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Pietro Fré

Publications and source records attributed to Pietro Fré.

At least 19 recordsLinked to original sources
arXiv2026

Homogeneous Non Symmetric Special Kähler Geometries as Broken Isometry Metrics on Symmetric CV {K}ähler Manifolds:a new tool for $r=2$ CaNNs

In this paper we prove in full detail the isomorphism between the solvable group $\mathcal{S}_{2,2+p}$, metric equivalent to the Calabi Vesentini symmetric space $\mathrm{SO(2,2+p)/SO(2)\times SO(2+p)}$, and the solvable group $\mathcal{S}_{\mathrm{L}(-1,p)}$ supporting the Kähler metric of the homogeneous non symmetric special manifold L$(-1,p)$. From an Alekseevskyan point of view the two spaces simply correspond to two different quadratic forms on the same solvable Lie algebra that we present and compare in detail. Furthermore considering the full group of isometries $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ of L$(-1,p)$ we show that $\mathrm{Iso_{L(-1,p)}}\subset \mathrm{SO}(2,2+p)$ is a non-semisimple subgroup of the simple isometry group of Calabi-Vesentini manifolds. Altogether the Special Kähler manifold L$(-1,p)$ can be seen as a coset manifold $\mathrm{Iso_{L(-1,p)}}/\mathrm{H}$ where $\mathrm{H}=\mathrm{U(1)}_L\times \mathrm{SO(p)}$, the generator of $\mathrm{U(1)}_L$ being in $\so(2,2+p)$, yet not in the canonical $\so(2)\oplus\so(2+p)$ subalgebra. The action of $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ on the solvable coordinates of the CV manifold can be constructed directly. This identification provides a valuable tool for Cartan Neural Networks, introducing additional non linear transformations in every map from one layer to the next one of an $r=2$ Neural Network based on the CV Tits Satake universality class; in perspective, this new tool increases expressivity.

hep-th↗
arXiv2026

The paint group Tits Satake theory of hyperbolic symmetric spaces: the distance function, paint invariants and discrete subgroups

The present paper, which is partially a review, but also contains several completely new results, aims at presenting, in a unified mathematical framework, a complex and articulated lore regarding non-compact symmetric spaces, with negative curvature, whose isometry group is a non-compact, real simple Lie group. All such manifolds are Riemannian normal manifolds, according to Alekseevsky's definition, in the sense that they are metrically equivalent to a solvable Lie group manifold. This identification provides a vision in which, on one side one can derive quite explicit and challenging formulae for the unique distance function between points of the manifold, on the other one, one can organize the entire set of the available manifolds in universality classes distinguished by their common Tits Satake submanifold and, correspondingly, by their non-compact rank. The members of the class are distinguished by their different Paint Groups, the latter notion having been introduced by two of the present authors in an earlier collaboration. In relation to the construction of neural networks, these mathematical structures offer unique possibilities of replacing ad hoc activation functions with the naturally defined non-linear operations that relate Lie algebras to Lie Groups and vice-versa. The Paint Group invariants offer new tokens both to construct algorithms and inspect (hopefully to control) their working. A conspicuous part of the paper is devoted to the study and systematic construction of parabolic/elliptic discrete subgroups of the Lie groups SO(r,r+q), in view of discretization and/or tessellations of the space to which data are to be mapped. Furthermore, it is shown how the ingredients of Special Kähler Geometry and the c-map, well known in the supergravity literature, provide a unified classification scheme of the relevant Tits Satake universality classes with non-compact rank r<5.

math.DG↗
arXiv2026

The macroscopic Kaehler metric of Geometric Thermodynamics versus the microscopic one on the Event Manifold: Exact Partition Functions on CV manifolds. Extended Souriau temperatures and spontaneous magnetizations

