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Paolo Rossi

Publications and source records attributed to Paolo Rossi.

At least 37 records · Page 2Linked to original sources
arXiv2021

A generalization of Witten's conjecture for the Pixton class and the noncommutative KdV hierarchy

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative KdV hierarchy, which we introduced in a previous work. The specialization of this conjecture to the top degree part of Pixton's class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the Double Ramification/Dubrovin--Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

math.AG↗
arXiv2021

Extended $r$-spin theory in all genera and the discrete KdV hierarchy

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

math.AG↗
arXiv2021

Moduli spaces of residueless meromorphic differentials and the KP hierarchy

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.

math.AG↗
arXiv2021

Riemannian F-manifolds, bi-flat F-manifolds, and flat pencils of metrics

In this paper we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat pencil of metrics satisfying a certain non-degeneracy assumption. In the last part of the paper we construct Legendre transformations for Riemannian F-manifolds.

math.DG↗
arXiv2021

Equivariant and supersymmetric localization in QFT

Equivariant localization theory is a powerful tool that has been extensively used in the past thirty years to elegantly obtain exact integration formulas, in both mathematics and physics. These integration formulas are proved within the mathematical formalism of equivariant cohomology, a variant of standard cohomology theory that incorporates the presence of a symmetry group acting on the space at hand. A suitable infinite-dimensional generalization of this formalism is applicable to a certain class of Quantum Field Theories (QFT) endowed with supersymmetry. In this thesis we review the formalism of equivariant localization and some of its applications in Quantum Mechanics (QM) and QFT. We start from the mathematical description of equivariant cohomology and related localization theorems of finite-dimensional integrals in the case of an Abelian group action, and then we discuss their formal application to infinite-dimensional path integrals in QFT. We summarize some examples from the literature of computations of partition functions and expectation values of supersymmetric operators in various dimensions. For 1-dimensional QFT, that is QM, we review the application of the localization principle to the derivation of the Atiyah-Singer index theorem applied to the Dirac operator on a twisted spinor bundle. In 3 and 4 dimensions, we examine the computation of expectation values of certain Wilson loops in supersymmetric gauge theories and their relation to 0-dimensional theories described by "matrix models". Finally, we review the formalism of non-Abelian localization applied to 2-dimensional Yang-Mills theory and its application in the mapping between the standard "physical" theory and a related "cohomological" formulation.

hep-th↗
arXiv2021

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(μ^2 Θ)$ where $μ$ is a parameter and $Θ$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.

math.AG↗
arXiv2020

Flat F-manifolds, F-CohFTs, and integrable hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus $1$, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank $1$ hierarchies of DR type at the order $9$ approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all $2$-dimensional homogeneous flat F-manifolds at genus $1$ approximation.

math-ph↗
arXiv2020

Integrable systems of double ramification type

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the first author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus $1$ quantum correction and, as an application, compute completely the quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for Witten's $3$- and $4$-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems which we call of double ramification type, as they satisfy all of the main properties of the DR hierarchy. In the second part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang hierarchies, via a geometric interpretation of the correlators forming the double ramification tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the Dubrovin-Zhang Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-$1$ cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus $5$ in this context.

math-ph↗
arXiv2020

Bibliometrics for collaboration works

An important issue in bibliometrics is the weighing of co-authorship in the production of scientific collaborations, which are becoming the standard modality of research activity in many disciplines. The problem is especially relevant in the field of high-energy physics, where collaborations reach 3000 authors, but it can no longer be ignored also in other domains, like medicine or biology. We present theoretical and numerical arguments in favour of weighing the individual contributions as $1/N_{\rm aut}^α$ where $N_{\rm aut}$ is the number of co-authors. When counting citations we suggest the exponent $α\approx 1$, that corresponds to fractional counting. When counting the number of papers we suggest $α\approx 1/3 - 1/2$, with the former (latter) value more appropriate for larger (smaller) collaborations. We expect and verify that the $h$ index scales as the square root of the average number of co-authors, and define a fractionalized $h$ index that does not scale with collaboration size.

cs.DL↗
arXiv2019

Topological effects in continuum 2d $U(N)$ gauge theories

We study the $θ$ dependence of the continuum limit of 2d $U(N)$ gauge theories defined on compact manifolds, with special emphasis on spherical ($g=0$) and toroidal ($g=1$) topologies. We find that the coupling between $U(1)$ and $SU(N)$ degrees of freedom survives the continuum limit, leading to observable deviations of the continuum topological susceptibility from the $U(1)$ behavior, especially for $g=0$, in which case deviations remain even in the large $N$ limit.

