VOAs and Rank-Two Instanton SCFTs

Null Field Construction in Conformal and Superconformal Algebras

We find that multiple vertex algebras can arise from a single 4d \mathcal{N}=2 𝒩 = 2 superconformal field theory (SCFT). The connection is given by the BPS monodromy operator M M , which is a wall-crossing invariant quantity that captures the BPS spectrum on the Coulomb branch. For a class of low-rank Argyres-Douglas theories, we find that the trace of the multiple powers of the monodromy operator Tr M^N M N yield modular functions that can be identified with the vacuum characters of certain vertex algebra for each N N . In particular, we realize unitary VOAs of the Deligne-Cvitanović exceptional series type (A_2)_1 ( A 2 ) 1 , (G_2)_1 ( G 2 ) 1 , (D_4)_1 ( D 4 ) 1 , (F_4)_1 ( F 4 ) 1 , (E_6)_1 ( E 6 ) 1 from Argyres-Douglas theories. We also find the modular invariant characters of the ‘intermediate vertex algebras’ (E_{7\frac{1}{2}})_1 ( E 7 1 2 ) 1 and (X_1)_1 ( X 1 ) 1 . Our analysis allows us to construct 3d \mathcal{N}=2 𝒩 = 2 gauge theories that flow to \mathcal{N}=4 𝒩 = 4 SCFTs in the IR, whose specialized half-index can be identified with these modular invariant characters.

General solutions’ laws of linear partial differential equations II

The Lie algebra comes from P. Deligne’s work on the exceptional series of Lie groups. Using the triality algebra, J. Landsberg and L. Manivel construct this Lie algebra in 2006. In this paper, we study the structure of following B. Gross and N. Wallach’s work on the highest root and Heisenberg parabolic subalgebra. The process of removing the lowest root from the extended Dynkin diagram of will contribute to the components of . We also study the branching rule of to and . We use the computer algebra system SageMath to carry out the branching rule calculations. Then we calculate the Dynkin indices for . We find that the number 24 behaves like the ‘dual Coxeter number’ of .

Oscillation theorems for a class of linear fourth order differential equations

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.

In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x(4)(t)+p(t)x(τ(t))=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x^{(4)}(t) + p(t)x\\bigl(\ au (t)\\bigr) = 0 $$\\end{document} by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

The category of weight modules of the simple affine vertex algebra of at an admissible level is neither finite nor semisimple and modules are usually not lower‐bounded and have infinite‐dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with , the simple principal ‐algebra of at level (the super conformal algebra) where the levels are related via . Every weight module of is lower‐bounded and has finite‐dimensional conformal weight spaces. The main technical result is that every weight module of is ‐cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on for any admissible level . As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that is a ribbon category if and only if is, in particular, it follows that for admissible levels and and , the category is a ribbon category.

A large class of supersymmetric quantum field theories, including all theories with $\mathcal{N} = 2$ supersymmetry in three dimensions and theories with $\mathcal{N} = 2$ supersymmetry in four dimensions, possess topological-holomorphic sectors. We formulate Poisson vertex algebras in such topological-holomorphic sectors and discuss some examples. For a four-dimensional $\mathcal{N} = 2$ superconformal field theory, the associated Poisson vertex algebra is the classical limit of a vertex algebra generated by a subset of local operators of the theory.

We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on $$ \mathbb{R}\times {S}^1 $$ , semi-classical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on $$ {\mathbb{R}}^2 $$ , which we call 2d-saddles. On $$ \mathbb{R}\times {S}^1 $$ , the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the o(N ) algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in O(N ) models in compactified space are in one-to-one correspondence with monopole-instanton saddles in SO(N ) gauge theory on $$ {\mathbb{R}}^3\times {S}^1 $$ . The Grassmannian sigma models Gr(N, M ) have 2d instantons, which fractionalize into N kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes, providing a microscopic argument for Haldane’s conjecture.

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S ~ TrWαWα, with the relation Sh = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.

With the aim of extending the gauge theory -- matrix model connection to more general matter representations, we prove that for various two-index tensors of the classical gauge groups, the perturbative contributions to the glueball superpotential reduce to matrix integrals. Contributing diagrams consist of certain combinations of spheres, disks, and projective planes, which we evaluate to four and five loop order. In the case of $Sp(N)$ with antisymmetric matter, independent results are obtained by computing the nonperturbative superpotential for $N=4,6$ and 8. Comparison with the Dijkgraaf-Vafa approach reveals agreement up to $N/2$ loops in matrix model perturbation theory, with disagreement setting in at $h=N/2+1$ loops, $h$ being the dual Coxeter number. At this order, the glueball superfield $S$ begins to obey nontrivial relations due to its underlying structure as a product of fermionic superfields. We therefore find a relatively simple example of an ${\cal N}=1$ gauge theory admitting a large $N$ expansion, whose dynamically generated superpotential differs from the one obtained in the matrix model approach.

Bosonic construction of an N = 2 extended superconformal theory in two dimensions

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.