On 2-form gauge models of topological phases

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by cohomology classes of $B^2G$. Discrete topological gauge theories can typically be embedded into continuous quantum field theories. In the 2-form case, the continuous theory is shown to be a strict 2-group gauge theory. This embedding is studied by carefully constructing the space of $q$-form connections using the technology of Deligne-Beilinson cohomology. The same techniques can then be used to study more general models built from Postnikov towers. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of $B^2G$ as provided by the so-called $W$-construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of $G$-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.


Introduction
Over the last several decades, quantum field theories have emerged as the central language in which modern theoretical physics is formulated. For instance, quantum phases of matter may succinctly be defined as equivalence classes of quantum field theories, and a given quantum model is a concrete realization of a phase. Topological quantum field theories (TQFTs) form a subclass of quantum field theories that are particularly tractable. Indeed, topological theories are much simpler than conventional theories as they associate finite dimensional Hilbert spaces to codimension-one submanifolds and have trivial Hamiltonian evolution. From a mathematical point of view, TQFTs can usually be reformulated algebraically in terms of finite sets of data. Such a reformulation, which bears a strong category theoretical flavor, was initially pioneered by Atiyah in [1] who defined a TQFT as a symmetric monoidal functor from a certain category of bordisms to the category of finite dimensional vector spaces. 1 This proposal was further developed by Baez and Dolan in [2] who suggested that higher category theory was the correct framework to capture the local structure inherent to quantum theory. More precisely, they proposed that a (d+1)-dimensional fully extended TQFT, which is capable of capturing locality all the way down to points, should be understood as a (d+1)-functor between a higher (d+1)-category of bordisms 2 and a higher symmetric monoidal (d+1)-category. This came to be known as the cobordism hypothesis [3][4][5]. These mathematical definitions that are motivated by topological invariance on the one hand and locality on the other hand severely constrain the structure of TQFTs, and can therefore be used as a classifying tool for topological theories in a given spacetime dimension.
It is believed that at long wavelengths gapped phases of matter, i.e. phases that have a spectral gap above the ground state that persists in the thermodynamics limit, are described by equivalence classes of topological quantum field theories. 3 Therefore, the above mentioned mathematical constraints turn out to have profound physical consequences and serve as an organizational tool for the space of gapped phases of matter. Furthermore, given a TQFT describing deep infrared physics, it is often possible to construct an exactly solvable model in terms of a lattice Hamiltonian projector. The model may then be deformed away from its exactly solvable projector in order study dynamical properties within the corresponding phase. This is one of the reasons why understanding topological theories and building the corresponding exactly solvable models is a worthwhile endeavor.
Naturally, the map from the space of ultraviolet models to the space of TQFTs is surjective. Since quantum models are understood in terms of correlation functions of the observables that they 1 For example, a (d+1)-dimensional TQFT Z is a symmetric monoidal functor that assigns to every oriented closed This data is subject to some coherence relations that ensure the topological nature of the theory. Moreover, it can be readily generalized to accommodate manifolds with additional structure such as spin structure or framing by suitably replacing the category of oriented bordisms. 2 It is a category of extended bordisms whose objects are points, 1-morphisms are 1-bordisms between disjoint union of points, 2-morphisms are bordisms between 1-bordisms, and so on and so forth. 3 Nevertheless, it is not completely clear whether there is a bijection between physically realizable gapped phase of matter and TQFTs. The subtle relation between TQFTs and gapped phases was carefully studied in [6] for theories displaying a global symmetry. furnish, going from the ultraviolet to the topological infrared is performed by a map that only retains the topological part of the correlation functions. As a matter of fact, it is a defining feature of topological theories to be blind to operators that are irrelevant under the renormalization group. Therefore, perturbing a TQFT away from its deep infrared fixed point, while maintaining its gap, may be thought of as going towards the ultraviolet regime.
There is a particular class of fully extended TQFTs, known as Dijkgraaf-Witten theories [7], that are mathematically well-defined in all dimensions. These theories are constructed from finite groups and have a topological gauge theory interpretation. Given a (d+1)-manifold M and a finite group G, they depend on a single datum, namely a cohomology class [ω] ∈ H d+1 (BG, R/Z) where BG is the classifying space of the group G, which has the property that its only non-vanishing homotopy group is the fundamental group and it equals the group G itself. Dijkgraaf-Witten theories can be cast in two equivalent ways: (i) as topological sigma models whose target space is BG and the sum in the partition function being performed over homotopy classes of maps from the spacetime manifold M to BG, (ii) as topological lattice gauge theories defined on a triangulation of the spacetime manifold together with a G-coloring, i.e. an assignment of group elements in G to every 1-simplex of the triangulation that satisfies compatibility conditions. Although, the first approach (i) is more mathematically succinct, the latter point of view (ii) has the advantage of being more physically transparent, i.e the fields, observables and gauge transformations can be more explicitly defined and studied. This happens to be very useful when studying for instance the excitations of the theory and their properties.