In this paper we clarify the relation between Geometric Thermodynamics and Information Geometry based on the Fisher matrix. On the macroscopic odd-dimensional contact manifold of thermodynamic variables, we introduce for the first time a metric, whose pull-back on the isoentropic symplectic submanifolds transverse to the Reeb field is Kählerian. The pull-back of such metric on equilibrium states, that are lagrangian submanifolds, is the Fisher Hessian. Then we consider the Souriau-like Thermodynamics that uses Calabi-Vesentini (CV) manifolds as Kaehlerian microscopic event manifolds and the Killing moment maps as observable functions. A systematic use of the theory of compact abelian structures and the setup of Special Kähler Geometry in which CV manifolds are encoded allows us to perform the explicit integration defining the partition function for any entry in the CV Tits Satake universality class. The additional actions completing the abelian structure are non linear Casimir functions of the Killing moment-maps and suggest a generalization of Souriau thermodynamics that partially breaks the isometry group symmetry by means of the non vanishing mean values of the Casimir functions in a manner similar to the spontaneous magnetization in ferromagnetism. Our new exact Gibbs distributions provide the analogue for Cartan Neural Networks of the Gaussian probability distributions in flat space used in conventional Machine Learning.

hep-th↗
arXiv2025

Axial Symmetric Navier Stokes Equations and the Beltrami /anti Beltrami spectrum in view of Physics Informed Neural Networks

In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact $3$-torus $T^3=\mathbb{R}^3/Λ$ where $Λ$ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic $1$-forms that, for each energy level, decomposes into six components: two Beltrami, two anti-Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an $L^2$ space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication.

physics.flu-dyn↗
arXiv2025

Tessellation Groups, Harmonic Analysis on Non-compact Symmetric Spaces and the Heat Kernel in view of Cartan Convolutional Neural Networks

In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The mathematical and conceptual results are diverse and span various mathematical fields, but the inspiring motivation is unified. The aim is to introduce layers that are mathematically modeled as non-compact symmetric spaces, each mapped onto the next one by solvable group homomorphisms. In particular, in the spirit of Convolutional neural networks, we have introduced the notion of Tits Satake (TS) vector bundles where the TS submanifold is the base space. Within this framework, the tiling of the base manifold, the representation of bundle sections using harmonics, and the need for a general theory of separator walls motivated a series of mathematical investigations that produced both definite and partial results. Specifically, we present the group theoretical construction of the separators for all non-compact symmetric spaces $\mathrm{U/H}$, as well as of the $Δ_{8,3,2}$ tiling group and its normal Fuchsian subgroups, respectively yielding the uniformization of the genus $g=3$ Fermat Quartic and of the genus $g=2$ Bolza surface. The quotient automorphic groups are studied. Furthermore, we found a new representation of the Laplacian Green function and the Heat Kernel on Hyperbolic Spaces $\mathbb{H}^{n}$, and a setup for the construction of the harmonic functions in terms of the spinor representation of pseudo-orthogonal groups. Finally, to obtain an explicit construction of the Laplacian eigenfunctions on the Bolza Riemann surface, we propose and conjecture a new strategy relying on the Abel-Jacobi map of the Riemann surface to its Jacobian variety and the Siegel Theta function.

cs.LG↗
arXiv2023

D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler-Einstein surfaces

D3-brane solutions of type IIB supergravity can be obtained by means a classical ansatz involving a harmonic warp factor and two summands, the first being the flat Minkowskian metric of the D3 brane world-sheet and the second a Ricci flat metric on a suitable 6-dimensional transverse space, both twisted by the warp factor. Of particular interest is the case of the total space of thecanonical bundle over a complex Kähler 2-fold. This situation emerges in many cases while considering the resolution of finite quotient singulaties. When the group is $\mathbb{Z}_4$, the complex 2-fold is the second Hirzebruch surface endowed with a Kähler metric having SU(2)xU(1) isometry. There is actually an entire class of such metrics parameterized by a single function, and best described in the AMSY symplectic formalism. We recover the existence of a two parameter subclass of Kähler-Einstein metrics on manifolds that are homeorphic to $S^2\times S^2$, and study in detail this class. The K"ahler-Einstein nature of these manifolds allows the construction of the Ricci flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. Furthermore we show the full integrability of the differential system of geodesics equations thanks to an additional conserved quantity that we unveil and which is similar to the Carter constant in the case of the Kerr metric.

