Extended Abelian Chern–Simons TQFTs

• Extended Abelian Chern–Simons theories are unitary (2+1)-dimensional TQFTs characterized by a gauge torus U(1)^n and an even integral form K that yields a finite discriminant quadratic module.
• They employ real-polarization quantization and Maslov-index corrected gluing to construct finite-dimensional Hilbert spaces from continuum gauge data.
• Their classification via finite quadratic modules connects these theories to pointed modular tensor categories, unifying diverse topological invariants and lattice constructions.

Extended Abelian Chern–Simons theories are unitary extended $(2+1)$-dimensional topological quantum field theories with gauge torus $\mathbb T=U(1)^n$ and even integral level form $K$. In the formulation developed for the Walker–Maslov–corrected extended bordism category, such a theory is a symmetric monoidal functor whose values on $0$-, $1$-, and $2$-dimensional bordism data encode finite-dimensional state spaces, bordism operators, and Maslov-corrected gluing. For an even integral nondegenerate lattice $(\Lambda,K)$, the decisive invariant is the discriminant quadratic module $(G_K,q_K)$; the theory is determined, up to symmetric monoidal natural isomorphism, by this finite quadratic module, and every finite quadratic module is realized by some even integral nondegenerate lattice. Consequently, extended Abelian Chern–Simons theories, pointed Abelian Reshetikhin–Turaev TQFTs, and pointed modular tensor categories are classified by the same data (Galviz, 3 Apr 2026, Galviz, 2 Apr 2026).

1. Extended $(2+1)$-TQFT structure

Let $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$ denote the standard Walker–Maslov–corrected extended bordism category in dimension $\mathbb T=U(1)^n$0. Its objects are $\mathbb T=U(1)^n$1-manifolds equipped with polarization data, its $\mathbb T=U(1)^n$2-morphisms are $\mathbb T=U(1)^n$3-manifolds with Lagrangian subspaces in first homology, and its $\mathbb T=U(1)^n$4-morphisms are compact oriented surfaces with Maslov-index weights. A unitary extended $\mathbb T=U(1)^n$5-TQFT is a symmetric monoidal functor

$\mathbb T=U(1)^n$6

satisfying the usual compatibility and unitarity axioms. An extended Abelian Chern–Simons theory with gauge torus $\mathbb T=U(1)^n$7 and even integral level form $\mathbb T=U(1)^n$8 is such a functor, realized either by real-polarization quantization or by a rigorous functional integral (Galviz, 3 Apr 2026).

For a closed oriented surface $\mathbb T=U(1)^n$9 of genus $K$0 together with a real Lagrangian $K$1, the theory assigns a finite-dimensional Hilbert space

$K$2

of dimension $K$3. A $K$4-bordism $K$5 with boundary data determines a linear map

$K$6

and the extended theory incorporates an additional Maslov-index weight $K$7 through

$K$8

where $K$9 is the canonical Maslov correction. This yields a unitary symmetric monoidal functor on $0$0 (Galviz, 3 Apr 2026).

In the geometric-quantization description, the classical phase space attached to a closed oriented surface $0$1 is

$0$2

equipped with the symplectic form

$0$3

The associated prequantum line bundle $0$4 satisfies $0$5. Choosing a rational Lagrangian subspace $0$6 defines a real polarization whose Bohr–Sommerfeld leaves form a torsor for $0$7, and geometric quantization in this polarization produces the same finite-dimensional Hilbert space. Canonical unitary identifications between different real polarizations are supplied by Blattner–Kostant–Sternberg operators

$0$8

with projective composition law

$0$9

where $1$0 is the $1$1-twisted surface Maslov index (Galviz, 2 Apr 2026).

2. Lattice input and the discriminant quadratic module

The algebraic input is a rank-$1$2 free Abelian group $1$3 together with an even integral nondegenerate symmetric bilinear form

$1$4

satisfying

$1$5

The dual lattice is

$1$6

and the discriminant group is

$1$7

a finite Abelian group of order $1$8 (Galviz, 3 Apr 2026).

