Arithmetic Dijkgraaf–Witten Theory

• Arithmetic Dijkgraaf–Witten theory is the arithmetic analogue of finite-group TQFTs, replacing 3-manifolds with arithmetic curves and flat bundles with continuous Galois representations.
• It employs arithmetic Chern–Simons functionals and explicit étale cohomological invariants, allowing mod 2 and p-adic refinements that parallel classical quantum field theories.
• The framework bridges number theory and topology through duality theories, cobordism categories on pro-p groups, and explicit formulas involving Legendre symbols and quadratic residue graphs.

Arithmetic Dijkgraaf–Witten theory is the arithmetic analogue of finite-group Dijkgraaf–Witten topological quantum field theory, obtained by replacing closed oriented $3$-manifolds with arithmetic curves such as $X=\operatorname{Spec}(\mathcal{O}_F)$ or $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$, replacing flat $G$-bundles by continuous Galois representations, and replacing evaluation on a fundamental class by étale or Galois cohomological invariant maps such as $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$ or $$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$ (Kim, 2015). In the literature this theme appears in several forms: Kim’s arithmetic Chern–Simons functionals on spaces of Galois representations (Kim, 2015), their boundary and $p$-adic refinements (Chung et al., 2016), Hirano’s extension to arbitrary number fields via modified étale cohomology (Hirano, 2019), explicit mod $2$ formulas for real quadratic fields (Deng et al., 2023), a pro-$p$ cobordism-theoretic arithmetic field theory with an arithmetic Dijkgraaf–Witten example (Gropper et al., 27 Apr 2025), and a duality theory parallel to Naidu’s classification of equivalent topological Dijkgraaf–Witten theories (Eichler, 26 Aug 2025).

1. Arithmetic replacement of the finite-group Dijkgraaf–Witten datum

In ordinary $3$-dimensional Dijkgraaf–Witten theory one fixes a finite group $X=\operatorname{Spec}(\mathcal{O}_F)$0 and a $X=\operatorname{Spec}(\mathcal{O}_F)$1-cocycle

$X=\operatorname{Spec}(\mathcal{O}_F)$2

and for a closed oriented $X=\operatorname{Spec}(\mathcal{O}_F)$3-manifold $X=\operatorname{Spec}(\mathcal{O}_F)$4 a classical gauge field is a flat $X=\operatorname{Spec}(\mathcal{O}_F)$5-bundle, equivalently a homomorphism

$X=\operatorname{Spec}(\mathcal{O}_F)$6

up to conjugacy. The partition function is

$X=\operatorname{Spec}(\mathcal{O}_F)$7

with $X=\operatorname{Spec}(\mathcal{O}_F)$8 obtained by pulling back $X=\operatorname{Spec}(\mathcal{O}_F)$9 and evaluating on the fundamental class of $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$0 (Chung et al., 2016).

The arithmetic analogue replaces $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$1 by

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$2

where $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$3 is a number field, often totally imaginary in the initial formulations. The étale fundamental group

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$4

plays the role of $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$5, and the cohomological input is the canonical isomorphism

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$6

together with the finite-coefficient variants

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$7

and

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$8

These are interpreted as arithmetic analogues of evaluation on the fundamental class of a compact oriented $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$9-manifold (Chung et al., 2016).

For a finite group $G$0, the arithmetic configuration space is the moduli set

$G$1

the set of isomorphism classes of principal $G$2-bundles in the étale topology. This is the arithmetic counterpart of the groupoid of flat $G$3-bundles in topological Dijkgraaf–Witten theory (Chung et al., 2016).

A decisive extension beyond totally imaginary fields is Hirano’s use of the Artin–Verdier site $G$4, the modified étale fundamental group

$G$5

and modified étale cohomology $G$6, so that arbitrary number fields, including those with real places, enter the theory through a $G$7-dimensional duality framework analogous to Poincaré duality (Hirano, 2019).

