Affine Kac-Moody Algebras Research Articles

Galilean free Lie algebras

We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that couple to them.

Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac–Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.

Higher-order Galilean contractions

A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinite-dimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, and W3 algebras.

Q-Virasoro algebra and affine Kac-Moody Lie algebras

In this paper, we introduce an infinite-dimensional Lie algebra DS for any abelian group S. If S is the additive group of integers, DS reduces to the q-Virasoro algebra Dq introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra gS with S as an automorphism group and we prove that DS is isomorphic to the S-covariant algebra of the affine Lie algebra gSˆ. We then relate restricted DS-modules of level ℓ∈C to equivariant quasi modules for the vertex algebra VgSˆ(ℓ,0) associated to gSˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q-Virasoro algebra Dq and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2l+1, DS is isomorphic to the affine Kac-Moody algebra of type Bl(1).

Representations of [formula omitted]-orbifold of the parafermion vertex operator algebra K(sl2,k)

In this paper, the irreducible modules for the Z2-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra A1(1) of level k are classified and constructed.

Geometric realizations of affine Kac–Moody algebras

The goal of the present paper is to obtain new free field realizations of affine Kac–Moody algebras motivated by geometric representation theory for generalized flag manifolds of finite-dimensional semisimple Lie groups. We provide an explicit construction of a large class of irreducible modules associated with certain parabolic subalgebras covering all known special cases.

MAXIMAL CLOSED SUBROOT SYSTEMS OF REAL AFFINE ROOT SYSTEMS

We completely classify and give explicit descriptions of all maximal closed subroot systems of real affine root systems. As an application, we describe a procedure to get the classification of all regular subalgebras of affine Kac-Moody algebras in terms of their root systems.

Integrable representations for toroidal extended affine Lie algebras

Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g˜(μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k-tuple λ=(λ1,⋯,λk) of weights of g, and k-tuple a=(a1,⋯,ak) of distinct non-zero complex numbers, we construct a class of modules V˜(λ,a) for the extended affine Lie algebra g˜(μ). We prove that the g˜(μ)-module V˜(λ,a) is completely reducible. We also prove that the g˜(μ)-module V˜(λ,a) is integrable when all weights λi in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g˜(μ).

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n1 and n2 from our list, that are positive parts of the affine Kac–Moody algebras A1(1) and A2(2), respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

Coproduct for Yangians of affine Kac–Moody algebras

Given an affine Kac–Moody algebra, we explain how to construct a coproduct on its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac–Moody algebra is, more generally, symmetrizable.

Tensor ideals, Deligne categories and invariant theory

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne’s universal categories $${\underline{\mathrm{Rep}}} O_\delta ,{\underline{\mathrm{Rep}}} GL_\delta $$ and $${\underline{\mathrm{Rep}}} P$$ . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P. We also find new short proofs for the classification of tensor ideals in $${\underline{\mathrm{Rep}}} S_t$$ and in the category of tilting modules for $${\text {SL}}_2(\Bbbk )$$ with $$\mathrm{char}(\Bbbk )>0$$ and for $$U_q(\mathfrak {sl}_2)$$ with q a root of unity. In general, for a simple Lie algebra $$\mathfrak {g}$$ of type ADE, we show that the lattice of such tensor ideals for $$U_q(\mathfrak {g})$$ corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac–Moody algebra.

Universal features of BPS strings in six-dimensional SCFTs

In theories with extended supersymmetry the protected observables of UV superconformal fixed points are found in a number of contexts to be encoded in the BPS solitons along an IR Coulomb-like phase. For six-dimensional SCFTs such a role is played by the BPS strings on the tensorial Coulomb branch. In this paper we develop a uniform description of the worldsheet theories of a BPS string for rank-one 6d SCFTs. These strings are the basic constituents of the BPS string spectrum of arbitrary rank six-dimensional models, which they generate by forming bound states. Motivated by geometric engineering in F-theory, we describe the worldsheet theories of the BPS strings in terms of topologically twisted 4d mathcal{N}=2 theories in the presence of 1/2-BPS 2d (0, 4) defects. As the superconformal point of a 6d theory with gauge group G is approached, the resulting worldsheet theory flows to an mathcal{N}=left(0, 4right) NLSM with target the moduli space of one G instanton, together with a nontrivial left moving bundle characterized by the matter content of the six-dimensional model. We compute the anomaly polynomial and central charges of the NLSM, and argue that the 6d flavor symmetry F is realized as a current algebra on the string, whose level we compute. We find evidence that for generic theories the G dependence is captured at the level of the elliptic genus by characters of an affine Kac-Moody algebra at negative level, which we interpret as a subsector of the chiral algebra of the BPS string worldsheet theory. We also find evidence for a spectral flow relating the R-R and NS-R elliptic genera. These properties of the string CFTs lead to constraints on their spectra, which in combination with modularity allow us to determine the elliptic genera of a vast number of string CFTs, leading also to novel results for 6d and 5d instanton partition functions.

Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in [FG].

On Integrable Field Theories as Dihedral Affine Gaudin Models

Abstract We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac–Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $\sigma $-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of its complexification. Most of the multi-parameter integrable deformations of these $\sigma $-models recently constructed in the literature provide further examples. The common feature shared by all these integrable field theories, which makes it possible to reformulate them as classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal formulation as another example of this construction. We propose that the interpretation of a given classical non-ultralocal integrable field theory as a classical dihedral affine Gaudin model provides a natural setting within which to address its quantisation. At the same time, it may also furnish a general framework for understanding the massive ordinary differential equations (ODE)/integrals of motion (IM) correspondence since the known examples of integrable field theories for which such a correspondence has been formulated can all be viewed as dihedral affine Gaudin models.

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories.

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

Representations of E7, equivalent combinatorics and algebraic varieties

In our earlier work on a new functor from [Formula: see text]-Mod to [Formula: see text]-Mod, we found a one-parameter ([Formula: see text]) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type [Formula: see text] in [Formula: see text] variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector [Formula: see text], we prove that the space forms an irreducible [Formula: see text]-module for any [Formula: see text] if [Formula: see text] is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of [Formula: see text] over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Darboux transformations for the Lax pairs related to two dimensional affine Toda equations

Apart from the simplest algebra $A_n^{(1)}$, for any infinite series of affine Kac-Moody algebra, the corresponding two dimensional Toda equations have a unitary symmetry, a cyclic symmetry and a reality symmetry. In this paper, we construct the Darboux transformations when the order of the cyclic symmetry is odd or both the order of the cyclic symmetry and the order of the matrices are even.

Positive energy representations of double extensions of Hilbert loop algebras

A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.

Local charges in involution and hierarchies in integrable sigma-models

Integrable σ-models, such as the principal chiral model, {mathbb{Z}}_T -coset models for Tin {mathbb{Z}}_{ge 2} and their various integrable deformations, are examples of non-ultralocal integrable field theories described by r/s-systems with twist function. In this general setting, and when the Lie algebra mathfrak{g} underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain ‘regular’ zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra widehat{mathfrak{g}} associated with mathfrak{g} . The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.

Macdonald index and chiral algebra

For any 4d mathcal{N} = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A1, A2n) and (A1, D2n+1) where the chiral algebras are given by Virasoro and widehat{mathfrak{su}}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.