The equivalence between the two aforementioned approaches is conceptually straightforward and yet slightly subtle: The topological action for the sigma model approach is provided by integrating the pullback of the cohomology class [ω] onto the manifold M, while in the lattice gauge theory picture, the topological action is provided by evaluating the cocycle on each G-colored (d+1)-simplices of the triangulation. But this relies implicitly on the fact that for discrete groups the cohomology H d+1 (BG, R/Z) as an algebraic description. More precisely, it uses the fact there is an equivalence between the cohomology H d+1 (BG, R/Z) of simplicial cocycles of BG and the cohomology H d+1 (G, R/Z) of algebraic group cocycles of G. Instead of representing (d+1)-cochains as simplices, they are then defined as functions from G d+1 to R/Z, and the coboundary operator is modified accordingly. This second approach in terms of group cohomology is naturally the one used in order to construct exactly solvable models that are lattice Hamiltonian realizations of Dijkgraaf-Witten theories [8,9]. It turns out that a similar correspondence can also be established for topological theories that have a higher gauge theory interpretation. It is however not as straightforward as we explain at length in the present manuscript.
It is possible to define different sigma models that generalize the Dijkgraaf-Witten construction by choosing different target spaces. The most natural generalization is obtained by replacing the classifying space BG of the discrete group G by the q-th classifying space B q G. 4 The q-th classifying space B q G is an example of Eilenberg-MacLane space K(G, q) which has the property that only its q-th homotopy group is non-vanishing and equals the group G itself, i.e. π n (K(G, q)) = δ q,n G [10,11]. 5 Interestingly, the same way Dijkgraaf-Witten theories have a lattice gauge theory interpretation, a topological sigma model whose target space is an Eilenberg-MacLane space K(G, q) can be interpreted as a q-form topological lattice gauge theory, i.e. a theory that contains (q−1)-dimensional symmetry operators instead of point-like ones. Theories displaying a (q−1)-form gauge invariance have a gauge field that is locally described by a q-form. A further generalization involves building topological sigma models whose target spaces are provided by Postnikov towers. A Postnikov tower is a topological space constructed as a sequence of fibrations of simpler topological spaces. In particular, in this manuscript we will be interested in Postnikov towers which are built as fibrations of Eilenberg-MacLane spaces. In analogy to Dijkgraaf-Witten theories, these may be understood as topological higher group gauge theories that contain several gauge fields. More specifically, for every Eilenberg-MacLane space K(G, q) contained in the Postnikov tower, the gauge theory will include a corresponding q-form gauge field. In the lattice gauge theory picture, a q-form gauge field is defined by coloring the q-simplices of the triangulation with elements of the group G that satisfy some consistency criteria in the form of cocycle conditions. The precise form of these cocycle conditions is obtained from the data that goes into building the Postnikov tower. The corresponding gauge transformations are built from the same data. These different generalizations are presented in sec. 2.
Throughout this manuscript, we focus most of our attention on (3+1)d topological sigma models with the second classifying space B 2 G as the target space where G is a finite abelian group, or equivalently discrete (3+1)d 2-form topological lattice gauge theories. As explained above, such higher form gauge theories arise naturally from a mathematical point of view. But they also happen to be physically motivated. For instance, it is known that Yang-Mills theory is confining and the gauge bosons are gapped at long wavelengths, and it was argued in [12] that the infrared physics of the confining phase is captured by a non-trivial 2-form topological gauge theory. The gauge group of this 2-form gauge theory is the magnetic gauge group that survives in the infra red [13,14]. These 2-form gauge theories have also appeared in various other contexts in the literature [15][16][17][18][19][20][21][22][23][24][25][26][27]. One particular reason for the interest in such TQFTs resides in the fact that they host a topologically ordered surface.
