Hodge Potential in Modern Analysis

• Hodge potential is a family of related constructs in Hodge theory, manifesting as a node potential in discrete settings, a proxy 1-form in semiclassical transport, or a Green-operator solution in PDE analysis.
• In discrete frameworks, it generates curl-free edge flows on simplicial complexes by linking node potentials to gradient fields, ensuring identifiability modulo constants.
• In semiclassical and analytic contexts, it regularizes topological flux and serves as a gauge-fixed representative in Hodge Laplacian decompositions and boundary-adapted Hodge theory.

Searching arXiv for recent and relevant papers on “Hodge Potential” and closely related usages. Hodge potential denotes different but structurally related objects in several Hodge-theoretic literatures. In the setting of edge flows on simplicial complexes, it refers primarily to a node potential $\mathbf f_0$ whose discrete gradient $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$ is the curl-free component of an edge function (Yang et al., 2023). In semiclassical transport on topological bands, it denotes a globally defined smooth $1$-form proxy potential $\mathcal A$ on the Brillouin torus, characterized by

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$

after the quantized cohomological flux has been isolated (Wang et al., 30 Jun 2026). In elliptic and Hodge-Laplacian analysis, it denotes Dirichlet and Neumann potentials, exact and coexact primitives, and gauge-fixing operators associated with decompositions of the form

$\omega=h+d\alpha+\delta\beta$

(Balci et al., 29 Apr 2025). Taken together, these usages suggest not a single universal definition but a family of potential objects tied to Hodge decomposition, Hodge Laplacians, and Hodge-theoretic regularization.

1. Discrete Hodge potentials on simplicial complexes

On a simplicial $2$-complex

$SC=(V,E,T),$

with oriented edges and triangular faces, an edge flow is an alternating $1$-cochain $f_1:E\to\mathbb R$ satisfying $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$0. The paper on Hodge-compositional edge Gaussian processes uses the discrete Hodge decomposition

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$1

with

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$2

Within this decomposition, the Hodge potential in the scalar-potential sense is the node function $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$3 for which

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$4

and the paper explicitly calls $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$5 a node potential (Yang et al., 2023).

The induced edge flow is the gradient or curl-free component. At an edge $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$6,

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$7

and the discrete curl identity

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$8

shows that any such potential-generated flow is curl-free. The paper separates this from the face-potential contribution

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$9

which is divergence-free, and from the harmonic part $1$0, which is both divergence-free and curl-free but not generated by either $1$1 or $1$2.

The same work turns this potential component into a probabilistic object. If

$1$3

then

$1$4

is an edge Gaussian process with covariance

$1$5

The full Hodge-compositional edge Gaussian process is

$1$6

with independent gradient, curl, and harmonic components. The paper also gives separate posterior formulas for $1$7, $1$8, and $1$9, so the potential-driven/curl-free part can be inferred directly. In the forex example, arbitrage-free exchange rates satisfy

$\mathcal A$0

so the learned flow is essentially the gradient/potential component; in the water-supply example, hydraulic head on nodes acts as a scalar potential generating pipe flow through a gradient relation.

A further structural point is identifiability. Because

$\mathcal A$1

the edge gradient flow is identifiable, whereas the underlying node potential is only identifiable modulo constants.

2. Proxy Hodge potentials in semiclassical transport

In the differential-geometric treatment of anomalous transport on a $\mathcal A$2-dimensional Brillouin zone, the Hodge potential is a globally smooth $\mathcal A$3-form

$\mathcal A$4

on the torus $\mathcal A$5, where the Berry curvature is the globally defined $\mathcal A$6-form

$\mathcal A$7

Because $\mathcal A$8 is a top-form on a closed $\mathcal A$9-manifold, the Hodge-de Rham decomposition takes the form

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$0

with

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$1

Thus $\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$2 is defined by

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$3

so it captures the exact, non-topological part of the curvature after removal of the harmonic/topological flux (Wang et al., 30 Jun 2026).

