135 research outputs found

    On the fundamental representation of Borcherds algebras with one imaginary simple root

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    Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ``by hand'' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.Comment: 8 page

    Borcherds symmetries in M-theory

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    It is well known but rather mysterious that root spaces of the EkE_k Lie groups appear in the second integral cohomology of regular, complex, compact, del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms) of toroidal compactifications of M theory. Their Borel subgroups are actually subgroups of supergroups of finite dimension over the Grassmann algebra of differential forms on spacetime that have been shown to preserve the self-duality equation obeyed by all bosonic form-fields of the theory. We show here that the corresponding duality superalgebras are nothing but Borcherds superalgebras truncated by the above choice of Grassmann coefficients. The full Borcherds' root lattices are the second integral cohomology of the del Pezzo surfaces. Our choice of simple roots uses the anti-canonical form and its known orthogonal complement. Another result is the determination of del Pezzo surfaces associated to other string and field theory models. Dimensional reduction on TkT^k corresponds to blow-up of kk points in general position with respect to each other. All theories of the Magic triangle that reduce to the EnE_n sigma model in three dimensions correspond to singular del Pezzo surfaces with A8nA_{8-n} (normal) singularity at a point. The case of type I and heterotic theories if one drops their gauge sector corresponds to non-normal (singular along a curve) del Pezzo's. We comment on previous encounters with Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real fermionic simple roots when they would naively aris

    Symmetries in M-theory: Monsters, Inc

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    We will review the algebras which have been conjectured as symmetries in M-theory. The Borcherds algebras, which are the most general Lie algebras under control, seem natural candidates.Comment: 6 pages, talk given by PHL at Cargese 200

    On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations

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    It is proved that if the Schr\"odinger equation Lψ=λψL \psi = \lambda \psi of Calogero-Moser-Sutherland type with L=Δ+αA+mα(mα+1)(α,α)sin2(α,x)L = -\Delta + \sum\limits_{\alpha\in{\cal A}_{+}} \frac{m_{\alpha}(m_{\alpha}+1) (\alpha,\alpha)}{\sin^{2}(\alpha,x)} has a solution of the product form ψ0=αA+sinmα(α,x),\psi_0 = \prod_{\alpha \in {\cal {A}_+}} \sin^{-m_{\alpha}}(\alpha,x), then the function F(x)=αA+mα(α,x)2log(α,x)2F(x) =\sum\limits_{\alpha \in \cal {A}_{+}} m_{\alpha} (\alpha,x)^2 {\rm log} (\alpha,x)^2 satisfies the generalised WDVV equations.Comment: 10 page

    The Tate conjecture for K3 surfaces over finite fields

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    Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p>3. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p>3. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p>3.Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality, but proofs don't change. Comments still welcom

    Conformal Field Theories, Representations and Lattice Constructions

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    An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and Z2Z_2-twisted theories, H(Λ)H(\Lambda) and H~(Λ)\tilde H(\Lambda) respectively, which may be constructed from a suitable even Euclidean lattice Λ\Lambda. Similarly, one may construct lattices ΛC\Lambda_C and Λ~C\tilde\Lambda_C by analogous constructions from a doubly-even binary code CC. In the case when CC is self-dual, the corresponding lattices are also. Similarly, H(Λ)H(\Lambda) and H~(Λ)\tilde H(\Lambda) are self-dual if and only if Λ\Lambda is. We show that H(ΛC)H(\Lambda_C) has a natural ``triality'' structure, which induces an isomorphism H(Λ~C)H~(ΛC)H(\tilde\Lambda_C)\equiv\tilde H(\Lambda_C) and also a triality structure on H~(Λ~C)\tilde H(\tilde\Lambda_C). For CC the Golay code, Λ~C\tilde\Lambda_C is the Leech lattice, and the triality on H~(Λ~C)\tilde H(\tilde\Lambda_C) is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories H(Λ)H(\Lambda) and H~(Λ)\tilde H(\Lambda) with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.Comment: 65 page

    Jacobi Identity for Vertex Algebras in Higher Dimensions

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    Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.Comment: 35 pages, references adde

    Quantum W-algebras and Elliptic Algebras

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    We define quantum W-algebras generalizing the results of Reshetikhin and the second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated to sl_N is an associative algebra depending on two parameters. For special values of parameters it becomes the ordinary W-algebra of sl_N, or the q-deformed classical W-algebra of sl_N. We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U_q(n^).Comment: 26 pages, AMSLATE

    A pp-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices

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    Suppose that MM is an even lattice with dual MM^{*} and level NN. Then the group Mp2(Z)Mp_{2}(\mathbb{Z}), which is the unique non-trivial double cover of SL2(Z)SL_{2}(\mathbb{Z}), admits a representation ρM\rho_{M}, called the Weil representation, on the space C[M/M]\mathbb{C}[M^{*}/M]. The main aim of this paper is to show how the formulae for the ρM\rho_{M}-action of a general element of Mp2(Z)Mp_{2}(\mathbb{Z}) can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation ρM\rho_{M} into pp-parts, in which each pp-part can be seen as subspace of the Schwartz functions on the pp-adic vector space MQpM_{\mathbb{Q}_{p}}. Then we consider the Weil representation of Mp2(Qp)Mp_{2}(\mathbb{Q}_{p}) on the space of Schwartz functions on MQpM_{\mathbb{Q}_{p}}, and see that restricting to Mp2(Z)Mp_{2}(\mathbb{Z}) just gives the pp-part of ρM\rho_{M} again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in SL2(Qp)SL_{2}(\mathbb{Q}_{p}), belong to the metaplectic double cover. Some other properties are also investigated.Comment: 29 pages, shortened a lo

    The structure of parafermion vertex operator algebras

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    It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k coincides with a certain W-algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.Comment: 12 page
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