In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces U/H that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on U/H provided by generalized thermodynamics à la Souriau. Our main result is the proof that U/H.s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of U of a positivity domain in the Cartan subalgebra Cc⊂H of the maximal compact subalgebra H⊂U. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao's, Chentsov's, and Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against U transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group H⊂U.

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.

When fermions are taken to be anti-periodic along a spacelike S^1 in AdS_4, certain ground states of the supergravity theory are described by AdS-soliton-like spacetimes. We study these geometries within the purely dilatonic STU model obtained from the compactification of M-theory on S^7 in the presence of non-trivial gauge fields. The resulting configurations define a rich family of everywhere regular supersymmetric vacua that holographically describe strongly coupled “confining” gauge theories in three dimensions. We provide a complete characterization of the moduli space of supersymmetric solutions in terms of the vacuum expectation values of the dimension-one operators of the truncation.

A bstract This paper is an extension of the results presented in [1]. We study G S -invariant subsectors of maximal gauged supergravities and show that such models can provide consistent truncations even when G S is not a symmetry of the original supergravity. We show that this construction is key to building pure supergravities around a supersymmetric AdS D solution. We illustrate this construction by building a consistent 𝒩 = 4 subsector of the D = 4 𝒩 = 8 [SO(6) × SO(1, 1)] ⋉ ℝ 12 gauged supergravity. We use this result to build the uplift of the multicharge spindle solutions in type IIB and we define a simple criterion for assessing the regularity of the uplift. We show that the type IIB uplift of the spindle is always non-regular, admitting eight codimension-six orbifold singularities. We apply the same criterion to other spindle uplifts, recovering known results and making predictions on the regularity of spindles on (quasi-)regular SE 7 manifolds.

We discuss some new results on the construction of supersymmetric solutions of Type IIB supergravity of the form WAdS_3× WS^3× T^4, WAdS_3 and WS^3 denoting warped anti-de Sitter spacetime and sphere, respectively. The distinctive feature of these backgrounds is that, in spite of them being supersymmetric, the warpings of the two factors are described by independent parameters. We illustrate how some of these geometries, characterised by a lightlike warping of the anti-de Sitter factor, arise in the near-horizon limit of a regular, asymptotically locally flat configuration of D-branes and fluxes. Central to the construction of the latter solutions is the use of two independent TsT transformations. We also give a new class of supersymmetric solutions of the general form WAdS_3× WS^3× T^4, which has not been published yet. They feature warpings of the anti-de Sitter factor of the lightlike, spacelike and timelike types. We discuss their properties.

We review recent progress in constructing maximal, classical supergravity models and their applications.

Following previous results recently obtained by us, on Information Geometry versus Geometrical Thermodynamics and on the exact calculation of partition functions for extended Souriau Gibbs distributions on Calabi Vesentini manifolds, we study the differential geometry of the corresponding thermo-metrics. A general intriguing scheme is discovered and put into evidence. A small yet significant difference, distinguishes the even from the odd dimensional instance of the microscopic CV manifolds. Apart from that the complete thermo-space is flat when no constraint is introduced. Freezing the magnetic fields, which can be done according to complicated combinatorials, forces the thermo-system to evolve on curved submanifolds of the thermo--space that have a structure depending only on the length of the $n-1$ chain of frozen contiguous magnetic fields. The behavior of Riemann tensor components for such spaces is codified by a symmetric matrix with peculiar behavior along special symmetrically arranged submanifolds that, might be responsible for the generation of curvature walls and for the categorical partitioning of the thermo space. The embedding of this curved submanifold into $\mathbb{R}^{2n-1}$ can be traced back to the vanishing of magnetic fields and, in this case, the flat metric on $\mathbb{R}^{2n-1}$ is the $\mathfrak{a}_{2n-1}$ simple Lie algebra Cartan matrix. In another version the flat embedding reveals the geometric interpretation of the $n$-manifold as a generalized translation hypersurface. The boundary at infinity has a hypercube structure whose face central points and vertices appear, numerically, to be the end-points of all geodesics depending only on their angular slope at the start. This general feature is reminiscent of the causal structure at infinity of Lorentzian space-times and of Penrose diagrams.

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 , as well as of the tiling group and its normal Fuchsian subgroups, respectively, yielding the uniformization of the genus Fermat quartic and of the genus 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 , 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.

We review a general paradigm for constructing U-fold backgrounds in (dimensionally reduced) Type IIB superstring theory, of the form AdS_d-1× S^1× S^d, with a monodromy along S^1 in the string-duality group. We also consider a special instance with d=3 in Type IIB superstring theory, discuss its ten-dimensional uplift and assess its supersymmetry.

