Topological Gauge Theories and Symmetry TFT

A topological action is not yet a topological field theory. A TQFT must assign states to spatial boundaries and composable maps to bordisms, respect disjoint union, orientation, cutting, and gluing, and carry whatever spin, framing, or other tangential data the quantum theory needs. Compact Abelian Chern–Simons, compact BF, and finite Dijkgraaf–Witten theories make those requirements calculable; their operators and admissible boundaries then lead to relative theories and, under additional existence hypotheses, to a symmetry-TFT sandwich.

Choose the definition-and-gluing route when the question is whether a metric-independent construction is a TQFT at all. Choose a model route for exact line, linking, state-space, or finite-bundle data. Choose the boundary-and-relative route when operators may end, interfaces are folded, or partition functions become vector valued. Choose the SymTFT route only after the auxiliary bulk, topological symmetry boundary, physical boundary, and allowed gauging interfaces are all specified.

Helpful background. When Is a Topological Term Well Defined? supplies the global phase, field-role, boundary, and tangential-structure tests. Fusion, Junctions, and Endpoints separates a fusion rule from the junction and coherence data that realize it. Neither is a hard prerequisite merely to choose a route.

Parent volume. Symmetry and Gauge Structure

Jump to: fix the TQFT criterion · choose a route · compare the three compact models · open the exact guide · review the chapter

A TQFT is a gluing assignment, not merely an action

In the ordinary finite-dimensional unextended setting used for the chapter’s first tests, an absolute anomaly-free $d$ -dimensional TQFT is a symmetric monoidal assignment

Z:\operatorname{Bord}^{\mathfrak s}_d \longrightarrow \operatorname{FinVect}_{\mathbb C},

where $\mathfrak s$ records the declared orientation, spin, framing, or other tangential structure. A closed spatial $(d-1)$ -manifold $\Sigma$ receives $\mathcal H(\Sigma)=Z(\Sigma)$ , and a bordism $M:\Sigma_{\mathrm{in}}\to\Sigma_{\mathrm{out}}$ receives a linear map

Z(M):\mathcal H(\Sigma_{\mathrm{in}}) \longrightarrow \mathcal H(\Sigma_{\mathrm{out}}).

Composition is geometric gluing,

Z(M_2\circ_{\Sigma}M_1)=Z(M_2)\circ Z(M_1),

the cylinder is the identity, disjoint union becomes tensor product, and orientation reversal produces the dual state space. Closing an endobordism contracts its incoming and outgoing boundaries, so

Z(\operatorname{cl}_{\Sigma}M) = \operatorname{Tr}_{\mathcal H(\Sigma)}Z(M), \qquad Z(\Sigma\times S^1)=\dim\mathcal H(\Sigma)

for the ordinary vector-valued product closure. Spin-graded theories can distinguish trace from supertrace through the spin structure on the closing circle, and a framing anomaly requires a fixed framing or relative gravitational convention. Atiyah’s axioms and trace construction are stated in Atiyah 1988, § 2, printed pp. 177–181, especially axioms (A)–(B) and (1)–(4c), with the trace on p. 180, PDF.

Metric independence of one term, one partition function, or one finite state space does not supply this whole assignment. The governed five-row comparison separates a generic QFT, topological term, invertible response, noninvertible TQFT, and intrinsic topological order. It is linked rather than copied so that the decisive tests and terminology have one canonical table.

Check your preparation

These questions identify the first missing capability. They are routing prompts, not a score.

Repair the first missing input before entering a technical route
Can you do this?	Ready	If unsure	Repair route
Distinguish a globally defined action phase from a completed quantum field theory	Enter the definition-and-gluing route	Ask where state spaces, measure, cylinder, and sewing maps come from	When Is a Topological Term Well Defined?
Orient a full braid, a self-twist, and a boundary seam	Enter the Abelian Chern–Simons or boundary route	Separate mutual monodromy from exchange and declare framing before using a spin	Linking, Braiding, and Framing
Distinguish a fixed background bundle from a field summed with gauge-groupoid measure	Enter the compact BF or finite-gauge route	Identify which bundles are integrated and how automorphisms enter	Gauging Continuous and Finite Symmetries
Distinguish a sewing cut from a physical boundary	Enter the relative or SymTFT route	Ask whether the full cut state space is contracted or a boundary condition selects endable operators	Edge Modes, Subregions, and Factorization

Choose a route from the physical question

The routes preserve hard prerequisites but are not additional hierarchy. The exact guide below states the actual hard background for every leaf.

