709 research outputs found

### Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum
gravity whose basic object is the Liouville action on the Riemann sphere
$\Sigma_{0,m+n}$ with both parabolic and elliptic points. The identification of
the classical limit of the conformal Ward identity with the Fuchsian projective
connection on $\Sigma_{0,m+n}$ implies a relation between conformal weights and
ramification indices. This formulation works for arbitrary $d$ and admits a
standard representation only for $d\le 1$. Furthermore, it turns out that the
integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for
$n=2m+1$ coincides with the unitary minimal series of CFT.Comment: DFPD/93/TH/62. Remarks on the d=1 barrier and references adde

### The Higgs model for anyons and Liouville action: Chaotic spectrum, energy gap and exclusion principle

Geodesic completness and self-adjointness imply that the Hamiltonian for
anyons is the Laplacian with respect to the Weil-Petersson metric. This metric
is complete on the Deligne-Mumford compactification of moduli (configuration)
space. The structure of this compactification fixes the possible anyon
configurations. This allows us to identify anyons with singularities (elliptic
points with ramification $q^{-1}$) in the Poincar\'e metric implying that anyon
spectrum is chaotic for $n\ge 3$. Furthermore, the bound on the holomorphic
sectional curvature of moduli spaces implies a gap in the energy spectrum. For
$q=0$ (punctures) anyons are infinitely separated in the Poincar\'e metric
(hard-core). This indicates that the exclusion principle has a geometrical
intepretation. Finally we give the differential equation satisfied by the
generating function for volumes of the configuration space of anyons.Comment: DFPD/93/TH/69, LaTe

### Uniformization theory and 2D gravity I. Liouville action and intersection numbers

This is the first part of an investigation concerning the formulation of 2D
gravity in the framework of the uniformization theory of Riemann surfaces. As a
first step in this direction we show that the classical Liouville action
appears in the expression of the correlators of topological gravity. Next we
derive an inequality involving the cutoff of 2D gravity and the background
geometry. Another result, always related to uniformization theory, concerns a
relation between the higher genus normal ordering and the Liouville action.
Furthermore, we show that the chirally split anomaly of CFT is equivalent to
the Krichever-Novikov cocycle. By means of the inverse map of uniformization we
give a realization of the Virasoro algebra on arbitrary Riemann surfaces and
find the eigenfunctions for {\it holomorphic} covariant operators defining
higher order cocycles and anomalies which are related to $W$-algebras. Finally
we attack the problem of considering the positivity of $e^\sigma$, with
$\sigma$ the Liouville field, by proposing an explicit construction for the
Fourier modes on compact Riemann surfaces.Comment: 53 pages. '95 publ. version, contains Eq.(5.23), independently
derived in hep-th/0004194 studying the null compactification of
type-IIA-strin

### N=2 SYM RG Scale as Modulus for WDVV Equations

We derive a new set of WDVV equations for N=2 SYM in which the
renormalization scale $\Lambda$ is identified with the distinguished modulus
which naturally arises in topological field theories.Comment: 6 pages, LaTe

### Koebe 1/4-Theorem and Inequalities in N=2 Super-QCD

The critical curve ${\cal C}$ on which ${\rm Im}\,\hat\tau =0$,
$\hat\tau=a_D/a$, determines hyperbolic domains whose Poincar\'e metric is
constructed in terms of $a_D$ and $a$. We describe ${\cal C}$ in a parametric
form related to a Schwarzian equation and prove new relations for $N=2$ Super
$SU(2)$ Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz's
lemma, we obtain inequalities involving $u$, $a_D$ and $a$, which seem related
to the Renormalization Group. Furthermore, we obtain a closed form for the
prepotential as function of $a$. Finally, we show that $\partial_{\hat\tau}
\langle {\rm tr}\,\phi^2\rangle_{\hat \tau}={1\over 8\pi i b_1}\langle
\phi\rangle_{\hat\tau}^2$, where $b_1$ is the one-loop coefficient of the beta
function.Comment: 11 pages, LaTex file, Expanded version: new results, technical
details explained, misprints corrected and references adde

### Reply to Comment on "Duality of x and psi in Quantum Mechanics"

The content of the comment [hep-th/9712219] is the derivation of Eq.(13) in
Phys. Rev. Lett. 78 (1997) 163 by direct differential calculus: which is
precisely the same method we used to derive it (it is in fact difficult to
imagine any other possible derivation).Comment: 2 pages, LaTe

### Solving N=2 SYM by Reflection Symmetry of Quantum Vacua

The recently rigorously proved nonperturbative relation between u and the
prepotential, underlying N=2 SYM with gauge group SU(2), implies both the
reflection symmetry $\overline{u(\tau)}=u(-\bar\tau)$ and $u(\tau+1)=-u(\tau)$
which hold exactly. The relation also implies that $\tau$ is the inverse of the
uniformizing coordinate u of the moduli space of quantum vacua. In this
context, the above quantum symmetries are the key points to determine the
structure of the moduli space. It turns out that the functions a(u) and a_D(u),
which we derive from first principles, actually coincide with the solution
proposed by Seiberg and Witten. We also consider some relevant generalizations.Comment: 12 pg. LaTex, Discussion of the generalization to higher rank groups
added. To be published in Phys. Rev.

### On the Structure of Noncommutative N=2 Super Yang-Mills Theory

We show that the recently proposed formulation of noncommutative N=2 Super
Yang-Mills theory implies that the commutative and noncommutative effective
coupling constants \tau(u) and \tau_{nc}(u) coincide. We then introduce a key
relation which allows to find a nontrivial solution of such equation, thus
fixing the form of the low-energy effective action. The dependence on the
noncommutative parameter arises from a rational function deforming the
Seiberg-Witten differential.Comment: 1+5 pages, LaTe

### Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

A long-standing question in string theory is to find the explicit expression
of the bosonic measure, a crucial issue also in determining the superstring
measure. Such a measure was known up to genus three. Belavin and Knizhnik
conjectured an expression for genus four which has been proved in the framework
of the recently introduced vector-valued Teichmueller modular forms. It turns
out that for g>3 the bosonic measure is expressed in terms of such forms. In
particular, the genus four Belavin-Knizhnik "wonderful formula" has a
remarkable extension to arbitrary genus whose structure is deeply related to
the characterization of the Jacobian locus. Furthermore, it turns out that the
bosonic string measure has an elegant geometrical interpretation as generating
the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to
identify forms on the Siegel upper half-space that, if certain conditions
related to the characterization of the Jacobian are satisfied, express the
bosonic measure as a multiresidue in the Siegel upper half-space. We also
suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde

### Noncommutative Riemann Surfaces

We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1.
Following [1], we construct a projective unitary representation of pi_1(Sigma)
realized on L^2(H), with H the upper half-plane. As a first step we introduce a
suitably gauged sl_2(R) algebra. Then a uniquely determined gauge connection
provides the central extension which is a 2-cocycle of the 2nd Hochschild
cohomology group. Our construction is the double-scaling limit N\to\infty,
k\to-\infty of the representation considered in the Narasimhan-Seshadri
theorem, which represents the higher-genus analog of 't Hooft's clock and shift
matrices of QCD. The concept of a noncommutative Riemann surface Sigma_\theta
is introduced as a certain C^\star-algebra. Finally we investigate the Morita
equivalence.Comment: LaTeX, 1+14 pages. Contribution to the TMR meeting ``Quantum aspects
of gauge theories, supersymmetry and unification'', Paris 1-7 September 199

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