: see text]-modules in terms of admissible [Formula: see text]-modules. We also give a complete set of irreducible inequivalent admissible [Formula: see text]-modules.

Reconstruction of Vertex Algebras in Even Higher Dimensions

Reconstruction of Vertex Algebras in Even Higher Dimensions

Gravity from holomorphic discs and celestial Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} symmetries

The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the analogous framework for Yang–Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.

Vertex algebra of extended operators in 4d N=2 superconformal field theories. Part I

We construct a class of extended operators in the cohomology of a pair of twisted Schur supercharges of 4d mathcal{N} =2 SCFTs. The extended operators are constructed from the local operators in this cohomology — the Schur operators — by a version of topological descent. They are line, surface, and domain wall world volume integrals of certain super descendants of Schur operators. Their world volumes extend in directions transverse to a spatial plane in Minkowski space-time. As operators in the cohomology of these twisted Schur supercharges, their correlators are (locally) meromorphic functions only of the positions where they intersect this plane. This implies the extended operators enlarge the vertex operator algebra of the Schur operators. We illustrate this enlarged vertex algebra by computing some extended-operator product expansions within a subalgebra of it for the free hypermultiplet SCFT.

PBW bases of irreducible Ising modules

To every h+N-graded module M over an N-graded conformal vertex algebra V, we associate an increasing filtration (GpM)p∈Z, which is compatible with the filtrations introduced by Haisheng Li. The associated graded vector space grG(M) is naturally a module over the vertex Poisson algebra grG(V). We study grG(M) for the three irreducible modules over the Ising model Vir3,4, namely Vir3,4=L(1/2,0), L(1/2,1/2) and L(1/2,1/16). We obtain an explicit PBW basis of each of these modules and a formula for their refined characters, which are related to Nahm sums for the matrix (8332).

Permutation orbifolds of vertex operator superalgebras and associative algebras

Permutation orbifolds of vertex operator superalgebras and associative algebras

A note on Cauchy's formula

We use the correlation functions of vertex operators to give a proof of Cauchy's formula∏i=1K∏j=1N(1−xiyj)=∑μ⊆[K×N](−1)|μ|sμ{x}sμ′{y}. As an application of the interpretation, we obtain an expansion of ∏i=1∞(1−qi)i−1 in terms of half plane partitions.

Semi-infinite construction for the double Yangian of type [formula omitted

We consider certain infinite dimensional modules of level 1 for the double Yangian DY(gl2) which are based on the Iohara–Kohno realization. We show that they possess topological bases of Feigin–Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give some applications of these bases to the representation theory of the corresponding quantum affine vertex algebra.

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Associative algebras and the representation theory of grading-restricted vertex algebras

We introduce an associative algebra [Formula: see text] using infinite matrices with entries in a grading-restricted vertex algebra [Formula: see text] such that the associated graded space [Formula: see text] of a filtration of a lower-bounded generalized [Formula: see text]-module [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. For [Formula: see text], there is a subalgebra [Formula: see text] of [Formula: see text] such that the subspace [Formula: see text] of [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that [Formula: see text] are finite-dimensional when [Formula: see text] is of positive energy (CFT type) and [Formula: see text]-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. In the case that [Formula: see text] is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized [Formula: see text]-modules are less than or equal to [Formula: see text], we prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] of finite length is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively.

Emergent Strings at an Infinite Distance with Broken Supersymmetry

We investigate the infinite-distance properties of families of unstable flux vacua in string theory with broken supersymmetry. To this end, we employ a generalized notion of distance in the moduli space and we build a holographic description for the non-perturbative regime of the tunneling cascade in terms of a renormalization group flow. In one limit, we recover an exponentially-light tower of Kaluza-Klein states, while in the opposite limit, we find a tower of higher-spin excitations of D1-branes, realizing the emergent string proposal. In particular, the holographic description includes a free sector, whose emergent superconformal symmetry resonates with supersymmetric stability, the CFT distance conjecture and S-duality. We compute the anomalous dimensions of scalar vertex operators and single-trace higher-spin currents, finding an exponential suppression with the distance which is not generic from the renormalization group perspective, but appears specific to our settings.