Four routes from a definition or model question to a controlled conclusion
Starting question	Route	Hard inputs along the route	Licensed conclusion
Is this construction a TQFT, and how does sewing work?	What Is a TQFT? → state spaces, cobordisms, and gluing	A globally defined topological term; the finite-dimensional bordism assignment	A cylinder, duality, trace, pair-of-pants, and cut-independence test
Which exact compact model carries the desired line or state data?	Enter Abelian Chern–Simons directly after its action and braiding inputs; for compact BF or finite gauge theory, first complete state spaces and gluing	Chern–Simons: level quantization plus linking and braiding; BF: compact BF couplings plus gluing; finite gauge theory: finite gauging plus gluing	A model-specific operator algebra, global sum, state count, and gluing check
Which lines may end, condense, or cross an interface?	Abelian line data + folding + junctions → operators, boundaries, and relative theories	Quadratic line data, physical boundary conditions, and typed junction spaces	An elementary condensability or interface test, plus a cut-versus-boundary verdict
Can one higher-dimensional topological bulk organize symmetry and gauging?	Relative boundary data + inflow + coupled gaugeability → Symmetry TFT	A specified auxiliary bulk, topological symmetry boundary, physical boundary, and admissible interface	A bounded sandwich description of symmetry, anomaly, global form, and conditional gauging—not a reconstruction of dynamics

One three-dimensional thread tests states, operators, and gluing

The chapter repeatedly compares three compact $2+1$ -dimensional theories. The dimensions, field variables, and global presentations differ; what stays fixed is the checklist: define the global sum, identify protected operators, compute a state space, sew a cylinder or solid torus, and state the boundary data that remain.

Abelian Chern–Simons gives a finite discriminant form

For a bosonic oriented compact Abelian theory with even, symmetric, nonsingular integral $K$ and a fixed quantum framing convention,

S_K= \frac{1}{4\pi} \int_{M^3}K_{IJ}a^I\wedge\mathrm da^J, \qquad \mathcal A=\mathbb Z^r/K\mathbb Z^r.

Fusion is addition in $\mathcal A$ . In the positive full-braid convention,

b([\ell],[m])=\ell^{\mathsf T}K^{-1}m\pmod1, \qquad M_{\ell m}=e^{2\pi i b([\ell],[m])},

and the even lattice gives

\theta_{\ell} = \exp\!\left(\pi i\ell^{\mathsf T}K^{-1}\ell\right), \qquad \dim\mathcal H(\Sigma_g)=\lvert\det K\rvert^g.

Odd $K$ requires spin-refined data; reversing orientation conjugates the phases. The discriminant group, lattice parity, and genus- $g$ state count are developed in Belov and Moore 2005, § 1, arXiv v1, printed pp. 3–5, eqs. (1.1)–(1.7), and § 5.3, printed p. 26 after eq. (5.17), PDF.

For $N\in\mathbb Z_{>0}$ , compact three-dimensional BF theory is the off-diagonal Abelian theory

K_{BF}= \begin{pmatrix} 0&N\\ N&0 \end{pmatrix}, \qquad \mathcal A_{BF}=\mathbb Z_N^{e}\oplus\mathbb Z_N^{m}, \qquad \dim\mathcal H(\Sigma_g)=N^{2g}.

Its electric and magnetic lines have full mutual phase $e^{2\pi i/N}$ . The compact global completion and finite operator algebra are given in Kapustin and Seiberg 2014, § 3, arXiv v2, printed pp. 9–13, eqs. (3.1)–(3.16), PDF.

Untwisted finite $\mathbb Z_N$ gauge theory instead sums principal bundles,

Z_{\mathbb Z_N,0}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}_{\mathbb Z_N}(M)} \frac{1}{\lvert\operatorname{Aut}(P)\rvert}.

