Finite Gauge Theory and Dijkgraaf–Witten Twists

A finite gauge theory does not integrate over a Lie-algebra-valued connection. It sums over the groupoid of principal bundles for a finite group $G$, with each isomorphism class weighted by the reciprocal of its automorphism group. On a closed oriented $D$-manifold $M$, a normalized cocycle $\omega\in Z^D(BG;U(1))$ assigns each bundle $P$ the multiplicative holonomy

$$\operatorname{Hol}_\omega(P) := \left\langle f_P^*[\omega],[M]\right\rangle \in U(1).$$

The resulting Dijkgraaf–Witten partition function is

$$\boxed{ Z_{G,\omega}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}_G(M)} \frac{ \operatorname{Hol}_\omega(P) }{ \left\lvert\operatorname{Aut}(P)\right\rvert } }.$$

Here $f_P:M\to BG$ classifies the bundle. Keeping $P$ fixed gives one invertible background phase; summing $P$ makes the finite gauge field dynamical and generally produces a noninvertible topological quantum field theory (TQFT). On a boundary the phase is not an absolute number: it belongs to a transgressed line, so a boundary condition or trivialization is additional data.

The main examples below are bosonic oriented theories. The untwisted construction works in every dimension $D\geq1$; the detailed line and cyclic twist calculations specialize to $2+1$ dimensions. Orientation reversal complex-conjugates the cocycle weight.

Required background. Gauging Continuous and Finite Symmetries supplies the orbit–stabilizer derivation of the automorphism weight and the distinction between fixed backgrounds and gauging. State Spaces, Cobordisms, and Gluing supplies the finite-state, trace, and sewing tests used below.

Helpful background. Chains, Homology, Cohomology, and Exact Sequences supplies the cocycle, coboundary, and evaluation language.

Finite bundles are the dynamical fields

For a discrete structure group, a principal bundle is already a flat local system; there is no continuous connection degree of freedom. Flat does not mean trivial: on a connected manifold its global data can be represented by

$$\phi:\pi_1(M)\longrightarrow G$$

up to conjugation. The stabilizer of $\phi$ under conjugation is precisely the automorphism group of the corresponding bundle. Therefore the partition function can also be written

$$Z_{G,\omega}(M) = \frac{1}{\lvert G\rvert} \sum_{\phi\in\operatorname{Hom}(\pi_1M,G)} W_\omega(\phi), \qquad W_\omega(\phi)\in U(1),$$

when $M$ is connected. Orbit–stabilizer counting turns the contribution of one conjugacy orbit into $W_\omega(\phi)/\lvert\operatorname{Aut}(\phi)\rvert$. Replacing this weight by one copy per orbit would break the canonical cylinder kernel and hence cutting and gluing.

For the trivial cocycle, two immediate controls are

$$Z_{G,0}(S^D)=\frac{1}{\lvert G\rvert}, \qquad D\geq2,$$

and

$$Z_{G,0}\!\left(\Sigma\times S^1\right) = \dim\mathcal H_{G,0}(\Sigma).$$

The first formula has one trivial bundle with automorphism group $G$. The second is the trace of the identity cylinder. Dijkgraaf and Witten construct the bundle sum, state spaces, and sewing maps in Dijkgraaf and Witten 1990, §§ 6.1–6.3, printed pp. 414–417, especially eqs. (6.1)–(6.17), version-of-record PDF.

Cocycle evaluation defines the twist

Choose an ordered triangulation of $M$. A flat lattice field assigns $g_{ij}\in G$ to each oriented edge and obeys

$$g_{ij}g_{jk}=g_{ik}$$

on every ordered triangle. If $\sigma=[v_0\cdots v_D]$ is a $D$-simplex and $\epsilon_\sigma=\pm1$ compares its vertex ordering with the orientation of $M$, then

$$W_\omega(g) = \prod_{\sigma} \omega\!\left( g_{01},g_{12},\ldots,g_{D-1,D} \right)^{\epsilon_\sigma}.$$

This is the action weight, not yet the normalized sum. If $K$ denotes the closed triangulation and $K^j$ its set of $j$-simplices, then summing the flat edge labels and dividing by the vertex gauge group gives

$$Z_{G,\omega}(K) = \frac{1}{\lvert G\rvert^{\lvert K^0\rvert}} \sum_{\substack{ g\in G^{K^1}\\ g_{ij}g_{jk}=g_{ik} }} W_\omega(g).$$

Gauge fixing along a maximal tree reduces this formula to the connected $\operatorname{Hom}(\pi_1M,G)/\lvert G\rvert$ expression above. With a boundary, boundary labels are retained and only the interior gauge data are summed; the result is a vector in the boundary state line rather than the closed scalar displayed here.

