Cohomological Holonomy

• Cohomological holonomy is a framework that extracts holonomy information from cohomological and homological data, such as differential characters and quadratic relations.
• It underpins key results in diverse settings, including hyperplane arrangements (yielding holonomy Lie algebras), moduli spaces of flat connections, and invariant theory on Ricci-flat varieties.
• Recent developments extend the concept to higher holonomy via Hochschild cycles, loop-space signatures, and even speculative models in martingale theory and twisted cohomology.

Cohomological holonomy denotes a family of constructions in which holonomy is either extracted from cohomological data, represented by cohomological or homological cycles, or used to determine cohomological invariants. The term is therefore not confined to a single formalism. In the literature it includes the holonomy Lie algebra obtained from the quadratic structure of the Orlik–Solomon algebra of a hyperplane arrangement complement, Chern–Simons differential characters on the moduli of flat connections whose second-order case is realized as line-bundle holonomy, higher Hochschild cycles encoding $2$-holonomy, and settings in which spaces of holomorphic tensors are recovered as invariants of holonomy representations (Löfwall, 2020, López et al., 2017, Abbaspour et al., 2012, Greb et al., 2017).

1. Scope and principal formulations

The expression appears in several mathematically distinct settings. This suggests that it functions less as a single standardized definition than as a recurring principle: holonomy is regarded as cohomological when it is canonically determined by cohomology, homology, differential characters, or related derived structures.

Setting	Cohomological input	Holonomy output
Hyperplane arrangements and matroids	Orlik–Solomon algebra $OS(M)$	Holonomy Lie algebra
Moduli of flat connections	Chern–Simons differential characters	$\mathbb R/\mathbb Z$-valued pairing; line-bundle holonomy
Singular Ricci-flat varieties	Invariant theory on reflexive tensors and forms	Holonomy determines $H^0$-spaces
Gerbes and $2$-bundles	Higher Hochschild homology	Cycle representing $2$-holonomy
Loop spaces in $\mathbb R^n$	Iterated integrals and shuffle relations	Holonomy identified with Chen signature
Categorical filtrations	First cohomology of gain systems	Cohomological holonomy along loops

A recurrent source of ambiguity is that some authors begin with a connection and recover cohomological data from its holonomy, while others begin with cohomology and reconstruct a holonomy-type object from it. Both directions occur in current usage (Alonso et al., 2024, Adachi, 2 May 2026, Popko et al., 2013).

2. Arrangement complements, matroids, and the holonomy Lie algebra

In the arrangement-theoretic setting, cohomological holonomy is realized most explicitly by the holonomy Lie algebra of a complex hyperplane arrangement complement. If $\mathcal A$ is a central hyperplane arrangement, the complement

$X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$

has cohomology ring given by the Orlik–Solomon algebra. For a matroid $M$, this algebra is presented as

$OS(M)$0

where $OS(M)$1 is the exterior algebra on degree-$OS(M)$2 generators and the ideal is generated by the boundary-type relations attached to dependent sets. For each dependent triple $OS(M)$3, the associated quadratic relation is

$OS(M)$4

These quadratic relations are the defining input for the holonomy Lie algebra (Löfwall, 2020).

The essential combinatorial datum is the collection of $OS(M)$5-flats, namely the rank-$OS(M)$6 flats of the matroid. The paper emphasizes that the holonomy Lie algebra depends only on this rank-$OS(M)$7 information. For a general simple matroid $OS(M)$8 on $OS(M)$9, one starts with the free Lie algebra $\mathbb R/\mathbb Z$0, imposes the relations coming from every $\mathbb R/\mathbb Z$1-flat $\mathbb R/\mathbb Z$2 with at least three elements, and also imposes $\mathbb R/\mathbb Z$3 whenever $\mathbb R/\mathbb Z$4 is not contained in any $\mathbb R/\mathbb Z$5-flat. The resulting quotient is the holonomy Lie algebra. In this sense the global object is assembled from local rank-$\mathbb R/\mathbb Z$6 pieces.

The rank-$\mathbb R/\mathbb Z$7 case exhibits the mechanism in its simplest form. When there is only one $\mathbb R/\mathbb Z$8-flat, one obtains

$\mathbb R/\mathbb Z$9

This formula shows that the quotient by the “diagonal” element $H^0$0 yields a free Lie algebra on $H^0$1 generators. The same local-to-global pattern is extended in the paper through set-arrangements and local Lie algebras $H^0$2.

The construction is cohomological in a precise sense. The holonomy Lie algebra is defined from the orthogonal complement of the quadratic relations in the Orlik–Solomon algebra, and the Orlik–Solomon algebra is the cohomology algebra of the arrangement complement. Hence the holonomy Lie algebra is determined by the quadratic part of $H^0$3. Kohno’s theorem supplies the topological bridge: $H^0$4 Thus the holonomy Lie algebra is simultaneously a Lie-algebraic shadow of the lower central series of $H^0$5, a cohomological invariant of the complement, and a matroid invariant in the representable case (Löfwall, 2020).

