1,857 research outputs found
From finite geometry exact quantities to (elliptic) scattering amplitudes for spin chains: the 1/2-XYZ
Initially, we derive a nonlinear integral equation for the vacuum counting
function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus
paralleling similar results by Kl\"umper \cite{KLU}, achieved through a
different technique in the {\it antiferroelectric regime}. In terms of the
counting function we obtain the usual physical quantities, like the energy and
the transfer matrix (eigenvalues). Then, we introduce a double scaling limit
which appears to describe the sine-Gordon theory on cylindrical geometry, so
generalising famous results in the plane by Luther \cite{LUT} and Johnson et
al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to
excitations, we derive scattering amplitudes involving solitons/antisolitons
first, and bound states later. The latter case comes out as manifestly related
to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this
nonlinear integral equations framework was contrived to deal with finite
geometries, we prove it to be effective for discovering or rediscovering
S-matrices. As a particular example, we prove that this unique model furnishes
explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe}
and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description
of unknown integrable field theories.Comment: Article, 41 pages, Late
On the Integrable Structure of the Ising Model
Starting from the lattice realization of the Ising model defined on a
strip with integrable boundary conditions, the exact spectrum (including
excited states) of all the local integrals of motion is derived in the
continuum limit by means of TBA techniques. It is also possible to follow the
massive flow of this spectrum between the UV conformal fixed point and
the massive IR theory. The UV expression of the eigenstates of such integrals
of motion in terms of Virasoro modes is found to have only rational
coefficients and their fermionic representation turns out to be simply related
to the quantum numbers describing the spectrum.Comment: 18 pages, no figure
Cross ratios and cubulations of hyperbolic groups
Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups . Under weak assumptions, we show that the space of cubulations of naturally injects into the space of -invariant cross ratios on the Gromov boundary . A consequence of our results is that essential, hyperplane-essential cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work in arXiv:1903.02447. Along the way, we describe the relationship between the Roller boundary of a cube complex, its Gromov boundary and - in the non-hyperbolic case - the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths. <br
Cross ratios on cube complexes and marked length-spectrum rigidity
We show that group actions on irreducible cube complexes with no free faces are uniquely determined by their length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of cube complexes. Inspired by results in negative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths. As a special case of our work, we construct a compactification of the Charney-Stambaugh-Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler-Vogtmann Outer Space
TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT
We consider high spin, , long twist, , planar operators (asymptotic
Bethe Ansatz) of strong SYM. Precisely, we compute the minimal
anomalous dimensions for large 't Hooft coupling to the lowest order
of the (string) scaling variable with GKP string size . At the leading order ,
we can confirm the O(6) non-linear sigma model description for this bulk term,
without boundary term . Going further, we derive,
extending the O(6) regime, the exact effect of the size finiteness. In
particular, we compute, at all loops, the first Casimir correction (in terms of the infinite size O(6) NLSM), which reveals only one
massless mode (out of five), as predictable once the O(6) description has been
extended. Consequently, upon comparing with string theory expansion, at one
loop our findings agree for large twist, while reveal for negligible twist,
already at this order, the appearance of wrapping. At two loops, as well as for
next loops and orders, we can produce predictions, which may guide future
string computations.Comment: Version 2 with: new exact expression for the Casimir energy derived
(beyond the first two loops of the previous version); UV theory formulated
and analysed extensively in the Appendix C; origin of the O(6) NLSM
scattering clarified; typos correct and references adde
Algebro-geometric analysis of bisectors of two algebraic plane curves
In this paper, a general theoretical study, from the perspective of the algebraic geometry, of the untrimmed bisector of two real algebraic plane curves is presented. The curves are considered in C2, and the real bisector is obtained by restriction to R2. If the implicit equations of the curves are given, the equation of the bisector is obtained by projection from a variety contained in C7, called the incidence variety, into C2. It is proved that all the components of the bisector have dimension 1. A similar method is used when the curves are given by parametrizations, but in this case, the incidence variety is in C5. In addition, a parametric representation of the bisector is introduced, as well as a method for its computation. Our parametric representation extends the representation in Farouki and Johnstone (1994b) to the case of rational curves
From the braided to the usual Yang-Baxter relation
Quantum monodromy matrices coming from a theory of two coupled (m)KdV
equations are modified in order to satisfy the usual Yang-Baxter relation. As a
consequence, a general connection between braided and {\it unbraided} (usual)
Yang-Baxter algebras is derived and also analysed.Comment: 13 Latex page
A braided Yang-Baxter Algebra in a Theory of two coupled Lattice Quantum KdV: algebraic properties and ABA representations
A generalization of the Yang-Baxter algebra is found in quantizing the
monodromy matrix of two (m)KdV equations discretized on a space lattice. This
braided Yang-Baxter equation still ensures that the transfer matrix generates
operators in involution which form the Cartan sub-algebra of the braided
quantum group. Representations diagonalizing these operators are described
through relying on an easy generalization of Algebraic Bethe Ansatz techniques.
The conjecture that this monodromy matrix algebra leads, {\it in the cylinder
continuum limit}, to a Perturbed Minimal Conformal Field Theory description is
analysed and supported.Comment: Latex file, 46 page
Three realizations of quantum affine algebra
In this article we establish explicit isomorphisms between three realizations
of quantum twisted affine algebra : the Drinfeld ("current")
realization, the Chevalley realization and the so-called realization,
investigated by Faddeev, Reshetikhin and Takhtajan.Comment: 15 page
- …