Given a finite abelian group G, 2-form topological theories are classified by a single datum, namely a cohomology class [ω] ∈ H 4 (B 2 G, R/Z). It was shown by Eilenberg and MacLane in a series of seminal papers [10,11] that the cohomology group H 4 (B 2 G, R/Z) is isomorphic to the group of (possibly degenerate) R/Z-valued quadratic functions on G. This result allows for an explicit expression of the topological action in terms of a quadratic form and a quadratic operation known as the Pontrjagin square on H 2 (M, G) that is the space of fields of the 2-form theory [14,28]. Moreover, the topological order living at the surface can be described in terms of a categorical structure whose input data is the same as the one labeling the bulk theory, namely a finite abelian group and a quadratic form. If the quadratic form is degenerate, then the topological order is non-trivial. 6 Furthermore, abelian Chern-Simons theories are labeled by precisely the same data. As a matter of fact, it was shown in [16] that the 2-form theory is precisely the anomaly theory for the framing anomaly within the abelian Chern-Simons theory. Therefore, we may interpret abelian Chern-simons as a framed topological quantum field theory or as a TQFT along with the corresponding (3+1)d 2-form topological gauge theory.
Besides topological gauge theories, there exist other TQFTs which have been extensively studied. For instance, in (2+1)d it is possible to define a topological theory from any modular tensor category using the Turaev-Viro construction [29][30][31] and the corresponding Hamiltonian realization is provided by the Levin-Wen models [32]. Similarly, in (3+1)d it is possible to define a topological theory for any premodular tensor category 7 using the Crane-Yetter construction [33][34][35] and the corresponding Hamiltonian realization is provided by the Walker-Wang models [17]. But, when the input data of the 1 |G| b0 where A is a representative in a homotopy class [γ] and ω(A) the evaluation of γ * ω on A. When the group G is abelian the partition sum is over a cohomology group which is the natural abelianization of the homotopy group. In other words, maps γ become G-valued 1-cocycles and the null homotopic maps are G-valued 1-coboundaries (written as dφ) so that the configuration space of the sigma model is H 1 (M, G).
Alternatively, (2.3) can be recast as a lattice gauge theory. In order to do so, let us endow M with a triangulation . Thanks to the path-connectedness of BG, one can smoothly deform maps γ so that the space of paths in BG that is G up to homotopy can be mapped to the 1-simplices of . The contractible paths are then mapped to the identity group element. In practice, this means that we assign to every 1-simplex (xy) ⊂ a group element g xy such that for every 2-simplex (xyz) whose boundary is associated with a contractible path, the flatness condition (or 1-cocycle condition) dg, (xyz) ≡ g yz · g −1 xz · g xy = 1 is imposed. This is merely the statement that a flat G-connection can 9 Although their paper only discusses (2+1)d, generalization to any dimension is very straightforward.
have non-trivial holonomies along non-contractible closed paths only. Non-trivial group elements are thus assigned to non-contractible cycles of M so that each assignment is an element of Hom(π 1 (M), G).
We refer to such an assignment of group elements as a G-coloring and we denote by Col(M, G) the set of G-colorings. The Dijkgraaf-Witten partition function then reads where | 0 | is the number of 0-simplices. The topological action is provided by S ω [g, d+1 ] := ( d+1 ) ω(g), d+1 such that ( d+1 ) = ±1 is determined by the orientation of the (d+1)-simplex and ω(g), d+1 is the evaluation of the cocycle ω on the G-colored simplex d+1 .  (2.4). The relation between g ∈ Col(M, G) and A ∈ Hom(π 1 (M), G)/ ∼ is that A corresponds to an equivalence class of g's where the equivalence relations are gauge transformations.

Generalized topological gauge theories
The compact expression (2.2) for the Dijkgraaf-Witten partition function can be readily generalized to the scenario where BG is replaced by some other space X. For several different choices of X, the space of homotopy classes of maps [M, X] is isomorphic to a generalized cohomology group on M.