hep-th↗
arXiv2019

The topological susceptibility of two-dimensional $U(N)$ gauge theories

In this paper we study the topological susceptibility of two-dimensional $U(N)$ gauge theories. We provide explicit expressions for the partition function and the topological susceptibility at finite lattice spacing and finite volume. We then examine the particularly simple case of the abelian $U(1)$ theory, the continuum limit, the infinite volume limit, and we finally discuss the large $N$ limit of our results.

hep-lat↗
arXiv2018

Quantum ${D_4}$ Drinfeld-Sokolov hierarchy and quantum singularity theory

In this paper we compute explicitly the double ramification hierarchy and its quantization for the $D_4$ Dubrovin-Saito cohomological field theory obtained applying the Givental-Teleman reconstruction theorem to the $D_4$ Coxeter group Frobenius manifold, or equivalently the $D_4$ Fan-Jarvis-Ruan-Witten cohomological field theory (with respect to the non-maximal diagonal symmetry group $\langle J\rangle = \mathbb{Z}_3$). We then prove its equivalence to the corresponding Dubrovin-Zhang hierarchy, which was known to coincide with the $D_4$ Drinfeld-Sokolov hierarchy. Our techniques provide hence an explicit quantization of the $D_4$ Drinfeld-Sokolov hierarchy. Moreover, since the DR hierarchy is well defined for partial CohFTs too, our approach immediately computes the DR hierarchies associated to the invariant sectors of the $D_4$ CohFT with respect to folding of the Dynkin diagram, the $B_3$ and $G_2$ Drinfeld-Sokolov hierarchies.

math-ph↗
arXiv2018

DR/DZ equivalence conjecture and tautological relations

In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin-Zhang equivalence conjecture. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus $0$ and $1$ and also when first pushed forward from $\overline{\mathcal{M}}_{g,n+m}$ to $\overline{\mathcal{M}}_{g,n}$ and then restricted to $\mathcal{M}_{g,n}$, for any $g,n,m\geq 0$. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for $g\leq 2$. As an application we find a new formula for the class $λ_g$ as a linear combination of dual trees intersected with kappa and psi classes, and we check it for $g \leq 3$.

math.AG↗
arXiv2018

Simple Lax Description of the ILW Hierarchy

In this note we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structure in terms of a single Lax difference-differential operator.

math-ph↗
arXiv2017

Integrability, Quantization and Moduli Spaces of Curves

This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré.

math-ph↗
arXiv2017

Theta dependence in the large N limit

Studies of the large $N$ behaviour of the topological properties of gauge theories typically focused on the large $N$ scaling of the topological susceptibility. A much more difficult task is the study of the behaviour of higher cumulants of the topological charge in the large $N$ limit, which up to now remained elusive. We will present first results confirming the expected large $N$ scaling of the coefficient commonly denoted by $b_2$, related to the kurtosis of the topological charge.

hep-lat↗
arXiv2016

$θ$ dependence of 4D $SU(N)$ gauge theories in the large-$N$ limit

We study the large-$N$ scaling behavior of the $θ$ dependence of the ground-state energy density $E(θ)$ of four-dimensional (4D) $SU(N)$ gauge theories and two-dimensional (2D) $CP^{N-1}$ models, where $θ$ is the parameter associated with the Lagrangian topological term. We consider its $θ$ expansion around $θ=0$, $E(θ)-E(0) = {1\over 2}χ\,θ^2 ( 1 + b_2 θ^2 + b_4θ^4 +\cdots)$ where $χ$ is the topological susceptibility and $b_{2n}$ are dimensionless coefficients. We focus on the first few coefficients $b_{2n}$, which parametrize the deviation from a simple Gaussian distribution of the topological charge at $θ=0$. We present a numerical analysis of Monte Carlo simulations of 4D $SU(N)$ lattice gauge theories for $N=3,\,4,\,6$ in the presence of an imaginary $θ$ term. The results provide a robust evidence of the large-$N$ behavior predicted by standard large-$N$ scaling arguments, i.e. $b_{2n}= O(N^{-2n})$. In particular, we obtain $b_2=\bar{b}_2/N^2 + O(1/N^4)$ with $\bar{b}_2=-0.23(3)$. We also show that the large-$N$ scaling scenario applies to 2D $CP^{N-1}$ models as well, by an analytic computation of the leading large-$N$ dependence.

hep-lat↗