On a closed genus- $g$ surface it also has $N^{2g}$ states. The agreement with compact BF requires the global sectors and groupoid normalization; one state count does not prove full equivalence. Dijkgraaf and Witten give the finite sum and state spaces in 1990, §§ 6.2–6.3, printed pp. 415–419, eqs. (6.8)–(6.21), Open PDF, while Freed and Quinn formulate the boundary line, measure, and sewing law in 1993, §§ 1–2, printed pp. 438–446, eqs. (1.1)–(1.2), (2.1), and (2.9)–(2.18), Theorem 2.13, Open PDF.

The product trace makes the comparison visible at $N=k=2$ :

Z_{U(1)_2}(T^3)=2, \qquad Z_{BF_2}(T^3)=4, \qquad Z_{\mathrm{DW}(\mathbb Z_2,0)}(T^3)=4.

The last equality is the automorphism-weighted count $\lvert\operatorname{Hom}(\mathbb Z^3,\mathbb Z_2)\rvert/2=8/2$ . The trace check confirms sewing normalization; it does not identify the theories at every codimension.

Boundaries and relative theories add new assignments

A sewing cut exposes the full state space and contracts it once. A physical boundary instead restricts the allowed fields, declares which lines may end or condense, and can carry its own degrees of freedom. Replacing a cut by a physical boundary therefore need not reproduce the identity cylinder.

In the controlled bosonic Abelian Chern–Simons class, let $q([\ell])=\tfrac12\ell^{\mathsf T}K^{-1}\ell\pmod1$ and let $L\subset\mathcal A$ . An elementary topological boundary candidate obeys

q|_L=0, \qquad L=L^{\perp}, \qquad \lvert L\rvert^2=\lvert\mathcal A\rvert.

The lines in $L$ may end. This finite Lagrangian test is not a universal boundary classification, and a topological vacuum boundary also requires the chiral or signature obstruction to vanish in this model. Kapustin and Saulina derive the controlled construction and its limitations in 2011, §§ 4.1, 5.1, and 6.2, arXiv v2, printed pp. 10–11, 15–17, and 30–33, PDF.

A relative $d$ -theory goes further: in the covector convention it assigns $\widetilde F_X\in\alpha(X)^*$ for a closed $d$ -manifold $X$ , where $\alpha$ is an extended $(d+1)$ -theory. A bulk filling gives $Z_\alpha(Y)\in\alpha(X)$ , and only the pairing

\widetilde F_X\!\left(Z_\alpha(Y)\right)\in\mathbb C

is an absolute number. A pointwise functional is not enough; the assignments must be compatible with bordisms and gluing. Freed and Teleman state the vector and covector conventions in 2014, Definition 2.1 and §§ 2–3, arXiv v3, printed pp. 3–4 and 8, eqs. (2.2)–(2.4), PDF.

The SymTFT capstone specializes that relative viewpoint. Let $I$ be the interval between a specified topological symmetry boundary $B_{\mathrm{sym}}^{\mathcal S}$ and physical boundary $B_{\mathrm{phys}}^{\mathcal T,\sigma}$ carrying the QFT $\mathcal T$ with symmetry coupling $\sigma$ , and let the auxiliary bulk $Z(\mathcal S)$ fill the region between them. Compactification along $I$ , denoted $\operatorname{Comp}_I$ , gives

\mathcal T_\sigma = \operatorname{Comp}_I\!\left( B_{\mathrm{sym}}^{\mathcal S} \mid Z(\mathcal S)\mid B_{\mathrm{phys}}^{\mathcal T,\sigma} \right).

Changing the symmetry boundary represents gauging or condensation only when the requisite topological interface exists while the physical boundary is held fixed. The auxiliary bulk organizes topological symmetry and anomaly data; it does not reconstruct the physical Hamiltonian, local spectrum, OPEs, correlators, couplings, or RG flow. The bounded sandwich and this conditional gauging statement are developed in Bhardwaj and Schäfer-Nameki 2025, §§ 2.4, 2.6–2.7, arXiv v3, printed pp. 21–32, Statement 2.1 and eqs. (2.40), (2.68)–(2.72), PDF.

Exact chapter guide

The seven pages appear below in chapter order. Each row names its actual hard preparation and the strongest conclusion the page licenses.

