Functional Berry Connection Overview

• Functional Berry Connection is a gauge potential defined over continuum fields and many-body states, extending beyond a finite set of external parameters.
• It unifies approaches from curved spacetime, electromagnetic modes, and many-body superconductivity while preserving a gauge-theoretic structure.
• Its formulation impacts transport phenomena, topological invariants, and experimental observables through the analysis of curvature, holonomy, and emergent gauge effects.

Functional Berry connection denotes a class of Berry connections whose dependence is carried by continuum fields, many-body wave functions, modular Hamiltonians, or background geometry rather than only by a finite list of external parameters. In the cited literature, the same geometric idea appears as a covariant connection for Dirac and Weyl spinors in curved spacetime, a momentum-space connection functional of electromagnetic mode fields and material tensors, a real-space gauge field extracted from many-body wave functions, a non-Abelian connection on bundles of modular zero modes over the space of regions, and a connection on state bundles over conformal manifolds (Kumar et al., 2022, Gangaraj et al., 2016, Koizumi, 2023, Czech et al., 2017, Baggio et al., 2017).

1. Definition and scope

In its standard form, the Berry connection is the parameter-space gauge potential associated with a smoothly varying family of normalized eigenstates. For a nondegenerate band or level one writes

$$\mathcal{A}_\mu(\lambda)= i\,\langle \psi(\lambda)|\partial_{\lambda^\mu}\psi(\lambda)\rangle,$$

while in Bloch problems this becomes

$$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$

For degenerate subspaces the connection is non-Abelian,

$$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$

so it acts on a vector bundle of states rather than on a single phase line (Kartik et al., 2019, Thacker et al., 2014, Cole et al., 21 Feb 2026).

The adjective “functional” becomes appropriate when the connection is presented as an explicit functional of fields or states. In curved-spacetime Dirac theory the covariant Berry connection depends on the metric, tetrads, spin connection, and local momentum. In classical electromagnetics it is written as a functional of continuum mode envelopes and constitutive tensors. In many-body superconductivity it is defined directly from the many-electron wave function and interpreted as an emergent gauge field in real space. In modular and conformal-field-theoretic settings, the relevant parameter is the spatial region or the marginal coupling, so the connection lives over the space of regions or over the conformal manifold rather than over ordinary momentum space (Kumar et al., 2022, Gangaraj et al., 2016, Koizumi, 2022, Czech et al., 2017, Baggio et al., 2017).

This breadth has two immediate consequences. First, functional Berry connections are not tied to a single ontological setting: they occur in single-particle quantum mechanics, continuum wave physics, interacting many-body systems, and quantum field theory. Second, the base space on which the connection is defined can be spacetime, phase space, momentum space, real space, coupling space, complex rapidity space, or the kinematic space of subregions, depending on the problem (Thacker et al., 2014, Ohya, 2020, Czech et al., 2017).

2. Gauge structure and covariance

A functional Berry connection retains the gauge-theoretic structure of the ordinary Berry connection. Under a local phase transformation

$$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$

the connection transforms as

$$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$

and in the non-Abelian case

$$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$

The cited works repeatedly emphasize that the connection is gauge dependent, whereas holonomies, curvatures, and appropriately defined Wilson loops are the invariant data (Gangaraj et al., 2016, Kartik et al., 2019, Cole et al., 21 Feb 2026).

In covariant gravitational formulations, gauge structure is supplemented by spacetime covariance. The Dirac equation in curved spacetime,

$$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$

contains the spinor covariant derivative

$\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$

with $\Omega_\mu$ built from the spin connection. The resulting Berry-like connection is therefore covariant under general coordinate transformations and local Lorentz transformations, not merely under phase redefinitions of the eigenbasis (Kumar et al., 2022).

A related extension occurs in modular Berry theory. There the parameter is the region $A$ and the fiber is the space of modular zero modes commuting with the modular Hamiltonian $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$0. The gauge freedom is the freedom to redefine the zero-mode basis at each point of the space of regions. In the vacuum of a $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$1 CFT, global conformal symmetry singles out a specific modular Berry connection, but even there the connection is defined only up to a gauge transformation $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$2 (Czech et al., 2017).

