Higher Berry Connection Overview

• Higher Berry connection is a generalized framework that extends conventional Berry phase concepts using higher-form and non-Abelian gauge structures.
• It organizes differential forms whose integrals yield topological invariants such as Chern numbers, Chern–Simons terms, and higher holonomies in various quantum systems.
• It finds practical applications in band theory, many-body systems, boundary conformal field theory, and string field theory to probe gapped phases and anomaly inflow.

Higher Berry connection denotes a family of generalizations of the ordinary Berry connection that arise when one passes from a $U(1)$ line-bundle connection on parameter space to either higher-form geometry of families of gapped many-body systems or non-Abelian connections on multi-band occupied subspaces. In the many-body and QFT literature, the central object is typically a $2$-form connection $B$ with closed $3$-form curvature $H=dB$, or, for a $D$-dimensional gapped system, a closed $(D+2)$-form on parameter space whose cohomology class is a topological invariant. In crystalline-band settings, the same theme is tied to the non-Abelian Berry connection of occupied Bloch bands and to higher response functionals such as Berry phases, Chern numbers, second Chern numbers, and Chern–Simons terms (Kapustin et al., 2020, Lo et al., 24 Feb 2026, Cole et al., 21 Feb 2026).

1. Ordinary Berry geometry and its higher-form extension

For a nondegenerate family of states $|\psi(\lambda)\rangle$, the ordinary Berry connection is the $1$-form

$$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$

with curvature $2$0. In the standard geometric picture, the ground states form a complex line bundle over parameter space, and $2$1 represents the first Chern class. The higher Berry program replaces this line-bundle geometry by higher-form geometry attached to families of gapped many-body systems, where field theory predicts that a $2$2-dimensional system naturally gives a closed $2$3-form on parameter space (Kapustin et al., 2020).

In the $2$4-dimensional many-body case studied in several recent works, the basic higher object is a $2$5-form connection with $2$6-form curvature. One formulation writes

$2$7

with $2$8 a closed $2$9-form whose periods define higher Berry invariants. A nontrivial higher Berry invariant means that the family of ground states does not form an ordinary line bundle, but a more general higher object, such as a bundle gerbe; in this language, the higher Berry connection is a connection on a gerbe or higher bundle, and $B$0 is its curvature (Lo et al., 24 Feb 2026, Choi et al., 16 Jul 2025).

This higher-form usage coexists with another established usage in band theory. There, the non-Abelian Berry connection of an occupied multiplet is still a $B$1-form, but it is matrix-valued, and higher topological response functionals are built from it. The two usages are not identical, but they are linked by the fact that both organize geometric information of families of gapped states into gauge-structured differential forms whose integrals are topological observables (Cole et al., 21 Feb 2026).

2. Non-Abelian Berry connection and gauge structure

For an insulator with $B$2 occupied Bloch bands, the non-Abelian Berry connection is

$B$3

with gauge transformation

$B$4

Because of the inhomogeneous term, $B$5 is not gauge covariant, so any direct use of it as a smooth field on the Brillouin zone requires a smooth global gauge. In Wannier and projection-gauge constructions, one obtains such a gauge by projecting localized trial orbitals onto the occupied subspace and then performing symmetric Löwdin orthonormalization. Within this framework, an exact expression has been derived for the projection-gauge non-Abelian Berry connection that is strictly local in $B$6, uses only $B$7, $B$8, $B$9, position matrix elements $3$0, and the Kubo-like matrix $3$1, and avoids finite differences, neighboring-$3$2 overlaps, and matrix logarithms (Cole et al., 21 Feb 2026).

That exact local formulation matters because higher response functionals such as the Chern–Simons axion angle are non-Abelian and non-gauge-covariant at the integrand level. Errors in the underlying $3$3 therefore propagate delicately. The same paper shows that the exact local projection-gauge connection provides a stable route for Berry phase and Chern–Simons calculations in tight-binding benchmarks, including the SSH model and the Fu–Kane–Mele model (Cole et al., 21 Feb 2026).

