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

Abstract: 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.

• The paper develops a unified framework by constructing a macroscopic Kähler metric whose Legendre transform recovers the Fisher information metric at equilibrium.
• The methodology involves explicit computation of Gibbs partition functions on Calabi-Vesentini manifolds using solvable Lie group parameterizations and moment-map techniques.
• The study demonstrates that extending Souriau temperatures induces spontaneous magnetization, offering insights into symmetry breaking in both thermodynamic phase transitions and machine learning models.

Macroscopic Kähler Geometry and Microscopic Structures in Geometric Thermodynamics: Exact Partition Functions, Extended Souriau Temperatures, and Magnetization Phenomena on Calabi-Vesentini Manifolds

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Introduction and Framework

The paper "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" (2606.09438) offers a comprehensive reconciliation of Geometric Thermodynamics and Information Geometry, fundamentally reformulating the statistical mechanical and geometric underpinnings of thermodynamic equilibrium for complex, non-Euclidean event manifolds. The authors explicitly construct a Kähler metric on the macroscopic (thermodynamic) space and clarify its reduction to the Fisher information metric in equilibrium, connecting macroscopic geometric structures directly with the microscopic symplectic-Kählerian geometry of the event manifold. Central to their methodology is the explicit calculation of partition functions for Gibbs distributions on Calabi-Vesentini (CV) manifolds, the extension of Souriau's approach to group thermodynamics, and the generalization to extended temperature vectors leading to spontaneous symmetry breaking phenomena analogous to magnetization in statistical mechanics.

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Synthesis of Geometric Thermodynamics and Information Geometry

Contact and Symplectic Reductions

The authors rigorously embed Jaynes' maximum entropy principle and the Fisher information metric into the modern language of contact geometry. The macroscopic thermodynamic space $\mho$ is realized as an odd-dimensional contact manifold, with equilibrium states forming Legendrian submanifolds (where the contact form vanishes). Transverse reduction to the Reeb field generates even-dimensional symplectic leaves, upon which the Fisher information metric arises as the pullback of a uniquely specified Kähler metric.

Notably,

• The macroscopic metric is obtained as the Hessian of the stochastic Hamiltonian, i.e., the Legendre transform of the log-partition function.
• This construction rigorously justifies the standard information geometry metric as a Kähler metric restricted to equilibrium submanifolds.

Microscopic Manifolds and Observables

The microscopic event manifold $\Omega$ is allowed to be a non-compact Kähler symmetric space $U/H$. The observable functions are taken as moment-maps of Killing fields associated with the isometry group $U$, enabling a group-theoretical formulation of thermodynamics (Souriau thermodynamics) amenable to exact partition function evaluation under the right geometric conditions.

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Generalized Gibbs Distributions on Calabi-Vesentini Manifolds

Mathematical Structure

Calabi-Vesentini (CV) manifolds, given by $$\mathcal{M}_{CV}^{[2,q]} = SO(2,2+q)/SO(2) \times SO(2+q)$$, are argued to be the essential mathematical models for layers in Cartan Neural Networks (CaNNs) due to their covariance properties, explicit solvable group realization, and compatibility with special Kähler geometry. The analysis exploits:

• The explicit solvable Lie group parameterization (solvable coordinates);
• Construction and manipulation of the Kähler potential, revealing a nontrivial abelian structure where only half the Cartan 'angles' correspond to global isometries, while the remainder are non-isometric but required for a complete Darboux basis.

