Geometric Perspective of Entropy Function: Embedding, Spectrum and Convexity

From the perspective of Sen entropy function, we study the geometric and algebraic properties of a class of (extremal) black holes in $ D \geq 4 $ spacetimes. For a given moduli space manifold, we describe the thermodynamic geometry away from attractor fixed point configurations with and without higher derivative corrections. From the notion of embedding theory, the present investigation offers a set of generalized complex structures and associated properties of differentiable manifolds. We have shown that the convexity of arbitrary entropy function can be realized in an extended subfield of the eigenvalues of the Hessian $\mathcal B$ of Sen entropy function. Thus, the spectra of $\mathcal B$ are analyzed by defining Krull of the corresponding semisymplectic algebras. From the framework of commutative algebra, we find that the convex hull of the eigenvalues defines a generalized spectrum of $\mathcal B$. The corresponding complexification is established for finitely many eigenvalues of $ \mathcal B $. For the minimally extended subfield, we show that the spectrum of $ \mathcal B $ reduces to the thermodynamic type spectra, at the attractor fixed point(s). In the limit of $ AdS_2 \times S^{D-2} $ near horizon geometry, the attractor flow analysis offers the stability of arbitrary extremal black hole. From the perspective of string compactifications, our investigation implies a set of deformed S-duality transformations, which contain both the duality invariant charges and monodromy invariant parameters. The role of the algebraic geometry is discussed towards the viewpoints of attractor stability conditions, rational conformal field theory, elliptic curves, deformed quantization(s), moduli manifolds and Calabi-Yau.