We consider the static planar black hole solutions in the STU model of the gauged 𝒩=8 supergravity in four dimensions. We give a straightforward derivation of the equation of state of the purely electric and purely magnetic solutions with four charges. Then we give a simple proof that the determinant of the Hessian of the energy is always negative below some critical finite temperature for the purely electric solutions. We compute the spinodal line for the usual planar Reissner-Nordström solution in four dimensions. Inspired by the magnetic superalgebra we show that the supersymmetric solutions are metastable if the energy is restricted to satisfy the topological twist condition ab initio and it is shifted to be zero on the BPS solutions.

A fluid described by an Abelian Chern-Simons action principle in 4+1 dimensions is considered. Letting 3+1 dimensions correspond to the usual space and time, and assuming the fields to be independent of the fifth coordinate, the free theory provides an interpretation as a system of advection equations, where the advecting velocity field is defined as the null vector of the field strength tensor (curvature). The free theory possesses a number of conservation laws which turn out to be prototypical forms of helicity and entropy conservation. Coupling the Chern-Simons field to an external source, a new conserved charge density is obtained which has the form of the Rossby-Ertel's potential vorticity (PV). Finally, by identifying the external current with the Chern-Simons field in a gauge-invariant setting, based on non-relativistic ideas, a self-interacting action principle is obtained whose Euler-Lagrange equations correspond precisely to a classical dissipationless compressible (3+1)-dimensional fluid endowed with thermodynamics, with only one extra condition: a constraint on the initial profile of the PV. After analysing this constraint of the "Chern-Simons fluid formulation", we investigate the helicity conservation of general fluids, going beyond classical analyses of barotropic fluids and no-cross boundary conditions for vorticity (Moffatt 1969). A new fluid helicity invariant for barotropic fluids under generic boundary conditions is obtained and the role of baroclinity in the helicity production is clarified. Inside a region bounded by an isentropic surface, the theory's constraint on the PV gives an integral formula for the mass, and for the evolution of fluid helicity in the baroclinic case. Finally, for an ideal gas exact, steady solutions of the equations of motion are found in a rotating scenario, showing that Ferrel-cell like patterns are produced in a rotating planet.

Near horizon geometries of Dp-branes with p ≠ 3 are singular with a running dilaton. Bound states of Dp branes with their magnetic cousins, D(6 − p) branes, can stabilise the dilaton such that an AdS factor might appear in the near horizon region, potentially leading to a chain of AdS vacua of the form AdSp+2 × Sp+2 × 𝕋6−2p. The solutions with p = −1, 1, 3 are supersymmetric with the cases p = 1, 3 being well-known examples already. We construct explicit (partially smeared) brane bound state solutions for all such configurations. The D2-D4 and D(−1)-D7 cases are entirely novel, but they do not have a near-horizon AdS geometry. The two novel classes of solutions feature ghost branes (negative tension branes), and we suggest they are physical for the D(−1)-D7 solutions but unphysical for the D2-D4 solutions. The bound state of a D(−1) and a D7 brane in supergravity was only hinted upon recently in [1]. We correct the solution here in order to preserve supersymmetry, and find that the dilaton can indeed be stabilized. This points to a possible dual matrix theory, generalizing the IKKT matrix model to allow for conformal invariance.

Fetching techniques from generalized geometry and exceptional field theory, we develop a new method to identify consistent subsectors of four-dimensional gauged maximal supergravities that possess a (locally) geometric embedding in type IIB or 11D supergravity. We show that a subsector that is invariant under a structure group GS subset of E7(7) can define a consistent truncation, even when GS is not part of the symmetry of the gauged maximal supergravity. As an illustration of the method, type IIB supergravity on S1 x S5 is shown to admit a consistent truncation to pure Al = 4, D = 4 gauged supergravity. Explicit uplift formulae are presented which provide a type IIB alternative to the M-theory embedding constructed by Cvetic, Lu, and Pope 25 years ago.

Fetching techniques from generalized geometry and exceptional field theory, we develop a new method to identify consistent subsectors of four-dimensional gauged maximal supergravities that possess a (locally) geometric embedding in type IIB or 11D supergravity. We show that a subsector that is invariant under a structure group GS⊂E7(7) can define a consistent truncation, even when GS is not part of the symmetry of the gauged maximal supergravity. As an illustration of the method, type IIB supergravity on S1×S5 is shown to admit a consistent truncation to pure N=4, D=4 gauged supergravity. Explicit uplift formulae are presented which provide a type IIB alternative to the M-theory embedding constructed by Cvetic, Lu, and Pope 25 years ago. Published by the American Physical Society 2025

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.