math-ph↗
arXiv2022

Gauged $\mathcal{N}=3,D=4$ Supergravity: a new web of marginally connected vacua

We analyze the vacuum structure of $\mathcal{N}=3,D=4$ supergravity coupled to 9 vector multiplets with gauge group ${\rm SO}(3)\times {\rm SU}(3)$. Aside from the central $\mathcal{N}=3$ AdS$_4$ vacuum at the origin, on which the supermultiplet structure reproduces the massless sector of M-theory compactified on $\mathrm{N^{0,1,0}}$, we find a rich structure of AdS$_4$ vacua preserving $\mathcal{N}=0,1,2,3$ supersymmetry. These new vacua are arranged in a manifold spanned by scalar fields corresponding to exactly marginal deformations of the dual CFT. This manifold has the form $T^3/K$, where $K$ is a discrete subgroup of the gauge group: $\mathcal{N}=3,2$ and $1$ vacua correspond, respectively, to a point, a line and a surface in the three-dimensional vacuum manifold. We study RG flows from the central $\mathcal{N}=3$ vacuum and elaborate on the possible higher dimensional origin of the new vacua. For the reader's convenience we also provide a review of the embedding tensor formulation of $D=4$, $\mathcal{N}=3$ gauged supergravities. In particular we provide formulas involving the fermion shift tensors and mass matrices in $\mathcal{N}=3$ theories, which can be applied to a generic gauging.

hep-th↗
arXiv2021

Resolution a la Kronheimer of $\mathbb{C}^3/Γ$ singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity

We analyze the relevance of the generalized Kronheimer construction for the gauge-gravity correspondence. We study the general structure of IIB supergravity D3-brane solutions on crepant resolutions $Y$ of singularities $\mathbb{C}^3/Γ$ with $Γ$ a finite subgroup of $SU(3)$. Next we concentrate on another essential item for the D3-brane construction, i.e., the existence of a Ricci-flat metric on $Y$, with particular attention to the case $Γ=\mathbb{Z}_4$. We conjecture that on the exceptional divisor the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on ${\rm tot} K_{{\mathbb WP}[112]}$ that we construct, which is a partial resolution of $\mathbb{C}^3/\mathbb{Z}_4$. For the full resolution we have $Y=\operatorname{tot} K_{\mathbb{F}_{2}}$, where $\mathbb{F}_2$ is the second Hizebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $\mathbb{F}_2$ produced by the Kronheimer construction as initial datum in a Monge-Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential; in all cases one can establish a series solution in powers of the fiber variable of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, but uniquely determines all the subsequent terms as local functionals of the initial datum. While a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation we have identified some new properties of this type of MA equations that we believe to be so far unknown.

math.DG↗
arXiv2019

Crepant Resolutions of $\mathbb{C}^3/\mathbb{Z}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold $\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface $\mathbb F_2$. We determine the explicit Kähler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of $\mathbb F_2$. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space $\mathbb P[1,1,2]$, while the other is the product of the ALE space $A_1$ by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.

hep-th↗
arXiv2019

The role of PSL(2,7) in M-theory: M2-branes, Englert equation and the septuples

Reconsidering the M2-brane solutions of d=11 supergravity with a transverse Englert flux introduced by one of us in 2016, we present a new purely group theoretical algorithm to solve Englert equation based on a specific embedding of the PSL(2,7) group into $\mathrm{Weyl[\mathfrak{e}_7]}$. The aforementioned embedding is singled out by the identification of $\mathrm{PSL(2,7)}$ with the automorphism group of the Fano plane. Relying on the revealed intrinsic PSL(2,7) symmetry of Englert equation and on the new algorithm we present an exhaustive classification of Englert fluxes. The residual supersymmetries of the corresponding M2-brane solutions associated with the first of the 8 classes into which we have partitioned Englert fluxes are exhaustively analyzed and we show that all residual d=3 supersymmetries with $\mathcal{N} \in \left\{1,2,3,4,5,6\right\}$ are available. Our constructions correspond to a particular case in the category of M2-brane solutions with transverse self-dual fluxes.