3. Differential characters and holonomy on moduli of flat connections

A second major formulation arises from Chern–Simons differential characters. Let $H^0$6 be a principal bundle and let $H^0$7 be an invariant polynomial of degree $H^0$8 on the Lie algebra of the structure group. For a connection $H^0$9 on $2$0, Cheeger–Simons theory associates a differential character

$2$1

with curvature

$2$2

and satisfying

$2$3

for any $2$4-chain $2$5. The relevant moduli-space construction is built on

$2$6

followed by the quotient by a subgroup $2$7 of the gauge group acting freely on $2$8, producing

$2$9

A connection $2$0 on $2$1 yields a differential character

$2$2

and its restriction to products $2$3 is the source of the homology pairing (López et al., 2017).

When one restricts from $2$4 to the moduli of flat connections $2$5, the construction descends to

$2$6

The paper states that this map is well defined and independent of the auxiliary connection $2$7. The mechanism is that, on $2$8, curvature terms simplify and a bigrading argument forces the relevant transgression terms to vanish in the indicated degree range.

The low-order cases clarify the geometric content. For $2$9, the character is first order and corresponds to a map $\mathbb R^n$0; the paper identifies this with the Dijkgraaf–Witten formulation of the Chern–Simons action. For $\mathbb R^n$1, one has a second-order differential character

$\mathbb R^n$2

and the paper invokes the general theorem that any second-order character is the logarithm of holonomy of a $\mathbb R^n$3-bundle with connection. In its specific setting, this yields a $\mathbb R^n$4-bundle $\mathbb R^n$5 with connection $\mathbb R^n$6 such that

$\mathbb R^n$7

On the moduli of flat connections, the construction acquires a sharper cohomological character. For $\mathbb R^n$8, the connection form on the restricted line bundle does not depend on the chosen connection on $\mathbb R^n$9; for $\mathcal A$0, its curvature vanishes, so the bundle is flat and its holonomy defines a class in

$\mathcal A$1

In this setting the phrase “cohomological holonomy” is literal: holonomy is the geometric realization of a differential cohomology class on the moduli space (López et al., 2017).

4. Holonomy representations and cohomology on singular Ricci-flat varieties

In birational and Kähler geometry, cohomological holonomy appears in the converse direction: holonomy representations determine spaces of differential forms and tensor fields. Let $\mathcal A$2 be a projective klt variety with $\mathcal A$3 numerically trivial and $\mathcal A$4 an ample Cartier divisor. By the theorem of Eyssidieux–Guedj–Zeriahi, $\mathcal A$5 determines a singular Ricci-flat Kähler current $\mathcal A$6, smooth on $\mathcal A$7, with associated Riemannian metric $\mathcal A$8. The full holonomy and restricted holonomy are

$\mathcal A$9

Because the metric is Kähler, $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$0 lies in the unitary group, and because it is Ricci-flat on the smooth locus, $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$1 lies in $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$2. The paper further proves that the isomorphism class of $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$3 is independent of the polarization $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$4 (Greb et al., 2017).

A central structural result is that the full holonomy has only finitely many connected components in the relevant setting. After passing to a quasi-étale holonomy cover $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$5, one can arrange that full holonomy and restricted holonomy coincide. More precisely, there exists a quasi-étale cover

$X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$6

with $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$7 an Abelian variety of dimension $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$8, $X=\mathbb C^n\setminus \bigcup_{H\in\mathcal A} H$9 having canonical singularities, $M$0, and $M$1, such that the pulled-back metric splits and the corresponding holonomy on $M$2 is connected. If $M$3, then $M$4 is finite.

The decisive cohomological statement is the singular Bochner principle. For reflexive tensor sheaves $M$5 of the type treated in the paper, every global holomorphic tensor on $M$6 is parallel with respect to $M$7, and

$M$8

For differential forms and symmetric powers this specializes to

$M$9

where $OS(M)$00. Holomorphic forms are therefore identified with holonomy-invariant tensors.

The same invariant-theoretic principle controls stability and decomposition. The paper proves

$OS(M)$01

and

$OS(M)$02

After passing to a holonomy cover, the tangent sheaf decomposes into strongly stable foliations whose holonomy groups are $OS(M)$03 and $OS(M)$04, together with a flat summand corresponding to the Abelian factor. In the strongly stable case, classification of irreducible Ricci-flat holonomy yields

$OS(M)$05

and after a suitable quasi-étale cover the variety is either Calabi–Yau or irreducible holomorphic symplectic. Here “cohomological holonomy” refers to the fact that global reflexive forms, symmetric differentials, and structural properties of $OS(M)$06 are read off from holonomy invariants (Greb et al., 2017).