We construct the universal AdS4 black hole that asymptotes to the (φ, χ)-family of type IIB S-fold backgrounds dual to the conformal manifold of 𝒩 = 2 S-fold CFT’s. We present the explicit type IIB embedding of such a universal black hole for two particular asymptotics: the 𝒩 = 2 S-fold with U(2) symmetry at (φ, χ) = (0, 0) and the 𝒩 = 4 S-fold with SO(4) symmetry at (φ, χ) = (1, 0). As a byproduct, we also present a novel two-parameter family of AdS2 × M8 supersymmetric S-fold backgrounds with M8 = ℍ2 × S5 × S1 that features a parametrically-controlled scale separation involving some of the internal directions.

Fetching techniques from Generalised Geometry and Exceptional Field Theory, we develop a new method to identify consistent subsectors of four-dimensional gauged maximal supergravities that possess a (locally) geometric embedding in type IIB or 11D supergravity. We show that a subsector that is invariant under a structure group G_S⊂E_7(7) can define a consistent truncation, even when G_S is not part of the symmetry of the gauged maximal supergravity. As an illustration of the method, type IIB supergravity on S^1×S^5 is shown to admit a consistent truncation to pure 𝒩=4, D=4 gauged supergravity. Explicit uplift formulae are presented which provide a type IIB alternative to the M-theory embedding constructed by Cvetic, Lu and Pope 25 years ago.

Fetching techniques from Generalised Geometry and Exceptional Field Theory, we develop a new method to identify consistent subsectors of four-dimensional gauged maximal supergravities that possess a (locally) geometric embedding in type IIB or 11D supergravity. We show that a subsector that is invariant under a structure group $\textrm{G}_\textrm{S} \subset \textrm{E}_{7(7)}$ can define a consistent truncation, even when $\textrm{G}_\textrm{S}$ is not part of the symmetry of the gauged maximal supergravity. As an illustration of the method, type IIB supergravity on $\textrm{S}^{1} \times \textrm{S}^{5}$ is shown to admit a consistent truncation to pure $\mathcal{N}=4$, $D=4$ gauged supergravity. Explicit uplift formulae are presented which provide a type IIB alternative to the M-theory embedding constructed by Cvetic, Lu and Pope $25$ years ago.

In this Letter, we construct a supersymmetric model, obtained by deforming N = 2 anti-de Sitter D = 3 supergravity through a chiral vector component of the torsion. Moreover, we study the existence of supersymmetric states of such theory by inspecting the presence of Killing spinors on a specific bosonic solution.

We exploit the presence of moduli fields in the ${\rm AdS}_3\times { S}^3\times CY_2$, where $CY_2=T^4$ or $K3$, solution to Type IIB superstring theory, to construct a U-fold solution with geometry ${\rm AdS}_2\times S^1\times {\rm S}^3\times CY_2$. This is achieved by giving a non-trivial dependence of the moduli fields in ${\rm SO}(4,n)/{\rm SO}(4)\times {\rm SO}(n)$ ($n=4$ for $CY_2=T^4$ and $n=20$ for $CY_2=K3$ ), on the coordinate $\eta$ of a compact direction $S^1$ along the boundary of ${\rm AdS}_3$, so that these scalars, as functions of $\eta$, describe a geodesic on the corresponding moduli space. The back-reaction of these evolving scalars on spacetime amounts to a splitting of ${\rm AdS}_3$ into ${\rm AdS}_2\times S^1$ with a non-trivial monodromy along $S^1$ defined by the geodesic. Choosing the monodromy matrix in ${\rm SO}(4,n;\,\mathbb{Z})$, this supergravity solution is conjectured to be a consistent superstring background. We generalize this construction starting from an ungauged theory in $D=2d$, $d$ odd, describing scalar fields non-minimally coupled to $(d-1)$-forms and featuring solutions with topology ${\rm AdS}_d\times S^d$, and moduli scalar fields. We show, in this general setting, that giving the moduli fields a geodesic dependence on the $\eta $ coordinate of an $S^1$ at the boundary of ${\rm AdS}_d$ is sufficient to split this space into ${\rm AdS}_{d-1}\times S^1$, with a monodromy along $S^1$ defined by the starting and ending points of the geodesic. This mechanism seems to be at work in the known J-fold solutions in $D=10$ Type IIB theory and hints towards the existence of similar solutions in the Type IIB theory compactified on $CY_2$. We argue that the holographic dual theory on these backgrounds is a 1+0 CFT on an interface in the 1+1 theory at the boundary of the original ${\rm AdS}_3$.