This construction is explicitly not the ordinary Berry connection in nontrivial bands. When $\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$4, no globally smooth Berry connection exists on the torus; the obstruction is exactly the nonzero total flux

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$5

The Hodge potential $\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$6 is instead a globally smooth geometric proxy potential. Its ambiguity is fixed by imposing the Coulomb-Hodge gauge

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$7

together with vanishing holonomies on the fundamental cycles,

$\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$8

Under these conditions, $\Omega=\frac{2\pi c_1}{A_T}\,vol+d\mathcal A,$9 solves

$\omega=h+d\alpha+\delta\beta$0

where

$\omega=h+d\alpha+\delta\beta$1

The potential reorganizes transport formulas. For linear anomalous Hall response,

$\omega=h+d\alpha+\delta\beta$2

so the current splits into a topological background and an exact-sector geometric term. At zero temperature,

$\omega=h+d\alpha+\delta\beta$3

and the exact contribution reduces by the co-area formula to

$\omega=h+d\alpha+\delta\beta$4

For nonlinear transport,

$\omega=h+d\alpha+\delta\beta$5

which reproduces the derivative-shifting structure of scalar integration by parts without differentiating noisy Berry-curvature data numerically.

The Fourier-space form makes the regularization especially explicit. The equations

$\omega=h+d\alpha+\delta\beta$6

$\omega=h+d\alpha+\delta\beta$7

show that solving for $\omega=h+d\alpha+\delta\beta$8 removes the uniform $\omega=h+d\alpha+\delta\beta$9 topological mode before inverting the Laplacian. The imposed zero-holonomy condition then sets $2$0.

3. Hodge potentials for the Hodge Laplacian and decomposition theory

In variable exponent Lebesgue and Sobolev spaces, the term potential is used directly for solution operators of the Hodge Laplacian and for the exact and coexact primitives they generate. On a compact finite-dimensional Riemannian manifold $2$1 with boundary, the Dirichlet potential $2$2, Neumann potential $2$3, and full Dirichlet potential $2$4 are defined variationally for the bilinear form

$2$5

The Dirichlet potential is the unique minimizer of

$2$6

and similarly for the Neumann and full Dirichlet cases (Balci et al., 29 Apr 2025).

These potentials feed directly into Hodge decomposition. The paper proves tangential and normal decompositions

$2$7

with

$2$8

or

$2$9

The corresponding gauge conditions are

$SC=(V,E,T),$0

or

$SC=(V,E,T),$1

Thus the Hodge potential may be the Laplacian solution $SC=(V,E,T),$2 or $SC=(V,E,T),$3, or the derived potentials $SC=(V,E,T),$4 and $SC=(V,E,T),$5 furnishing the exact and coexact parts.

The same framework yields canonical primitives for first-order systems. For closed data,

$SC=(V,E,T),$6

solves

$SC=(V,E,T),$7

For co-closed data,

$SC=(V,E,T),$8

solves

$SC=(V,E,T),$9

For the tangential div-curl system, the solution formula is

$1$0

and the Neumann analogue is

$1$1

For the Hodge-Dirac operator

$1$2

the paper gives the explicit inverse formula

$1$3

and its Neumann analogue.

A closely related use is gauge fixing. If

$1$4

then

$1$5

satisfies

$1$6

In this sense, a Hodge potential is also a gauge-fixed representative of a prescribed exact or coexact field.

4. Boundary-adapted and noncompact Hodge potentials

On compact smooth Riemannian manifolds with smooth boundary, boundary-adapted Hodge theory yields potential operators explicitly named the Neumann potential and Dirichlet potential. Writing

$1$7

for the Hodge Laplacian and using absolute Neumann and relative Dirichlet boundary conditions, the injective operators

$1$8

have inverses

$1$9

which the paper explicitly calls the Neumann potential and Dirichlet potential (Huynh, 2019).

These inverses enter a boundary-adapted Hodge-Morrey decomposition algorithm,

$f_1:E\to\mathbb R$0

Here the exact part is represented through a Dirichlet potential, the coexact part through a Neumann potential, and the harmonic part through $f_1:E\to\mathbb R$1. The same paper defines a “potential for $f_1:E\to\mathbb R$2,”

$f_1:E\to\mathbb R$3

which is used to reconstruct the pressure in the Euler equation from the nonlinear term.

On $f_1:E\to\mathbb R$4-dimensional $f_1:E\to\mathbb R$5 manifolds, weighted Fredholm theory for the Hodge Laplacian provides another noncompact notion of Hodge potential. With

$f_1:E\to\mathbb R$6

weighted norms

$f_1:E\to\mathbb R$7

and noninteger weights $f_1:E\to\mathbb R$8, the map

$f_1:E\to\mathbb R$9

is Fredholm, and

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$00

is solvable iff

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$01

(Chen et al., 2021). The same theory yields harmonic functions with prescribed asymptotics:

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$02

and a harmonic description of $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$03:

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$04

In this weighted noncompact setting, the Hodge potential is the solution of a weighted Laplace equation with prescribed asymptotic mode or prescribed cohomological class.

