Complex Legendre Transform

• Complex Legendre Transform is a generalization of the classical Legendre transformation that extends duality from convex analysis to the realm of Kähler manifolds using strictly plurisubharmonic potentials.
• It establishes an involutive, isometric mapping on the space of Kähler metrics, playing a crucial role in symplectic geometry, complex Monge–Ampère equations, and quantum mechanics.
• The framework also leads to generalized dualities through deformed Legendre transforms for non-additive entropies like Tsallis and Rényi, impacting statistical mechanics and differential geometry.

The complex Legendre transform is a generalization of the classical Legendre transformation, extending the duality framework from real affine geometry and convex analysis to the setting of Kähler manifolds and complex differential geometry. It provides a local isometric symmetry on the space of Kähler metrics and serves as a foundational tool for understanding symplectic geometry, complex Monge–Ampère equations, and statistical manifolds with non-Euclidean or non-dually-flat structures. The complex Legendre transform is tightly connected to the theory of real-analytic Kähler potentials, the geometry of infinite-dimensional symmetric spaces, generalized divergences such as the Rényi divergence, and physical applications in quantum and statistical mechanics.

1. Classical Legendre Transform and Its Extension

The classical Legendre transform of a convex function $$\psi : \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$$ is defined as

$$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$

This involutive transformation underpins the duality structure of convex analysis, thermodynamics, and classical mechanics.

In the complex setting, particularly for Kähler manifolds, this duality is generalized by replacing the real pairing (dot product) with an appropriate polarization or kernel derived from a strictly plurisubharmonic Kähler potential. The geometrization of the Legendre transform replaces the Euclidean scalar product with a more general link function or diastasis (Berndtsson et al., 2016, Lempert, 2017).

2. Complex Legendre Transform on Kähler Manifolds

Let $M$ be a compact complex manifold of complex dimension $n$, equipped with a real-analytic Kähler form $\omega$. For each chart $U \subset M$, there exists a real-analytic, strictly plurisubharmonic potential $\phi : U \to \mathbb{R}$ such that $\omega|_U = i \partial \bar{\partial} \phi$. The diastasis function $D_\phi$ associated with $\phi$ takes the role of a canonical “distance-like” function: $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$0 where $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$1 denotes the analytic continuation (polarization) of $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$2 in a neighborhood of the diagonal.

The local complex Legendre transform $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$3 of a function $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$4 is defined as

$$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$5

or, equivalently,

$$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$6

When $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$7 is sufficiently regular and small in the $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$8-norm, $$\psi^*(y) = \sup_{x \in \mathbb{R}^n} \{ x \cdot y - \psi(x) \}.$$9 is involutive, i.e., $M$0 in a neighborhood of $M$1 (Berndtsson et al., 2016, Lempert, 2017).

3. Geometric and Analytic Properties

The complex Legendre transform globally assembles a map $M$2 on a compact Kähler manifold $M$3, independent of the choice of chart and local potential. For $M$4 a small enough element in the space $M$5 of Kähler potentials,

$M$6

Key theorems established for $M$7 are:

• Involution: $M$8, and $M$9.
• Gradient Map and Uniqueness: The supremum is uniquely attained at $n$0, with $n$1 a $n$2-diffeomorphism satisfying $n$3.
• Complex Monge–Ampère Pullback: $n$4.
• Isometry for the Mabuchi Metric: $n$5 is an isometry of the infinite-dimensional Mabuchi Riemannian metric on $n$6, sending geodesics to geodesics (Berndtsson et al., 2016, Lempert, 2017).

The analyticity of the reference Kähler potential is both necessary and sufficient for the existence of fixed points of the complex Legendre transform—a fixed point is real-analytic if and only if it is left unchanged by a local Legendre symmetry (Lempert, 2017).

4. Symplectic and Complex Geometric Framework

The complex Legendre transform can be derived via symplectic geometry through the complexification of a real divergence $n$7, which induces a non-canonical symplectic two-form $n$8: $n$9 where $\omega$0 are dual coordinates.

A Kähler complexification requires the existence of an integrable almost-complex structure $\omega$1 for which the cross-partial symmetry and certain curvature-type conditions are satisfied. The Kähler potential $\omega$2 is constructed from the divergence $\omega$3, and the complex Legendre duality is recast as a pair of conjugate mappings between holomorphic potentials $\omega$4 and $\omega$5, interconnected via the link function $\omega$6 (Morales et al., 2022).

5. Generalized Dualities and Deformed Legendre Transforms

Beyond the classical and complex cases, the Legendre transform can be further generalized for non-additive entropies, such as Tsallis and Rényi, leading to deformed dualities applicable to systems with power-law statistics or non-dually flat information geometries.

For the Rényi divergence of order $\omega$7, the link function becomes

$\omega$8

and potentials $\omega$9 and $U \subset M$0 are constructed using integration against power-law kernels. In this context, the (deformed) complex Legendre transform encodes the curvature of the statistical manifold, with involutivity and duality properties adapted to $U \subset M$1-concave functions and corresponding non-Euclidean structures (Kalogeropoulos, 2017, Morales et al., 2022).

6. Examples and Illustrative Cases

• Flat Euclidean Case: For $U \subset M$2 with the flat metric $U \subset M$3, the complex Legendre transform reduces to the standard real-part Legendre transform, with the fixed point at the quadratic potential $U \subset M$4.
• Fubini–Study and Spin Coherent States: Taking $U \subset M$5 yields the Fubini–Study metric on complex projective space, connecting to quantization and coherent state models (Morales et al., 2022).
• $U \subset M$6-Gaussian Equilibria: In non-additive thermodynamics, minimizers of generalized free energies derived from the $U \subset M$7-Legendre transform yield $U \subset M$8-Gaussian distributions, consistent with systems exhibiting power-law tails (Kalogeropoulos, 2017).

7. Physical and Mathematical Significance

The complex Legendre transform establishes a unified language for duality in curved statistical manifolds, connecting Kähler geometry, information geometry, and physical models beyond the reach of classical Bregman or exponential-family machinery. Applications include:

• Symmetry and isometry structures in the space of Kähler potentials, vital for the study of complex Monge–Ampère equations and moduli of Kähler metrics.
• Deformed Hamiltonian and symplectic flows in phase space corresponding to non-canonical statistical and physical systems, such as those described by Rényi or Tsallis theories.
• Explicit construction of finite-dimensional quantization models and links to field theory, condensed matter physics, and anomalous transport phenomena (Morales et al., 2022, Berndtsson et al., 2016).

The construction crucially depends on analyticity conditions, integrability requirements for complex structures, and the algebraic properties of the underlying divergence or kernel function. The framework provides a platform for further exploration of generalized dualities, non-equilibrium thermodynamics, and statistical geometry in both mathematical and physical domains.