135 research outputs found
On the fundamental representation of Borcherds algebras with one imaginary simple root
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
It is well known but rather mysterious that root spaces of the 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 corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (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
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
It is proved that if the Schr\"odinger equation of
Calogero-Moser-Sutherland type with
has a solution of the product form then the function satisfies the
generalised WDVV equations.Comment: 10 page
The Tate conjecture for K3 surfaces over finite fields
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
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 -twisted theories, and respectively,
which may be constructed from a suitable even Euclidean lattice .
Similarly, one may construct lattices and by
analogous constructions from a doubly-even binary code . In the case when
is self-dual, the corresponding lattices are also. Similarly,
and are self-dual if and only if is. We show that
has a natural ``triality'' structure, which induces an
isomorphism and also a triality
structure on . For the Golay code,
is the Leech lattice, and the triality on 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
and 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
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
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 -adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices
Suppose that is an even lattice with dual and level . Then the
group , which is the unique non-trivial double cover of
, admits a representation , called the Weil
representation, on the space . The main aim of this paper
is to show how the formulae for the -action of a general element of
can be obtained by a direct evaluation which does not
depend on ``external objects'' such as theta functions. We decompose the Weil
representation into -parts, in which each -part can be seen as
subspace of the Schwartz functions on the -adic vector space
. Then we consider the Weil representation of
on the space of Schwartz functions on
, and see that restricting to just
gives the -part of 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 , 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
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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