Exceptional Algebras Research Articles

Polyvector deformations of Type IIB backgrounds

We develop a formalism of poly-vector deformations for Type IIB backgrounds with a block diagonal metric and non-vanishing self-dual 5-form RR field strength. Making use of the embedding of the Type IIB theory into the E6(6)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{E}_{6(6)}$$\\end{document} exceptional field theory we derive explicit transformation rules for four-vector deformations. We prove that the algebraic condition following from the Type IIB realization of exceptional Drinfeld algebras is sufficient for the transformation to generate a solution.

𝐵𝑉-structure on Hochschild cohomology for exceptional local algebras of quaternion type. The case of an even parameter

A full description is provided for the B V BV -structure on the Hochschild cohomology of exceptional local algebras of quaternion type determined by parameters ( k , 0 , d ) (k,0,d) in the case of an even parameter k ≥ 3 k \geq 3 , according to Erdmann’s classification. The method of comparison morphisms and the weak self-homotopy method are developed and used. This article may be viewed as a generalization of similar results about the B V BV -structures on the Hochschild cohomology of algebras of quaternion type.

On 10-dimensional Exceptional Drinfeld algebras

Abstract Based on Mubarakzyanov’s classification of four-dimensional real Lie algebras, we classify ten-dimensional Exceptional Drinfeld algebras (EDAs). The classification is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot be represented as a product of a 3D Lie algebra and a 1D abelian factor. We collect the obtained algebras into families depending on the dualities found between them. Despite algebras related by a generalized Yang–Baxter deformation we find two algebras related by a different Nambu–Lie U-duality transformation. We show that this duality relates two Type IIA backgrounds.

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored.

Hochschild Cohomology for Algebras of Semidihedral Type. X. Cohomology Algebra for the Exceptional Local Algebras

Hochschild cohomology algebra is described in term of generators and relations for a family of local algebras of semidihedral type. This family appears in famous K. Erdmann’s classification only if the characteristic of the base field is equal to 2.

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.

The center of the universal enveloping algebras of small-dimensional nilpotent Lie algebras in prime characteristic

We describe the centers of the universal enveloping algebras of nilpotent Lie algebras of dimension at most six over fields of prime characteristic. If the characteristic is not smaller than the nilpontency class, then the center is the integral closure of the algebra generated over the p-center by the same generators that also occur in characteristic zero. Except for three examples, two of which are standard filiform, this algebra is already integrally closed and hence it coincides with the center. In the case of these three exceptional algebras, the center has further generators. Then we show that the center of the universal enveloping algebra of the algebras investigated in this paper is isomorphic to the Poisson center (the algebra of invariants under the adjoint representation). This shows that Braun’s conjecture is valid for this class of Lie algebras.

On the Non-Abelian U-Duality of 11D Backgrounds

In this work, we generalise the procedure of the non-abelian T-duality based on a B-shift and a sequence of formal abelian T-dualities in non-isometric directions to 11-dimensional backgrounds. This consists of a C-shift followed by either a formal (abelian) U-duality transformation or taking an IIB section. By construction, this is a solution generating transformation. We investigate the restrictions and applicability of the procedure and find that it can provide supergravity solutions for the SL(5) exceptional Drinfeld algebra only when the isometry algebra of the sigma-model target space decomposes into a direct sum. This is consistent with examples known in the literature.

Folding orthosymplectic quivers

Folding identical legs of a simply-laced quiver creates a quiver with a non-simply laced edge. So far, this has been explored for quivers containing unitary gauge groups. In this paper, orthosymplectic quivers are folded, giving rise to a new family of quivers. This is realised by intersecting orientifolds in the brane system. The monopole formula for these non-simply laced orthosymplectic quivers is introduced. Some of the folded quivers have Coulomb branches that are closures of minimal nilpotent orbits of exceptional algebras, thus providing a new construction of these fundamental moduli spaces. Moreover, a general family of folded orthosymplectic quivers is shown to be a new magnetic quiver realisation of Higgs branches of 4d mathcal{N} = 2 theories. The Hasse (phase) diagrams of certain families are derived via quiver subtraction as well as Kraft-Procesi transitions in the brane system.

CLUSTER ALGEBRAS FROM SURFACES AND EXTENDED AFFINE WEYL GROUPS

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

Polyvector deformations in eleven-dimensional supergravity

We consider 3- and 6-vector deformations of 11-dimensional supergravity backgrounds of the form $M_5\times M_6$ admitting at least 3 Killing vectors. Using flux formulation of the E${}_{6(6)}$ exceptional field theory we derive (sufficient) conditions for the deformations to generate a solution. In the group manifold case these generalisations of the classical Yang-Baxter equation for the case of r-matrices with 3 and 6 indices are shown to reproduce those obtained from exceptional Drinfeld algebra for E${}_{6(6)}$. In general we see an additional constraint, which might be related to higher exceptional Drinfeld algebras.

On generalized Melvin solutions for Lie algebras of rank 4

We consider generalized Melvin-like solutions associated with Lie algebras of rank 4 (namely, A 4, B 4, C 4, D 4, and the exceptional algebra F 4 ) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically-symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs (z) ( s = 1,…,4) of squared radial coordinate z = ρ 2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n 1, n 2, n 3, n 4) = (4, 6, 6, 4), (8, 14, 18, 10), (7, 12, 15, 16), (6, 10, 6, 6), (22, 42, 30, 16) for Lie algebras A 4, B 4, C 4, D 4, F 4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4 × 4 matrix v connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A 4 case) the matrix representing a generator of the ℤ2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained. We also presented 2-form flux integrals over 2-dimensional discs.

