Twisted Affine Lie Algebras Research Articles

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Twisted elliptic genera

We study the twisted elliptic genera of 2d (0, 4) SCFTs associated with the BPS strings in the twisted circle compactification of 6d rank-one (1, 0) SCFTs. Such objects can arise when the 6d gauge algebra allows outer automorphism, thus are classified by twisted affine Lie algebras. We study several fascinating aspects of the twisted elliptic genera including 2d localization, twisted elliptic blowup equations, Higgsing and spectral flow symmetry. We derive a recursion formula with respect to the number of strings to exactly compute the twisted elliptic genera. We also investigate the modular bootstrap of twisted one-string elliptic genera and find the modularity of congruence subgroups Γ1(N) naturally appears with possible N = 2, 3, 4. Geometrically, our study solves the refined BPS partition functions of the underlying genus-one fibered Calabi-Yau threefolds with N-section.

Combinatorial bases of standard modules of twisted affine Lie algebras in types and : rectangular highest weights

We consider the standard modules of rectangular highest weights of affine Lie algebras in types and . By using vertex algebraic techniques we construct the combinatorial bases for standard modules and their principal subspaces and parafermionic spaces. Finally, we compute the corresponding character formulae and, as an application, we obtain two new families of combinatorial identities.

Quadratic Relations of the Deformed W-Algebra for the Twisted Affine Lie Algebra of Type A<sup>(2)</sup><sub>2N</sub>

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

Parafermionic Bases of Standard Modules for Twisted Affine Lie Algebras of Type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_{6}^{(2)}$ and $D_{4}^{(3)}$

Using the bases of principal subspaces for twisted affine Lie algebras except \(A_{2l}^{(2)}\) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001).

Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras

We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules .

Poincaré series, exponents of affine Lie algebras, and McKay-Slodowy correspondence

Let N be a normal subgroup of a finite group G and V a fixed finite-dimensional G-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra T(V)=⊕k≥0V⊗k are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of SU2 give rise to the exponents of all untwisted and twisted affine Lie algebras except A2n(1).

Algebraic complete integrability of the [formula omitted] Toda lattice

The aim of this paper is to study the algebraic complete integrability of an integrable system, namely the Toda lattice associated to the twisted affine Lie algebra c2(1). We show that the generic fiber of the momentum map of this system is an affine part of an Abelian surface and that the flows of integrable vector fields are linear on this surface. Finally, using the formal Laurent solutions of the system, we give a detailed geometric description of these Abelian surfaces and of the divisor at infinity.

Demazure flags, q-Fibonacci polynomials and hypergeometric series

We study a family of finite-dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $$\mathtt A_2^{(2)}$$ . We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $$m'$$ Demazure module admits a filtration by level m Demazure modules for all $$m\ge m'$$ . We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish a recursive formulae. In the case when $$m'=1,2$$ and $$m=2,3$$ , we determine these generating functions completely and show that they define hypergeometric series and that they are related to the q-Fibonacci polynomials defined by Carlitz.

Generalized Rogers–Ramanujan identities for twisted affine algebras

The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

Twisted Γ-Lie algebras and their vertex operator representations

Let Γ be a generic subgroup of the multiplicative group C⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of ZN and an arbitrary finite order isometry of ZN preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As an application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras.

Weyl Modules for the Hyperspecial Current Algebra

We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type $A_{2n}^{(2)}$. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type $A_{2n}^{(2)}$.

Complete Reducibility for the Twisted Affine Lie Algebras

We prove complete reducibility theorem for integrable modules for the twisted affine Lie algebras where the central element acts non-trivially.

Algebraic integrability and geometry of the $\mathfrak{d}_3^{(2)}$ toda lattice

In this paper, we consider the Toda lattice associated to the twisted affine Lie algebra \(\mathfrak{d}_3^{(2)}\). We show that the generic fiber of the momentum map of this system is an affine part of an Abelian surface and that the flows of integrable vector fields are linear on this surface, so that the system is algebraic completely integrable. We also give a detailed geometric description of these Abelian surfaces and of the divisor at infinity. As an application, we show that the lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation with spectral parameter for this Toda lattice and we construct an explicit linearization of the system.

VERTEX-ALGEBRAIC STRUCTURE OF THE PRINCIPAL SUBSPACES OF CERTAIN $A_{1}^{(1)}$-MODULES, I: LEVEL ONE CASE

This is the first in a series of papers in which we study vertex-algebraic structure of Feigin–Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even "small" spanning sets of these spaces. In this paper, we prove presentations of the principal subspaces of the basic [Formula: see text]-modules. These convenient presentations were previously used in work of Capparelli–Lepowsky–Milas for the purpose of obtaining the classical Rogers–Ramanujan recursion for the graded dimensions of the principal subspaces.

On certain generalizations of twisted affine Lie algebras and quasimodules for [formula omitted]-vertex algebras

We continue a previous study on Γ -vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any Z -graded vertex algebra V and a positive integer N , the category of V -modules is naturally isomorphic to the category of quasimodules of a certain type for V ⊗ N . We also study certain generalizations of twisted affine Lie algebras and we relate such Lie algebras to vertex algebras and their quasimodules, in a way similar to how twisted affine Lie algebras are related to vertex algebras and their twisted modules.

Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395–412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A 2 ℓ ( 2 ) and of type D ℓ + 1 ( 2 ) . We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of Hom ( W , W ( ( x ) ) ) for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of Hom ( W , W ( ( x ) ) ) . A notion of Γ -vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C × of nonzero complex numbers. It is proved that any maximal quasilocal subspace of Hom ( W , W ( ( x ) ) ) is naturally a Γ -vertex algebra and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a Γ -vertex algebra. It is also proved that a Γ -vertex algebra exactly amounts to a vertex algebra equipped with a Γ -module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.

TheN=1* Theories onR1 2 S1with Twisted Boundary Conditions

We explore the N=1* theories compactified on a circle with twisted boundary conditions. The gauge algebra of these theories are the so-called twisted affine Lie algebra. We propose the exact superpotentials by guessing the sum of all monopole-instanton contributions and also by requiring SL(2,Z) modular properties. The latter is inherited from the N=4 theory, which will be justified in the M theory setting. Interestingly all twisted theories possess full SL(2,Z) invariance, even though none of them are simply-laced. We further notice that these superpotentials are associated with certain integrable models widely known as elliptic Calogero-Moser models. Finally, we argue that the glueball superpotential must be independent of the compactification radius, and thus of the twisting, and confirm this by expanding it in terms of glueball superfield in weak coupling expansion.

Virasoro Algebra, Dedekind ?-function and Specialized Macdonald Identities

We motivate and prove a series of identities which form a generalization of the Euler pentagonal number theorem, and are closely related to specialized Macdonald identities for powers of the Dedekind η-function. More precisely, we show that what we call “denominator formula” for the Virasoro algebra has “higher analogue” for all cs,t-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as fusion and modular invariance that require all the irreducible modules of the series. In particular, in the case of c2,2k+1-minimal models we give a new proof of a family of specialized Macdonald identities associated with twisted affine Lie algebras of type A(2) 2k k ≥ 2 (i.e., BCk affine root system) which involve (2k2 - k)-th powers of the Dedekind η-function.