For $D=3$, the cocycle identity is

$$\begin{aligned} 1 = (\delta\omega)(g,h,k,\ell) ={}& \omega(h,k,\ell)\, \omega(gh,k,\ell)^{-1}\\ &\times \omega(g,hk,\ell)\, \omega(g,h,k\ell)^{-1}\, \omega(g,h,k). \end{aligned}$$

It is the local consistency equation behind retriangulation and gauge invariance on a closed manifold. Normalization means that $\omega$ is one whenever any argument is the identity.

Changing representatives by a coboundary,

$$\omega' = \omega\,\delta\beta, \qquad \beta\in C^{D-1}(BG;U(1)),$$

does not change a closed-manifold amplitude: factors on internal faces cancel. On a manifold with boundary, the uncancelled $\beta$ factors rephase the boundary state. Thus cohomologous cocycles give equivalent bulk theories together with the induced identification of their boundary lines; they are not literally the same boundary action before that identification.

The lattice evaluation and its boundary factors are derived in Dijkgraaf and Witten 1990, §§ 6.4–6.5, printed pp. 419–422, eqs. (6.22)–(6.31), version-of-record PDF.

The groupoid measure makes cutting and gluing work

Let $\Sigma$ be a closed oriented $(D-1)$-manifold. Transgressing $\omega$ to the groupoid of boundary bundles assigns a one-dimensional line $L_{\omega,Q}$ to each $G$-bundle $Q$ on $\Sigma$, together with an action of $\operatorname{Aut}(Q)$. The state space is

$$\mathcal H_{G,\omega}(\Sigma) \cong \bigoplus_{[Q]\in\pi_0\operatorname{Bun}_G(\Sigma)} L_{\omega,Q}^{\,\operatorname{Aut}(Q)}.$$

This formula matters: in a twisted theory, automorphism holonomy can remove a summand, so the state space is not always the space of ordinary functions on bundle classes. For $\omega=0$,

$$\mathcal H_{G,0}(\Sigma) \cong \operatorname{Fun}\!\left( \pi_0\operatorname{Bun}_G(\Sigma),\mathbb C \right).$$

The same measure appears in the boundary pairing,

$$\langle\Psi_2,\Psi_1\rangle_\Sigma = \sum_{[Q]} \frac{ \left\langle \Psi_2(Q),\Psi_1(Q) \right\rangle }{ \lvert\operatorname{Aut}(Q)\rvert },$$

and hence in the sewing formula

$$Z(M_2\circ_\Sigma M_1) = \sum_{[Q]} \frac{ \left\langle Z(M_2;Q),Z(M_1;Q) \right\rangle }{ \lvert\operatorname{Aut}(Q)\rvert }.$$

Freed and Quinn construct the boundary lines, groupoid measure, inner product, and gluing theorem in Freed and Quinn 1993, §§ 1–2, current arXiv v3, internal printed pp. 4–13, especially eqs. (1.2), (1.6), (2.1), (2.9)–(2.12), and Theorem 2.13 with eq. (2.17), PDF. The current v3 should be used; v1 and v2 are withdrawn.

State spaces and line operators are transgressed data

Now specialize to $D=3$. An ordinary Wilson line in a representation $R$ of $G$ is

$$W_R(\gamma) = \operatorname{Tr}_R \operatorname{Hol}_\gamma(P).$$

It measures electric charge. A magnetic line instead fixes the meridian holonomy around the line to a conjugacy class $C=[g]$. In the untwisted theory, a simple dyonic line is labeled by

$$\left(C,\pi\right), \qquad \pi\in\operatorname{Irr} C_G(g),$$

where $C_G(g)$ is the centralizer of $g$. Its quantum dimension and spin are

$$d_{(C,\pi)} = \lvert C\rvert\,\dim\pi, \qquad \theta_{(C,\pi)} = \frac{\chi_\pi(g)}{\dim\pi}.$$

For nontrivial $\omega$, transgression supplies a two-cocycle $\alpha_g$ on $C_G(g)$, and $\pi$ must be an irreducible $\alpha_g$-projective representation. In one common convention, for $x,y\in C_G(g)$,

$$\alpha_g(x,y) = \frac{ \omega(x,y,g)\,\omega(g,x,y) }{ \omega(x,g,y) }.$$

The convention-independent statement is that $\alpha_g$ is the slant transgression of $\omega$. Changing the cocycle convention changes this representative and the projective matrices together, not the resulting line theory. Willerton derives the loop-groupoid transgression and the twisted centralizer representation theory in Willerton 2008, § 1.4.3, printed p. 1438, and § 3.3, printed p. 1454, especially Theorem 22, PDF.