Magnetic lattices for orthosymplectic quivers

For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the choice of the precise gauge group becomes crucial when the magnetic lattice of the theory is considered. This question is addressed in the context of Coulomb branches for 3d mathcal{N} = 4 quiver gauge theories, which are moduli spaces of dressed monopole operators. We compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers for different choices of gauge groups, including diagonal quotients of the product gauge group of individual factors, where the quotient is by a trivially acting subgroup. Choosing different such diagonal groups results in distinct Coulomb branches, related as orbifolds. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions 4, 5, and 6. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

Spectral theories and topological strings on del Pezzo geometries

Motivated by understanding M2-branes, we propose to reformulate partition functions of M2-branes by quantum curves. Especially, we focus on the backgrounds of del Pezzo geometries, which enjoy Weyl group symmetries of exceptional algebras. We construct quantum curves explicitly and turn to the analysis of classical phase space areas and quantum mirror maps. We find that the group structure helps in clarifying previous subtleties, such as the shift of the chemical potential in the area and the identification of the overall factor of the spectral operator in the mirror map. We list the multiplicities characterizing the quantum mirror maps and find that the decoupling relation known for the BPS indices works for the mirror maps. As a result, with the group structure we can present explicitly the statement for the correspondence between spectral theories and topological strings on del Pezzo geometries.

On (ad)n(X2)k series of universal quantum dimensions

We present a universal, in Vogel’s sense, expression for the quantum dimension of the Cartan product of arbitrary powers of the adjoint and X2 representations of simple Lie algebras. The same formula mysteriously yields quantum dimensions of some other representations of the same Lie algebra for permuted universal parameters, provided linear resolvability of singularities is applied. We list these representations for the exceptional algebras and their stable versions for the classical algebras (when the rank of the classical algebra is sufficiently large with regard to the powers of representations). Universal formulas may have singularities at the points in Vogel’s plane, corresponding to some simple Lie algebras. We prove that our formula is linearly resolvable at all those singular points, i.e., yields finite answers when restricted either on the classical or the exceptional lines, and make a conjecture that these answers coincide with (quantum) dimensions of some irreducible representations. In a number of cases, particularly, in the case of so(8) algebra, which belongs both to the orthogonal and the exceptional lines, it is confirmed that both resolutions yield relevant answers. We note that an irreducible representation may have several universal formulas for its (quantum) dimension and discuss the impact of this phenomenon to the method presented by Cohen and de Man [C. R.Acad. Sci., Ser. 1 322(5), 427–432 (1996)] for derivation of universal formulas.

Poisson-Lie U-duality in exceptional field theory

Poisson-Lie duality provides an algebraic extension of conventional Abelian and non-Abelian target space dualities of string theory and has seen recent applications in constructing quantum group deformations of holography. Here we demonstrate a natural upgrading of Poisson-Lie to the context of M-theory using the tools of exceptional field theory. In particular, we propose how the underlying idea of a Drinfeld double can be generalised to an algebra we call an exceptional Drinfeld algebra. These admit a notion of “maximally isotropic subalgebras” and we show how to define a generalised Scherk-Schwarz truncation on the associated group manifold to such a subalgebra. This allows us to define a notion of Poisson-Lie U-duality. Moreover, the closure conditions of the exceptional Drinfeld algebra define natural analogues of the cocycle and co-Jacobi conditions arising in Drinfeld double. We show that upon making a further coboundary restriction to the cocycle that an M-theoretic extension of Yang-Baxter deformations arise. We remark on the application of this construction as a solution-generating technique within supergravity.

Classification of exceptional train algebras of rank 3 and type (4, 2)

Classification of exceptional train algebras of rank 3 and type (4, 2)

An algebraic classification of exceptional EFTs. Part II. Supersymmetry

We present a novel approach to classify supersymmetric effective field theories (EFTs) whose scattering amplitudes exhibit enhanced soft limits. These enhancements arise due to non-linearly realised symmetries on the Goldstone modes of such EFTs and we classify the algebras that these symmetries can form. Our main focus is on so-called exceptional algebras which lead to field-dependent transformation rules and EFTs with the maximum possible soft enhancement at a given derivative power counting. We adapt existing techniques for Poincaré invariant theories to the supersymmetric case, and introduce superspace inverse Higgs constraints as a method of reducing the number of Goldstone modes while maintaining all symmetries. Restricting to the case of a single Goldstone supermultiplet in four dimensions, we classify the exceptional algebras and EFTs for a chiral, Maxwell or real linear supermultiplet. Moreover, we show how our algebraic approach allows one to read off the soft weights of the different component fields from superspace inverse Higgs trees, which are the algebraic cousin of the on-shell soft data one provides to soft bootstrap EFTs using on-shell recursion. Our Lie-superalgebraic approach extends the results of on-shell methods and provides a complementary perspective on non-linear realisations.

Exceptional Chern-Simons-Matter dualities

We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is (E_{N})_{1}\leftrightarrow SU(9-N)_{-1}(EN)1↔SU(9−N)−1. Others such as SO(3)_{8}\leftrightarrow PSU(3)_{-6},SO(3)8↔PSU(3)−6, are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons-matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant G_{2}G2 gauge theory coupled to either an adjoint fermion, or two fundamental fermions.

Gradings on tensor products of composition algebras and on the Smirnov algebra

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra C and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of C⊗n and Cn are isomorphic.On the other hand, we prove that the automorphism group schemes of a Smirnov algebra T(C) (a 35-dimensional simple exceptional structurable algebra constructed from a Cayley algebra C) and C are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras.