Geometric Trace Formulas

• Geometric trace formulas are a set of identities that express operator traces in terms of geometric or topological data, linking spectral invariants with closed orbits and symmetry groups.
• They unify techniques from quantum mechanics, spectral geometry, and arithmetic via explicit orbital integrals and Fourier expansions of operators.
• Recent developments extend these formulas using localization and sheaf-theoretic frameworks, impacting analysis in number theory, representation theory, and mathematical physics.

Geometric trace formulas provide a unified framework for expressing spectral invariants of differential, difference, or integral operators in terms of sums over geometric or topological data. These formulas arise naturally in the study of quantum mechanics, spectral geometry, automorphic forms, representation theory, and mathematical physics, offering deep links between spectra, dynamical invariants, and the geometry of underlying spaces—be they manifolds, discrete graphs, or moduli stacks. Across these contexts, “geometric” means that the orbital side of the formula involves explicit sums or integrals over closed geodesics, conjugacy classes, cycles, or algebraic data derived from the structure of the space or group under consideration. The following sections survey foundational constructions, archetypal examples, contemporary extensions, and the unifying principles of geometric trace formulas.

1. Fundamental Constructions and Paradigms

The geometric trace formula refers to a spectral identity that expresses a trace of an operator on a Hilbert (or derived) space as a sum (or integral) over orbit-type geometric data. The archetype is the Selberg trace formula, which for compact hyperbolic surfaces equates the spectral data (eigenvalues of the Laplacian) with explicit geometric terms indexed by conjugacy classes of the fundamental group, corresponding to closed geodesics:

$$\sum_n e^{-t\lambda_n} = \frac{\operatorname{Vol}(\Sigma)}{4\pi}\int_{-\infty}^\infty e^{-t(r^2+\frac{1}{4})} \tanh(\pi r) dr + \sum_{[\gamma]\neq e} \frac{\ell_0(\gamma)}{2\sinh(\ell(\gamma)/2)} e^{-t(\ell(\gamma)^2+\omega_w(\gamma))}$$

with $\lambda_n$ the Laplace eigenvalues and the second sum indexed by nontrivial conjugacy classes in $\Gamma$ (the fundamental group), each corresponding to a primitive closed geodesic (Choi et al., 2023). This paradigm recurs in:

• The Arthur–Selberg trace formula for reductive groups over local and global fields, connecting automorphic spectra to weighted orbital integrals over conjugacy classes (Hoffmann, 2014, Wong, 2019, Xinghua et al., 29 Aug 2025, Wang et al., 29 Aug 2025).
• The relative trace formula, relating periods of automorphic forms or eigenfunctions to double coset or special cycle data (Martin et al., 2015, Delorme et al., 2015).
• Variants appearing for discrete graphs, quantum systems, and moduli spaces, where geometric objects include cycles, paths, and moduli points (Korotyaev et al., 2022, Frenkel et al., 2010).

Key functional ingredients are: a global symmetry group (or graph automorphism group), a suitable operator (Laplace, Schrödinger, convolution, or transfer), and a decomposition of the space into geometric orbits under this group (Frenkel, 2012).

2. Geometric Trace Formulas on Periodic Graphs

For periodic magnetic Schrödinger operators on discrete graphs, the trace formula assumes a combinatorial–Fourier form capturing the presence of magnetic and electric potentials and their interaction with the graph's geometry. Consider a $\Gamma\cong\mathbb{Z}^d$-periodic graph $G$, with finite quotient $G_0=G/\Gamma$; cycles in $G_0$ encode all periodic, topologically distinct traversals.

For the fiber operator $H(k)$ (after Floquet decomposition), the powers admit an explicit geometric Fourier expansion (Theorem 2.6, (Korotyaev et al., 2022)):

$$\operatorname{Tr}[H(k)^n] = \sum_{c\in C_n}w(c)\exp\bigl\{-i[\alpha(c)+\langle\tau(c),k\rangle]\bigr\} = \sum_{m\in\mathbb{Z}^d,\,|m|\leq nT_+}T_{n,m}e^{i\langle m,k\rangle}$$

where the index $\tau(c)$ tracks the net lattice displacement of a cycle, $\alpha(c)$ is the total magnetic flux, and $w(c)$ is the product of edge weights modified by potentials. The Fourier coefficients are given by

$T_{n,m} = \sum_{c\in C_{n,m}}w(c)e^{-i\alpha(c)}$

Spectral invariants such as total bandwidth are bounded in terms of geometric and magnetic parameters via these expansions, leading to sharp inequalities and characterization of flat-band/degenerate spectrum phenomena. The geometric trace data is built explicitly from cycles and their associated indices (cf. §3 of (Korotyaev et al., 2022)).

3. Geometric Trace Formulas for Locally Symmetric and Arithmetic Spaces

Geometric trace formulas in the context of locally symmetric spaces, such as the Selberg trace formula, are constructed via harmonic analysis on quotients $\Gamma\backslash G/K$. A pivotal technical advance is the use of supersymmetric localization and path integral methods to provide conceptual and calculational access to these formulas, as developed in (Choi et al., 2023) and (Choi et al., 14 Feb 2025).