Because $1$9 is nondegenerate, it admits a rational inverse $2$0 on $2$1. This induces the discriminant quadratic form

$2$2

or, multiplicatively,

$2$3

The pair $2$4 is a finite quadratic module, also described as a metric group. Its associated bicharacter is

$2$5

and nondegeneracy of $2$6 implies nondegeneracy of this pairing (Galviz, 3 Apr 2026, Galviz, 2 Apr 2026).

This reduction from an integral lattice to a finite quadratic module is the central structural simplification of the subject. The lattice $2$7 controls the continuum gauge theory and its geometric quantization, but the resulting fully extended TQFT depends only on the finite discriminant data. A plausible implication is that, at the level of extended topological content, the infinite lattice input survives only through its finite residual quadratic module.

3. Quantization, state spaces, and bordism amplitudes

On a closed oriented $2$8-manifold $2$9 with $(\Lambda,K)$0-connection $(\Lambda,K)$1, the classical Abelian Chern–Simons action may be written as

$(\Lambda,K)$2

Formally, the partition function is

$(\Lambda,K)$3

When $(\Lambda,K)$4, geometric quantization of the moduli torus of flat connections on $(\Lambda,K)$5 in a real polarization produces a finite Hilbert space of dimension $(\Lambda,K)$6. The Bohr–Sommerfeld points are labeled by $(\Lambda,K)$7, and the mapping-class-group action is given by the usual $(\Lambda,K)$8 matrices of the metric group $(\Lambda,K)$9 (Galviz, 3 Apr 2026).

For a compact oriented $(G_K,q_K)$0-manifold $(G_K,q_K)$1 with boundary $(G_K,q_K)$2, flat $(G_K,q_K)$3-connections decompose by torsion class

$(G_K,q_K)$4

Each torsion sector is an affine torus mapping to a translated Lagrangian leaf $(G_K,q_K)$5 carrying a covariantly constant Chern–Simons section $(G_K,q_K)$6 of the prequantum bundle and a translation-invariant Reidemeister-torsion half-density $(G_K,q_K)$7. With the normalization exponent

$(G_K,q_K)$8

the boundary state is

$(G_K,q_K)$9

For closed $(2+1)$0, this specializes to

$(2+1)$1

After absorbing the Maslov anomaly via

$(2+1)$2

one obtains a unitary extended $(2+1)$3-dimensional TQFT satisfying cylinder normalization and cutting–gluing laws (Galviz, 2 Apr 2026).

These constructions make explicit why the state spaces are finite despite the continuum origin of the theory. The moduli space of flat torus-connections on a surface is itself a torus, but the combination of the even integral form $(2+1)$4, real polarization, and Bohr–Sommerfeld quantization produces a discrete finite set of quantum states indexed by the finite discriminant group.

4. Classification by finite quadratic modules

The main classification statement is the theorem that two even integral nondegenerate lattices $(2+1)$5 and $(2+1)$6 define equivalent extended Abelian Chern–Simons theories

$(2+1)$7

as symmetric monoidal functors

$(2+1)$8

if and only if their discriminant quadratic modules $(2+1)$9 and $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$0 are isomorphic. Moreover every finite quadratic module arises from some even integral lattice. Hence there is a bijection

$\mathrm{Cob}^{\mathrm{ext}}_{2+1}$1

(Galviz, 3 Apr 2026).

The proof has two principal ingredients. First, one shows that for each lattice $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$2, Abelian Chern–Simons TQFT is naturally isomorphic to the Reshetikhin–Turaev theory attached to the pointed modular tensor category $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$3. This establishes that the extended theory depends only on $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$4. Second, one uses the realization theorem for finite quadratic modules: any finite quadratic module $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$5 splits orthogonally into indecomposables of type

$\mathrm{Cob}^{\mathrm{ext}}_{2+1}$6

each realized by an explicit even lattice, and direct sums realize $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$7 as $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$8 for some $\mathrm{Cob}^{\mathrm{ext}}_{2+1}$9 (Galviz, 3 Apr 2026).