5. Higher holonomy, Hochschild models, and loop-space signatures

A homological version of cohomological holonomy is developed for non-abelian gerbes and principal $OS(M)$07-bundles. For an ordinary principal $OS(M)$08-bundle with a flat connection $OS(M)$09-form $OS(M)$10, the paper recalls the classical Hochschild chain

$OS(M)$11

in the Hochschild complex. The decisive proposition is that $OS(M)$12 satisfies the Maurer–Cartan equation

$OS(M)$13

if and only if $OS(M)$14 is a cycle. Flatness is therefore reformulated as closedness in Hochschild homology (Abbaspour et al., 2012).

The higher-categorical analogue begins with a crossed module of Lie groups $OS(M)$15 and its Lie algebra crossed module $OS(M)$16. A crucial technical step is that, up to equivalence of crossed modules, one may assume $OS(M)$17 is abelian. With Baez–Schreiber $OS(M)$18-connection data $OS(M)$19, satisfying the fake curvature condition

$OS(M)$20

the gerbe determines a loop-space connection $OS(M)$21-form $OS(M)$22. The paper computes its curvature in terms of $OS(M)$23 and $OS(M)$24, so the Maurer–Cartan and fake-curvature conditions imply that $OS(M)$25 is flat. Consequently

$OS(M)$26

is a cycle. Because the torus satisfies the higher-Hochschild identification recalled in the paper, this cycle becomes

$OS(M)$27

representing $OS(M)$28-holonomy of the gerbe. The cohomological content lies in the passage from surface holonomy to a torus-indexed higher Hochschild class.

A related but distinct loop-space formulation identifies Chen’s signature with ordinary holonomy for a universal connection. The relevant loop group is the group of piecewise smoothly immersed closed curves based at $OS(M)$29, modulo retrace relation; the paper also proves that in this class the retrace relation equals thin homotopy. Using the holonomy embedding, one equips the loop group with the initial topology pulled back from a connected, simply connected, infinite-dimensional Banach Lie group $OS(M)$30, constructed as the completion of the free Lie algebra on $OS(M)$31 generators under the pro-nilpotent topology. On the trivial principal bundle

$OS(M)$32

the connection form is

$OS(M)$33

and the resulting holonomy map

$OS(M)$34

is a monomorphism that coincides with the Chen signature map. The signature is

$OS(M)$35

The paper does not build a full cohomology theory, but it explicitly places holonomy and signature in a framework of noncommutative cohomological invariants of path spaces, with image closure characterized by shuffle relations and vanishing degree-one part (Alonso et al., 2024).

6. Converse, generalized, and conjectural directions

One converse form of the theme appears in the study of oriented Hantzsche–Wendt manifolds. Such a manifold is an oriented flat manifold of odd dimension $OS(M)$36 with holonomy group

$OS(M)$37

The paper studies the torus quotient model $OS(M)$38 and the Lyndon–Hochschild–Serre spectral sequence of the covering. The key transgression map

$OS(M)$39

produces quadratic classes $OS(M)$40 forming a basis of $OS(M)$41. The structure of these classes, together with the set of decomposable elements in $OS(M)$42, allows reconstruction of the HW-matrix encoding the diagonal holonomy action. The conclusion is cohomological rigidity: the graded ring $OS(M)$43 determines the homeomorphism type within the HW class. Here cohomology remembers holonomy strongly enough to recover the manifold (Popko et al., 2013).

A nonclassical extension is proposed in martingale theory with categorical time. Time is modeled by a small category $OS(M)$44, a filtration is a contravariant functor

$OS(M)$45

and the density distortion is removed by a simplexwise normalization called the $OS(M)$46-gauge. This yields a genuine cochain complex $OS(M)$47 with

$OS(M)$48

For a closed gain system $OS(M)$49 and a loop $OS(M)$50, additive holonomy is defined by conditional-expectation transport around the loop. The cohomological refinement is

$OS(M)$51

where $OS(M)$52 is the transport defect space. The paper proves that this depends only on the cohomology class $OS(M)$53, so it isolates the intrinsic loop-level arbitrage component from transport artifacts (Adachi, 2 May 2026).

Another recent usage occurs for foliated manifolds with stratified boundaries. The paper compresses foliation-boundary intersection data into a path-connected topological multigraph $OS(M)$54, equips it with a vector-bundle-like structure and spinor fields, and defines cycle holonomy by a sign rule

$OS(M)$55

A twist map $OS(M)$56 induces twisted cohomology $OS(M)$57, while an Ihara zeta function is attached to primitive cycles of the $OS(M)$58-set. The paper frames as a conjecture a duality between holonomy fixed points and poles or zeros of the zeta function, extended to twisted cohomology classes. Because the central statement is conjectural, this direction is best regarded as a speculative extension of the broader cohomological-holonomy motif rather than a settled theorem (Zimmerman et al., 16 Jan 2025).

Taken together, these formulations show that cohomological holonomy is not restricted to ordinary parallel transport. It may denote a Lie algebra recovered from the quadratic part of a cohomology ring, a differential character realized as line-bundle holonomy, a homological cycle encoding higher holonomy, a holonomy representation whose invariants are spaces of forms, or a loop observable defined only up to a cohomology class. The stable common feature is the passage between transport data and cohomological structure, in either direction.