2. Arithmetic Chern–Simons functionals and the boundary formalism

Kim’s foundational construction begins with a finite group $G$8, a class

$G$9

a totally imaginary number field $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$0 containing $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$1, and a trivialization

$$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$2

For $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$3, the arithmetic Chern–Simons functional is

$$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$4

This is the direct arithmetic analogue of the Dijkgraaf–Witten action obtained by pulling back a group $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$5-cocycle and integrating it against a $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$6-dimensional fundamental class (Kim, 2015).

When one removes a finite set of primes $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$7, the open arithmetic curve

$$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$8

plays the role of a $$\operatorname{inv}:H^3(X,\mu_n)\simeq \tfrac{1}{n}\mathbb{Z}/\mathbb{Z}$$9-manifold with boundary. Its arithmetic fundamental group is

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$0

and for each $$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$1 there is a local Galois group

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$2

with maps $$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$3. The local and global representation groupoids are

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$4

together with the restriction functor

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$5

This is the arithmetic replacement of restricting a flat bundle to boundary components (Chung et al., 2016).

The boundary theory is organized by an arithmetic Chern–Simons line. For local boundary data

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$6

one defines

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$7

then

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$8

a torsor under

$$\operatorname{inv}:H^3(X,\mathbf{G}_m)\simeq \mathbb{Q}/\mathbb{Z}$$9

Pushing out by

$p$0

gives the arithmetic Chern–Simons line

$p$1

a $p$2-torsor. The global Chern–Simons action with boundary becomes a section

$p$3

rather than merely a scalar-valued function (Chung et al., 2016).

A $p$4-adic version replaces the finite group $p$5 by a $p$6-adic Lie group with an open surjection

$p$7

takes

$p$8

and uses a $p$9-equivariant isomorphism

$2$0

to define a $2$1-adic Chern–Simons invariant

$2$2

The subsequent quantization is explicitly deferred; the papers focus on the classical action-functional side (Chung et al., 2016).

3. General number fields and explicit mod $2$3 arithmetic Dijkgraaf–Witten invariants

For arbitrary number fields $2$4, Hirano defines the arithmetic Chern–Simons functional using modified étale cohomology on the Artin–Verdier site. With

$2$5

and

$2$6

the arithmetic Chern–Simons invariant of $2$7 is the image of $2$8 under

$2$9

where $p$0 is the edge map of the modified Hochschild–Serre spectral sequence

$p$1

The arithmetic Dijkgraaf–Witten invariant is then

$p$2

(Hirano, 2019).

In the basic case

$p$3

with $p$4 the Bockstein from

$p$5

Hirano proves that for a surjective

$p$6

if

$p$7

is the corresponding unramified Kummer extension and $p$8 satisfies

$p$9

then

$3$0

where $3$1 is the Artin symbol and

$3$2

is the canonical character (Hirano, 2019).

The mod $3$3 theory for real quadratic fields becomes especially explicit. For

$3$4

let

$3$5

and let $3$6 be the vector with $3$7’s in the $3$8-th and $3$9-th positions. Then Hirano shows

$X=\operatorname{Spec}(\mathcal{O}_F)$00

where the mod $X=\operatorname{Spec}(\mathcal{O}_F)$01 linking number is defined by

$X=\operatorname{Spec}(\mathcal{O}_F)$02

and consequently

$X=\operatorname{Spec}(\mathcal{O}_F)$03

This identifies the arithmetic Dijkgraaf–Witten invariant with an explicit Legendre-symbol expression (Hirano, 2019).

Deng, Kurimaru, and Matsusaka then simplify Hirano’s formula by introducing the quadratic residue graph $X=\operatorname{Spec}(\mathcal{O}_F)$04 for

$X=\operatorname{Spec}(\mathcal{O}_F)$05

with an edge $X=\operatorname{Spec}(\mathcal{O}_F)$06 exactly when

$X=\operatorname{Spec}(\mathcal{O}_F)$07

They prove the closed formula

$X=\operatorname{Spec}(\mathcal{O}_F)$08

for

$X=\operatorname{Spec}(\mathcal{O}_F)$09

thereby answering Ken Ono’s question about whether Hirano’s right-hand side admits a simple expression (Deng et al., 2023). They also prove the density formula

$X=\operatorname{Spec}(\mathcal{O}_F)$10

for the proportion of such real quadratic fields with nonzero mod $X=\operatorname{Spec}(\mathcal{O}_F)$11 arithmetic Dijkgraaf–Witten invariant (Deng et al., 2023).