hep-th↗
arXiv2018

The Kähler Quotient Resolution of $\mathbb{C}^3/Γ$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories

We advocate that a generalized Kronheimer construction of the Kähler quotient crepant resolution $\mathcal{M}_ζ\longrightarrow \mathbb{C}^3/Γ$ of an orbifold singularity where $Γ\subset \mathrm{SU(3)}$ is a finite subgroup naturally defines the field content and interaction structure of a superconformal Chern-Simons Gauge Theory. This is supposedly the dual of an M2-brane solution of $D=11$ supergravity with $\mathbb{C}\times\mathcal{M}_ζ$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $\pmb{p}\wedge\pmb{p}=0$ which provides the Kähler analogue of the holomorphic sector in the hyperKähler moment map equations canonically defines the structure of a universal superpotential in the CS theory. The kernel of the above equation can be described as the orbit with respect to a quiver Lie group $\mathcal{G}_Γ$ of a locus $L_Γ\subset \mathrm{Hom}_Γ(\mathcal{Q}\otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold $\mathcal{G}_Γ/\mathcal{F}_Γ$, the gauge group $\mathcal{F}_Γ$ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the tautological vector bundles that are in a one-to-one correspondence with the nontrivial irreps of $Γ$. These first Chern classes provide a basis for the cohomology group $H^2(\mathcal{M}_ζ)$. We discuss the relation with conjugacy classes of $Γ$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of $\mathbb{C}^2/Γ$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.

hep-th↗
arXiv2017

$\mathrm{D=10}$ Super-Yang-Mills Theory and Poincaré Duality in Supermanifolds

We consider super Yang-Mills theory on supermanifolds $\mathcal{M}^{(D|m)}$ using integral forms. The latter are used to define a geometric theory of integration and are essential for a consistent action principle. The construction relies on Picture Changing Operators $\mathbb{Y}^{(0|m)}$, analogous to those introduced in String Theory, that admit the geometric interpretation of Poincaré duals of closed submanifolds of superspace $\mathcal{S}^{(D|0)} \subset \mathcal{M}^{(D|m)}$ having maximal bosonic dimension $D$. We discuss the case of Super-Yang-Mills theory in $D=10$ with $\mathcal{N}=1$ supersymmetry and we show how to retrieve its pure-spinor formulation from the rheonomic lagrangian $\mathcal{L}_{rheo}$ of D'Auria, Fré and Da Silva, choosing a suitable $\mathbb{Y}^{(0|m)}_{ps}$. From the same lagrangian $\mathcal{L}_{rheo}$, with another choice $\mathbb{Y}^{(0|m)}_{comp}$ of the PCO, one retrieves the component form of the SYM action. Equivalence of the formulations is ensured when the corresponding PCO.s are cohomologous, which is true, in this case, of $\mathbb{Y}^{(0|m)}_{ps}$ and $\mathbb{Y}^{(0|m)}_{comp}$.

hep-th↗
arXiv2011

Black Hole Nilpotent Orbits and Tits Satake Universality Classes

In this paper we consider the problem of classification of nilpotent orbits for the pseudo-quaternionic coset manifolds U/H* obtained in the time-like dimensional reduction of N = 2 supergravity models based on homogeneous symmetric special geometries. Within the D=3 approach this classification amounts to a classification of regular and singular extremal black hole solutions of supergravity. We show that the pattern of such orbits is a universal property depending only on the Tits-Satake universality class of the considered model, the number of such classes being five. We present a new algorithm for the classification and construction of the nilpotent orbits for each universality class which is based on an essential use of the Weyl group W of the Tits Satake subalgebra U_{TS} of U and on a certain subgroup W_H thereof. The splitting of orbits of the full group Uinto suborbits with respect to the stability subgroup H* is shown to be governed by the structure of the discrete coset W/W_H. For the case of the universality class SO(4,5) /[SO(2,3) x SO(2,2)] we derive the complete list of nilpotent orbits which happens to contain 37 elements. We also show how the universal orbits are regularly embedded in all the members of the class that are infinite in number. As a matter of check we apply our new algorithm also to the Tits Satake class G_(2,2)/[SL(2)x SL(2)] confirming the previously obtained result encompassing 7 nilpotent orbits. Perspectives for future developments based on the obtained results are outlined.