Partition Functions and Abelian Structures

For Gibbs-type distributions on these manifolds, classical Gaussian integrals are replaced by integrals over moment-maps of compact Cartan and Casimir invariants:

• Compact abelian structures: The Darboux basis is completed by inclusion of square roots of Casimir invariants of a nested sequence of subalgebras, which, while not corresponding to Killing isometries, still commute in the Poisson sense.
• Exact partition functions: For temperature vectors restricted to the compact Cartan, the partition function solves exactly to a product formula involving $(\beta_0^2 - \beta_i^2)$ and exponential terms, with explicit domains of convergence:

$$Z^{\mathfrak{b},\mathfrak{d}}(\beta_0,\beta_{i=1,...,\nu+1}) = c^{\mathfrak{b},\mathfrak{d}} \frac{ (8\pi^2)^{\nu+1} e^{-\beta_0}}{\prod_{i=1}^{\nu+1} (\beta_0^2 - \beta_i^2)},$$

with domain $\{\beta_0 > 0,\, \beta_0 > |\beta_i|\}$ (see eq. (zolotayaformula)).

Notably, these results apply uniformly across the CV Tits Satake universality class, showing high computational tractability in evaluating normalization constants for nontrivial Gibbs distributions on symmetric non-Euclidean event spaces.

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Extended Souriau Temperatures and Spontaneous Magnetization

The abelian structure completion motivates the introduction of extended temperature vectors, coupling to Casimir-derived moment maps beyond the compact Cartan. This generalization is structurally analogous to adding magnetic fields that break the $U$-isometry group:

• Turning on 'magnetic' temperatures causes nonvanishing expectation values of the corresponding non-Killing action variables. The system exhibits spontaneous symmetry breaking: the expectation values $m_i$ of the new Casimir functions remain nonzero even as the conjugate 'fields' vanish, directly paralleling spontaneous magnetization in ferromagnets.
• The precise pattern of symmetry breaking depends on the sequence of invariant Casimirs derived from the nested compact subalgebra hierarchy.

Consequently, the thermodynamic phase diagram is extraordinarily rich, with a correspondence between phase transitions (singularities in the Kähler metric curvature) and the nonanalytic locus of the partition function.

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Theoretical and Practical Implications

Implications for Machine Learning and AI

• Probability Distributions: The explicit form of the Gibbs measures derived here, with exact partition functions, provides well-defined, group-covariant alternatives to multiparametric Gaussian distributions for architectures (e.g., Cartan Neural Networks) where the underlying space cannot be treated as flat.
• Geometric Priors: Thermodynamic metrics and their singularities encode geometric priors and phase structure, offering a geometric approach to regularization, prior selection, and learning dynamics in models operating on data structurally modeled by symmetric spaces.
• Interpretability and Symmetry Breaking: The spontaneous magnetization mechanism corresponds in the machine learning context to cluster formation or category boundary delineation, suggesting a deep theoretical link between geometric thermodynamics and representation learning, especially in contexts with high symmetry (e.g., neural architecture layers with Lie group symmetries).

Broader Theoretical Consequences

• The framework assigns a central role to Kähler, and more generally, special Kähler geometry, accentuating the connection to mathematical structures originally developed for supergravity moduli spaces.
• The breakdown of isometry upon activation of Casimir-linked temperatures serves as a precise mathematical model for symmetry breaking transitions in geometrically-structured probability spaces.
• The explicit link between macroscopic (thermodynamic) and microscopic (event manifold) metrics enables a unified study of geometry-driven phenomena across physics and statistical learning.

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Conclusion

This work solidifies the geometrical foundations of thermodynamics and information geometry for statistically correlated systems on nontrivial symmetric spaces, offering exact analytic tools for the computation of partition functions and the analysis of symmetry breaking. The extended Souriau thermodynamics formulated here is not only mathematically robust—providing formulas for normalization, explicit construction of abelian Darboux structures, and tractable calculation of expectation values—but also has immediate analogs in statistical learning theory for complex data manifolds. The spontaneous emergence of magnetizations, interpreted as non-vanishing means of Casimir observables, points toward a new intersection of phase transition theory, geometric probability, and interpretability in AI models built atop non-Euclidean spaces. The prospect for further exploration includes detailed study of symmetry-breaking phase diagrams, metric geometry of the resulting Kähler structures, and explicit implementation in neural network frameworks leveraging these novel, geometry-driven probabilistic measures.