The classical-electromagnetic literature also removes a common misconception: Berry connection is not inherently quantum. In continuum bianisotropic media, Maxwell’s equations define a Hermitian eigenproblem with a generalized inner product involving the material matrix $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$3, and the Berry connection emerges as a gauge potential on the bundle of classical mode envelopes over $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$4-space (Gangaraj et al., 2016).

3. Spacetime and real-space constructions

In the covariant gravitational construction for spin-$$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$5 particles, a WKB ansatz

$$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$6

leads, at order $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$7, to a transport equation for the amplitudes $$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$8 and identifies the matrix-valued connection

$$\mathbf{A}_n(\mathbf{k})= i\,\langle u_n(\mathbf{k})|\nabla_{\mathbf{k}}u_n(\mathbf{k})\rangle.$$9

Because the local eigenspinors $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$0 depend on $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$1, on the spin connection $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$2, and on the local momentum $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$3, the connection is explicitly a functional of the gravitational field. The same paper extends the construction to massless Weyl particles, where the analogous connection contains an explicit $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$4 and the effective transport velocity becomes observer dependent (Kumar et al., 2022).

That gravitational connection admits an internal decomposition,

$$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$5

The cited interpretation assigns the $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$6-term to a Pancharatnam–Berry-like contribution and the $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$7-term to an Aharonov–Bohm-like contribution; for spherically symmetric metrics the latter vanishes because only $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$8 components survive and $$(\mathcal{A}_\mu)_{mn}(\mathbf{k})= i\,\langle u_{m\mathbf{k}}|\partial_\mu u_{n\mathbf{k}}\rangle,$$9 when $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$0 (Kumar et al., 2022).

A second real-space construction appears in many-body superconductivity. One definition is

$$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$1

which is a vector potential functional of the many-body wave function. In related formulations the same real-space connection is written as $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$2, so that the superconducting phase variable is reinterpreted as a Berry phase field and enters the effective gauge potential through $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$3 (Koizumi, 2023, Koizumi, 2022, Koizumi, 2021, Koizumi, 2021, Koizumi et al., 2021).

These many-body constructions are explicitly functional in two senses. The connection is computed from a many-body state or density matrix, and the associated energy or transport equations become functionals of the phase field $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$4 or of $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$5. The cited papers then use that field to formulate supercurrent generation, flux quantization, London response, electromotive-force-like effects, and particle-number-conserving Bogoliubov–de Gennes equations (Koizumi, 2023, Koizumi, 2021, Koizumi et al., 2021).

4. Momentum-space, modular, and field-space constructions

In continuum electromagnetics, the source-free Maxwell equations in a homogeneous lossless bianisotropic medium can be written as a generalized Hermitian eigenproblem. For a six-component field envelope $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$6 and material matrix $$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$7, the Berry connection is

$$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$8

or, in dispersive media,

$$|u_n(\mathbf{k})\rangle \to e^{i\phi_n(\mathbf{k})}|u_n(\mathbf{k})\rangle,$$9

Here the connection is a functional of the field configuration and of the constitutive tensors $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$0 (Gangaraj et al., 2016).

In crystalline band theory, the functional viewpoint reappears in the projection gauge. Starting from projected and orthonormalized Bloch-like states, the non-Abelian Berry connection in the projection gauge is derived in a closed $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$1-local form, expressed entirely through local projectors, overlap matrices, velocity matrix elements, and position matrix elements. This eliminates finite-difference momentum derivatives and recasts the connection as an explicit functional of $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$2, trial orbitals, and Bloch data (Cole et al., 21 Feb 2026).

Chiral lattice systems provide a related momentum-space realization. For a two-band chiral model with Bloch vector $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$3 in the equatorial plane, a convenient gauge gives

$$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$4

The cited work shows that, in this gauge, the Berry connection itself controls an observable mean chiral displacement of delocalized wave packets (Colandrea et al., 2024).