4. Pro-$X=\operatorname{Spec}(\mathcal{O}_F)$12 arithmetic field theory and an arithmetic Dijkgraaf–Witten TQFT

A different but related development replaces manifolds by pro-$X=\operatorname{Spec}(\mathcal{O}_F)$13 Poincaré duality groups and pairs, producing a cobordism category suited to arithmetic topology. In this framework a profinite group pair $X=\operatorname{Spec}(\mathcal{O}_F)$14 is a pro-$X=\operatorname{Spec}(\mathcal{O}_F)$15 $X=\operatorname{Spec}(\mathcal{O}_F)$16 pair if it has type $X=\operatorname{Spec}(\mathcal{O}_F)$17, cohomological dimension $X=\operatorname{Spec}(\mathcal{O}_F)$18, and duality

$X=\operatorname{Spec}(\mathcal{O}_F)$19

The top cohomology yields an orientation character

$X=\operatorname{Spec}(\mathcal{O}_F)$20

and an orientability level $X=\operatorname{Spec}(\mathcal{O}_F)$21 (Gropper et al., 27 Apr 2025).

In dimension $X=\operatorname{Spec}(\mathcal{O}_F)$22, Wilkes’ classification of pro-$X=\operatorname{Spec}(\mathcal{O}_F)$23 $X=\operatorname{Spec}(\mathcal{O}_F)$24 pairs gives explicit surface-group-type presentations, allowing the construction of a pro-$X=\operatorname{Spec}(\mathcal{O}_F)$25 cobordism category

$X=\operatorname{Spec}(\mathcal{O}_F)$26

Its objects are finite collections of pro-$X=\operatorname{Spec}(\mathcal{O}_F)$27 $X=\operatorname{Spec}(\mathcal{O}_F)$28 groups, and up to isomorphism these are finite multisets of $X=\operatorname{Spec}(\mathcal{O}_F)$29. Morphisms are equivalence classes of pro-$X=\operatorname{Spec}(\mathcal{O}_F)$30 $X=\operatorname{Spec}(\mathcal{O}_F)$31-cobordisms, assembled from $X=\operatorname{Spec}(\mathcal{O}_F)$32 group triples and gluing along boundary subgroups (Gropper et al., 27 Apr 2025).

The resulting $X=\operatorname{Spec}(\mathcal{O}_F)$33-dimensional pro-$X=\operatorname{Spec}(\mathcal{O}_F)$34 TQFTs are classified by $X=\operatorname{Spec}(\mathcal{O}_F)$35-extended Frobenius algebras, where

$X=\operatorname{Spec}(\mathcal{O}_F)$36

More precisely,

$X=\operatorname{Spec}(\mathcal{O}_F)$37

an equivalence between the groupoid of pro-$X=\operatorname{Spec}(\mathcal{O}_F)$38 TQFTs and the groupoid of $X=\operatorname{Spec}(\mathcal{O}_F)$39-extended $X=\operatorname{Spec}(\mathcal{O}_F)$40-Frobenius algebras (Gropper et al., 27 Apr 2025). This is the arithmetic-topology analogue of the classical relation between $X=\operatorname{Spec}(\mathcal{O}_F)$41-dimensional TQFTs and Frobenius algebras.

Within this framework the authors define a $X=\operatorname{Spec}(\mathcal{O}_F)$42-dimensional pro-$X=\operatorname{Spec}(\mathcal{O}_F)$43 Dijkgraaf–Witten theory for a finite gauge $X=\operatorname{Spec}(\mathcal{O}_F)$44-group $X=\operatorname{Spec}(\mathcal{O}_F)$45. For a boundary object

$X=\operatorname{Spec}(\mathcal{O}_F)$46

the state space is

$X=\operatorname{Spec}(\mathcal{O}_F)$47

For a $X=\operatorname{Spec}(\mathcal{O}_F)$48-cobordism $X=\operatorname{Spec}(\mathcal{O}_F)$49, the linear map

$X=\operatorname{Spec}(\mathcal{O}_F)$50

is defined by summing over conjugacy classes of homomorphisms $X=\operatorname{Spec}(\mathcal{O}_F)$51 with prescribed boundary restrictions, weighted by automorphism-group factors exactly as in classical finite-group Dijkgraaf–Witten theory (Gropper et al., 27 Apr 2025).

