8,246 research outputs found
Differential Calculus on h-Deformed Spaces
We construct the rings of generalized differential operators on the -deformed vector space of -type. In contrast to the -deformed
vector space, where the ring of differential operators is unique up to an
isomorphism, the general ring of -deformed differential operators
is labeled by a rational function
in variables, satisfying an over-determined system of
finite-difference equations. We obtain the general solution of the system and
describe some properties of the rings
Dressing chain equations associated to difference soliton systems
The dressing chain equations for factorizing operators of a spectral problem
are derived. The chain equations itselves yield nonlinear systems which closure
generates solutions of the equations as well as of the nonlinear system if both
operators of the correspondent Hirota bilinearization are covariant with
respect to Darboux transformation which hence defines a symmetry of the
nonlinear system as well as of these closed chains. Examples of Hirota and Nahm
equations are specified.Comment: 12 page
The Elliptic curves in gauge theory, string theory, and cohomology
Elliptic curves play a natural and important role in elliptic cohomology. In
earlier work with I. Kriz, thes elliptic curves were interpreted physically in
two ways: as corresponding to the intersection of M2 and M5 in the context of
(the reduction of M-theory to) type IIA and as the elliptic fiber leading to
F-theory for type IIB. In this paper we elaborate on the physical setting for
various generalized cohomology theories, including elliptic cohomology, and we
note that the above two seemingly unrelated descriptions can be unified using
Sen's picture of the orientifold limit of F-theory compactification on K3,
which unifies the Seiberg-Witten curve with the F-theory curve, and through
which we naturally explain the constancy of the modulus that emerges from
elliptic cohomology. This also clarifies the orbifolding performed in the
previous work and justifies the appearance of the w_4 condition in the elliptic
refinement of the mod 2 part of the partition function. We comment on the
cohomology theory needed for the case when the modular parameter varies in the
base of the elliptic fibration.Comment: 23 pages, typos corrected, minor clarification
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