Seven pages from the bordism assignment to the qualified SymTFT sandwich
Page	Question and capability	Required background	Stop condition
What Is a Topological Field Theory?	Define the finite unextended assignment and distinguish it from a term, response, and microscopic topological order	Globally defined topological terms	Metric independence and finite state spaces are useful tests, not universal definitions or converses
State Spaces, Cobordisms, and Gluing	Derive duality, evaluation, trace, sewing, pair-of-pants operations, and cut independence	The unextended TQFT definition	A finite gluing check does not supply lower-codimension extension data or a physical boundary
Abelian Chern–Simons Theory	Compute line labels, fusion, braiding, spin, framing dependence, and genus-g states from an integral K matrix	Chern–Simons quantization; linking, braiding, and framing	Odd K requires spin refinement; degenerate K and non-Abelian classification lie outside this finite card
BF Theory as a Topological Gauge Theory	Derive finite holonomy, complementary Wilson linking, Heisenberg operators, torsion-sensitive states, and boundary polarizations	Compact BF couplings; state spaces and gluing	The local real-form action does not by itself construct compact differential-cohomology sectors or the quantum finite theory
Finite Gauge Theory and Dijkgraaf–Witten Twists	Build the automorphism-weighted bundle sum, transgressed state spaces, flux–charge lines, cocycle twists, and elementary gauging boundaries	Finite gauging; state spaces and gluing	A fixed cocycle weight is not the dynamical sum, and group cohomology does not classify every finite topological phase
Operators, Boundaries, and Relative Topological Theories	Distinguish line data, physical boundaries, folded interfaces, sewing cuts, and vector- or covector-valued relative amplitudes	Abelian Chern–Simons data; boundaries and folding; fusion junctions	Elementary Lagrangian and cocycle tests do not classify all boundaries, modules, or fully extended relative theories
Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging	Separate auxiliary bulk, topological symmetry boundary, and physical boundary; encode symmetry, anomaly, global form, and conditional gauging	Relative boundary data; anomaly inflow; coupled gaugeability	No universal existence theorem or reconstruction of physical boundary dynamics is asserted

Keep the model-specific nonimplications visible

The canonical five-row comparison already separates terms, responses, TQFTs, and microscopic phases. The models add further stop conditions:

equal state dimensions or one partition function do not prove equivalence;
a finite line quotient does not determine associators, boundaries, or a fully extended theory;
a sewing cut is not a physical boundary condition;
a Lagrangian subgroup test does not cancel a nonzero chiral or framing obstruction;
a fixed finite-bundle phase is not the automorphism-weighted gauge theory;
agreement between compact BF and untwisted cyclic gauge theory in selected closed checks is not a proof at every codimension; and
a SymTFT bulk does not determine the local dynamics or unique infrared phase of its physical boundary.

For the chapter-wide object types and decisive tests, return to the governed comparison.

What the chapter establishes—and where it stops

The chapter supplies a bounded physical workflow:

demand a structured state-and-bordism assignment with gluing;
compute exact operator and state data in compact Abelian and finite models;
test the cylinder, trace, Fourier kernel, or groupoid sewing measure;
distinguish a cut from a physical boundary and state which operators may end;
type relative amplitudes before pairing them into numbers; and
use a specified one-higher-dimensional bulk and boundary pair to organize symmetry and conditional gauging.

It does not prove the cobordism hypothesis, classify fully dualizable objects, construct every higher-categorical target, or classify non-Abelian modular tensor categories and their boundaries. It also does not establish that every formal TQFT has a unitary microscopic realization or that every relative QFT admits a SymTFT presentation. Those theorem-level questions require declared higher targets and coherence data beyond the finite physical tests used here.

Review the chapter

These prompts test retrieval, translation, and transfer without creating a separate formal assessment. A successful response states its hypotheses and passes the listed invariant.