hep-th↗
arXiv2011

Integrability of Supergravity Black Holes and New Tensor Classifiers of Regular and Nilpotent Orbits

In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of N=2 supergravities with symmetric Special Geometry. Our main goal is the classification of these black-holes according to the H*-orbits in which the space of possible Lax operators decomposes, H* being the isotropy group of scalar manifold originating from time-like dimensional reduction of supergravity from D=4 to D=3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H* orbits of Lax operators. Our tensor based classification is compared, in the case of the simple one-field model S^3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple and practical discriminating method. We present a detailed analysis of the S^3 model and its black hole solutions, discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostant-representation of a generic Lie algebra element, we were able to develop an algorithm which produces the necessary number of hamiltonians in involution required by Liouville integrability of generic orbits. The degenerate orbits correspond to extremal black-holes and are nilpotent. We analyze these orbits in some detail working out different representatives thereof and showing that the relation between H* orbits and critical points of the geodesic potential is not one-to-one. Finally we present the conjecture that our newly identified tensor classifiers are universal and able to label all regular and nilpotent orbits in all homogeneous symmetric Special Geometries.

hep-th↗
arXiv2009

Theory of Superdualities and the Orthosymplectic Supergroup

We study the dualities for sigma models with fermions and bosons. We found that the generalization of the SO(m,m) duality for D=2 sigma models and the Sp(2n) duality for D=4 sigma models is the orthosymplectic duality OSp(m,m|2 n). We study the implications of this and we derive the most general D=2 sigma model, coupled to fermionic and bosonic one-forms, with such dualities. To achieve this we generalize Gaillard-Zumino analysis to orthosymplectic dualities, which requires to define embedding of the superisometry group of the target space into the duality group. We finally discuss the recently proposed fermionic dualities as a by-product of our construction.

hep-th↗
arXiv2008

Pure Spinor Formalism for Osp(N|4) backgrounds

We start from the Maurer-Cartan (MC) equations of the Osp(N|4) superalgebras satisfied by the left-invariant super-forms realized on supercoset manifolds of the corresponding supergroups and we derive some new pure spinor constraints. They are obtained by "ghostifying" the MC forms and extending the differential d to a BRST differential. From the superalgebras G =Osp(N|4) we single out different subalgebras H contained in G associated with the different cosets G/H: each choice of H leads to a different weakening of the pure spinor constraints. In each case, the number of parameter is counted and we show that in the cases of Osp(6|4)/U(3) x SO(1,3), Osp(4|4)/SO(3) x SO(1,3) and finally Osp(4|4) U(2)} x SO(1,3) the bosonic and fermionic degrees of freedom match in order to provide a c=0 superconformal field theory. We construct both the Green-Schwarz and the pure spinor sigma model for the case Osp(6|4)/U(3)x SO(1,3) corresponding to AdS_4 x P^3. The pure spinor sigma model can be consistently quantized.

hep-th↗
arXiv2008

On full-fledged supergravity cosmologies and their Weyl group asymptotics

We consider a class of cosmological solutions of d=4, N=2 supergravity theories coupled to vector multiplets. The solutions result from performing a compactification to three dimensions, where the theory reduces to a symmetric space sigma model coupled to gravity and where the resulting equations of motion are integrable. We describe in detail how the three-dimensional solutions can be uplifted to four dimensions again. The four-dimensional cosmologies are generically characterized by an algebra of translational isometries that is of Heisenberg type. We give explicit examples of these cosmologies for the S-T-U model and comment also on their uplift to 10 dimensions by interpreting them as solutions of a truncation of type IIB supergravity on a K3 x T^2/Z_2 orientifold.

hep-th↗