arXiv subjects
Publications and source records attributed to Paolo Rossi.
The effective low energy Lagrangian of $CP^{N-1}$ models in $d < 4$ dimensions can be constructed in the large $N$ limit by solving the saddle point equations in the presence of a constant field strength. The two dimensional case is explicitly worked out and possible applications are briefly discussed.
We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1 + 1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed Gelfand-Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki-Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz-Ladik hierarchy to Gromov-Witten theory by proving an analogous mirror theorem for rational reductions: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi-Yau threefolds obtained from minimal resolutions of the local orbifold line.
In this paper we investigate the algebraic structure related to a new type of correlator associated to the moduli spaces of $S^1$-parametrized curves in contact homology and rational symplectic field theory. Such correlators are the natural generalization of the non-equivariant linearized contact homology differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis (or hereditary) operator (à la Magri-Fuchssteiner) in contact homology which recovers the descendant theory from the primaries. We also sketch how such structure generalizes to the full SFT Poisson homology algebra to a (graded symmetric) bivector. The descendant hamiltonians satisfy to recursion relations, analogous to bihamiltonian recursion, with respect to the pair formed by the natural Poisson structure in SFT and such bivector. In case the target manifold is the product stable Hamiltonian structure $S^1\times M$, with $M$ a symplectic manifold, the recursion coincides with genus $0$ topological recursion relations in the Gromov-Witten theory of $M$.
The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is discussed. A wide set of measurable quantities ("invariant moments") whose expectation value does not in general depend on the size of the sample is constructed and illustrated by applying the results to Ewens sampling formula. Invariant moments are especially useful in the sampling of systems characterized by the absence of an intrinsic scale. Distribution functions that may parametrize the samples of scale-free distributions are considered and their invariant expectation values are computed. The conditions under which the scaling limit of such distributions may exist are described.
The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is esablished. A set of combinations of expectation values whose value does not in general depend on the size of the sample is constructed. Distribution functions such that the distribution of the expectation values of their samples is invariant in form are found and studied. The conditions under which the scaling limit of such distributions may exist are described.
Based on the localization result for descendants in rational SFT moduli spaces from our last joint paper, we prove topological recursion relations for the Hamiltonian in SFT of symplectic mapping tori and in local SFT. Combined with the dilaton equation in SFT, we use them to prove a reconstruction theorem for descendants from primaries. While it turns out that (in contrast to Gromov-Witten theory and non-equivariant cylindrical contact homology) the descendant Hamiltonian cannot be computed from the Hamiltonian without descendants alone, we get that the only additional piece of information needed is the first descendant Hamiltonian, which counts holomorphic curves tangent to the symplectic fibre. As already known from the explicit computations in local SFT, it follows that the descendant SFT invariants in general contain more geometric information than the primary SFT invariants.
The time dependence of the $h$-index is analyzed by considering the average behaviour of $h$ as a function of the academic age $A_A$ for about 1400 Italian physicists, with career lengths spanning from 3 to 46 years. The individual $h$-index is strongly correlated with the square root of the total citations $N_C$: $h \approx 0.53 \sqrt{N_C}$. For academic ages ranging from 12 to 24 years, the distribution of the time scaled index $h/\sqrt{A_A}$ is approximately time-independent and it is well described by the Gompertz function. The time scaled index $h/\sqrt{A_A}$ has an average approximately equal to 3.8 and a standard deviation approximately equal to 1.6. Finally, the time scaled index $h/\sqrt{A_A}$ appears to be strongly correlated with the contemporary $h$-index $h_c$.
We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of local CP1. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two main strands. First of all, we exhibit a manifestly local bi-Hamiltonian structure of the Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make precise the relation between this canonical Frobenius structure and the one that underlies the Gromov-Witten theory of the resolved conifold in the equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of "almost duality" of Frobenius manifolds. As a consequence, we obtain a derivation of genus zero mirror symmetry for local CP1 in terms of a dual logarithmic Landau-Ginzburg model.
Infinite dimensional Hamiltonian systems appear naturally in the rich algebraic structure of Symplectic Field Theory. Carefully defining a generalization of gravitational descendants and adding them to the picture, one can produce an infinite number of symmetries of such systems . As in Gromov-Witten theory, the study of the topological meaning of gravitational descendants yields new differential equations for the SFT Hamiltonian, where the key point is to understand the dependence of the algebraic constructions on choices of auxiliary data like contact form, cylindrical almost complex structure, abstract perturbations, differential forms and coherent collections of sections used to define gravitational descendants.