Eight ways to demonstrate the chapter's central capabilities
Mode and task	Owner pages	Successful response and invariant	Characteristic repair
Retrieval — state the finite unextended TQFT assignment	What Is a TQFT?	Names structured closed spatial manifolds, bordisms, vector spaces, linear maps, disjoint union, cylinder, orientation duality, and gluing	Return to states and composable evolution if a closed partition function is the only datum
Explanation — explain why a cut is not a physical boundary	State spaces and gluing; physical boundaries and relative theories	Contrasts contraction over the full cut state space with a boundary condition that restricts fields and endable operators	Repair at the cut-versus-boundary test
Derivation check — recover the product trace identity	State Spaces, Cobordisms, and Gluing	Closes the cylinder with compatible tangential data and obtains Z(Σ×S¹)=dim H(Σ), while stating the spin/supertrace caveat	Return to cutting contracts the shared boundary if the seam is summed twice
Representation change — translate three-dimensional compact BF into K-matrix language	BF theory; Abelian Chern–Simons	Obtains the off-diagonal K matrix, ℤ_N² line group, mutual phase, and N^2g states; retains the boundary-term/polarization caveat	Repair the compact normalization at BF Couplings and Discrete Topological Data
Comparison — distinguish U(1)₄, BF₂, and double semion despite four bulk line labels	Abelian Chern–Simons; finite twists; boundary test	Uses quadratic spin and signature data to find respectively zero, two, and one elementary bosonic Lagrangian vacuum-boundary choices	Return to the three-model boundary comparison if line count is treated as complete data
Transfer — analyze compact BF with K having off-diagonal entry 3	K-matrix data; compact BF	Finds nine line classes, H(Σ_g) dimension 9^g, two complementary elementary boundary subgroups, and conjugate phases under orientation reversal	Repair the state count at canonical quantization
Failure diagnosis — locate the error in “equal torus degeneracy proves equivalent TQFTs”	Gluing checks; finite twists	Names fusion, spin, braiding, mapping-class action, boundary, framing, and higher-codimension data that the one count omits	Return to what gluing can fail to prove
Synthesis — decide whether a symmetry operation is a boundary change inside one SymTFT	Relative boundaries; Symmetry TFT	Specifies the auxiliary bulk, holds the physical boundary fixed, and exhibits the topological interface changing the symmetry boundary; otherwise withholds the claim	Return to the conditional gauging test

Continue from topological gauge data

For the action-level global and boundary tests that precede the TQFT construction, return to Topological Terms and Invertible Responses.
For anomaly representatives and their cancellation by a complete bulk–boundary system, continue to Anomaly Polynomials and Inflow.
For coupled higher backgrounds and the gaugeability of a substructure, continue to Higher-Group Operators, Gauging, and Anomalies.
For topological defects whose fusion has no two-sided inverse, continue to Non-Invertible Topological Defects and Fusion.
For theorem-level bordism categories, fully extended TQFTs, higher targets, dualizability, non-Abelian modular or categorical classification, universal SymTFT constructions, and the cobordism hypothesis, the canonical continuation is Mathematical QFT. Microscopic realizations and matter diagnostics belong to Many-Body QFT and Quantum Matter. This chapter stops at the finite physical tests needed to state those questions with their dimension and tangential structure fixed.

References

Atiyah, Michael F. “Topological Quantum Field Theory.” Publications Mathématiques de l’IHÉS 68 (1988): 175–186. DOI. Official PDF.
Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 [hep-th] (2005). Stable record.
Bhardwaj, Lakshya, and Sakura Schäfer-Nameki. “Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT.” SciPost Physics 19, no. 4 (2025): 098. DOI. Open PDF, arXiv v3.
Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129, no. 2 (1990): 393–429. DOI. Open PDF.
Freed, Daniel S., and Frank Quinn. “Chern–Simons Theory with Finite Gauge Group.” Communications in Mathematical Physics 156, no. 3 (1993): 435–472. DOI. Open published PDF.
Freed, Daniel S., and Constantin Teleman. “Relative Quantum Field Theory.” Communications in Mathematical Physics 326, no. 2 (2014): 459–476. DOI. Open PDF, arXiv v3.
Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. DOI. Open PDF, arXiv v2.
Kapustin, Anton, and Natalia Saulina. “Topological Boundary Conditions in Abelian Chern–Simons Theory.” Nuclear Physics B 845, no. 3 (2011): 393–435. DOI. Open PDF, arXiv v2.