A different gauge problem appears in non-Hermitian systems. There, left and right eigenvectors are independent, biorthogonal normalization leaves a $3$4 frame freedom, and the standard Berry connection admits four inequivalent definitions depending on how left and right eigenvectors are paired. A metric-covariant formalism built from a Hilbert-space metric tensor and a covariant derivative yields a unique Berry connection

$3$5

which remains real-valued under arbitrary $3$6 frame changes and reduces to the conventional Berry or Wilczek–Zee connection in the Hermitian limit (Arkhipov, 27 Jan 2026).

3. Wave-function, Green’s-function, and tensor-network constructions

A general interacting-lattice construction associates to a $3$7-dimensional family of gapped Hamiltonians a tower of local forms satisfying descent relations and produces a closed $3$8-form $3$9 on parameter space. In this formalism, cutoff functions $H=dB$0 encode the coarse geometry of the infinite lattice, and the resulting cohomology class $H=dB$1 is independent of the detailed choice of $H=dB$2. For short-range entangled families, integrals of these forms over spherical cycles are argued to be quantized, providing higher analogues of Chern numbers (Kapustin et al., 2020).

A wave-function formulation for locally parameterized states rewrites the same structure directly in terms of local exterior derivatives $H=dB$3, simplex chains, and the simplex-exterior derivative $H=dB$4. For a $H=dB$5-dimensional system, the higher Berry curvature becomes a closed $H=dB$6-form

$H=dB$7

where $H=dB$8 is obtained from repeated application of $H=dB$9 to the ordinary Berry connection and $D$0 is a $D$1-cochain. In $D$2, $D$3 describes the flow of Berry curvature from one boundary at infinity to the other; in general, $D$4 characterizes a flow of $D$5-form higher Berry curvature (Sommer et al., 2024).

For translationally invariant matrix product states, the discrete higher Berry connection is attached to oriented triangles in parameter space. If $D$6 denotes the dominant eigenvector of the mixed transfer matrix between neighboring MPS and $D$7 the Schmidt matrix, then

$D$8

The higher Berry curvature on a tetrahedron is the coboundary

$D$9

and the sum over a closed triangulated $(D+2)$0-manifold yields an integer. This is the MPS realization of the $(D+2)$1 class associated with one-dimensional invertible phases (Shiozaki et al., 2023).

4. Boundary scattering, BCFT, and curvature flow

A physically transparent realization of higher Berry invariants appears in one-dimensional gapped free-fermion systems coupled to a gapless lead. The reflection matrix $(D+2)$2 defines a smooth map from parameter space $(D+2)$3 into $(D+2)$4, and the higher invariant on a $(D+2)$5-dimensional parameter space is the winding number

$(D+2)$6

For $(D+2)$7, this computes the degree of the map $(D+2)$8. The resulting invariant is robust against perturbations such as disorder and provides a boundary-scattering probe of higher Berry phase; in the explicit $(D+2)$9 continuum and lattice examples analyzed in that work, $|\psi(\lambda)\rangle$0 (Lo et al., 24 Feb 2026).

A closely related BCFT formulation interprets higher Berry curvature as a flow of ordinary Berry curvature in Fock space. For parametrized conformal boundary conditions inherited from a family of gapped systems, one defines a total $|\psi(\lambda)\rangle$1-form Berry curvature $|\psi(\lambda)\rangle$2 of the filled modes and then the higher Berry curvature

$|\psi(\lambda)\rangle$3

In the $|\psi(\lambda)\rangle$4 example built from a two-species Dirac BCFT,

$|\psi(\lambda)\rangle$5

The integral measures a Chern number pump in the Fock space of the BCFT, and the same structure appears in entanglement Hamiltonians obtained from regularized boundary states (Wen, 16 Jul 2025).

These constructions make the “higher” character concrete. The ordinary Berry curvature of individual modes is not discarded; instead, its transport through parameter space is organized into a closed $|\psi(\lambda)\rangle$6-form, and boundary reflection data or BCFT spectral flow provide direct observables for that higher geometry.