With $\chi_\pi$ interpreted as the corresponding projective character in this same convention, the displayed quantum-dimension and spin formulas continue to hold in the twisted theory.

Non-Abelian fusion is not addition of two labels. It involves induction, restriction, and projective representation data. The formulas above identify the simple sectors and basic invariants; the full braided-category classification is outside this page.

An untwisted symmetric-group control

Take $G=S_3$. Its three conjugacy classes give the following simple lines:

• the identity class has centralizer $S_3$ and its three irreducible representations give dimensions $1,1,2$;
• the transposition class has size $3$, centralizer $\mathbb Z_2$, and two lines of dimension $3$, with spins $+1$ and $-1$;
• the three-cycle class has size $2$, centralizer $\mathbb Z_3$, and three lines of dimension $2$, with spins $1,e^{2\pi i/3},e^{4\pi i/3}$.

There are eight torus states, and their squared quantum dimensions check

$$\sum_a d_a^2 = 1+1+4+9+9+4+4+4 = 36 = \lvert S_3\rvert^2.$$

The same eight appear directly from the groupoid trace on $T^3$. The number of commuting pairs in the centralizer of the first holonomy is $18$ for the identity, $4$ for each of the three transpositions, and $9$ for each of the two three-cycles. Hence

$$Z_{S_3,0}(T^3) = \frac{18+3\cdot4+2\cdot9}{6} = 8 = \dim\mathcal H_{S_3,0}(T^2),$$

while $Z_{S_3,0}(S^3)=1/6$. Dijkgraaf and Witten relate the three-torus sum to projective centralizer representations in Dijkgraaf and Witten 1990, § 6.6, printed pp. 423–425, eqs. (6.34)–(6.41), version-of-record PDF.

Cyclic twists change line data without changing the count

Let $G=\mathbb Z_N$, written additively, and let $[x]_N\in\{0,\ldots,N-1\}$ denote the standard residue. A normalized set of representatives for

$$H^3(B\mathbb Z_N;U(1)) \cong \mathbb Z_N$$

is

$$\omega_s(a,b,c) = \exp\!\left[ \frac{2\pi i s}{N^2}\, a\bigl(b+c-[b+c]_N\bigr) \right], \qquad s\in\mathbb Z_N.$$

The carry $b+c-[b+c]_N$ is either $0$ or $N$; it is the finite cochain datum that changes the topological spins.

To compare with the preceding compact BF and Abelian Chern–Simons pages, use the continuum convention

$$K_{N,r} = \begin{pmatrix} 2r&N\\ N&0 \end{pmatrix}, \qquad r\in\mathbb Z_N.$$

With the positive orientation and Chern–Simons phase convention used here, this $K$-matrix realizes $\omega_{-r}$, not $\omega_r$. Reversing the orientation complex-conjugates the cocycle and sends $r\mapsto-r$. The inverse matrix is

$$K_{N,r}^{-1} = \begin{pmatrix} 0&1/N\\ 1/N&-2r/N^2 \end{pmatrix}.$$

Line labels are

$$\mathcal A_{N,r} = \mathbb Z^2/K_{N,r}\mathbb Z^2.$$

Writing a representative as $\ell=(e,m)$, the relations are

$$(e,m) \sim (e+N,m) \sim (e+2r,m+N).$$

Canonical representatives have $0\leq e,m<N$, but their fusion contains a carry. If

$$t=\left\lfloor\frac{m+m'}{N}\right\rfloor,$$

then

$$(e,m)\otimes(e',m') = \left( [e+e'-2rt]_N,\, [m+m']_N \right).$$

The spin and full mutual monodromy are

$$\boxed{ \theta_{(e,m)} = \exp\!\left[ 2\pi i \left( \frac{em}{N} - \frac{r m^2}{N^2} \right) \right] }$$

and

$$M_{(e,m),(e',m')} = \exp\!\left[ 2\pi i \left( \frac{em'+me'}{N} - \frac{2rmm'}{N^2} \right) \right].$$