In non-Hermitian band theory the single-band object splits into four biorthogonal connections,

$$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$5

Specific gauge-invariant combinations of $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$6, $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$7, and $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$8 enter the semiclassical equations of motion, so the functional Berry connection becomes part of the transport law rather than only of a topological classification (Silberstein et al., 2020).

The modular Berry connection changes the base space altogether. For a region $$\mathbf{A}_n(\mathbf{k})\to \mathbf{A}_n(\mathbf{k})+\nabla_{\mathbf{k}}\phi_n(\mathbf{k}),$$9 with modular Hamiltonian $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$0, the parameter is the region itself, and the fiber is the space of modular zero modes. In the vacuum of a $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$1 CFT the modular Berry connection on kinematic space is

$$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$2

with $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$3 the interval entanglement entropy. In AdS$$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$4, Wilson loops of this connection reproduce lengths of bulk curves via differential entropy (Czech et al., 2017).

A further field-space construction occurs in QFT on coupling space. When couplings $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$5 vary adiabatically, the Berry connection on the bundle of states over the conformal manifold coincides with the connection obtained in conformal perturbation theory. For chiral primary states in $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$6 $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$7 and $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$8 $$\tilde{\mathcal{A}}_\mu = U^\dagger \mathcal{A}_\mu U + i U^\dagger \partial_\mu U.$$9 SCFTs, the corresponding Berry curvature is governed by the $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$0 equations (Baggio et al., 2017). An analytically distinct but structurally allied realization is the BPS Berry connection, where a family of Schrödinger Hamiltonians is engineered so that the ground-state non-Abelian Berry connection becomes the $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$1 BPS monopole over $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$2 (Ohya, 2020).

5. Curvature, holonomy, and topological order

Once a functional Berry connection is given, the associated curvature follows the usual pattern. In abstract form,

$$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$3

for non-Abelian bundles, while the Berry phase along a closed loop $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$4 is

$$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$5

The gravitational literature writes the corresponding closed-loop phase as $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$6, and the modular literature interprets the Wilson loop of $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$7 as a geometric holonomy in kinematic space (Kumar et al., 2022, Czech et al., 2017).

In photonics and continuum electromagnetics, the curvature $$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$8 and the Chern number

$$\big(i\hbar \gamma^\mu \nabla_\mu - m\big)\Psi(x)=0,$$9

are computed directly from classical modes. The curved-waveguide example yields a Berry monopole field $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$0, while the biased magneto-plasma example gives explicit Berry connection and Berry curvature formulas in terms of the dispersive permittivity tensor (Gangaraj et al., 2016).

In one-dimensional topological superconductors and related chiral systems, the Berry connection reduces to a winding problem. For the Kitaev chain, the winding number can be written as

$\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$1

and the paper uses the argument principle in complex analysis to characterize the topological phase by the position of zeros of an associated meromorphic function relative to the unit circle. In that framework, singularities of the Berry connection at gap-closing points mark topological transitions and make winding numbers ill-defined (Kartik et al., 2019).

In $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$2 QED and in the QCD analogy developed from it, the Berry phase serves as a topological order parameter rather than merely as a transport correction. For the Bloch form $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$3, the Berry connection

$\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$4

has Wilson loop

$\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$5

which is interpreted as electric polarization of the vacuum. The $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$6 QED paper identifies $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$7 with transport of an integer unit of charge across the cell, while the QCD paper interprets the analogous Berry phase as the polarization of Chern–Simons membrane–antimembrane pairs and the label of discrete quasivacua differing by $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$8 (Thacker et al., 2014, Thacker, 2014).

6. Dynamical, transport, and experimental manifestations

The cited literature repeatedly assigns direct dynamical consequences to functional Berry connections. In curved spacetime, the covariant $\nabla_\mu\Psi=(\partial_\mu+\Omega_\mu)\Psi,$9 governs geometric phases of massive Dirac and massless Weyl particles, and the decomposition into Pancharatnam–Berry-like and Aharonov–Bohm-like pieces provides a covariant route to gravitational geometric phases (Kumar et al., 2022).

