Exploring the landscape of heterotic strings on Td

Compactifications of the heterotic string on Td are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in IId,d+16. In agreement with the duality between heterotic on T2 and F-theory on K3, all possible gauge groups on T2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

On the complete integrability of the periodic quantum Toda lattice

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and

RECURRENCE RELATIONS, DYNKIN DIAGRAMS AND PLÜCKER FORMULAE

We prove a generalisation of an observation of N. Iwahori concerning the coefficients of the extended Dynkin diagram of a complex simple Lie algebra. We relate the combinatorics of these coefficients to the orders of finite groups that act discontinuously on the Riemann sphere and to the Plücker formulae.

Tensor Product Structure of Affine Demazure Modules and Limit Constructions

Let g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g. Let Λ0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g-coweight λ∨, the Demazure submodule V_λ∨ (mΛ0) is a g-module. We provide a description of the g-module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ∨ = λ∑j as a sum of dominant integral g-coweights, the Demazure module is (as g-module) isomorphic to ⊗jV_ (mΛ0). For the “smallest” case, λ∨ = ω∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ω∨(mΛ0) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the Uq(g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_λ∨,q(mΛ0) can be naturally endowed with the structure of a -module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the “smallest” Demazure modules are, when viewed as g-modules, isomorphic to some KR-modules. For an integral dominant ĝ-weight Λ let V(Λ) be the corresponding irreducible ĝ-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g-module structure of V(Λ) as a semi-infinite tensor product of finite dimensional g-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

BRST EXACTNESS OF STRESS-ENERGY TENSORS

BRST commutators in the topological conformal field theories obtained by twisting N=2 theories are evaluated explicitly. By our systematic calculations of the multiple integrals which contain screening operators, the BRST exactness of the twisted stress-energy tensors is deduced for classical simple Lie algebras and general level k. We can see that the paths of integrations do not affect the result, and further, the N=2 coset theories are obtained by deleting two simple roots with Kac-label 1 from the extended Dynkin diagram; in other words, by not performing the integrations over the variables corresponding to the two simple roots of Kac-Moody algebras. It is also shown that a series of N=1 theories are generated in the same way by deleting one simple root with Kac-label 2.

Infinite dimensional representations of

By a representation of the extended Dynkin diagram we shall mean a list of 5 vector spaces P, E1, E2, E3, E4 over an algebraically closed field K, and 4 linear maps a1, a2, a3, a4 as shown.The spaces need not be of finite dimension.In their solution of the 4-subspace problem [6], Gelfand and Ponomarev have classified such representations when the spaces are finite dimensional. A representation like (1) can also be viewed as a module over the K-algebra R4 consisting of all 5 × 5 matrices having zeros off the first row and off the main diagonal.