The twist can change fusion as well as phases. Smith normal form gives

$$\mathcal A_{N,r} \cong \mathbb Z_d \times \mathbb Z_{N^2/d}, \qquad d=\gcd(N,2r).$$

For example, $N=3,r=1$ gives $\mathbb Z_9$, whereas $r=0$ gives $\mathbb Z_3\times\mathbb Z_3$. Nevertheless,

$$\lvert\det K_{N,r}\rvert=N^2, \qquad \dim\mathcal H_{N,r}(\Sigma_g)=N^{2g}.$$

This invariance of the state count is special to the cyclic family. For a generic finite group, a twist can change the number and dimensions of projective centralizer representations and hence the torus-state count.

The standard cyclic cocycle, its $N^2$ torus ground states, and its modular data are given in Hu, Wan, and Wu 2013, § VII.A, current arXiv v2, printed p. 17, eqs. (89)–(92), PDF. The continuum action, the identification $p_{\mathrm{KS}}=2r$, the periodicity $p_{\mathrm{KS}}\sim p_{\mathrm{KS}}+2N$, and the odd-$p_{\mathrm{KS}}$ spin refinement are in Kapustin and Seiberg 2014, § 5, arXiv v2, printed pp. 20–21, eqs. (5.1)–(5.3), PDF. The displayed $p_{\mathrm{KS}}=2r$ subfamily is the ordinary oriented bosonic Dijkgraaf–Witten theory.

The $N=2$ discriminator

At $N=2,r=0$,

$$K_{2,0} = \begin{pmatrix}0&2\\2&0\end{pmatrix},$$

and the four spins are

$$1,\quad 1,\quad 1,\quad -1.$$

This is the toric-code line data: electric and magnetic generators are bosons, while their composite is a fermion. At $N=2,r=1$,

$$K_{2,1} = \begin{pmatrix}2&2\\2&0\end{pmatrix},$$

and the spins are

$$1,\quad 1,\quad i,\quad -i$$

up to orientation reversal. This is the double-semion discriminator. Both theories have four torus states, so state count alone cannot detect the twist. The carry fusion and sign-conjugate braid formulas are worked out in de Wild Propitius 1995, § 2.6.3, arXiv v1, internal printed pp. 102–104, especially eqs. (2.6.33)–(2.6.42), PDF.

First application: the finite row completes the three-model thread

The three models below share finite topological data only after every field and global sector named in the second column is summed. A fixed connection or fixed finite bundle gives a background response instead. The rows compare a diagnostic; they do not assert a cross-dimensional or fully extended equivalence.

Three dynamical 2+1-dimensional models under one line-and-state test; matching cardinalities do not erase their different global definitions or twists

Dynamical model	Global data summed	Coefficient or twist	Finite line datum	Genus-g state check	Decisive caveat
Compact bosonic U(1) Chern–Simons	One compact one-form connection	Nonzero even level k	Cyclic group of order equal to the absolute value of k	Absolute value of k to the power g	A self-pairing and framing phase remain; this is not a finite-bundle sum
Untwisted compact BF at positive level N	Two compact one-form connections	Off-diagonal integer level N	Z/NZ × Z/NZ with electric–magnetic pairing	N to the power 2g	It matches the finite row only after compact sectors and measure are included
Cyclic Dijkgraaf–Witten gauge theory	The full groupoid of finite Z/NZ bundles	Continuum parameter r modulo N; cocycle class s = −r in this convention	Z/dZ × Z/(N²/d)Z, where d is gcd(N, 2r)	N to the power 2g	At r = 0 it is untwisted BF; nonzero r changes spin and braiding and can change fusion

For $r=0$, the cyclic bundle theory and compact $K=\left(\begin{smallmatrix}0&N\\N&0\end{smallmatrix}\right)$ BF theory agree only after the compact differential-cocycle sectors and the automorphism measure are included. Local equations such as flatness do not establish that global equivalence. For $r\ne0$, the diagonal entry $2r$ is the continuum memory of the finite cocycle.