Noncommutative Poisson vertex algebras and Courant–Dorfman algebras

We introduce the notion of double Courant–Dorfman algebra and prove that it satisfies the so-called Kontsevich–Rosenberg principle, that is, a double Courant–Dorfman algebra induces Roytenberg's Courant–Dorfman algebras on the affine schemes parametrizing finite-dimensional representations of a noncommutative algebra. The main example is given by the direct sum of double derivations and noncommutative differential 1-forms, possibly twisted by a closed Karoubi–de Rham 3-form. To show that this basic example satisfies the required axioms, we first prove a variant of the Cartan identity [LX,LY]=L[X,Y] for double derivations and Van den Bergh's double Schouten–Nijenhuis bracket. This new identity, together with noncommutative versions of the other Cartan identities already proved by Crawley-Boevey–Etingof–Ginzburg and Van den Bergh, establishes the differential calculus on noncommutative differential forms and double derivations and should be of independent interest. Motivated by applications in the theory of noncommutative Hamiltonian PDEs, we also prove a one-to-one correspondence between double Courant–Dorfman algebras and double Poisson vertex algebras, introduced by De Sole–Kac–Valeri, that are freely generated in degrees 0 and 1.

Cosets from equivariant 𝒲-algebras

The equivariant W \mathcal W -algebra of a simple Lie algebra g \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of g \mathfrak {g} . We construct a family of vertex algebras A [ g , κ , n ] A[\mathfrak {g}, \kappa , n] as subalgebras of the equivariant W \mathcal W -algebra of g \mathfrak {g} tensored with the integrable affine vertex algebra L n ( g ˇ ) L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra g ˇ \check {\mathfrak {g}} at level n ∈ Z > 0 n\in \mathbb {Z}_{>0} . They are conformal extensions of the tensor product of an affine vertex algebra and the principal W \mathcal {W} -algebra whose levels satisfy a specific relation. When g \mathfrak {g} is of type A D E ADE , we identify A [ g , κ , 1 ] A[\mathfrak {g}, \kappa , 1] with the affine vertex algebra V κ − 1 ( g ) ⊗ L 1 ( g ) V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}) , giving a new and efficient proof of the coset realization of the principal W \mathcal W -algebras of type A D E ADE .

Logarithmic Vertex Algebras and Non-local Poisson Vertex Algebras

Logarithmic Vertex Algebras and Non-local Poisson Vertex Algebras

Symmetric universal character, integrable hierarchy, and affine Yangian of [formula omitted

We define a symmetric universal character by using the affine Yangian of gl(1), which is related with a pair of 2D Young diagram which is considered as 3D Young diagrams with only one layer in z-axis direction. We also define the symmetric universal character hierarchy and prove that all symmetric universal characters are tau functions of this integrable hierarchy. Then, we construct the Boson-Fermion correspondence in the symmetric universal character hierarchy and symmetric universal character can be realized through them. Finally, the (m,n)-soliton solution of the symmetric universal character hierarchy is given by vertex operators.

Superspace expansion of the 11D linearized superfields in the pure spinor formalism, and the covariant vertex operator

11D pure spinors have been shown to successfully describe 11D supergravity in a manifestly super-Poincaré covariant manner. The feasibility of its actual usage for scattering amplitude computations requires an efficient manipulation of the superfields defining linearized 11D supergravity. In this paper, we directly address this problem by finding the superspace expansions of these superfields, at all orders in θ, from recursive relations their equations of motion obey in Harnad-Shnider-like gauges. After introducing the 11D analogue of the 10D ABC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{ABC} $$\\end{document} superparticle, we construct, for the first time, a fully covariant vertex operator for 11D supergravity by making use of the linearized 11D superfields. Notably, we show that this vertex reproduces the Green-Gutperle-Kwon 11D operators in light-cone gauge.