5. Discrete, DEC, and boundary-integral realizations

For classical collocated finite-difference summation-by-parts operators, exact scalar and vector potentials do not exist for all discrete curl-free or divergence-free fields. The obstruction is an explicit family of grid oscillations lying in adjoint nullspaces, and the discrete Helmholtz–Hodge decomposition takes the form

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$05

in $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$06D, or

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$07

in $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$08D, with a nonzero oscillatory remainder $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$09 in general. The paper therefore defines practical discrete Hodge potentials as least-norm least-squares projection potentials, computed by LSQR, LSMR, or LSLQ after $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$10-scaling (Ranocha et al., 2019). In this usage, a Hodge potential is not always an exact discrete potential in the algebraic sense, but it is still a computable projection potential.

In Discrete Exterior Calculus, a new discrete Hodge operator is constructed for $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$11D primal $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$12-forms without requiring a well-centered circumcentric dual. For a triangle, the local Hodge matrix is

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$13

and it is exact on piecewise constant forms (Ayoub et al., 2020). The clearest potential interpretation there is the Poisson equation

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$14

and the stream-function formulation

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$15

where $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$16 is a scalar potential for the Hodge dual of the velocity $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$17-form.

For coupled domain–boundary formulations of Hodge–Helmholtz operators, the exterior field is represented entirely through boundary Hodge potentials. Any radiating exterior solution satisfies

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$18

where the single-layer and double-layer potentials are

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$19

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$20

These layer potentials solve the homogeneous exterior Hodge–Helmholtz equation and are assembled into Calderón projectors

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$21

which permit a symmetric domain–boundary coupling proved stable by a generalized Gårding inequality, that is, by T-coercivity (Schulz et al., 2020).

6. Disambiguations and adjacent usages

Several papers are relevant precisely because they do not define a quantity literally called “Hodge potential.” In the FLPR quantum-mechanical model, the paper states that it does not define a quantity called a “Hodge potential”; the rotationally invariant mechanical potential

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$22

is a symmetry-compatible background ingredient, while the Hodge-theoretic structure comes from BRST, co-BRST, bosonic, and discrete duality symmetries, together with the Hilbert-space decomposition

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$23

(Krishna et al., 2023). In the interacting $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$24D Stückelberg-modified Proca theory, the closest analogue is not a named Hodge potential but the pair of combinations

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$25

together with the identifications

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$26

(Tripathi et al., 2021). In the massive $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$27D Abelian $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$28-form model, the closest interpretation is that the $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$29-form gauge field

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$30

and its Stückelberg partners are the potential-level objects on which the BRST/co-BRST realization of Hodge theory acts (Krishna et al., 2018).

Other nearby literatures use the word potential only indirectly. For monodromic mixed Hodge modules, the Fourier–Laplace transform and exponential twist $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$31 produce irregular Hodge filtrations, but the paper explicitly states that it does not define a notion called “Hodge potential”; its main formulas are instead

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$32

(Saito, 2022). In logarithmic Hodge theory on toroidal varieties, the paper again does not introduce a literal Hodge potential; the nearest analogues are the weight function $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$33, the weighted divisor $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$34, the weighted Hodge filtration

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$35

and the obstruction complex

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$36

(Luo, 9 Sep 2025). A further source of ambiguity is the Hodge operator itself: the Berezin–Fourier reformulation of the Hodge star,

$\mathbf f_G=\mathbf B_1^\top \mathbf f_0$37

is about the Hodge operator rather than a Hodge potential (Castellani et al., 2015).

Taken together, these disambiguations show that “Hodge potential” is stable only within a local context. In discrete Hodge decomposition it is a scalar node potential; in semiclassical transport it is a smooth proxy $\mathbf f_G=\mathbf B_1^\top \mathbf f_0$38-form; in PDE and geometric analysis it is a Green-operator solution, an exact or coexact primitive, or a gauge-fixing operator; and in several adjacent Hodge-theoretic literatures the phrase is absent altogether, with nearby roles played instead by gauge fields, weighted filtrations, obstruction complexes, or the Hodge star itself.