Kac-Moody Algebra Research Articles

Real-space entanglement spectra of parton states in fractional quantum Hall systems

Real-space entanglement spectra (RSES) capture characteristic features of the topological order encoded in the fractional quantum Hall (FQH) states. In this work, we numerically compute, using Monte Carlo methods, the RSES and the counting of edge excitations of non-Abelian FQH states constructed using the parton theory. Efficient numerical computation of RSES of parton states is possible, thanks to their product-of-Slater-determinant structure, allowing us to compute the spectra in systems of up to 80 particles. Specifically, we compute the RSES of the parton states $\phi_2^2$, $\phi_2^3$, and $\phi_3^2$, where $\phi_n$ is the wave function of $n$ filled Landau levels, in the ground state as well as in the presence of bulk quasihole states. We then explicitly demonstrate a one-to-one correspondence of RSES of the parton states with representations of the Kac-Moody algebras satisfied by their edge currents. We also show that for the lowest Landau level projected version of these parton states, the spectra match with that obtained from the edge current algebra. We also perform a computation of spectra of the overlap matrices corresponding to the edge excitations of the parton states with a constrained number of particles in the different parton Landau levels. Counting in these matches the individual branches present in RSES, providing insight about how different branches are formed.

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Folded quantum integrable models and deformed W-algebras

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra \({{\mathfrak {g}}}\), which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra \(U_q(\widehat{{{\mathfrak {g}}}'})\) of the simply laced Lie algebra \({{\mathfrak {g}}}'\) corresponding to \({{\mathfrak {g}}}\). Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of \({{\mathfrak {g}}}\), which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of \(U_q(\widehat{{{\mathfrak {g}}}'})\). We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of \(U_q({}^L{\widehat{{{\mathfrak {g}}}}})\), where \(^L{\widehat{{{\mathfrak {g}}}}}\) is the (twisted) affine Kac–Moody algebra Langlands dual to \({\widehat{{{\mathfrak {g}}}}}\). We discuss the analogous structures in the Gaudin model which appears in the limit \(q \rightarrow 1\). Finally, we describe a conjectural construction of the simple \({{\mathfrak {g}}}\)-crystals in terms of the folded q-characters.

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

Edge observables of the Maxwell-Chern-Simons theory

We analyze the Lagrangian and Hamiltonian formulations of the Maxwell-Chern-Simons theory defined on a manifold with boundary for two different sets of boundary equations derived from a variational principle. We pay special attention to the identification of the infinite chains of boundary constraints and their resolution. We identify edge observables and their algebra [which corresponds to the well-known $U(1)$ Kac-Moody algebra]. Without performing any gauge fixing, and using the Hodge-Morrey theorem, we solve the Hamilton equations whenever possible. In order to give explicit solutions, we consider the particular case in which the fields are defined on a $2$-disk. Finally, we study the Fock quantization of the system and discuss the quantum edge observables and states.

An overview of generalised Kac-Moody algebras on compact real manifolds

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expansion. The Peter–Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space L2(M) of square-integrable functions are constructed. It is shown that such bases are characterised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed.

New meromorphic CFTs from cosets

In recent years it has been understood that new rational CFTs can be discovered by applying the coset construction to meromorphic CFTs. Here we turn this approach around and show that the coset construction, together with the classification of meromorphic CFT with c ≤ 24, can be used to predict the existence of new meromorphic CFTs with c ≥ 32 whose Kac-Moody algebras are non-simply-laced and/or at levels greater than 1. This implies they are non-lattice theories. Using three-character coset relations, we propose 34 infinite series of meromorphic theories with arbitrarily large central charge, as well as 46 theories at c = 32 and c = 40.

Rigid Reflections of Rank 3 Coxeter Groups and Reduced Roots of Rank 2 Kac–Moody Algebras

In a recent paper by K.-H. Lee and K. Lee, rigid reflections are defined for any Coxeter group via non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, the rigid reflections are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, it was conjectured in the same paper that there is a natural bijection from the set of reduced positive roots of a symmetric rank 2 Kac–Moody algebra onto the set of rigid reflections of the corresponding rank 3 Coxeter group. In this paper, we prove the conjecture.

Virasoro constraints for Drinfeld–Sokolov hierarchies and equations of Painlevé type

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra with gradations s ⩽ 1 $\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld–Sokolov hierarchy of Witten–Kontsevich and of Brezin–Gross–Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on the space of solutions of such ordinary differential equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations.

The weights of simple modules in Category [formula omitted] for Kac–Moody algebras

We give the first positive formulas for the weights of every simple highest weight module L(λ) over an arbitrary Kac–Moody algebra. Under a mild condition on the highest weight, we also express the weights of L(λ) as an alternating sum similar to the Weyl–Kac character formula.To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding D-module on certain divisors, which answers a question of Brion.Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group.