Gauging requires an anomaly-free symmetry

Let a $D$-dimensional QFT $\mathcal T$ have a finite global symmetry $G$, and let $Z_{\mathcal T}(M;P)$ be its partition function in a fixed background bundle. When the symmetry is gaugeable,

$$Z_{\mathcal T/G,\omega}(M) = \sum_{[P]} \frac{ W_\omega(P) }{ \lvert\operatorname{Aut}(P)\rvert } Z_{\mathcal T}(M;P).$$

Before the sum, $W_\omega(P)$ is a same-dimensional invertible background counterterm or symmetry-protected response. After the sum, it is the Dijkgraaf–Witten twist of the dynamical finite gauge field. These are two field roles for the same local phase, not two interchangeable theories.

A same-dimensional cocycle does not cancel an arbitrary anomaly. In the group-cohomological sector, a $D$-dimensional Dijkgraaf–Witten term lies in $H^D(BG;U(1))$, whereas an ordinary $D$-dimensional ’t Hooft anomaly lies one degree higher, in $H^{D+1}(BG;U(1))$. A nontrivial anomaly obstructs an absolute groupoid sum unless it is trivialized or cancelled by inflow or additional degrees of freedom. Gaiotto, Kapustin, Seiberg, and Willett explain the background-versus-summed distinction in Gaiotto et al. 2015, § 1, printed pp. 3–4, especially eq. (1.3), and § 2, printed pp. 7–10, especially p. 9, arXiv v2 PDF. Kapustin and Thorngren identify the $H^{D+1}(BG;U(1))$ Dijkgraaf–Witten inflow class and also exhibit more severe discrete-symmetry obstructions that are not captured by such an inflow in Kapustin and Thorngren 2014, §§ 1–2, current arXiv v2, internal printed pp. 1–5, PDF.

A boundary needs a cocycle trivialization

On a boundary, the transgressed phase must be cancelled or represented by boundary data. A simple class of $2+1$-dimensional boundaries chooses a subgroup inclusion

$$\iota:H\hookrightarrow G$$

and a two-cochain $\vartheta\in C^2(BH;U(1))$ satisfying

$$\delta\vartheta=\iota^*\omega.$$

If $[\iota^*\omega]\ne0$, this elementary $H$-boundary does not exist. Other boundary categories may exist, but they require additional structure; the subgroup condition is not a universal classification. This is a physical boundary condition, not the artificial cut surface $\Sigma$ used in the sewing formula: at a cut one retains the full boundary bundle state and contracts its action line instead of selecting a subgroup $H$. Fuchs, Schweigert, and Valentino develop the relative-bundle and cocycle data in Fuchs, Schweigert, and Valentino 2014, §§ 2.5 and 3.2, arXiv v3, internal printed pp. 13–18, especially eq. (3.13), PDF.

What the finite construction does not classify

The finite bundle sum is exact inside its domain, but several stronger conclusions do not follow.

• A finite gauge TQFT has no local photon or continuum curvature mode. It is not a weak-coupling approximation to a generic Yang–Mills theory.
• Group cohomology supplies Dijkgraaf–Witten twists, not every possible TQFT, anomaly, or gapped phase.
• The cyclic $K$-matrix is a special Abelian continuum presentation. A generic finite group, especially with a nontrivial twist, need not admit such a presentation.
• Matching $Z(S^3)$, $Z(T^3)$, or a state-space dimension does not prove equality of all bordism maps, line categories, boundaries, or fully extended data.
• The oriented bosonic construction does not automatically extend to unorientable or spin-dependent theories. Those require the corresponding tangential refinement.
• A lattice Hamiltonian realization requires its own local Hilbert spaces, constraints, and continuum/topological-limit argument. The state sum alone does not provide that physical realization.

For a general finite group, even the torus-state count can depend on $\omega$: projective centralizer representations can merge into higher-dimensional blocks. This can happen even for an Abelian product. For $G=(\mathbb Z_2)^3$, a type-III twist has 22 simple lines—eight one-dimensional and fourteen two-dimensional—rather than the 64 untwisted lines Willerton 2008, example following Theorem 24, printed p. 1456, PDF. The fixed $N^{2g}$ count established above is a property of the cyclic family, not a theorem about all finite groups.

Check your understanding

1. Recover the normalization on a sphere

Show that the untwisted finite $G$ theory has $Z(S^3)=1/\lvert G\rvert$.