In chiral lattice systems, the mean chiral displacement

$\Omega_\mu$0

obeys an asymptotic relation in which it becomes the convolution of the initial momentum distribution with $\Omega_\mu$1. For a narrow wave packet, $\Omega_\mu$2. A photonic topological quantum walk then measures the Berry connection over the Brillouin zone and recovers the winding number by integration (Colandrea et al., 2024).

In non-Hermitian systems, the semiclassical wave-packet equations acquire Berry-connection-induced anomalous terms. For a dominant band,

$\Omega_\mu$3

while the norm and center-of-mass velocity depend on the effective complex energy

$\Omega_\mu$4

This yields an anomalous weight rate and, already in one dimension, an anomalous drift velocity proportional to $\Omega_\mu$5 (Silberstein et al., 2020).

The many-body superconductivity papers make the strongest claim for direct macroscopic observability. There the Berry connection enters the velocity field,

$\Omega_\mu$6

or equivalently through $\Omega_\mu$7, and is used to explain supercurrent generation, flux quantization, the London moment, Andreev–Saint-James reflection, the Josephson effect, and Berry-connection motive force. One example is the generalized Faraday-like law

$\Omega_\mu$8

and another is the particle-number-conserving Bogoliubov–de Gennes formulation in which the pairing field carries explicit $\Omega_\mu$9 factors derived from the many-body Berry connection (Koizumi, 2023, Koizumi, 2021, Koizumi, 2021, Koizumi, 2022, Koizumi et al., 2021).

The BPS construction adds a parameter-space realization with non-Abelian holonomy. There the ground-state Berry connection of a family of spin-$A$0 Schrödinger Hamiltonians is exactly the $A$1 BPS monopole, so adiabatic transport in parameter space realizes monopole Wilson lines and Higgs-field data through the Berry bundle itself (Ohya, 2020).

7. Limitations, obstructions, and unsettled points

Functional Berry connections inherit the usual adiabatic and smoothness assumptions, but each construction adds further restrictions. The gravitational formalism is explicitly semiclassical and WKB based; its Berry connection is derived at leading nontrivial order in $A$2, and the massless case introduces an observer-dependent effective velocity because there is no rest frame for Weyl particles (Kumar et al., 2022).

In projection-gauge band theory, the exact local formula removes finite-difference errors but requires a smooth gauge built from projected trial orbitals. Small singular values of the overlap matrix signal ill-conditioned projections, and topologically nontrivial cases may obstruct symmetry-preserving smooth gauges, forcing a gauge choice that breaks the obstructing symmetry (Cole et al., 21 Feb 2026). In non-Hermitian systems, the single-band semiclassical description is valid only when one band dominates by having the largest imaginary part of the energy; crossings in $A$3 and exceptional points undermine the adiabatic single-band picture (Silberstein et al., 2020).

The superconductivity papers themselves state important limitations. The constructions rely on spin-twisting itinerant motion, on a well-defined many-body wave function or density matrix, and on hydrodynamic treatment of the emergent gauge field. Practical evaluation of $A$4 in realistic materials is described as nontrivial, and the explicit model calculations are tied to specific microscopic pictures such as spin-vortices or Rashba-coupled lattices (Koizumi, 2021, Koizumi et al., 2021).

A persistent conceptual issue is gauge dependence. The Berry connection is not directly observable as a gauge-invariant scalar field in general; what is invariant are curvatures, Wilson loops, winding numbers, Chern numbers, and gauge-invariant combinations selected by a physical measurement protocol. The chiral-lattice and non-Hermitian papers show that the connection itself can enter directly once the measurement basis or the biorthogonal structure fixes the relevant gauge combination, but this does not erase its status as a connection rather than an absolute observable (Gangaraj et al., 2016, Colandrea et al., 2024, Silberstein et al., 2020).

Taken together, these results suggest a unifying interpretation: a functional Berry connection is a connection 1-form on a bundle of states, modes, or zero modes whose coefficients are explicit functionals of continuum data—metrics, tetrads, constitutive tensors, many-body wave functions, projectors, regions, or couplings. Its central role is not merely classificatory. It organizes transport equations, geometric phases, Wilson loops, polarization responses, topological invariants, and, in several settings, directly measurable dynamics.