A note on parabolic Verma module homomorphisms over Kac-Moody algebras

We consider a general parabolic category O S over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms .

Triple derivations on parabolic subalgebras of Kac-Moody algebras

A linear map on a Lie algebra is called a triple derivation if for all Let be a Kac-Moody algebra over an algebraically closed field of characteristic 0, and an arbitrary parabolic subalgebra of In this paper, we prove that any triple derivation of is a derivation, which generalizes the corresponding existing result for finite-dimensional simple Lie algebra.

Representations of Involutory Subalgebras of Affine Kac–Moody Algebras

We consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebras that are fixed by the Cartan–Chevalley involution. These infinite-dimensional Lie algebras are not of Kac–Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of {mathbb {N}}-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity and work out a detailed example in the case of the affine extension of {mathfrak {e}}_8.

Colored black holes and Kac-Moody algebra

We demonstrate that the near horizon symmetries of black holes in Einstein-Yang-Mills (EYM) theory are generated by an infinite-dimensional algebra that contains, in addition to supertranslations and superrotations, a non-Abelian loop algebra. This means that the Virasoro-Kac-Moody structure of EYM in asymptotically flat spacetimes has an exact analog in the near horizon region.

Multi-grounded partitions and character formulas

We introduce a new generalisation of partitions, multi-grounded partitions, related to ground state paths indexed by dominant weights of Lie algebras. We use these to express characters of irreducible highest weight modules of Kac–Moody algebras of affine type as generating functions for multi-grounded partitions. This generalises the approach of our previous paper, where only irreducible highest weight modules with constant ground state paths were considered, to all ground state paths. As an application, we compute the characters of the level 1 modules of the affine Lie algebras A2n(2)(n≥2), Dn+1(2)(n≥2), A2n−1(2)(n≥3), Bn(1)(n≥3), and Dn(1)(n≥4).

Skew-symmetric biderivations and linear commuting maps of Kac-Moody algebras

A map φ on a Lie algebra L is called commuting if for all . Let be a Kac-Moody algebra over an algebraically closed field of characteristic 0. In this paper, we determine the skew-symmetric biderivations and the linear commuting maps of . As applications, the commuting derivations and the commuting automorphisms of are gained.

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by the crystal of all Lakshmibai-Seshadri paths of shape λ. Let be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector of weight λ, and let be the crystal basis of with the element corresponding to We prove that the connected component of containing is isomorphic, as a crystal, to the connected component of containing the straight line Furthermore, we prove that if λ satisfies a special condition, then the crystal basis is isomorphic, as a crystal, to the crystal As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in where are the fundamental weights, in the case that is symmetric.

Continuum Kac–Moody Algebras

We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac-Moody algebras. Their construction is closely related to that of usual Kac-Moody algebras, but they feature a continuum root system with no simple roots. Their Cartan datum encodes the topology of a one-dimensional real space and can be thought of as a generalization of a quiver, where vertices are replaced by connected intervals. For these Lie algebras, we prove an analogue of the Gabber-Kac-Serre theorem, providing a complete set of defining relations featuring only quadratic Serre relations. Moreover, we provide an alternative realization as continuum colimits of symmetric Borcherds-Kac-Moody algebras with at most isotropic simple roots. The approach we follow deeply relies on the more general notion of a semigroup Lie algebra and its structural properties.

Nonlinear evolution equations related to Kac-Moody algebras $A_r^{(1)}$: spectral aspects

We analyze three types of integrable nonlinear evolution equations (NLEE) related to the Kac-Moody algebras $A_r^{(1)}$. These are $\mathbb{Z}_h$-reduced derivative NLS equations (DNLS), multicomponent mKdV equations and 2-dimensional Toda field theories (2dTFT). We outline the basic tools of this analysis: i) the gradings of the simple Lie algebras using their Coxeter automorphisms; ii) the construction of the relevant Lax representations; and iii) the spectral properties of the Lax operators and their reduction to Riemann-Hilbert problems. We also formulate the minimal set of scattering data which allow one to recover the asymptotics of the fundamental analytic solutions to $L$ and its potential.

From 2d droplets to 2d Yang-Mills

We establish a connection between time evolution of free Fermi droplets and partition function of generalised q-deformed Yang-Mills theories on Riemann surfaces. Classical phases of (0+1) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces H+ and H− associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large N limit, equal to the quadratic Casimir of u(N). We establish an exact mapping between states in H± and geometries of droplets. In particular, coherent states in H± correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in H± is equal to the chiral and anti-chiral partition function of 2d Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space H+⊗H− admits a composite basis, we show that correlation between two classical droplet geometries is equal to the full U(N) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in H± and higher point correlators in 2d Yang-Mills on Riemann surface. There are special states in H± whose transition amplitudes are equal to the partition function of 2d q-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the q-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.