Moody Algebras Research Articles

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.

On cohomological Hall algebras of quivers: Generators

Abstract We study the cohomological Hall algebra Y ♭ {\operatorname{Y}\nolimits^{\flat}} of a Lagrangian substack Λ ♭ {\Lambda^{\flat}} of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y ♭ {\operatorname{Y}\nolimits^{\flat}} is pure and we compute its Poincaré polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y ♭ {\operatorname{Y}\nolimits^{\flat}} is equal, after a suitable extension of scalars, to the Yangian 𝕐 {\mathbb{Y}} introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov’s conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac–Moody algebras.

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.

Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras

Abstract We prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

Root multiplicities for Borcherds algebras and graph coloring

We establish a connection between root multiplicities for Borcherds–Kac–Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces corresponding to these roots and determine the Hilbert series in the case when all simple roots are imaginary. As an application we give a Lie theoretic proof of Stanley's reciprocity theorem of chromatic polynomials.

Lakshmibai–Seshadri paths for hyperbolic Kac–Moody algebras of rank 2

ABSTRACTLet 𝔤 be a hyperbolic Kac–Moody algebra of rank 2, and set , where are the fundamental weights for 𝔤; note that λ is neither dominant nor antidominant. Let 𝔹(λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. We prove that (the crystal graph of) 𝔹(λ) is connected. Furthermore, we give an explicit description of Lakshmibai–Seshadri paths of shape λ.

Monoidal categorification of cluster algebras

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ringAq(n(w))A_q(\mathfrak {n}(w)), associated with a symmetric Kac–Moody algebra and its Weyl group elementww, admits a monoidal categorification via the representations of symmetric Khovanov–Lauda–Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional gradedRR-modules to become a monoidal categorification, whereRRis a symmetric Khovanov–Lauda–Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed ofAq(n(w))A_q(\mathfrak {n}(w))which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power ofq1/2q^{1/2}. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac–Moody algebras

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant.

Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

BPS algebras, genus zero and the heterotic Monster

In this note, we expand on some technical issues raised in (Paquette et al 2016 Commun. Number Theory Phys. 10 433–526) by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0 + 1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras —some of the first and most prominent examples of Generalized Kac–Moody algebras—constructed by Borcherds and Carnahan. In particular, we clarify the structure of the module present in the second-quantized string theory. We also sketch a proof of our methods in the language of vertex operator algebras, for the interested mathematician.

Faces of highest weight modules and the universal Weyl polyhedron

Let V be a highest weight module over a Kac–Moody algebra g, and let convV denote the convex hull of its weights. We determine the combinatorial isomorphism type of convV, i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini and Marietti (2015) [7] for the adjoint representation, and by Khare (2016, 2017) [17,18] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous works of Satake (1960) [25], Borel and Tits (1965) [3], Vinberg (1990) [26], and Casselman (1997) [6].For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type.We further answer a question of Michel Brion, by showing that the localization of convV along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.

On Brane Solutions with Intersection Rules Related to Lie Algebras

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M0 of dimension d0 ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented.

Nonsupersymmetric magic theories and Ehlers truncations

We consider the nonsupersymmetric “magic” theories based on the split quaternion and the split complex division algebras. We show that these theories arise as “Ehlers” [Formula: see text] and [Formula: see text] truncations of the maximal supergravity theory, exploiting techniques related to the very-extended Kac–Moody algebras. We also generalize the procedure to other [Formula: see text] truncations, resulting in additional classes of nonsupersymmetric theories, as well as to truncations of nonmaximal theories. Finally, we discuss duality orbits of extremal black hole solutions in some of these nonsupersymmetric theories.

Weyl groups of some hyperbolic Kac–Moody algebras

We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac–Moody algebras Tn++, obtained by a process of double extension from a Cartan matrix of finite type Tn, whose corresponding generalized Cartan matrices are symmetric.

Structure of parabolically induced modules for affine Kac–Moody algebras

The main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over Affine Lie algebras for any non standard parabolic subalgebra. It generalizes all previously known partial results and provides a construction of many new irreducible modules.

A new involution for quantum loop algebras

In this article, we introduce a completion Uˆv+(Lg) of the positive half of the quantum affinization Uv+(Lg) of a symmetrizable Kac–Moody algebra g. On Uˆv+(L(g)), we define a new “bar-involution” and construct the analogue Kashiwara's operators. We conjecture that the resulting pair (Lˆ,Bˆ) is a crystal basis which provides the existence of the “canonical basis” on the (completion of the) of the positive half of the quantum affinization.