The equivalence with pointed Reshetikhin–Turaev theories is constructive. Given the finite quadratic module $\mathbb T=U(1)^n$00, the pointed ribbon category $\mathbb T=U(1)^n$01 has simple objects $\mathbb T=U(1)^n$02 with tensor product

$\mathbb T=U(1)^n$03

duals $\mathbb T=U(1)^n$04, twist

$\mathbb T=U(1)^n$05

and Hopf-link pairing $\mathbb T=U(1)^n$06. Nondegeneracy of $\mathbb T=U(1)^n$07 makes $\mathbb T=U(1)^n$08 modular. The resulting Reshetikhin–Turaev theory

$\mathbb T=U(1)^n$09

is canonically symmetrically monoidally naturally isomorphic to the toral Chern–Simons theory

$\mathbb T=U(1)^n$10

both for closed $\mathbb T=U(1)^n$11-manifold invariants and for bordisms with boundary (Galviz, 2 Apr 2026).

A standard algebraic reformulation then follows. Pointed braided fusion categories are equivalent to finite Abelian groups equipped with quadratic forms, and nondegenerate metric groups correspond exactly to pointed modular categories. The same finite quadratic module $\mathbb T=U(1)^n$12 therefore classifies finite metric groups, pointed modular tensor categories $\mathbb T=U(1)^n$13, and extended Abelian $\mathbb T=U(1)^n$14 Chern–Simons TQFTs (Galviz, 3 Apr 2026).

5. Boundaries, surface defects, and alternative classification criteria

The extended structure is enriched by topological boundaries, boundary line operators, and codimension-one defects. In the formulation of topological boundary conditions for Abelian Chern–Simons theory, bulk Wilson lines are labeled by the finite Abelian group

$\mathbb T=U(1)^n$15

and the inverse form $\mathbb T=U(1)^n$16 induces the quadratic refinement

$\mathbb T=U(1)^n$17

Topological boundary conditions correspond to Lagrangian subgroups

$\mathbb T=U(1)^n$18

characterized by isotropy, $\mathbb T=U(1)^n$19, and co-isotropy: if $\mathbb T=U(1)^n$20 for all $\mathbb T=U(1)^n$21, then $\mathbb T=U(1)^n$22. A topological boundary exists only if $\mathbb T=U(1)^n$23. Bulk lines with charge in $\mathbb T=U(1)^n$24 can end on the boundary and become trivial, while genuine boundary lines are labeled by $\mathbb T=U(1)^n$25 (Kapustin et al., 2010).

These data organize into a $\mathbb T=U(1)^n$26-category. Its objects are Lagrangian subgroups $\mathbb T=U(1)^n$27, its $\mathbb T=U(1)^n$28-morphisms are boundary-changing line operators, and its $\mathbb T=U(1)^n$29-morphisms are local junction operators. For fixed $\mathbb T=U(1)^n$30, boundary lines form a twisted group category $\mathbb T=U(1)^n$31 with canonical boundary associator determined by a $\mathbb T=U(1)^n$32-cocycle

$\mathbb T=U(1)^n$33

Surface operators arise by the folding trick: a surface defect in the $\mathbb T=U(1)^n$34 theory is equivalent to a topological boundary in the theory with bilinear form $\mathbb T=U(1)^n$35. The diagonal Lagrangian in $\mathbb T=U(1)^n$36 gives the identity defect, while the anti-diagonal gives the sign-reversal defect $\mathbb T=U(1)^n$37 (Kapustin et al., 2010).

The literature records several classification statements with different targets. One result states that two even Abelian Chern–Simons theories $\mathbb T=U(1)^n$38 and $\mathbb T=U(1)^n$39 are equivalent in the sense that all mapping-class-group representations on any Riemann surface agree if and only if

$\mathbb T=U(1)^n$40

A stronger duality-wall refinement requires exact equality of signature and stable equivalence of lattices; equivalently, there must exist an invertible surface operator between the two theories (Kapustin et al., 2010). By contrast, the classification of extended Abelian Chern–Simons theories as symmetric monoidal extended functors is stated purely in terms of isomorphism classes of finite quadratic modules (Galviz, 3 Apr 2026). This suggests that the precise notion of equivalence—mapping-class-group data, invertible surface operators, or symmetric monoidal natural isomorphism of extended TQFTs—matters in comparing classification theorems.

