151 research outputs found
Modular Invariance and Structure of the Exact Wilsonian Action of N=2 SYM
We construct modular invariants on the moduli space of quantum vacua of N=2
SYM with gauge group SU(2). We also introduce a nonchiral function K which is
expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out
that K has all the expected properties of the next to leading term in the
Wilsonian effective action whose modular properties are considered in the
framework of the dimensional regularization.Comment: 10 pages, LaTeX file, misprints and a factor 2 in the derivation of
alpha correcte
Quantum Field Perturbation Theory Revisited
Schwinger's formalism in quantum field theory can be easily implemented in
the case of scalar theories in dimension with exponential interactions,
such as . In particular, we use the relation
with the external source, and . Such a
shift is strictly related to the normal ordering of and to a
scaling relation which follows by renormalizing . Next, we derive a new
formulation of perturbation theory for the potentials , using the generating functional associated to
. The -terms related to the normal ordering are
absorbed at once. The functional derivatives with respect to to compute the
generating functional are replaced by ordinary derivatives with respect to
auxiliary parameters. We focus on scalar theories, but the method is general
and similar investigations extend to other theories.Comment: 21 pages. Includes a modified Feynman propagator which is massless in
D=4 and scaling relations for the generating functional. References added.
PRD versio
A Surprising Relation for the Effective Coupling Constants of N=2 Super Yang-Mills Theories
We show that the effective coupling constants of supersymmetric gauge
theories described by hyperelliptic curves do not distinguish between the
lattices of the two kinds of heterotic string. In particular, the following
relation holds. This is
reminiscent of the relation, by -duality, of the two heterotic strings. We
suggest that such a relation extends to all curves describing effective
supersymmetric gauge theories.Comment: 7 page
Modular Invariant Regularization of String Determinants and the Serre GAGA Principle
Since any string theory involves a path integration on the world-sheet
metric, their partition functions are volume forms on the moduli space of genus
g Riemann surfaces M_g, or on its super analog. It is well known that modular
invariance fixes strong constraints that in some cases appear only at higher
genus. Here we classify all the Weyl and modular invariant partition functions
given by the path integral on the world-sheet metric, together with space-time
coordinates, b-c and/or beta-gamma systems, that correspond to volume forms on
M_g. This was a long standing question, advocated by Belavin and Knizhnik,
inspired by the Serre GAGA principle and based on the properties of the Mumford
forms. The key observation is that the Bergman reproducing kernel provides a
Weyl and modular invariant way to remove the point dependence that appears in
the above string determinants, a property that should have its superanalog
based on the super Bergman reproducing kernel. This is strictly related to the
properties of the propagator associated to the space-time coordinates. Such
partition functions Z[J] have well-defined asymptotic behavior and can be
considered as a basis to represent a wide class of string theories. In
particular, since non-critical bosonic string partition functions Z_D are
volume forms on M_g, we suggest that there is a mapping, based on bosonization
and degeneration techniques, from the Liouville sector to first order systems
that may identify Z_D as a subclass of the Z[J]. The appearance of b-c and
beta-gamma systems of any conformal weight shows that such theories are related
to W algebras. The fact that in a large N 't Hooft-like limit 2D W_N minimal
models CFTs are related to higher spin gravitational theories on AdS_3,
suggests that the string partition functions introduced here may lead to a
formulation of higher spin theories in a string context.Comment: 30 pp. New results on Quillen metric and its relations with the
Mumford forms and the tautological classes. References adde
"Thermodynamique cach\'ee des particules" and the quantum potential
According to de Broglie, temperature plays a basic role in quantum
Hamilton-Jacobi theory. Here we show that a possible dependence on the
temperature of the integration constants of the relativistic quantum
Hamilton-Jacobi may lead to corrections to the dispersion relations. The change
of the relativistic equations is simply described by means of a thermal
coordinate.Comment: 8 page
An algorithm for the Baker-Campbell-Hausdorff formula
A simple algorithm, which exploits the associativity of the BCH formula, and
that can be generalized by iteration, extends the remarkable simplification of
the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and
Visser. We show that if
, , and, consistently with the Jacobi
identity, , then where , , and are
solutions of four equations. In particular, the Van-Brunt and Visser formula
extends to cases when
contains also elements different from and . Such a closed form of the
BCH formula may have interesting applications both in mathematics and physics.
As an application, we provide the closed form of the BCH formula in the case of
the exponentiation of the Virasoro algebra, with
following as a subcase. We also determine three-dimensional subalgebras of the
Virasoro algebra satisfying the Van-Brunt and Visser condition. It turns out
that the exponential form of has a nice representation in
terms of its eigenvalues and of the
fixed points of the corresponding M\"obius transformation. This may have
applications in Uniformization theory and Conformal Field Theories.Comment: 1+8 pages. Comments and refences added. Typos corrected. Version to
appear in JHE
Liouville Equation And Schottky Problem
An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed.
This implies that the Liouville equation reduces to an equation resembling a
non chiral analogous of the higher genus relationships (KP equation) arising in
the framework of Schottky's problem solution. This approach connects
uniformization (Fuchsian groups) and moduli space theories with KP hierarchy.
Besides its mathematical interest, the Ansatz has some applications in the
framework of quantum Riemann surfaces arising in 2D gravity.Comment: 12 pages, LaTex file. Expanded version and misprints correcte
Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas
We show that there are {\it 13 types} of commutator algebras leading to the
new closed forms of the Baker-Campbell-Hausdorff (BCH) formula
derived in arXiv:1502.06589,
JHEP {\bf 1505} (2015) 113. This includes, as a particular case, , with containing other elements in addition to and .
The algorithm exploits the associativity of the BCH formula and is based on the
decomposition , with fixed in such a way that it reduces to
, with and satisfying
the Van-Brunt and Visser condition . It turns out that satisfies, in the generic
case, an algebraic equation whose exponents depend on the parameters defining
the commutator algebra. In nine {\it types} of commutator algebras, such an
equation leads to rational solutions for . We find all the equations
that characterize the solution of the above decomposition problem by combining
it with the Jacobi identity.Comment: 15 pages. Typos corrected. JGP versio
Equivalence Postulate and the Quantum Potential of Two Free Particles
Commutativity of the diagram of the maps connecting three one--particle
state, implied by the Equivalence Postulate (EP), gives a cocycle condition
which unequivocally leads to the quantum Hamilton--Jacobi equation. Energy
quantization is a direct consequences of the local homeomorphicity of the
trivializing map. We review the EP and show that the quantum potential for two
free particles, which depends on constants which may have a geometrical
interpretation, plays the role of interaction term that admits solutions which
do not vanish in the classical limit.Comment: 7 pages, LaTeX. Talk at the First International Conference on String
Cosmology. Oxford, United Kingdom. July 200
Equivalence Postulate and Quantum Origin of Gravitation
We suggest that quantum mechanics and gravity are intimately related. In
particular, we investigate the quantum Hamilton-Jacobi equation in the case of
two free particles and show that the quantum potential, which is attractive,
may generate the gravitational potential. The investigation, related to the
formulation of quantum mechanics based on the equivalence postulate, is based
on the analysis of the reduced action. A consequence of this approach is that
the quantum potential is always non-trivial even in the case of the free
particle. It plays the role of intrinsic energy and may in fact be at the
origin of fundamental interactions. We pursue this idea, by making a
preliminary investigation of whether there exists a set of solutions for which
the quantum potential can be expressed with a gravitational potential leading
term which alone would remain in the limit hbar \to 0. A number of questions
are raised for further investigation.Comment: 1+19 pages, minor changes and typos corrected, to appear in Found.
Phys. Let
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