63 research outputs found
Equivalence Principle, Higher Dimensional Moebius Group and the Hidden Antisymmetric Tensor of Quantum Mechanics
We show that the recently formulated Equivalence Principle (EP) implies a
basic cocycle condition both in Euclidean and Minkowski spaces, which holds in
any dimension. This condition, that in one-dimension is sufficient to fix the
Schwarzian equation [6], implies a fundamental higher dimensional Moebius
invariance which in turn univocally fixes the quantum version of the
Hamilton-Jacobi equation. This holds also in the relativistic case, so that we
obtain both the time-dependent Schroedinger equation and the Klein-Gordon
equation in any dimension. We then show that the EP implies that masses are
related by maps induced by the coordinate transformations connecting different
physical systems. Furthermore, we show that the minimal coupling prescription,
and therefore gauge invariance, arises quite naturally in implementing the EP.
Finally, we show that there is an antisymmetric two-tensor which underlies
Quantum Mechanics and sheds new light on the nature of the Quantum
Hamilton-Jacobi equation.Comment: 1+48 pages, LaTeX. Expanded version, two appendices, several
comments, including comparison with Einstein Equivalence Principle, added.
Typos corrected, one reference added. To appear in CQ
Thermodynamics of black branes in asymptotically Lifshitz spacetimes
Recently, a class of gravitational backgrounds in 3+1 dimensions have been
proposed as holographic duals to a Lifshitz theory describing critical
phenomena in 2+1 dimensions with critical exponent . We continue our
earlier work \cite{Bertoldi:2009vn}, exploring the thermodynamic properties of
the "black brane" solutions with horizon topology . We find that
the black branes satisfy the relation where
is the energy density, is the temperature, and is the
entropy density. This matches the expected behavior for a 2+1 dimensional
theory with a scaling symmetry , .Comment: 8 pages, references added and regroupe
The Concept of a Noncommutative Riemann Surface
We consider the compactification M(atrix) theory on a Riemann surface Sigma
of genus g>1. A natural generalization of the case of the torus leads to
construct a projective unitary representation of pi_1(\Sigma), realized on the
Hilbert space of square integrable functions on the upper half--plane. A
uniquely determined gauge connection, which in turn defines a gauged sl_2(R)
algebra, provides the central extension. This has a geometric interpretation as
the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the
2nd Hochschild cohomology group of the Fuchsian group uniformizing Sigma. Our
construction can be seen as a suitable double-scaling limit N\to\infty,
k\to-\infty of a U(N) representation of pi_1(Sigma), where k is the degree of
the associated holomorphic vector bundle, which can be seen as the higher-genus
analog of 't Hooft's clock and shift matrices of QCD. We compare the above
mentioned uniqueness of the connection with the one considered in the
differential-geometric approach to the Narasimhan-Seshadri theorem provided by
Donaldson. We then use our infinite dimensional representation to construct a
C^\star-algebra which can be interpreted as a noncommutative Riemann surface
Sigma_\theta. Finally, we comment on the extension to higher genus of the
concept of Morita equivalence.Comment: 1+16 pages, LaTe
Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent
Recently, a class of gravitational backgrounds in 3+1 dimensions have been
proposed as holographic duals to a Lifshitz theory describing critical
phenomena in 2+1 dimensions with critical exponent . We numerically
explore black holes in these backgrounds for a range of values of . We find
drastically different behavior for and
() the Lifshitz fixed point is repulsive (attractive) when going to larger
radial parameter . For the repulsive backgrounds, we find a continuous
family of black holes satisfying a finite energy condition. However, for
we find that the finite energy condition is more restrictive, and we expect
only a discrete set of black hole solutions, unless some unexpected
cancellations occur. For all black holes, we plot temperature as a function
of horizon radius . For we find that this curve
develops a negative slope for certain values of possibly indicating a
thermodynamic instability.Comment: 23 pages, 6 figures, references corrected, graphs made readable in
greyscal
Large N gauge theories and topological cigars
We analyze the conjectured duality between a class of double-scaling limits
of a one-matrix model and the topological twist of non-critical superstring
backgrounds that contain the N=2 Kazama-Suzuki SL(2)/U(1) supercoset model. The
untwisted backgrounds are holographically dual to double-scaled Little String
Theories in four dimensions and to the large N double-scaling limit of certain
supersymmetric gauge theories. The matrix model in question is the auxiliary
Dijkgraaf-Vafa matrix model that encodes the F-terms of the above
supersymmetric gauge theories. We evaluate matrix model loop correlators with
the goal of extracting information on the spectrum of operators in the dual
non-critical bosonic string. The twisted coset at level one, the topological
cigar, is known to be equivalent to the c=1 non-critical string at self-dual
radius and to the topological theory on a deformed conifold. The spectrum and
wavefunctions of the operators that can be deduced from the matrix model
double-scaling limit are consistent with these expectations.Comment: 34 page
Double Scaling Limits in Gauge Theories and Matrix Models
We show that gauge theories with an adjoint chiral multiplet admit a
wide class of large-N double-scaling limits where is taken to infinity in a
way coordinated with a tuning of the bare superpotential. The tuning is such
that the theory is near an Argyres-Douglas-type singularity where a set of
non-local dibaryons becomes massless in conjunction with a set of confining
strings becoming tensionless. The doubly-scaled theory consists of two
decoupled sectors, one whose spectrum and interactions follow the usual large-N
scaling whilst the other has light states of fixed mass in the large-N limit
which subvert the usual large-N scaling and lead to an interacting theory in
the limit. -term properties of this interacting sector can be calculated
using a Dijkgraaf-Vafa matrix model and in this context the double-scaling
limit is precisely the kind investigated in the "old matrix model'' to describe
two-dimensional gravity coupled to conformal field theories. In
particular, the old matrix model double-scaling limit describes a sector of a
gauge theory with a mass gap and light meson-like composite states, the
approximate Goldstone boson of superconformal invariance, with a mass which is
fixed in the double-scaling limit. Consequently, the gravitational -terms in
these cases satisfy the string equation of the KdV hierarchy.Comment: 38 pages, 1 figure, reference adde
Double Scaling Limits and Twisted Non-Critical Superstrings
We consider double-scaling limits of multicut solutions of certain one matrix
models that are related to Calabi-Yau singularities of type A and the
respective topological B model via the Dijkgraaf-Vafa correspondence. These
double-scaling limits naturally lead to a bosonic string with c 1. We
argue that this non-critical string is given by the topologically twisted
non-critical superstring background which provides the dual description of the
double-scaled little string theory at the Calabi-Yau singularity. The
algorithms developed recently to solve a generic multicut matrix model by means
of the loop equations allow to show that the scaling of the higher genus terms
in the matrix model free energy matches the expected behaviour in the
topological B-model. This result applies to a generic matrix model singularity
and the relative double-scaling limit. We use these techniques to explicitly
evaluate the free energy at genus one and genus two.Comment: 32 pages, 3 figure
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