Current algebra formulation of M-theory based on E11 Kac–Moody algebra

Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac–Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the [Formula: see text] Kac–Moody algebra that was shown recently[Formula: see text] to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside [Formula: see text], so that, by constructing the current algebra of [Formula: see text], I obtain both internal gauge theory and gravity theory. The energy–momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein–Hilbert action.

Implications of heavy neutralino dark matter: The underlying structure in the hidden sector

The possibility of heavy neutralino dark matter (DM) in the gravity-mediation mechanism is explored. The appearance of the heavy lightest supersymmetric particle is seemingly suggested by Large Hadron Collider runs, which have not provided evidence of superparticles around the TeV region. On the basis of the so-called WIMPZILLA scenario, it is understood that the nonthermally produced DM has the larger mass than the reheating temperature. Hence, the expected DM mass should be more than 109 GeV so that thermal leptogenesis successfully occurs. In this paper, we first examine the generation of the Higgsino mass parameter [Formula: see text] in the context of gravity mediation, postulating that the resolution of the strong CP problem should be the criterion for arriving at a valid hypothesis for heavy neutralino DM. Accordingly, we address how the Peccei–Quinn (PQ) symmetry could influence dynamical supersymmetry breaking (DSB) models. It is found that as long as [Formula: see text] (the SUSY-breaking scale) approximately coincides with [Formula: see text] (the PQ-breaking scale), no DSB models can naturally account for the existence of the heavy neutralino DM, based upon the supersymmetric Dine–Fischler–Srednicki–Zhitinitski (DFSZ)-like mechanism. Thus, we attempt to construct a new model wherein hierarchical SUSY breakings occur. For this purpose, we propose gauge coupling unification in the hidden-sector dynamics at some high-energy scale, and we show that such a class of models can achieve [Formula: see text] through renormalization flow. As a consequence, the nonthermal neutralino, practically the wino-like one in our model, is shown to be a rather natural and viable DM candidate. Moreover, we argue that on the basis of Kac–Moody algebra, multiple breakdowns of supersymmetry may entail unified gauge dynamics. We also present a possible unified model. Finally, the heavy wino-like neutralino may be a DM candidate that will favor future direct DM detection experiments, mainly because its scattering on nuclei well conserves isospin symmetry.

A Preparation Theorem for the Kashiwara $$B(\infty )$$ B ( ∞ ) crystal

The Kashiwara \(B(\infty )\) crystal pertains to a Verma module for a Kac–Moody algebra. It has a rich and important combinatorial structure. A Preparation Theorem is established as a preliminary step in an explicit description \(B(\infty )\). It is noted that \(B(\infty )\) has in general infinitely many presentations as subsets of countably many copies of the natural numbers each given by successive reduced decompositions of Weyl group elements. In each presentation there is an action of Kashiwara operators determined by Kashiwara functions. These functions are linear in the entries. Thus a natural question is to show that in each presentation the subset \(B(\infty )\) is polyhedral and to describe explicitly the linear inequalities involved. Here an approach to this question is initiated based on constructing dual Kashiwara functions. Our basic hypothesis is that dual Kashiwara functions exist and can be expressed as differences of successive Kashiwara functions with non-negative integer coefficients. This last requirement is called the positivity condition. Here one starts from an explicit linear function called the “initial driving function”. Then in terms of the reduced decomposition one seeks an algorithm to provide further dual Kashiwara functions which in particular will provide “driving functions” for the next induction step. The Preparation Theorem resolves a main difficulty in this step-wise construction, namely that the resulting functions must exhibit an invariance with respect to the Kashiwara operators. This is expressed through a sum, or simply S, condition. It depends very subtly on inequalities between the coefficients occurring in the driving function obtained from the previous step. The proof of the Preparation Theorem depends on the construction of S-graphs constructed by a process of “binary fusion” based on the structure of the driving function.

A remark on boundary level admissible representations

We point out that it is immediate by our character formula that in the case of a boundary level the characters of admissible representations of affine Kac–Moody algebras and the corresponding W-algebras decompose in products in terms of the Jacobi form ϑ11(τ,z).