arXiv subjects
Publications and source records attributed to Paolo Valtancoli.
Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.
We show how to generate the AdS spacetime metric in general Lorentz covariant gauges. In particular we propose an iterative method for solving the Lorentz gauge.
We show how to solve the Dirac oscillator with a minimal length by using previous results on the harmonic oscillator in a Snyder algebra.
We find that, in presence of the Snyder geometry, the notion of translational invariance needs to be modified, allowing a momentum dependence of this symmetry. This step is necessary to build the maximally localized states and the Feynman rules of the corresponding quantum field theory.
We solve exactly the scalar box integral using the Mellin-Barnes representation. Firstly we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we perform an analytic continuation before applying the Laurent expansion in^2 = (d !' 4)=2 of the result.
We reconsider the perturbative expansion of the Wilson loop in 2d noncommutative gauge theories, using an improved integration method. For the class of maximally crossed diagrams in the $θ\to \infty$ limit we find an intriguing formula, easily generalizable to all orders in perturbation theory.
We solve (2+1) noncommutative gravity coupled to point-like sources. We find continuity with Einstein gravity since we recover the classical gravitational field in the $θ\to 0$ limit or at large distance from the source. It appears a limitation on the mass which is twice than expected. Since the distance is not gauge invariant, the measure of the deficit angle near the source is intrinsically ambiguous, with the gauge group playing the role of statistical ensemble. Einstein determinism can be recovered only at large distance from the source, compared with the scale of the noncommutative parameter $\sqrtθ$.
We compare the classical scattering of kinks in (1+1) Higgs model with its analogous noncommutative counterpart. While at a classical level we are able to solve the scattering at all orders finding a smooth solution, at a noncommutative level we present only perturbative results, suggesting the existence of a smooth solution also in this case.
We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions.
We introduce a matrix model for noncommutative gravity, based on the gauge group $U(2) \otimes U(2)$. The vierbein is encoded in a matrix $Y_μ$, having values in the coset space $U(4)/ (U(2) \otimes U(2))$, while the spin connection is encoded in a matrix $X_μ$, having values in $U(2) \otimes U(2)$. We show how to recover the Einstein equations from the $θ\to 0$ limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by $U(2) \otimes U(2)$ valued quasi-unitary operators acting on the background of the Matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections.
We describe how to reduce the fuzzy four-sphere algebra to a set of four independent raising and lowering oscillator operators. In terms of them we derive the projector valued operators for the fuzzy four-sphere, which are the global definition of k-instanton connections over this noncommutative base manifold.
In the case of an invertible coordinate commutator matrix $θ_{ij}$, we derive a general instanton solution of the noncommutative gauge theories on $d=2n$ planes given in terms of $n$ oscillators.
Recently we have introduced a matrix model depending on two coupling constants $g^2$ and $λ$, which contains the fuzzy sphere as a background; to obtain the classical limit $g^2$ must depend on $N$ in a precise way. In this paper we show how to obtain the classical solitons of the $N \to \infty$ limit imposing the development $λ= \half + \frac{λ_0}{N}$; as a consequence at finite $N$ one obtains a noncommutative version of the solitons for the fuzzy sphere.
We consider a matrix model depending on a parameter $λ$ which permits the fuzzy sphere as a classical background.By expanding the bosonic matrices around this background ones recovers a U(1) (U(n)) noncommutative gauge theory on the fuzzy sphere. To check classical stability of this background, we look for new classical solutions of this model and find them for $λ< 1$, that make the fuzzy sphere solution unstable for $λ< \half$ and stable otherwise. \\
We discuss a formalism for solving (2+1) AdS gravity on Riemann surfaces. In the torus case the equations of motion are solved by two functions f and g, solutions of two independent O(2,1) sigma models, which are distinct because their first integrals contain a different time dependent phase factor. We then show that with the gauge choice $k = \sqrtΛ/ tg (2 \sqrtΛt)$ the same couple of first integrals indeed solves exactly the Einstein equations for every Riemann surface. The $X^A=X^A(x^mu)$ polydromic mapping which extends the standard immersion of a constant curvature three-dimensional surface in a flat four-dimensional space to the case of external point sources or topology, is calculable with a simple algebraic formula in terms only of the two sigma model solutions f and g. A trivial time translation of this formalism allows us to introduce a new method which is suitable to study the scattering of black holes in (2+1) AdS gravity.
We derive a perturbative scheme to treat the interaction between point sources and AdS-gravity. The interaction problem is equivalent to the search of a polydromic mapping $X^A= X^A(x^μ)$, endowed with 0(2,2) monodromies, between the physical coordinate system and a Minkowskian 4-dimensional coordinate system, which is however constrained to live on a hypersurface. The physical motion of point sources is therefore mapped to a geodesic motion on this hypersuface. We impose an instantaneous gauge which induces a set of equations defining such a polydromic mapping. Their consistency leads naturally to the Einstein equations in the same gauge. We explore the restriction of the monodromy group to O(2,1), and we obtain the solution of the fields perturbatively in the cosmological constant.
The Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theories with matter is analysed as an isomonodromy problem. We show that the holomorphic section describing the effective action can be deformed by moving its singularities on the moduli space while keeping their monodromies invariant. Well-known examples of isomonodromic sections are given by the correlators of two-dimensional rational conformal field theories -- the conformal blocks. The Seiberg-Witten section similarly admits the operations of braiding and fusing of its singularities, which obey the Yang-Baxter and Pentagonal identities, respectively. Using them, we easily find the complete expressions of the monodromies with affine term, and the full quantum numbers of the BPS spectrum. While the braiding describes the quark-monopole transmutation, the fusing implies the superconformal points in the moduli space. In the simplest case of three singularities, the supersymmetric sections are directly related to the conformal blocks of the logarithmic minimal models.
We search a canonical basis of Dirac's observables for the classical Abelian Higgs model with fermions in the case of a trivial U(1) principal bundle. The study of the Gauss law first class constraint shows that the model has two disjoint sectors of solutions associated with two physically different phases. In the electromagnetic phase, the electromagnetic field remains massless: after the determination of the Dirac's observables we get that both the reduced physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the electromagnetic field becomes massive and in terms of Dirac's observables we get a local, but nonanalytic in the electric charge (or equivalently in the sum of the electromagnetic mass and of the residual Higgs field), physical Hamiltonian; however the associated Lagrangian is nonlocal. Some comments on the R-gauge-fixing, the possible elimination of the residual Higgs field and on the Nielsen-Olesen vortex solution close the paper.