10,332 research outputs found
(2+1) gravity for higher genus in the polygon model
We construct explicitly a (12g-12)-dimensional space P of unconstrained and
independent initial data for 't Hooft's polygon model of (2+1) gravity for
vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our
method relies on interpreting the boost parameters of the gluing data between
flat Minkowskian patches as the lengths of certain geodesic curves of an
associated smooth Riemann surface of the same genus. The appearance of an
initial big-bang or a final big-crunch singularity (but never both) is verified
for all configurations. Points in P correspond to spacetimes which admit a
one-polygon tessellation, and we conjecture that P is already the complete
physical phase space of the polygon model. Our results open the way for
numerical investigations of pure (2+1) gravity.Comment: 35 pages, 22 figure
GLSM realizations of maps and intersections of Grassmannians and Pfaffians
In this paper we give gauged linear sigma model (GLSM) realizations of a
number of geometries not previously presented in GLSMs. We begin by describing
GLSM realizations of maps including Veronese and Segre embeddings, which can be
applied to give GLSMs explicitly describing constructions such as the
intersection of one hypersurface with the image under some map of another. We
also discuss GLSMs for intersections of Grassmannians and Pfaffians with one
another, and with their images under various maps, which sometimes form exotic
constructions of Calabi-Yaus, as well as GLSMs for other exotic Calabi-Yau
constructions of Kanazawa. Much of this paper focuses on a specific set of
examples of GLSMs for intersections of Grassmannians G(2,N) with themselves
after a linear rotation, including the Calabi-Yau case N=5. One phase of the
GLSM realizes an intersection of two Grassmannians, the other phase realizes an
intersection of two Pfaffians. The GLSM has two nonabelian factors in its gauge
group, and we consider dualities in those factors. In both the original GLSM
and a double-dual, one geometric phase is realized perturbatively (as the
critical locus of a superpotential), and the other via quantum effects.
Dualizing on a single gauge group factor yields a model in which each geometry
is realized through a simultaneous combination of perturbative and quantum
effects.Comment: LaTeX, 50 pages; v2: typos fixed and a few comments on other
dualities adde
On spinodal points and Lee-Yang edge singularities
We address a number of outstanding questions associated with the analytic
properties of the universal equation of state of the theory, which
describes the critical behavior of the Ising model and ubiquitous critical
points of the liquid-gas type. We focus on the relation between spinodal points
that limit the domain of metastability for temperatures below the critical
temperature, i.e., , and Lee-Yang edge singularities that
restrict the domain of analyticity around the point of zero magnetic field
for . The extended analyticity conjecture (due to Fonseca and
Zamolodchikov) posits that, for , the Lee-Yang edge
singularities are the closest singularities to the real axis. This has
interesting implications, in particular, that the spinodal singularities must
lie off the real axis for , in contrast to the commonly known result
of the mean-field approximation. We find that the parametric representation of
the Ising equation of state obtained in the expansion, as
well as the equation of state of the -symmetric theory at
large , are both nontrivially consistent with the conjecture. We analyze the
reason for the difficulty of addressing this issue using the
expansion. It is related to the long-standing paradox associated with the fact
that the vicinity of the Lee-Yang edge singularity is described by Fisher's
theory, which remains nonperturbative even for , where the
equation of state of the theory is expected to approach the mean-field
result. We resolve this paradox by deriving the Ginzburg criterion that
determines the size of the region around the Lee-Yang edge singularity where
mean-field theory no longer applies.Comment: 26 pages, 8 figures; v2: shortened Sec. 4.1 and streamlined
arguments/notation in Sec. 4.2, details moved to appendix, added reference 1
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