Entropy of the Serre Functor

• Entropy of the Serre functor is a categorical invariant that measures the exponential growth rate of morphism complexity under repeated duality in triangulated, dg, or module categories.
• It is defined via the limit of logarithmic complexity of Hom spaces computed from a split generator, connecting indices like spectral radii and Serre dimensions.
• This analysis bridges representation theory, algebraic geometry, symplectic topology, and mathematical physics by categorifying dynamical invariants and informing stability conditions.

The entropy of the Serre functor is a categorical dynamical invariant measuring the exponential growth rate of complexity under iterates of the Serre functor (or its analogues) in various triangulated, dg, or module categories. This notion links deep duality properties of the Serre functor with dynamical behavior, often interpreted in terms of growth rates, spectral radii, or dimensions, and plays a central role in the interface between representation theory, algebraic geometry, symplectic topology, and mathematical physics.

1. Categorical Definition and Formalism

The entropy of an autoequivalence $F$ (in particular, a Serre functor $S$) on a triangulated or dg-category $\mathcal{T}$ is defined as: $$h_t(F) = \lim_{n\to\infty} \frac{1}{n} \log \mathcal{O}_t(G, F^n(G))$$ where $G$ is a split generator of $\mathcal{T}$ and $\mathcal{O}_t(G, F^n(G))$ is a complexity function (often encoding the dimension or structure of $\mathrm{Hom}^*(G, F^n(G))$) (Han, 2022, Kikuta et al., 2019). The entropy can be specialized at $t = 0$ to measure the growth rate of graded pieces or morphism spaces: $$h_0(F) = \lim_{n\to\infty} \frac{1}{n} \log \dim \mathrm{Hom}^\bullet (G, F^n G)$$ For categories with a Serre functor $S$, the entropy $h_t(S)$ thus probes the rate at which morphism spaces “spread” under repeated dualities.

The upper and lower Serre dimensions are then defined as

$$\mathrm{Sdim}^+(\mathcal{T}) = \limsup_{m\to\infty}\frac{-e_-(G, S^m G)}{m}, \quad \mathrm{Sdim}^-(\mathcal{T}) = \liminf_{m\to\infty}\frac{-e_+(G, S^m G)}{m}$$

where $e_-$/$e_+$ are the extremal degrees of nonvanishing morphisms.

2. Explicit Calculations and Main Results

Partially Wrapped Fukaya Category and Gentle Algebras

For the partially wrapped Fukaya category $\mathcal{W}(\Sigma, Z)$ of a graded surface $\Sigma$ with stops $Z$, the entropy of the Serre functor $\mathbb{S}$ is given by: $$h_t(\mathbb{S}) = \begin{cases} (1-\min \Omega)t, & t \geq 0 \ (1-\max \Omega)t, & t \leq 0 \end{cases}$$ where

$$\Omega = \left\{\frac{\omega_1}{m_1}, \ldots, \frac{\omega_b}{m_b}, 0 \right\}$$

with $\omega_i$ the winding number of the $i$-th boundary component $\partial_i\Sigma$ and $m_i$ the number of stops on $\partial_i\Sigma$ (Chang et al., 20 Aug 2025).

Thus,

• The upper Serre dimension is $1-\min \Omega$,
• The lower Serre dimension is $1-\max \Omega$.

For a finite-dimensional gentle algebra $A$, which models the endomorphism algebra of a generator of a partially wrapped Fukaya category, the categorical entropy of the Serre functor is related to classical linear algebraic invariants: $h_0(\mathbb{S}) = \log \rho([\mathbb{S}])$ where $\rho([\mathbb{S}])$ is the spectral radius of the Coxeter transformation acting on the Grothendieck group of $A$ (a Gromov–Yomdin–like equality) (Chang et al., 20 Aug 2025, Han, 2022).

Higher Hereditary Algebras

For higher hereditary algebras $A$, and their perfect derived or dg categories, the entropy of the Serre functor is governed by the global dimension $d$, the Calabi–Yau dimension $g$ (if A is twisted fractionally Calabi–Yau), and the spectral radius of the Coxeter matrix $\Phi$. Key results (Han, 2022):

• For twisted fractionally Calabi–Yau algebras, $h_t(S) = g t$.
• For higher representation-infinite algebras,

$h_t(S) = d t + \log p(\Phi),$

where $p(\Phi)$ is the spectral radius of the Coxeter matrix.

• The categorical entropy and Hochschild (co)homology entropy coincide: $h^{HH}(S) = h(S)$.

Monomial Algebras and Noncommutative Projective Schemes

For a monomial algebra $A = k\Gamma/(F)$, the entropy of the Serre twist functor $S$ on $\mathrm{D}^b(\mathrm{qgr}\,A)$ satisfies

$$h_t(\mathrm{D}^b(\mathrm{qgr}\,A), S) = \log h_{\mathrm{alg}}(A)$$

where $h_{\mathrm{alg}}(A)$ is the algebraic entropy---the exponential growth rate of the dimension of graded pieces---which coincides with the topological entropy of the Ufnarovski graph $Q_A$ and, for path algebras, with the logarithm of the spectral radius of the adjacency matrix (Lu et al., 2021).