Solution

The three-sphere is simply connected, so there is one bundle class. Its automorphisms are the constant gauge transformations $G$. Therefore its groupoid-cardinality contribution is

$$Z_{G,0}(S^3) = \frac{1}{\lvert\operatorname{Aut}(P_{\mathrm{triv}})\rvert} = \frac{1}{\lvert G\rvert}.$$

2. See why a coboundary is not invisible at a boundary

Let $\omega'=\omega\,\delta\beta$. Explain why the two closed amplitudes agree but boundary wavefunctions are rephased.

Solution

On a closed triangulation, every codimension-one face belongs to two top-dimensional simplices with opposite induced orientations, so the $\beta$ factors cancel. Boundary faces occur only once. Their surviving product is exactly the multiplication map that identifies the two transgression lines. Thus the closed numbers agree, while boundary states agree only after the induced $\beta$-dependent identification.

3. Count the untwisted $S_3$ torus states

Use the three conjugacy classes and their centralizers to recover eight simple lines.

Solution

The identity centralizer $S_3$ has three irreducible representations. A transposition has centralizer $\mathbb Z_2$, which has two. A three-cycle has centralizer $\mathbb Z_3$, which has three. Therefore

$$3+2+3=8.$$

The trace check gives the same answer:

$$\frac{18+3\cdot4+2\cdot9}{6}=8.$$

4. Distinguish toric code from double semion

For $N=2$, compare $r=0$ and $r=1$. Which invariant sees the difference if the torus-state count does not?

Solution

Both matrices have determinant $-4$, so both theories have four torus states. Their twists differ:

$$\{\theta_a\}_{r=0} = \{1,1,1,-1\}, \qquad \{\theta_a\}_{r=1} = \{1,1,i,-i\},$$

up to simultaneous complex conjugation by orientation reversal. Topological spin, and equivalently the quadratic refinement of the braiding pairing, detects the cocycle.

5. Keep a twist separate from an anomaly

In $D$ dimensions, why does a class in $H^D(BG;U(1))$ not by itself remove a nonzero class in $H^{D+1}(BG;U(1))$?

Solution

The first class defines a globally well-defined same-dimensional phase. It changes the weighting of allowed background sectors and becomes a Dijkgraaf–Witten twist after gauging. The second class is the obstruction to making the background partition function gauge invariant. Different cohomological degrees encode different consistency problems; an anomalous theory needs a trivialization, inflow, or additional degrees of freedom before the absolute finite-bundle sum is defined.

Continue to relative operators and symmetry TFT

The already developed BF Theory as a Topological Gauge Theory gives the compact continuum presentation of the untwisted cyclic model. Operators, Boundaries, and Relative Topological Theories will develop line condensation, boundary algebras, and relative amplitudes beyond the elementary cocycle trivialization used here.

Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging will use finite gauge theories as controlled bulk encodings of boundary symmetry data and alternative gauging choices. The theorem-level classification by group cohomology, twisted centers, and higher categories, as well as microscopic lattice realizations, requires additional structures not developed on this page.

References

• de Wild Propitius, Mark. Topological Interactions in Broken Gauge Theories. PhD thesis, University of Amsterdam, 1995. Stable record, arXiv:hep-th/9511195v1. Open PDF.
• Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129, no. 2 (1990): 393–429. DOI. Open version-of-record 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 PDF, current arXiv:hep-th/9111004v3; v1 and v2 are withdrawn.
• Fuchs, Jürgen, Christoph Schweigert, and Alessandro Valentino. “A Geometric Approach to Boundaries and Surface Defects in Dijkgraaf–Witten Theories.” Communications in Mathematical Physics 332, no. 3 (2014): 981–1015. DOI. Open PDF, arXiv:1307.3632v3.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF, arXiv:1412.5148v2.
• Hu, Yuting, Yidun Wan, and Yong-Shi Wu. “Twisted Quantum Double Model of Topological Phases in Two Dimensions.” Physical Review B 87 (2013): 125114. DOI. Open PDF, arXiv:1211.3695v2.
• 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:1401.0740v2.
• Kapustin, Anton, and Ryan Thorngren. “Anomalies of Discrete Symmetries in Various Dimensions and Group Cohomology.” arXiv:1404.3230v2 [hep-th] (2014). Stable record. Open PDF.
• Willerton, Simon. “The Twisted Drinfeld Double of a Finite Group via Gerbes and Finite Groupoids.” Algebraic & Geometric Topology 8 (2008): 1419–1457. DOI. Open PDF.