It was pointed out by Eliashberg in his ICM 2006 plenary talk that the integrable systems of rational Gromov-Witten theory very naturally appear in the rich algebraic formalism of symplectic field theory (SFT). Carefully generalizing the definition of gravitational descendants from Gromov-Witten theory to SFT, one can assign to every contact manifold a Hamiltonian system with symmetries on SFT homology and the question of its integrability arises. While we have shown how the well-known string, dilaton and divisor equations translate from Gromov-Witten theory to SFT, the next step is to show how genus-zero topological recursion translates to SFT. Compatible with the example of SFT of closed geodesics, it turns out that the corresponding localization theorem requires a non-equivariant version of SFT, which is generated by parametrized instead of unparametrized closed Reeb orbits. Since this non-equivariant version is so far only defined for cylindrical contact homology, we restrict ourselves to this special case. As an important result we show that, as in rational Gromov-Witten theory, all descendant invariants can be computed from primary invariants, i.e. without descendants.
In this paper we attempt a self-contained approach to infinite dimensional Hamiltonian systems appearing from holomorphic curve counting in Gromov-Witten theory. It consists of two parts. The first one is basically a survey of Dubrovin's approach to bihamiltonian tau-symmetric systems and their relation with Frobenius manifolds. We will mainly focus on the dispersionless case, with just some hints on Dubrovin's reconstruction of the dispersive tail. The second part deals with the relation of such systems to rational Gromov-Witten and Symplectic Field Theory. We will use Symplectic Field theory of $S^1\times M$ as a language for the Gromov-Witten theory of a closed symplectic manifold $M$. Such language is more natural from the integrable systems viewpoint. We will show how the integrable system arising from Symplectic Field Theory of $S^1\times M$ coincides with the one associated to the Frobenius structure of the quantum cohomology of $M$.
According to many phenomenological and theoretical studies the distribution of family name frequencies in a population can be asymptotically described by a power law. We show that the Galton-Watson process corresponding to the dynamics of a growing population can be represented in Hilbert space, and its time evolution may be analyzed by renormalization group techniques, thus explaining the origin of the power law and establishing the connection between its exponent and the ratio between the population growth and the name production rates.
We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains as special cases the ones constructed on the space of Laurent polynomials. We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial P1-orbifolds. Finally we link rational Symplectic Field Theory of Seifert fibrations over S^2 and three singular fibers with orbifold Gromov-Witten invariants of the base, extending a known result valid in the smooth case.
We compute the Gromov-Witten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. This amounts to impose (and possibly solve) different kinds of Schroedinger equations related to some quantization of the dispersionless KdV hierarchy. In particular we find very explicit formulas for the Gromov-Witten invariants of low degree of P1 with descendants of the Kaehler class.
The center and radius of perception associated with a written text are defined, and algorithms for their computation are presented. Indicators for anisotropy in large scale space perception are introduced. The relevance of these notions for the analysis of literary and historical records is briefly discussed and illustrated with an example taken from medieval historiography.
We discuss several examples of three-dimensional critical phenomena that can be described by Landau-Ginzburg-Wilson $ϕ^4$ theories. We present an overview of field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the $ε$ and of the fixed-dimension d=3 expansions. In particular, we discuss the stability of the O(N)-symmetric fixed point in a generic N-component theory, the critical behaviors of randomly dilute Ising-like systems and frustrated spin systems with noncollinear order, the multicritical behavior arising from the competition of two distinct types of ordering with symmetry O($n_1$) and O($n_2$) respectively.
In lattice Hamiltonian systems with a quartic coupling $γ$, a critical value $γ^*$ may exist such that, when $γ=γ^*$, the leading irrelevant operator decouples from the Hamiltonian and the leading nonscaling contribution to renormalization-group invariant physical quantities (evaluated in the critical region) vanishes. The 1/N expansion technique is applied to the evaluation of $γ^*$ for the lattice Hamiltonian of vector spin models with O(N) symmetry. As a byproduct, systematic asymptotic expansions for the relevant lattice massive one-loop integrals are obtained.
We study the spectrum of the confining strings in four-dimensional SU(N) gauge theories. We compute, for the SU(4) and SU(6) gauge theories formulated on a lattice, the string tensions sigma_k related to sources with Z_N charge k, using Monte Carlo simulations. Our results are consistent with the sine formula sigma_k/sigma = sin k pi/N / sin pi/N for the ratio between sigma_k and the standard string tension sigma. For the SU(4) and SU(6) cases the accuracy is approximately 1% and 2%, respectively. The sine formula is known to emerge in various realizations of supersymmetric SU(N) gauge theories. On the other hand, our results show deviations from Casimir scaling. We also discuss an analogous behavior exhibited by two-dimensional SU(N) x SU(N) chiral models.