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M. Matone

Publications and source records attributed to M. Matone.

16 recordsLinked to original sources
arXiv2003

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 $e^σ$, with $σ$ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces.

hep-th↗
arXiv2000

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(Σ), 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_θ. Finally, we comment on the extension to higher genus of the concept of Morita equivalence.

hep-th↗
arXiv2000

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_θis introduced as a certain C^\star-algebra. Finally we investigate the Morita equivalence.

hep-th↗
arXiv1998

RG Flow Irreversibility, C-Theorem and Topological Nature of 4D N=2 SYM

We determine the exact beta function and a RG flow Lyapunov function for N=2 SYM with gauge group SU(n). It turns out that the classical discriminants of the Seiberg-Witten curves determine the RG potential. The radial irreversibility of the RG flow in the SU(2) case and the non-perturbative identity relating the $u$-modulus and the superconformal anomaly, indicate the existence of a four dimensional analogue of the c-theorem for N=2 SYM which we formulate for the full SU(n) theory. Our investigation provides further evidence of the essentially topological nature of the theory.

hep-th↗
arXiv1998

Beta Function, C--Theorem and WDVV Equations in 4D N=2 SYM

We show that the exact $beta$--function of 4D N=2 SYM plays the role of the metric whose inverse satisfies the WDVV--like equations $\F_{ikl}β^{lm} \F_{mnj}=\F_{jkl}β^{lm}\F_{mni}$. The conjecture that the WDVV--like equations are equivalent to the identity involving the $u$--modulus and the prepotential $\F$, seen as a superconformal anomaly, sheds light on the recently considered c-theorem for the N=2 SYM field theories.

hep-th↗
arXiv1997

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).

hep-th↗
arXiv1997

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 $Λ$ is identified with the distinguished modulus which naturally arises in topological field theories.

hep-th↗
arXiv1997

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 $\overline{u(τ)}=u(-\barτ)$ and $u(τ+1)=-u(τ)$ which hold exactly. The relation also implies that $τ$ 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.

hep-th↗
arXiv1996

Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation

We find the nonperturbative relation between $\langle {\rm tr} ϕ^2 \rangle$, $\langle {\rm tr} ϕ^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle ϕ_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space. The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups.

hep-th↗
arXiv1996

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

The critical curve ${\cal C}$ on which ${\rm Im}\,\hatτ=0$, $\hatτ=a_D/a$, determines hyperbolic domains whose Poincaré 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 $\partial_{\hatτ} \langle {\rm tr}\,ϕ^2\rangle_{\hat τ}={1\over 8πi b_1}\langle ϕ\rangle_{\hatτ}^2$, where $b_1$ is the one-loop coefficient of the beta function.

hep-th↗
arXiv1995

Algebraic-geometrical formulation of two-dimensional quantum gravity

We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlevé I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem.

hep-th↗
arXiv1994

Nonperturbative 2D Gravity, Punctured Spheres and $Θ$-Vacua in String Theories

We consider a model of 2D gravity with the coefficient of the Einstein-Hilbert action having an imaginary part $π/2$. This is equivalent to introduce a $Θ$-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumably Borel summable. We show that the specific heat of the model has a physical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitable moduli space of infinitely punctured spheres. This is an explicit realization à la Friedan-Shenker of 2D quantum gravity. We conjecture that the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expansions in terms of punctured spheres also for multicritical models.

hep-th↗
arXiv1994

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 Poincaré 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 Poincaré 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.

hep-th↗
arXiv1994

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 $Σ_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $Σ_{0,m+n}$ 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.

hep-th↗
arXiv1994

Nonperturbative Model Of Liouville Gravity

We obtain nonperturbative results in the framework of continuous Liouville theory. In particular, we express the specific heat ${\cal Z}$ of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. The integrands are written in terms of the Liouville action. We show that ${\cal Z}$ satisfies the Painlevé I.

hep-th↗