A common ambiguity concerns the word “extended.” In the TQFT context it refers to extension to lower-codimension bordisms, boundaries, and defects. This is distinct from the use of “extended” in Abelian Chern–Simons–matter theories with $\mathbb T=U(1)^n$41 or $\mathbb T=U(1)^n$42 supersymmetry, where the extension refers to supersymmetry rather than to the bordism-theoretic locality structure [(Arai et al., 2011); (0805.3662)].

6. Canonical examples and related developments

The basic example is the $\mathbb T=U(1)^n$43 lattice. Taking $\mathbb T=U(1)^n$44 and

$\mathbb T=U(1)^n$45

one has $\mathbb T=U(1)^n$46 and

$\mathbb T=U(1)^n$47

The quadratic form satisfies

$\mathbb T=U(1)^n$48

or in exponential form $\mathbb T=U(1)^n$49. The resulting TQFT is the familiar semion theory with two anyon sectors (Galviz, 3 Apr 2026).

The root lattices $\mathbb T=U(1)^n$50 furnish a wider class of even integral lattices. For $\mathbb T=U(1)^n$51 one obtains

$\mathbb T=U(1)^n$52

The exceptional $\mathbb T=U(1)^n$53 lattice has trivial discriminant and therefore defines the trivial Abelian Chern–Simons theory (Galviz, 3 Apr 2026). This is the clearest illustration of the general principle that unimodular even contributions disappear from the extended classification because they contribute no nontrivial discriminant data.

Beyond the $\mathbb T=U(1)^n$54-dimensional classification problem, higher-dimensional Abelian Chern–Simons theories exist in dimensions $\mathbb T=U(1)^n$55, with gauge fields modeled by Deligne–Beilinson classes

$\mathbb T=U(1)^n$56

Their action is

$\mathbb T=U(1)^n$57

well defined precisely for $\mathbb T=U(1)^n$58, and Wilson observables of charge $\mathbb T=U(1)^n$59 require $\mathbb T=U(1)^n$60. In these theories the expectation values of generalized Wilson $\mathbb T=U(1)^n$61-loops are controlled by higher-dimensional linking numbers. Nontrivial Abelian Chern–Simons theories occur only in dimensions $\mathbb T=U(1)^n$62 (1207.1270). This broader setting is structurally parallel to the familiar $\mathbb T=U(1)^n$63-dimensional theory, but it is not the classification problem addressed by finite quadratic modules.

Lattice constructions provide complementary realizations of the same topological features. In the modified Villain formulation of $\mathbb T=U(1)^n$64 Chern–Simons theory on a Euclidean spacetime lattice, exact level quantization, the discrete $\mathbb T=U(1)^n$65 $\mathbb T=U(1)^n$66-form symmetry, its Pontryagin-square anomaly, framed ribbon operators, and anomaly inflow from a $\mathbb T=U(1)^n$67-dimensional bulk SPT are all manifest at finite lattice spacing (Jacobson et al., 2023). In doubled compact Abelian Chern–Simons–Maxwell theory on a spatial lattice, the compact group-space torus $\mathbb T=U(1)^n$68 carries $\mathbb T=U(1)^n$69 units of quantized magnetic flux, the local Hilbert space decomposes into representations of a magnetic translation group, and in the pure Chern–Simons limit the model reduces to a $\mathbb T=U(1)^n$70-symmetric variant of Kitaev’s toric code, extended by static background fields, with mutual anyon statistics angle $\mathbb T=U(1)^n$71 (Olesen et al., 2015).

Taken together, these developments place extended Abelian Chern–Simons theories at the intersection of geometric quantization, finite quadratic forms, modular tensor categories, boundary and defect $\mathbb T=U(1)^n$72-categories, and lattice realizations of topological order. Within $\mathbb T=U(1)^n$73 dimensions, the dominant conclusion is precise: the complete extended Abelian theory with gauge group $\mathbb T=U(1)^n$74 is controlled by the finite quadratic module $\mathbb T=U(1)^n$75 (Galviz, 3 Apr 2026, Galviz, 2 Apr 2026).