1,016 research outputs found
Kida's formula and congruences
We prove a formula (analogous to that of Kida in classical Iwasawa theory and
generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic
and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian
p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side
our methods, which make use of congruences between modular forms, yield a
Kida-type formula for a very general class of ordinary Galois representations.
We are further able to deduce a Kida-type formula for elliptic curves at
supersingular primes
Mazur-Tate elements of non-ordinary modular forms
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of
cuspidal eigenforms, generalizing known results in weight 2. Our first theorem
deals with forms of "medium" weight, and our second deals with forms of small
slope . We give examples illustrating the strange behavior which can occur in
the high weight, high slope case
Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis
We reconsider the open XXZ chain of Yang and Fendley. This model possesses a
lattice supersymmetry which changes the length of the chain by one site. We
perform an algebraic Bethe ansatz analysis of the model and derive the
commutation relations of the lattice SUSY operators with the four elements of
the open-chain monodromy matrix. Hence we give the action of the SUSY operator
on off-shell and on-shell Bethe states. We show that this action generally
takes one on-shell Bethe eigenstate to another. The exception is that a
zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe
roots we obtain are analogous to those found previously for closed chains by
Fendley and Hagendorf by analysing the Bethe equations.Comment: 13 pages; dimension counting of SUSY operator image and kernel spaces
added; references update
PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
We investigate the PT-symmetry of the quantum group invariant XXZ chain. We
show that the PT-operator commutes with the quantum group action and also
discuss the transformation properties of the Bethe wavefunction. We exploit the
fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order
to give an explicit and exact construction of an operator that ensures
quasi-Hermiticity of the model. This construction relys on earlier ideas
related to quantum group reduction. We then employ this result in connection
with the quantum analogue of Schur-Weyl duality to introduce a dual pair of
C-operators, both of which have closed algebraic expressions. These are novel,
exact results connecting the research areas of integrable lattice systems and
non-Hermitian Hamiltonians.Comment: 32 pages with figures, v2: some minor changes and added references,
version published in JP
A Free Field Representation of the Screening Currents of $U_q(\widehat{sl(3)})
We construct five independent screening currents associated with the
quantum current algebra. The screening currents are
expressed as exponentials of the eight basic deformed bosonic fields that are
required in the quantum analogue of the Wakimoto realization of the current
algebra. Four of the screening currents are `simple', in that each one is given
as a single exponential field. The fifth is expressed as an infinite sum of
exponential fields. For reasons we discuss, we expect that the structure of the
screening currents for a general quantum affine algebra will be similar to the
case.Comment: 21 pages (LaTeX), CRM-126
The Dynamical Correlation Function of the XXZ Model
We perform a spectral decomposition of the dynamical correlation function of
the spin XXZ model into an infinite sum of products of form factors.
Beneath the four-particle threshold in momentum space the only non-zero
contributions to this sum are the two-particle term and the trivial vacuum
term. We calculate the two-particle term by making use of the integral
expressions for form factors provided recently by the Kyoto school. We evaluate
the necessary integrals by expanding to twelfth order in . We show plots of
, for and at various values of the anisotropy parameter,
and for fixed anisotropy at various around and .Comment: 20 pages (LaTeX), CRM-219
Variation of Iwasawa invariants in Hida families
Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible
residual modular Galois representation. We show that the vanishing of the
algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r
implies the vanishing of the corresponding mu-invariant for all such forms.
Assuming that the mu-invariant vanishes, we also give explicit formulas for the
difference in the algebraic or analytic lambda-invariants of modular forms
lifting r. In particular, our formula shows that the lambda-invariant is
constant on branches of the Hida family of r. We further show that our formulas
are identical for the algebraic and analytic invariants, so that the truth of
the main conjecture of Iwasawa theory for one form in the Hida family of r
implies it for the entire Hida family
Vertex Operators and Matrix Elements of via Bosonization
We construct bosonized vertex operators (VOs) and conjugate vertex operators
(CVOs) of for arbitrary level and representation . Both are obtained directly as two solutions of the defining condition of
vertex operators - namely that they intertwine modules. We
construct the screening charge and present a formula for the n-point function.
Specializing to we construct all VOs and CVOs explicitly. The existence
of the CVO allows us to place the calculation of the two-point function on the
same footing as ; that is, it is obtained without screening currents and
involves only a single integral from the CVO. This integral is evaluated and
the resulting function is shown to obey the q-KZ equation and to reduce simply
to both the expected and limits.Comment: 20 pages, LaTex. Minor change
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