3. Conceptual and Categorical Interpretations

Calabi–Yau and Fractional Calabi–Yau Cases

In Calabi–Yau settings, where the Serre functor is a pure shift (i.e., $S \cong [n]$), the entropy is zero: the action is “periodic” up to shift, and Hom spaces are preserved up to degree shift after iteration. This is reflected both in modular representation theory and in strict polynomial functor categories, where on subcategories corresponding to basic blocks, the Serre functor acts as a shift (often $[2d(p^2-1)]$), indicating vanishing entropy (Chałupnik, 2016).

Stability Conditions and Gepner Type

For a triangulated category with a Serre functor $S$, the Serre dimension

$$\mathrm{Sdim} \,\mathcal{T} = \lim_{t \to +\infty} \frac{h_t(S)}{t}$$

is majorized by the infimum of global dimensions of Bridgeland stability conditions: $$\mathrm{Sdim}\,\mathcal{T} \leq \inf \{\mathrm{gldim}\,\sigma\}$$ (Kikuta et al., 2019). Gepner-type stability conditions, which satisfy $S \cdot \sigma = \sigma \cdot p$, realize equality when $\mathcal{T}$ is fractional Calabi–Yau.

Modular Tensor Categories and Spherical Morita Contexts

For module categories over finite tensor categories, the relative Serre functor $S_{\mathcal{M}}$ is isomorphic to the double dual, and in a spherical Morita context satisfies $S_M^2 \cong \mathrm{id}$ up to distinguished invertible objects (Radford isomorphism) (Fuchs et al., 2022). This periodicity forces zero entropy in the categorical dynamics.

Hybrid/Landau–Ginzburg Models

In the hybrid model description of residual categories, the Serre functor becomes an explicit line bundle twist followed by a shift: $S_{RX} \cong - \otimes \mathcal{O}(n+1-d)[n-k]$ Serre dimensions can then be computed by comparing the twist degree and the number of defining hypersurfaces, thereby controlling the growth rates of morphism spaces (Barbacovi et al., 2022).

4. Link to Representation Theory and Algebraic Geometry

The entropy invariant detects the dynamical (especially periodic or expanding) behavior of the Serre functor and is tightly constrained by representation-theoretic properties:

• In concealed-canonical, canonical, or derived-tame settings, the Serre functor often acts via cyclic permutation of summands, indicating zero or minimal entropy (Chan et al., 2015, Chan et al., 2017).
• Periodicity or twisted Calabi–Yau properties yield cases where minimal Auslander–Gorenstein algebras arise infinitely often, as the entropy is minimized due to predictable dynamics of the Serre functor (Chan et al., 2017).
• For gentle algebras and partially wrapped Fukaya categories, the categorical entropy is directly computable from combinatorial (winding, stop) data of the surface, and aligns with spectral radii of associated algebraic matrices (Chang et al., 20 Aug 2025).

5. Geometric and Topological Examples

In geometric representation theory and sheaf-theoretic contexts:

• On the constructible derived category $\mathrm{D}_c^b(\mathbb{P}^n)$, the inverse Serre functor is realized as a P-twist at the simple perverse sheaf of the open stratum; its entropy, controlled by the structure of P-objects and spherical twists, is typically zero or directly related to the combinatorics of the generating objects (Bonfert et al., 6 Jun 2025).
• In the dg-category of equivariant constructible sheaves on a flag variety $X$, the Serre functor is expressed via Matsuki functors and cohomological degree shift

$$\mathrm{Se}_{K\backslash X} \cong T_{GR \rightarrow K} \circ T_{K\rightarrow GR}[ \dim_\mathbb{R} X - \dim_\mathbb{R} K_R ]$$

suggesting, unless the Matsuki functors introduce expanding behavior on $K_0$, that the entropy is determined by the shift part (Chen, 2021).

• For microlocal and symplectic settings, as in the wrapped Fukaya category and sheaf-microlocalization, the Serre functor is identified (up to twist) with geometric wrapping functors (e.g., negative wrap-once functor), with the entropy tied to the underlying geometric flow (e.g., Reeb dynamics) (Kuo et al., 2022).

6. Broader Mathematical Significance

The entropy of the Serre functor categorifies classical dynamical invariants (such as topological entropy and spectral radii) and encodes how categorical duality interacts with growth phenomena and periodicity in abstract triangulated settings. Its computation and vanishing, often governed by Calabi–Yau, periodic, or spherical structures, facilitate the classification of categories with "tame" versus "wild" dynamics, inform the existence of stability conditions, and underpin links to homological mirror symmetry, topological field theories, and representation-theoretic phenomena.

The entropy invariant thus provides a unifying measure of complexity, linking moduli-theoretic, representation-theoretic, geometric, and dynamical perspectives in the modern theory of triangulated and dg-categories.