Localization identifies “saddle points” of the path integral—namely, classes of closed geodesics or torus-fixed configurations—which correspond directly to conjugacy classes in $\Gamma$. The orbital side is a sum over these classes, each term encoding the contribution from periodic (“geometric”) orbits, while the spectral side is the trace of an operator (e.g., $e^{-t\Delta}$) on appropriate function spaces. The method generalizes to higher rank locally symmetric spaces and compact Lie groups (Frenkel trace formula (Choi et al., 14 Feb 2025)):

• Contributions from constant loops correspond to continuous spectrum (identity class).
• Contributions from nontrivial conjugacy classes encode closed geodesics (Selberg) or windings in compact groups (Frenkel).
• Fluctuation determinants yield the precise prefactors, such as Weyl denominators or heat kernel factors.

In higher rank generalizations, the geometric contributions are indexed by more intricate orbit and conjugacy data, with explicit orbital integrals and weighting (Biró et al., 2023).

4. Trace Formulas for Reductive Groups and Their Geometric Side

For reductive algebraic groups over number fields, the Arthur–Selberg trace formula forms the backbone of modern trace formula theory. The geometric side decomposes into weighted orbital integrals indexed by (semi)simple conjugacy classes:

$$J_\mathrm{geom}(f) = \sum_{M} \sum_{\gamma\in\Gamma(M)} a^M(\gamma)J_M(\gamma,f)$$

with $J_M(\gamma,f)$ a weighted orbital integral, and $a^M(\gamma)$ an explicit volume factor. The geometric analysis is built upon:

• Detailed classification of semisimple, unipotent, and mixed conjugacy classes (e.g., for $\mathrm{GL}(n)$ (Xinghua et al., 29 Aug 2025, Wang et al., 29 Aug 2025)).
• Truncation procedures (Arthur truncation operator $\Lambda^T$) to regularize divergent terms and produce well-defined geometric expansions.
• Explicit computation of geometric contributions attached to ramified orbits as limiting distributions of unramified ones, with formulas in terms of zeta residues or higher derivatives of auxiliary functions (Xinghua et al., 29 Aug 2025).

Recent advances express the geometric side in terms of zeta integrals attached to prehomogeneous vector spaces (PVS), providing new explicit formulas for weighted orbital integrals and encapsulating the arithmetic structure of the coefficients in the trace formula (Hoffmann, 2014). This approach unifies and generalizes previous methods for classical groups of low rank (completely established for rank $\leq2$) and opens the way for further arithmetic applications.

5. Geometrization and Sheaf-Theoretic Frameworks

The geometric trace formula admits a categorified, sheaf-theoretic reformulation in the context of the geometric Langlands program (Frenkel et al., 2010, Frenkel, 2012). The classical sums of orbital integrals become traces (or RHom’s) of functors acting on derived categories of perverse sheaves or $D$-modules over moduli stacks of $G$-bundles ($\mathrm{Bun}_G$) on algebraic curves:

• For function fields ($k=\mathbb{F}_q$), orbital integrals are realized as Frobenius traces on the cohomology of moduli of $G$-pairs.
• For complex curves ($k=\mathbb{C}$), the cohomological orbital side is $\mathrm{RHom}$ over stacks, and functoriality is encoded in the equivalence of derived categories matching Hecke and Wilson functors.
• The spectral side is similarly geometrized via sheaf cohomology on the moduli of $^LG$-local systems.

This categorical perspective connects the classical trace formula to moduli-theoretic and representation-theoretic objects, leading to conjectural isomorphisms matching orbital and spectral cohomology, and giving rise to categorified versions of functoriality and spectral decomposition (Frenkel et al., 2010, Frenkel, 2012).

6. Specializations and Relative/Weighted Trace Formulas

Variants of geometric trace formulas arise in several directions:

• Relative trace formulas, which integrate kernel functions over subvarieties or subgroups (e.g., over a closed geodesic $C$ in a compact hyperbolic surface) (Martin et al., 2015), relate period spectra and ortholength spectra, producing identities between spectral sums over squared periods and sums over orthogonal geodesic segment lengths.
• Weighted variants, such as those including automorphic eigenfunction weights or characters, generalize to higher rank or to non-invariant settings, leading to sophisticated decompositions involving more subtle orbit classification and centralizer structure (Biró et al., 2023, Delorme et al., 2015).
• Twisted trace formulas incorporate automorphisms or involutive symmetries, replacing conjugacy classes by twisted conjugacy data, and yielding new geometric terms with direct spectral consequences (such as equivariant indices, $L_2$-torsion) (Liu, 2019).

Each instance retains the central theme: expressing spectral invariants as explicit geometric or cohomological sums/integrals, with the geometry encoding dynamical, arithmetic, or representation-theoretic information.

7. Applications, Impact, and Current Directions

Geometric trace formulas underpin a wide spectrum of results and conjectures:

• Weyl laws and spectral asymptotics,
• Prime geodesic theorems and spectral statistics in quantum chaos,
• Arthur’s endoscopic classification and functorial transfers,
• Lower bounds for spectral bands in quantum/graph models (Korotyaev et al., 2022),
• Explicit computations of L-functions via the $r$-trace formula and Beyond Endoscopy (Wong, 2019),
• Geometric Langlands dualities and categorification approaches to functoriality (Frenkel et al., 2010, Frenkel, 2012).

Ongoing developments include extending geometric trace techniques to higher rank and more general conjugacy classes (notably for prehomogeneous vector spaces), establishing finer control over convergence and truncation mechanisms, and further developing the sheaf-theoretic and categorical frameworks to provide a bridge between number theory, representation theory, and algebraic geometry.

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The geometric trace formula persists as a primary tool for extracting fine arithmetic, spectral, and dynamical invariants from elliptic, locally symmetric, or periodic structures, with methodology unifying combinatorial, analytic, and categorical techniques across mathematical physics, geometry, and arithmetic.