One then may study topological sigma models, with the space X as the target space, that provides generalizations of conventional topological gauge theories. Similar to the Dijkgraaf-Witten partition functions above (2.2)-(2.4), such generalized gauge theories can also be built as lattice (higher) gauge theories on triangulated space-time manifolds. We first describe the construction of these generalized gauge theories as sigma models and then as topological lattice theories. A topological sigma model can be constructed by generalizing the Dijkgraaf-Witten partition function as follows: where ω ∈ C d+1 (X, R/Z) is a (d+1)-cochain, M is a compact oriented (d+1)-manifold, [M] ∈ H d+1 (M, Z) its fundamental homology cycle and N X is a normalization constant that depends on the manifold and the choice of target space X. The sum in the partition function is over homotopy classes [γ] of maps γ from M to X. Naturally, the choice of (d+1)-cochain ω is constrained: Given an oriented (d+2)-bordism W : [7,44] where d : C d (X, R/Z) → C d+1 (X, R/Z) is the coboundary operator on the space of cochains. Condition (2.6) is required to hold for every bordism W which implies that ω must be a cocycle in Z d+1 (X, R/Z). When M is closed, modifying the cocycle ω by a coboundary dφ where φ ∈ C d (X, R/Z) has clearly no ∼ 8 ∼ effect. However, when M is an open manifold, this alters the action by a boundary term that can be absorbed into a U(1) phase upon quantization of the theory. Correspondingly, the redefined Hilbert space preserves amplitudes and as such describes the same theory. Putting everything together, we obtain that distinct topological sigma models are classified by cohomology classes [ω] ∈ H d+1 (X, R/Z).
Let us now consider several examples of sigma models that correspond to different choices of target space X: Example 2.1 (X is the q-th classifying space B q G of a finite abelian group G). This example is the immediate generalization of the Dijkgraaf-Witten theory obtained by considering the target space X to be the so-called q-th classifying space B q G of a finite abelian group G for q > 1. 10 This space satisfies the defining property π n (B q G) = δ n,q G. Similarly to the above construction, one can build models which have a higher-form topological gauge theory interpretation by providing a cohomology class [ω] ∈ H d+1 (B q G, R/Z). The partition sum looks almost identical to the Dijkgraaf-Witten partition function: is the i-th Betti number of the manifold M. Like the classifying space BG, the q-th classifying space B q G can be constructed as a simplicial complex so that the simplicial map γ : M → B q G is furnished by a G-valued q-cocycle A. The null homotopic maps can be extended to a cone above M and take the form dφ ∈ B q (M, G). These null homotopies represent the (q−1)-form gauge transformations in the q-form gauge theory, i.e A ∼ A + dφ. Hence the homotopy classes of maps are isomorphic to the cohomology classes In the following sections, we almost exclusively restrict ourselves to the study of the sigma models whose target space are provided by the second classifying space B 2 G of a finite abelian group G. Such models have a natural interpretation in terms of 2-form gauge theories. Nevertheless, before going into the details of these theories, it is enlightening to sketch out some further generalizations.
Example 2.2 (X is a two-stage Postnikov tower ). Following the theory of Postnikov towers [45], let us denote the q 1 -th classifying space of a finite abelian group G 1 by E 1 := B q1 G 1 . As described in the previous example, a topological (q 1 -form) gauge theory can be built wherein the local fields are cocycles A 1 ∈ Z q1 (M, G 1 ) which represent maps from γ : M → E 1 . Furthermore, the null homotopic maps are captured by coboundaries dφ 1 ∈ B q1 (M, G 1 ). Since the partition sum is over homotopy classes of maps, we must identify A 1 ∼ A 1 +dφ 1 which we recognize as the (q 1 −1)-form gauge invariance so that gauge inequivalent configurations are isomorphic to H q1 (M, G). The target space E 1 is referred to as a one-stage Postnikov tower. Things get more interesting if we consider a 2-stage Postnikov tower 10 Explicit constructions of classifying spaces are provided in sec. 4. in the kernel of a differential operator denoted by D E2 , i.e. D E2 A 2 = 0, such that In other words, A 1 and A 2 satisfy some cocycle conditions twisted by the extension class [α 2 ]. Similarly, a null homotopy is provided by the image of an operator D E2 that acts on a tuple We can easily check that D E2 • D E2 = 0 so that one can define a cohomology H q E2 (M) := kerD E2 /imD E2 where q = (q 1 , q 2 ). Homotopy classes of maps [γ] : M → E 2 are in one-to-one correspondence with the equivalence classes of the cohomology we just defined. Given a class [ω] ∈ H d+1 (E 2 , R/Z), we can thus define a topological gauge theory whose partition function reads (2.12) In the case where q 1 = 1 and q 2 = 2, the previous construction reduces to a (weak) 2-group bundle which has been recently studied in several papers, see for instance [20,21,46,47]. Topological gauge models built from 2-group connections can be found in [20,46,[48][49][50][51]. This construction can be even further generalized to so-called k-stage Postnikov towers (see app. A).