5. Higher Berry geometry in topological response theory

In crystalline insulators, the non-Abelian connection and its curvature feed directly into higher response functionals. In one dimension, the Berry phase is

$|\psi(\lambda)\rangle$7

while in three dimensions the Chern–Simons axion angle may be written as

$|\psi(\lambda)\rangle$8

The exact projection-gauge connection derived locally in $|\psi(\lambda)\rangle$9 space reproduces the quantized SSH Berry phase and gives an accurate Chern–Simons axion angle along the Fu–Kane–Mele adiabatic cycle that pumps $1$0 by $1$1 with second Chern number $1$2 (Cole et al., 21 Feb 2026).

A distinct, but related, program rewrites a four-dimensional Chern insulator as a family of translationally invariant one-dimensional chains over the three-dimensional Brillouin zone and computes its higher $1$3-form Berry curvature using iMPS. In that setting, one defines a discrete higher Berry phase on triangles, a higher Berry curvature on tetrahedra, and the Dixmier–Douady–Kapustin–Spodyneiko invariant

$1$4

For the lattice Dirac model, the phase diagram of $1$5 is exactly congruent to the known phase diagram of the second Chern number. The same work then compares the higher Berry phase on $1$6-dimensional parameter slices with the Chern–Simons magnetoelectric coupling $1$7, finding excellent agreement in the four-dimensional Dirac model, while also showing that the identification is not universal in the Hopf-insulator example (Heinsdorf et al., 24 Jun 2026).

Taken together, these results place higher Berry connection at the interface between wave-function geometry and response theory: second Chern numbers, Chern–Simons terms, and magnetoelectric couplings can all be reformulated in terms of higher Berry data of families of lower-dimensional ground states.

6. Boundary conformal manifolds, string field theory, modular geometry, and QFT couplings

On boundary conformal manifolds, higher Berry connection is defined directly from phases of boundary-condition-changing three-point functions. If $1$8 is the OPE coefficient for three nearby boundary conditions, then

$1$9

In free compact bosons with Dirichlet moduli $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$0, $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$1 reproduces the constant target-space Kalb–Ramond $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$2-field. In WZW models, $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$3 becomes the standard Wess–Zumino $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$4-form, and for D-brane moduli spaces the higher Berry connection coincides with the NS–NS $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$5-field (Choi et al., 16 Jul 2025).

Open string field theory admits an analogous construction on the space of classical solutions. For a family $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$6 satisfying $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$7, the proposed $$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$8-form connection is

$$A = A_i(\lambda)\, d\lambda^i,\qquad A_i(\lambda)= i\langle \psi(\lambda)|\partial_{\lambda^i}\psi(\lambda)\rangle,$$9

with curvature $2$00 and higher holonomy $2$01 on closed $2$02-cycles. This $2$03 transforms by $2$04 under the infinite-dimensional OSFT gauge algebra, and the paper suggests identifying it with the Kalb–Ramond $2$05-field in favorable situations (Choi, 14 Feb 2026).

A different generalization is the modular Berry connection, defined on the space of regions rather than the space of couplings. In the vacuum of a $2$06-dimensional CFT, the zero modes of modular Hamiltonians form fibers over kinematic space, and global conformal symmetry singles out a unique modular Berry connection equal to the $2$07 spin connection of that coset geometry. Its Wilson loops reproduce lengths of bulk curves in AdS$2$08 via the differential-entropy formula (Czech et al., 2017). More generally, in QFT the higher Berry connection can be viewed as a topological term in the effective action for background couplings, and via inflow this term corresponds to a boundary anomaly in the space of couplings (Dedushenko, 2022).

These developments show that higher Berry connection is not confined to one formalism. It appears in band theory, interacting lattice systems, boundary scattering, BCFT, modular Hamiltonian geometry, and string field theory, but the recurring structure is stable: a gauge object on parameter space, often higher-form, whose curvature is closed and whose integrals encode topological transport, anomaly inflow, or response coefficients.