The arithmetic force of this construction appears when $X=\operatorname{Spec}(\mathcal{O}_F)$52 is a Demuškin group or a maximal pro-$X=\operatorname{Spec}(\mathcal{O}_F)$53 Galois group of a local $X=\operatorname{Spec}(\mathcal{O}_F)$54-adic field. The theory yields character-sum formulas for $X=\operatorname{Spec}(\mathcal{O}_F)$55, and for a $X=\operatorname{Spec}(\mathcal{O}_F)$56-adic field $X=\operatorname{Spec}(\mathcal{O}_F)$57 of degree $X=\operatorname{Spec}(\mathcal{O}_F)$58, $X=\operatorname{Spec}(\mathcal{O}_F)$59, with $X=\operatorname{Spec}(\mathcal{O}_F)$60, $X=\operatorname{Spec}(\mathcal{O}_F)$61, it gives

$X=\operatorname{Spec}(\mathcal{O}_F)$62

Together with Hall’s Möbius inversion formula, this produces explicit counts of Galois extensions $X=\operatorname{Spec}(\mathcal{O}_F)$63 with prescribed finite $X=\operatorname{Spec}(\mathcal{O}_F)$64-group Galois group $X=\operatorname{Spec}(\mathcal{O}_F)$65 (Gropper et al., 27 Apr 2025).

5. Duality, equivalence, and failure of equivalence

Arithmetic Dijkgraaf–Witten theory also admits a duality problem parallel to Naidu’s classification of equivalent topological Dijkgraaf–Witten theories. Eichler formulates the arithmetic theory for

$X=\operatorname{Spec}(\mathcal{O}_F)$66

over a totally imaginary number field $X=\operatorname{Spec}(\mathcal{O}_F)$67 containing $X=\operatorname{Spec}(\mathcal{O}_F)$68, with $X=\operatorname{Spec}(\mathcal{O}_F)$69 invertible on $X=\operatorname{Spec}(\mathcal{O}_F)$70, and boundary

$X=\operatorname{Spec}(\mathcal{O}_F)$71

The groupoid of fields is

$X=\operatorname{Spec}(\mathcal{O}_F)$72

and the arithmetic partition function is a state sum over $X=\operatorname{Spec}(\mathcal{O}_F)$73 weighted by

$X=\operatorname{Spec}(\mathcal{O}_F)$74

for

$X=\operatorname{Spec}(\mathcal{O}_F)$75

In exact form,

$X=\operatorname{Spec}(\mathcal{O}_F)$76

equivalently

$X=\operatorname{Spec}(\mathcal{O}_F)$77

(Eichler, 26 Aug 2025).

The duality data consist of a finite group $X=\operatorname{Spec}(\mathcal{O}_F)$78, finite $X=\operatorname{Spec}(\mathcal{O}_F)$79-torsion $X=\operatorname{Spec}(\mathcal{O}_F)$80-modules $X=\operatorname{Spec}(\mathcal{O}_F)$81 and $X=\operatorname{Spec}(\mathcal{O}_F)$82, cocycles

$X=\operatorname{Spec}(\mathcal{O}_F)$83

and

$X=\operatorname{Spec}(\mathcal{O}_F)$84

with

$X=\operatorname{Spec}(\mathcal{O}_F)$85

One forms

$X=\operatorname{Spec}(\mathcal{O}_F)$86

and the $X=\operatorname{Spec}(\mathcal{O}_F)$87-cocycles

$X=\operatorname{Spec}(\mathcal{O}_F)$88

For local fields and for $X=\operatorname{Spec}(\mathcal{O}_F)$89, Eichler proves an arithmetic analogue of Naidu’s equivalence: $X=\operatorname{Spec}(\mathcal{O}_F)$90 The proof rests on local Tate duality and Artin–Verdier duality, which furnish the perfect pairings needed by the abstract duality theorem (Eichler, 26 Aug 2025).

