51 research outputs found
SUSY vertex algebras and supercurves
This article is a continuation of math.QA/0603633 Given a strongly conformal
SUSY vertex algebra V and a supercurve X we construct a vector bundle V_X on X,
the fiber of which, is isomorphic to V. Moreover, the state-field
correspondence of V canonically gives rise to (local) sections of these vector
bundles. We also define chiral algebras on any supercurve X, and show that the
vector bundle V_X, corresponding to a SUSY vertex algebra, carries the
structure of a chiral algebra.Comment: 50 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
Coherent states for Hopf algebras
Families of Perelomov coherent states are defined axiomatically in the
context of unitary representations of Hopf algebras possessing a Haar integral.
A global geometric picture involving locally trivial noncommutative fibre
bundles is involved in the construction. A noncommutative resolution of
identity formula is proved in that setup. Examples come from quantum groups.Comment: 19 pages, uses kluwer.cls; the exposition much improved; an example
of deriving the resolution of identity via coherent states for SUq(2) added;
the result differs from the proposals in literatur
Randomness in Classical Mechanics and Quantum Mechanics
The Copenhagen interpretation of quantum mechanics assumes the existence of
the classical deterministic Newtonian world. We argue that in fact the Newton
determinism in classical world does not hold and in classical mechanics there
is fundamental and irreducible randomness. The classical Newtonian trajectory
does not have a direct physical meaning since arbitrary real numbers are not
observable. There are classical uncertainty relations, i.e. the uncertainty
(errors of observation) in the determination of coordinate and momentum is
always positive (non zero).
A "functional" formulation of classical mechanics was suggested. The
fundamental equation of the microscopic dynamics in the functional approach is
not the Newton equation but the Liouville equation for the distribution
function of the single particle. Solutions of the Liouville equation have the
property of delocalization which accounts for irreversibility. The Newton
equation in this approach appears as an approximate equation describing the
dynamics of the average values of the position and momenta for not too long
time intervals. Corrections to the Newton trajectories are computed. An
interpretation of quantum mechanics is attempted in which both classical and
quantum mechanics contain fundamental randomness. Instead of an ensemble of
events one introduces an ensemble of observers.Comment: 12 pages, Late
On the Genus Two Free Energies for Semisimple Frobenius Manifolds
We represent the genus two free energy of an arbitrary semisimple Frobenius
manifold as a sum of contributions associated with dual graphs of certain
stable algebraic curves of genus two plus the so-called "genus two G-function".
Conjecturally the genus two G-function vanishes for a series of important
examples of Frobenius manifolds associated with simple singularities as well as
for -orbifolds with positive Euler characteristics. We explain the
reasons for such Conjecture and prove it in certain particular cases.Comment: 37 pages, 3 figures, V2: the published versio
Differential Calculus on the Quantum Superspace and Deformation of Phase Space
We investigate non-commutative differential calculus on the supersymmetric
version of quantum space where the non-commuting super-coordinates consist of
bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum
deformation of the general linear supergroup, , is studied and the
explicit form for the -matrix, which is the solution of the
Yang-Baxter equation, is presented. We derive the quantum-matrix commutation
relation of and the quantum superdeterminant. We apply these
results for the to the deformed phase-space of supercoordinates and
their momenta, from which we construct the -matrix of q-deformed
orthosymplectic group and calculate its -matrix. Some
detailed argument for quantum super-Clifford algebras and the explict
expression of the -matrix will be presented for the case of
.Comment: 17 pages, KUCP-4
On algebraic models of dynamical systems
We describe a universal algebraic model which, being read appropriately, yields (periodic and infinite) discrete dynamical systems, as well as their ‘continuous limits’, which cover all differential scalar Lax systems. For this model we give: Two different constructions of an infinity of integrals; modified equations; deformations; infinitesimal automorphisms. The basic tools are supplied by symbolic calculus and the abstract Hamiltonian formalism.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43214/1/11005_2004_Article_BF00401731.pd
Hidden Symmetries and Integrable Hierarchy of the N=4 Supersymmetric Yang-Mills Equations
We describe an infinite-dimensional algebra of hidden symmetries of N=4
supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a
generalization of the supertwistor correspondence. Using the latter, we
construct an infinite sequence of flows on the solution space of the N=4 SYM
equations. The dependence of the SYM fields on the parameters along the flows
can be recovered by solving the equations of the hierarchy. We embed the N=4
SYM equations in the infinite system of the hierarchy equations and show that
this SYM hierarchy is associated with an infinite set of graded symmetries
recursively generated from supertranslations. Presumably, the existence of such
nonlocal symmetries underlies the observed integrable structures in quantum N=4
SYM theory.Comment: 24 page
Quantum Deformed Algebra and Superconformal Algebra on Quantum Superspace
We study a deformed algebra on a quantum superspace. Some
interesting aspects of the deformed algebra are shown. As an application of the
deformed algebra we construct a deformed superconformal algebra. {}From the
deformed algebra, we derive deformed Lorentz, translation of
Minkowski space, and its supersymmetric algebras as closed
subalgebras with consistent automorphisms.Comment: 27 pages, KUCP-59, LaTeX fil
Nilpotent deformations of N=2 superspace
We investigate deformations of four-dimensional N=(1,1) euclidean superspace
induced by nonanticommuting fermionic coordinates. We essentially use the
harmonic superspace approach and consider nilpotent bi-differential Poisson
operators only. One variant of such deformations (termed chiral nilpotent)
directly generalizes the recently studied chiral deformation of N=(1/2,1/2)
superspace. It preserves chirality and harmonic analyticity but generically
breaks N=(1,1) to N=(1,0) supersymmetry. Yet, for degenerate choices of the
constant deformation matrix N=(1,1/2) supersymmetry can be retained, i.e. a
fraction of 3/4. An alternative version (termed analytic nilpotent) imposes
minimal nonanticommutativity on the analytic coordinates of harmonic
superspace. It does not affect the analytic subspace and respects all
supersymmetries, at the expense of chirality however. For a chiral nilpotent
deformation, we present non(anti)commutative euclidean analogs of N=2 Maxwell
and hypermultiplet off-shell actions.Comment: 1+16 pages; v2: discussion of (pseudo)conjugations extended, version
to appear in JHE
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