Topological lattice (higher) gauge theories
In order to build a lattice (higher) gauge theory which corresponds to a certain topological sigma model described above, we can proceed as follows: Let the target space of the sigma model be X and let us endow the space-time manifold M with a triangulation . For each non-vanishing homotopy group π qi (X) = G i , we introduce a G i -valued q i -cochain on M. Locally, this amounts to labeling the q i -simplices of the triangulation with elements in G i . Furthermore, we introduce constraints on the labelings of the different simplices that are analogous to the cocycle conditions satisfied by the data representing a homotopy class of a map from M to X. Labelings satisfying such constraints are referred to as X-colorings of the triangulation M and the set of all colorings is denoted by Col(M, X). 11 Denoting a given coloring by g ∈ Col(M, X), the partition function takes the form where N X is a normalization constant and S ω [g, d+1 ] is the topological action whose value depends on the local data g as well as a representative of the class [ω] ∈ H d+1 (X, R/Z). 11 Actually this set has a monoidal structure which makes it a group or a generalization thereof.
∼ 10 ∼ Example 2.3 (q-form lattice gauge theories). Let us construct the lattice realization of a qform topological gauge theory that corresponds to a topological sigma model with target space B q G. Flat q-form connections (dubbed flat G [q] -connections) can have non-trivial q-holonomies along non contractible closed q-paths only. Therefore, a flat G [q] -connection can be defined as a homomorphism from the q-th homotopy group π q (M) to G. Locally, this means that a flat G [q] -connection is fully characterized by a q-cochain valued in G satisfying dg = 0, with 0 ∈ G the unit element. In practice, we assign to every q- where the notation• indicates that the corresponding vertex is omitted from the list. Such a labeling is referred to as a G [q] -coloring and the set of Note that a (q−1)-form gauge transformation is defined as a gauge parameter φ which acts on such colorings as The topological action is provided by pulling back a class representative in a cohomology class ). The partition function finally looks like In the following sections, we focus our attention on 2-form topological gauge theories and their lattice realization as defined in the previous example. In particular, in sec. 4, we will carefully build the cohomology group H d+1 (B q G, R/Z) for the case q = 2 so as to provide a more explicit expression for (2.16) which can be used to construct a lattice Hamiltonian realization of this topological theory. As before, this lattice construction can be readily generalized to sigma models whose target space is provided by a Postnikov tower (see app. A).

2-form topological action
Let us explore in more detail topological sigma models that have a 2-form gauge theory interpretation.
In particular, we wish to emphasize the role played by the classification of the relevant cohomology group in terms of quadratic forms.
We explained above how 2-form topological gauge theories for a finite abelian group G can be built as topological sigma models with target space the second classifying space X = B 2 G of G. Homotopy classes of maps [M, B 2 G] can be labeled by B ∈ H 2 (M, G) and the homotopies of these maps are gauge transformations B ∼ B + dφ where φ ∈ C 1 (M, G). The partition function is provided by (2.7) ∼ 11 ∼ which we repeat below: (2.17) Restricting to (3+1)d, the topological actions are classified by [ω] ∈ H 4 (B 2 G, R/Z). But, since H 3 (B 2 G, Z) = 0, we may write is known as the universal quadratic group for G [10,11]. Before stating the defining property of Γ(G), let us first recall the definition of a quadratic form: Conversely, any lattice with a symmetric bilinear form b defines a quadratic form via q(x) := 1 2 b(x, x). Furthermore, it can be checked that the value of b and q on the generators of G completely determine these forms.