The full ring-of-integers case

$X=\operatorname{Spec}(\mathcal{O}_F)$91

is subtler. Here $X=\operatorname{Spec}(\mathcal{O}_F)$92 is not invertible on $X=\operatorname{Spec}(\mathcal{O}_F)$93, and the cup-product pairing is no longer perfect. Eichler introduces the orthogonal complement

$X=\operatorname{Spec}(\mathcal{O}_F)$94

and proves sufficient conditions under which arithmetic Dijkgraaf–Witten invariants still coincide: $X=\operatorname{Spec}(\mathcal{O}_F)$95 The paper simultaneously gives examples where the equivalence fails. For duality data involving the quaternion group $X=\operatorname{Spec}(\mathcal{O}_F)$96, the topological Dijkgraaf–Witten theories remain equivalent for closed oriented $X=\operatorname{Spec}(\mathcal{O}_F)$97-manifolds, but arithmetic invariants over certain quadratic imaginary fields do not. One explicit example is

$X=\operatorname{Spec}(\mathcal{O}_F)$98

for which

$X=\operatorname{Spec}(\mathcal{O}_F)$99

Thus arithmetic Dijkgraaf–Witten duality is not a formal copy of the topological theory; it depends essentially on arithmetic duality, linking forms, and the invertibility of $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$00 on $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$01 (Eichler, 26 Aug 2025).

6. Related templates and extensions

The arithmetic literature repeatedly treats standard Dijkgraaf–Witten constructions as templates rather than as completed arithmetic theories. Kim’s first paper explicitly states that it does not yet construct a full TQFT; it develops the classical Chern–Simons functionals and sketches the possibility of defining quantum partition functions or even relating them to $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$02-functions (Kim, 2015). This restriction is substantive rather than rhetorical: arithmetic Dijkgraaf–Witten theory is, in several sources, a programmatic extension of arithmetic Chern–Simons theory rather than a single universally fixed formalism.

Two broader topological frameworks function as prototypes. First, Monnier’s higher abelian Dijkgraaf–Witten theory replaces degree-$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$03 gauge fields by $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$04-form gauge fields classified by

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$05

with action determined by

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$06

and measure

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$07

This suggests arithmetic analogues in which singular cohomology is replaced by étale or Galois cohomology and the finite field-space sum is replaced by a sum over arithmetic cocycles (Monnier, 2015).

Second, Kim’s spectrum-based generalization of Dijkgraaf–Witten theory replaces ordinary cohomology by a pairing

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$08

a class

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$09

and an invariant

$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$10

The paper treats this as a template for an “arithmetic Dijkgraaf–Witten theory” in generalized cohomology or KK-theory, where arithmetic content would enter through the choice of spectra and coefficient systems (Kim, 2018).

A plausible implication is that “Arithmetic Dijkgraaf–Witten theory” now names a family of mathematically distinct but structurally allied constructions. One branch centers on arithmetic Chern–Simons actions and Artin–Verdier duality for arithmetic curves (Kim, 2015, Chung et al., 2016, Hirano, 2019); another studies explicit mod $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$11 partition functions through Legendre symbols and quadratic residue graphs (Deng et al., 2023); another builds a $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$12-dimensional arithmetic field theory on pro-$X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$13 duality groups and derives counting formulas for local Galois extensions (Gropper et al., 27 Apr 2025); and a further branch analyzes equivalence and failure of equivalence for arithmetic Dijkgraaf–Witten theories attached to different pairs $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$14 (Eichler, 26 Aug 2025). What unifies them is the Dijkgraaf–Witten pattern itself: finite or profinite gauge data, group or étale cohomology in degree $X=\operatorname{Spec}(\mathcal{O}_F)\setminus S$15 or its analogues, and state-sum or path-integral expressions whose arithmetic meaning is extracted from Galois representations, duality, and class field theory.