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Register a new userThe perturbations $phi_{2,1}$ and $phi_{1,5}$ of the minimal models $M(p,p)$ and the trinomial analogue of Baileys lemma
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We derive the fermionic polynomial generalizations of the characters of the integrable perturbations $phi_{2,1}$ and $phi_{1,5}$ of the general minimal $M(p,p)$ conformal field theory by use of the recently discovered trinomial analogue of Baileys lemma. For $phi_{2,1}$ perturbations results are given for all models with $2p>p$ and for $phi_{1,5}$ perturbations results for all models with ${pover 3}<p< {pover 2}$ are obtained. For the $phi_{2,1}$ perturbation of the unitary case $M(p,p+1)$ we use the incidence matrix obtained from these character polynomials to conjecture a set of TBA equations. We also find that for $phi_{1,5}$ with $2<p/p < 5/2$ and for $phi_{2,1}$ satisfying $3p<2p$ there are usually several different fermionic polynomials which lead to the identical bosonic polynomial. We interpret this to mean that in these cases the specification of the perturbing field is not sufficient to define the theory and that an independent statement of the choice of the proper vacuum must be made.
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The filtration of the Virasoro minimal series representations M^{(p,p)}_{r,s} induced by the (1,3)-primary field $phi_{1,3}(z)$ is studied. For 1< p/p< 2, a conjectural basis of M^{(p,p)}_{r,s} compatible with the filtration is given by using monomial vectors in terms of the Fourier coefficients of $phi_{1,3}(z)$. In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series (p=p+1), we establish for each $m$ the equality between the character of the degree $m$ monomial basis and the character of the degree $m$ component in the associated graded module Gr(M^{(p,p+1)}_{r,s}) with respect to the filtration defined by $phi_{1,3}(z)$.
We describe an extension of the nonlinear integral equation (NLIE) method to Virasoro minimal models perturbed by the relevant operator $Phi_{(1,3)$. Along the way, we also complete our previous studies of the finite volume spectrum of sine-Gordon theory by considering the attractive regime and more specifically, breather states. For the minimal models, we examine the states with zero topological charge in detail, and give numerical comparison to TBA and TCS results. We think that the evidence presented strongly supports the validity of the NLIE description of perturbed minimal models.
The scaled inverse of a nonzero element $a(x)in mathbb{Z}[x]/f(x)$, where $f(x)$ is an irreducible polynomial over $mathbb{Z}$, is the element $b(x)in mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c pmod{f(x)}$ for the smallest possible positive integer scale $c$. In this paper, we investigate the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $Phi_{p^s}(x)$ or $Phi_{p^s q^t}(x)$, where $p, q$ are primes with $p<q$ and $s, t$ are positive integers. Our main results are that the coefficient size of the scaled inverse of $(x^i-x^j)$ is bounded by $p-1$ with the scale $p$ modulo $Phi_{p^s}(x)$, and is bounded by $q-1$ with the scale not greater than $q$ modulo $Phi_{p^s q^t}(x)$. Previously, the analogous result on cyclotomic polynomials of the form $Phi_{2^n}(x)$ gave rise to many lattice-based cryptosystems, especially, zero-knowledge proofs. Our result provides more flexible choice of cyclotomic polynomials in such cryptosystems. Along the way of proving the theorems, we also prove several properties of ${x^k}_{kinmathbb{Z}}$ in $mathbb{Z}[x]/Phi_{pq}(x)$ which might be of independent interest.
We consider the tricritical Ising model on a strip or cylinder under the integrable perturbation by the thermal $phi_{1,3}$ boundary field. This perturbation induces five distinct renormalization group (RG) flows between Cardy type boundary conditions labelled by the Kac labels $(r,s)$. We study these boundary RG flows in detail for all excitations. Exact Thermodynamic Bethe Ansatz (TBA) equations are derived using the lattice approach by considering the continuum scaling limit of the $A_4$ lattice model with integrable boundary conditions. Fixing the bulk weights to their critical values, the integrable boundary weights admit a thermodynamic boundary field $xi$ which induces the flow and, in the continuum scaling limit, plays the role of the perturbing boundary field $phi_{1,3}$. The excitations are completely classified, in terms of string content, by $(m,n)$ systems and quantum numbers but the string content changes by either two or three well-defined mechanisms along the flow. We identify these mechanisms and obtain the induced maps between the relevant finitized Virasoro characters. We also solve the TBA equations numerically to determine the boundary flows for the leading excitations.
We consider black $p$-brane solutions of the low energy string action, computing scalar perturbations. Using standard methods, we derive the wave equations obeyed by the perturbations and treat them analytically and numerically. We have found that tensorial perturbations obtained via a gauge-invariant formalism leads to the same results as scalar perturbations. No instability has been found. Asymptotically, these solutions typically reduce to a $AdS_{(p+2)}times S^{(8-p)}$ space, which, in the framework of Maldacenas conjecture, can be regarded as a gravitational dual to a conformal field theory defined in a $(p+1)$-dimensional flat space-time. The results presented open the possibility of a better understanding the AdS/CFT correspondence, as originally formulated in terms of the relation among brane structures and gauge theories.
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