19 research outputs found
Some remarks on the GNS representations of topological -algebras
After an appropriate restatement of the GNS construction for topological
-algebras we prove that there exists an isomorphism among the set
\cycl(A) of weakly continuous strongly cyclic -representations of a
barreled dual-separable -algebra with unit , the space \hilb_A(A^*) of
the Hilbert spaces that are continuously embedded in and are
-invariant under the dual left regular action of and the set of the
corresponding reproducing kernels. We show that these isomorphisms are cone
morphisms and we prove many interesting results that follow from this fact. We
discuss how these results can be used to describe cyclic representations on
more general inner product spaces.Comment: 34 pages. Minor changes. To appear in J. Math. Phys. 49 (4) Apr-0
Some recursive formulas for Selberg-type integrals
A set of recursive relations satisfied by Selberg-type integrals involving
monomial symmetric polynomials are derived, generalizing previously known
results. These formulas provide a well-defined algorithm for computing
Selberg-Schur integrals whenever the Kostka numbers relating Schur functions
and the corresponding monomial polynomials are explicitly known. We illustrate
the usefulness of our results discussing some interesting examples.Comment: 11 pages. To appear in Jour. Phys.
Coulomb integrals for the SL(2,R) WZNW model
We review the Coulomb gas computation of three-point functions in the SL(2,R)
WZNW model and obtain explicit expressions for generic states. These amplitudes
have been computed in the past by this and other methods but the analytic
continuation in the number of screening charges required by the Coulomb gas
formalism had only been performed in particular cases. After showing that ghost
contributions to the correlators can be generally expressed in terms of Schur
polynomials we solve Aomoto integrals in the complex plane, a new set of
multiple integrals of Dotsenko-Fateev type. We then make use of monodromy
invariance to analytically continue the number of screening operators and prove
that this procedure gives results in complete agreement with the amplitudes
obtained from the bootstrap approach. We also compute a four-point function
involving a spectral flow operator and we verify that it leads to the one unit
spectral flow three-point function according to a prescription previously
proposed in the literature. In addition, we present an alternative method to
obtain spectral flow non-conserving n-point functions through well defined
operators and we prove that it reproduces the exact correlators for n=3.
Independence of the result on the insertion points of these operators suggests
that it is possible to violate winding number conservation modifying the
background charge.Comment: Improved presentation. New section on spectral flow violating
correlators and computation of a four-point functio
Constraints on the Variation of the Fine Structure Constant from Big Bang Nucleosynthesis
We put bounds on the variation of the value of the fine structure constant
, at the time of Big Bang nucleosynthesis. We study carefully all light
elements up to Li. We correct a previous upper limit on estimated from He primordial abundance and we find interesting new
potential limits (depending on the value of the baryon-to-photon ratio) from
Li, whose production is governed to a large extent by Coulomb barriers. The
presently unclear observational situation concerning the primordial abundances
preclude a better limit than |\Delta \alpha/\alpha| \lsim 2\cdot 10^{-2}, two
orders of magnitude less restrictive than previous bounds. In fact, each of the
(mutually exclusive) scenarios of standard Big Bang nucleosynthesis proposed,
one based on a high value of the measured deuterium primordial abundance and
one based on a low value, may describe some aspects of data better if a change
in of this magnitude is assumed.Comment: 21 pages, eps figures embedded using epsfig macr
Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
In the case of SU(2), associated by the AGT relation to the 2d Liouville
theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld
periods of 1d sine-Gordon model. If the same construction is literally applied
to monodromies of exact wave functions, the prepotential turns into the
one-parametric Nekrasov prepotential F(a,\epsilon_1) with the other epsilon
parameter vanishing, \epsilon_2=0, and \epsilon_1 playing the role of the
Planck constant in the sine-Gordon Shroedinger equation, \hbar=\epsilon_1. This
seems to be in accordance with the recent claim in arXiv:0908.4052 and poses a
problem of describing the full Nekrasov function as a seemingly straightforward
double-parametric quantization of sine-Gordon model. This also provides a new
link between the Liouville and sine-Gordon theories.Comment: 10 page
Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge
theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev
matrix model (beta-ensemble) representations the latter being polylinear
combinations of Selberg integrals. The "pure gauge" limit of these matrix
models is, however, a non-trivial multiscaling large-N limit, which requires a
separate investigation. We show that in this pure gauge limit the Selberg
integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the
Nekrasov function for pure SU(2) theory acquires a form very much reminiscent
of the AMM decomposition formula for some model X into a pair of the BGW
models. At the same time, X, which still has to be found, is the pure gauge
limit of the elliptic Selberg integral. Presumably, it is again a BGW model,
only in the Dijkgraaf-Vafa double cut phase.Comment: 21 page
Non-Perturbative Topological Strings And Conformal Blocks
We give a non-perturbative completion of a class of closed topological string
theories in terms of building blocks of dual open strings. In the specific case
where the open string is given by a matrix model these blocks correspond to a
choice of integration contour. We then apply this definition to the AGT setup
where the dual matrix model has logarithmic potential and is conjecturally
equivalent to Liouville conformal field theory. By studying the natural
contours of these matrix integrals and their monodromy properties, we propose a
precise map between topological string blocks and Liouville conformal blocks.
Remarkably, this description makes use of the light-cone diagrams of closed
string field theory, where the critical points of the matrix potential
correspond to string interaction points.Comment: 36 page
On "Dotsenko-Fateev" representation of the toric conformal blocks
We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the
original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal
blocks in the same sense that the spherical blocks are given by the integral
representation of arXiv:1001.0563 with a peculiar choice of open integration
contours for screening insertions. In other words, we provide some evidence
that the toric conformal blocks are reproduced by appropriate beta-ensembles
not only in the large-N limit, but also at finite N. The check is explicitly
performed at the first two levels for the 1-point toric functions.
Generalizations to higher genera are briefly discussed.Comment: 10 page
Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
We give a concise summary of the impressive recent development unifying a
number of different fundamental subjects. The quiver Nekrasov functions
(generalized hypergeometric series) form a full basis for all conformal blocks
of the Virasoro algebra and are sufficient to provide the same for some
(special) conformal blocks of W-algebras. They can be described in terms of
Seiberg-Witten theory, with the SW differential given by the 1-point resolvent
in the DV phase of the quiver (discrete or conformal) matrix model
(\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p}
\rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS
parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for
conformal blocks in terms of analytically continued contour integrals and
resolves the old puzzle of the free-field description of generic conformal
blocks through the Dotsenko-Fateev integrals. Most important, this completes
the GKMMM description of SW theory in terms of integrability theory with the
help of exact BS integrals, and provides an extended manifestation of the basic
principle which states that the effective actions are the tau-functions of
integrable hierarchies.Comment: 14 page