The universal quadratic group Γ(G) is uniquely defined by the property that any quadratic function q : G → R/Z may be written as the composition q = q • γ where γ : G → Γ(G) and q ∈ Hom(Γ(G), R/Z). For instance, the universal quadratic group of Z n is Γ(Z n ) = Z n or Z 2n for n an odd integer or an even integer, respectively. The universal quadratic group of any finite abelian group of the form G = ⊕ I Z n I is then where gcd(n I , n J ) is the greatest common divisor of n I and n J . It was shown by Eilenberg and MacLane [11] that the cohomology group H 4 (B 2 G, R/Z) is isomorphic to the group of quadratic functions. Following the above discussion, the topological action in (2.17) can thus be defined as the composition of a canonical quadratic operation P : H 2 (M, G) → H 4 (M, Γ(G)) known as the Pontrjagin square, with a homomorphism q from Γ(G) to R/Z, i.e. ω(B) ≡ q * P(B) ∈ H 4 (M, R/Z) (see app. B and [12,14,16] for more details). The form of the topological action ω(B) ≡ q * P(B) ∈ H 4 (M, R/Z) naturally depends on a choice of homomorphism q ∈ Hom(Γ(G), R/Z). Since the universal quadratic group for Z n depends on whether n is even or odd, without loss of generality let us write our gauge group G = I Z n I such that n I is even if I ≤ K and odd for I > K. Then, for a field configuration B I ∈ Z 2 (M, Z n I ) the action takes the form It can be checked that P(B I + dλ I ) − P(B I ) d = 0 (mod n I or 2n I ) when n I is odd or even, respectively. There is also a 2-form global symmetry B I → B I + β I where β I ∈ Z 2 (M, Z n I ) and Sq 2 (β I ) = 0. 12 The partition function for the above topological gauge theory was computed in [14,16] for the case where M has vanishing torsion in all its homology groups. 13 In that case, we may write where h a is a basis element in H 2 (M, Z). The topological action evaluates to where Tor refers to the torsion subgroup. If we consider the simpler case of isomorphism classes of 1-form Zn-bundles that fit in the sequence where b is the Bockstein map, we may evaluate the topological action 1 n M A dA on a manifold with non-vanishing torsion such as the lens space L(n, 1) for which H 1 (L(n, 1), Z) = Zn. Then, one has 1 n L(n,1) where b I ∈ (Z/2n I Z) b2(M) when n I is even and b I ∈ (Z/n I Z) b2(M) when n I is odd. The object I defined as (I) ab = M h a h b is the intersection pairing in H 2 (M, Z). The partition function finally reads (2.28) The topological theories defined so far are all constructed from finite groups. As such, these theories are naturally defined in the discrete on a lattice. However, it is often desirable to have a continuous formulation of a theory. Such a formulation, if it exists, may give one access to powerful and sometimes familiar tools of quantum field theory. It turns out that topological gauge theories for finite abelian groups can be naturally embedded into continuous toric gauge theories. The simplest example of this statement is the Z 2 topological gauge theory in (d+1)-dimensions, or equivalently the G = Z 2 Dijkgraaf-Witten theory with a trivial cohomology class in H d+1 (BG, R/Z). 14 The continuous topological gauge theory that embeds Z 2 gauge theory is the BF theory described by the action where A is a 1-form U(1) gauge field, B is a (d−1)-form U(1) gauge field, and d dR is the usual exterior derivative on differential forms. One obtains the Z 2 gauge theory by simply integrating over B in the path integral. Indeed, integrating over the globally defined field configurations imposes that d dR A = 0, i.e A is a locally flat U(1) connection while summing over the topological sectors (monopole configurations) of B imposes that the holonomies of A are Z 2 quantized. This makes A a Z 2 gauge field and reduces the BF theory to a cohomologically trivial Z 2 gauge theory. Such formulations of (3+1)-dimensional Dijkgraaf-Witten theories in terms of (muli-component coupled) BF theories have been studied at length in recent years [54][55][56][57][58]. Next we discuss embedding the above finite gauge theory into a continuous topological gauge theory built from toric U(1) 1-form and 2-form gauge fields. See for example [14,59] for earlier works studying this theory. For the above parameters {p I , p IJ }, the continuous action takes the form where lcm(n I , n J ) is the lowest common multiple of n I and n J . The partition function evaluated for (2.30) matches with (2.28). This can be shown quite explicitly, at least for manifolds with vanishing torsion: Integrating over A I enforces B I to be flat with holonomies on closed non-contractible surfaces restricted to integer multiples of 1/N I . In other words, B I ∈ Hom(H 2 (M, Z), Z N I ) which is simply a flat 2-form Z N I -bundle. But this continuous formulation of the 2-form gauge theory has an interesting gauge structure due to the presence of the cohomological twist. The conserved charges (or Gauß operators) that generate the gauge transformations take the form where λ I are circle-valued scalars and θ I are 1-form fields. Both these gauge transformations have quantized periods, i.e dλ I ∈ Ω 1 Z (M) and dθ I ∈ Ω 2 Z (M). 15 Hence we see that embedding the discrete 2-form theory into a continuous theory indeed has a non-trivial effect on the gauge structure. Numerous properties of the topological action (2.30) are reviewed in app. C. In the next section, we study in detail the configuration space of 1-form and 2-form U(1) gauge theories before returning to the discussion of strict 2-group bundles in sec. 3.3 that embeds a finite group bundle and also encodes the non-trivial cohomological twist.

Deligne-Beilinson cohomology and higher gauge theory
In this section, we describe the configuration space of twisted 2-form gauge theory for a finite abelian group G. As described above such 2-form gauge theories can be embedded into U(1) gauge theories that involve both 1-form and 2-form U(1) gauge fields. However, these different fields transform under gauge transformations in an unconventional way. In order to have a better understanding of this formulation, it is necessary to have a systematic understanding of the configuration space of gauge inequivalent configurations. Here we present such an understanding using the technology of Deligne-Beilinson (DB) cohomology [40]. An alternative approach is provided by Cheeger-Simons differential cohomology [52,60,61] that may be employed to systematize the configuration space of q-form U(1) gauge theory. The two approaches of DB cohomology and Cheeger-Simons differential cohomology are equivalent [62] however in this work we stick to the former. In order to be self-consistent we begin by assembling the necessary ingredients to describe q-form U(1) connections using DB cohomology [41,42]. 15 We use the notation Ω p Z (M) to denote the space of q-forms with integer periods on any p-cycle L (p) ∈ Zp(M, Z) i.e for some ξ ∈ Ω p Z (M), L (p) ξ ∈ Z. ∼ 15 ∼

Preliminaries and definitions
Let us briefly revisit the physical understanding of a 1-form U(1) connection. Locally a 1-form connection A is simply a 1-form field. There is an equivalence relation related to gauge transformations which are redundancies of the physical description. These gauge transformations act as A → A + dλ where dλ ∈ Ω 1 Z (M). Hence the gauge invariant information is encoded in holonomies Furthermore, for topologically non-trivial bundles, i.e those with non-vanishing Chern number, there is no globally defined 1-form connection. Instead, one has to work with a field strength F ∈ Ω 2 Z (M). On contractible patches, the field strength and holonomies agree via where ∂ −1 L (1) is a surface that bounds L (1) . We shall now see that all this data fits neatly together into the Deligne-Beilinson cohomology group. In order to do so, we need first to introduce the basic notions of oriented open cover,Čech-de Rham bicomplex and polyhedral decomposition: We denote overlaps of sets as The index of U i0i1i2...ip is referred to as theČech index of this intersection and p ∈ Z as theČech degree. We only consider overlaps whose indices are ordered i.e i 0 < i 1 < · · · < i p and refer to U as an ordered cover of M. Let the collection of all non-vanishing overlaps of ordered (p+1)-open sets be denoted by U p . Since M is compact, the cardinality of U p and of U is finite.
We denote by Ω r (U p ) the space of de Rham r-forms assigned to all elements in U p and µ r p ∈ Ω r (U p ) a generic element. The quantity n p ∈ Map(U p , Z) =: Ω −1 (U p ) denotes an assignment of integers to all elements of U p . One can define two independent differential operators that act on µ r p , namely the de Rham differential d r dR and theČech differential d p that satisfy the properties d r+1 dR • d r dR = 0 and d p+1 • d p = 0. The action of d r dR is simply given by the exterior derivative that acts locally on each open set, while theČech differential acts as   TheČech-de-Rham bicomplex is a bicomplex of cochains Ω r (U p ) labeled by two indices r and p which are the de Rham andČech degrees, respectively. The maps between cochains are provided by d r dR and d p as described above. Furthermore, we define a completion of the de Rham complex via the differential d −1 dR : Ω −1 (U p ) → Ω 0 (U p ) where d −1 dR is simply the injection of integers into the space of (constant) functions.
Let Z p (M, Z) denote the space of oriented p-cycles in M. In order to integrate p-cochains on M over p-cycles, we need to introduce the notion of polyhedral decomposition: We define a boundary map ∂ whose action reads The boundary operator further acts as i0i1i2 ⊂ U i0i1i2 . This process is iterative and after k iterations, we obtain where as before l Note that some of the entries in this sum vanish (e.g. l i0i2i1 ) since we only consider an ordered cover.
In the following, we work with four-manifolds, therefore we do not need to iterate this procedure defined above more than four times. It is important to note that it is always possible to find a good open cover with respect to which a given p-cycle admits a polyhedral decomposition. Let us consider a few simple examples to illustrate the previous definition: The boundary operator acts as In the third equality, we used the fact that we are working with an ordered cover, therefore 0-chains of the form l  (3.10) It is easy to check that ∂L (2) = 0 as it should be.
We now have all the ingredients to introduce theČech-de Rham construction of Deligne-Beilinson (DB) cohomology: We call a DB q-cochain a (q+2)-tuple of data of the form: and denote the space of DB q-cochains by C q DB (M, Z). We define two differential operators where the first index in the subscript is meant to denote the degree of DB cochain that the given codifferential operator acts on, and the second index denotes the maximum de Rham degree in the image of the given operator. It can easily be checked that D (q,q) • D (q−1,q) = 0. A DB q-cocycle is defined as a DB q-cochain in the kernel of the operator D (q,q) , while a DB q-coboundary is a q-cochain in the image of D (q−1,q) . We may then define the q-th Deligne-Beilinson cohomolgy as the following quotient 14) The DB cohomology as defined above has degree one lower than corresponding differential cohomology defined for example in [52,61]. Here, we follow the conventions of [41,42].

Configuration space for q-form U(1) connections
We defined above the Deligne-Beilinson cohomology of cochains on aČech-de-Rham bicomplex. We will now use this technology in order to define the configuration space of q-form connections. Below, we illustrate this construction with a couple of examples of U(1) connections at low form degree and check that they are indeed described by DB cohomology classes. But, before getting to this we provide some intuition about why this somewhat intricately defined cohomology group is isomorphic to the space of gauge inequivalent configurations of U(1) fields. A q-form U(1) connection is usually defined by specifying q-forms on open sets. However, for topologically non-trivial bundles, it is not possible to describe a connection via a globally defined q-form, in which case one works with a covering of open sets with representatives of the connection defined locally as q-forms on each of the open sets. On overlaps of open sets these q-forms need to be glued together via (q−1)-form gauge transformations. The (q−1)-form gauge transformation fields in turn are only defined on double overlaps of open sets and not globally. A gluing condition needs to be provided for them on triple overlaps via a (q−2)-form gauge field. This process continues iteratively until a specification of integers on (q+1)-overlaps of open sets and finally the consistency condition for this specification requires that the oriented sum of these integers must vanish on the corresponding overlap of q + 2 open sets. All this data defined on open sets as well as overlaps of open sets at various degrees can be succinctly described as a DB q-cochain. Furthermore, the various gluing conditions are nothing but the statement that the DB cochain must actually be a DB cocycle. Finally, there are some redundancies in this description that can very naturally be understood as the image of a DB codifferential operator acting on the space of DB (q−1)-cochains. Upon modding out by this redundancy, what we obtain are the isomorphism classes of gauge inequivalent q-form U(1) fields on M but defined as such this is nothing but the q-th DB cohomology group. We illustrate this idea through a few simple examples. Let us first consider the case of 1-form connections: . DB 1-cochains are defined by the data A ≡ (µ 1 0 , µ 0 1 , n A 2 ) where µ 1 0 are 1-forms defined on local contractible patches, µ 0 1 are functions defined on overlaps of open sets and n A 2 are integers defined on double overlaps. As described above, this is precisely the data one requires to build a connection for a 1-form U(1) bundle. All this data can be glued together by imposing that D (1,1) A = 0. This cocycle condition implies It remains to quotient by the redundancies which physically correspond to 0-form gauge transformations and mathematically correspond to DB 1-coboundaries.
which are nothing but 0-form U(1) gauge transformations. For completeness, we can check that where the third line follows from the fact that C 1 DB (M, Z) ⊂ ker(d 2 ). It is well-known that the field strength of a U(1) connection is quantized to have integer periods. This can be readily checked: we can use d 1 dR µ 1 0 as local representative of the field strength. Let the field strength corresponding to a connection A be denoted by F A . Then on an open set U i we may write the local representative of the field strength as (F A ) i0 := d 1 dR µ 1 0,i0 . Given a 2-cycle L (2) together with a polyhedral decomposition, we obtain which is obviously the expected quantization of field strength. Note finally that given a 1-cycle L (1) together with a polyhedral decomposition, the holonomy of A along L (1) takes the form which is invariant under (0-form) gauge transformations.
∼ 20 ∼ Following exactly the same steps, we define 2-form connections: . Similar to the case of 1-form connections, this is precisely the data one needs to construct/describe a 2-form U(1) connection in the most general case. However, in order to glue all this data together correctly we need to impose that B is in the kernel of D (2,2) . Writing D (2,2) B = 0 explicitly, we get It remains to quotient by 1-form gauge transformations which in the context of the DB construction implies modding out by coboundaries in the image of We could check explicitly that . Similar to 1-form connections, the field strength of a 2-form U(1) connection satisfies a generalized Dirac quantization condition which means that the monopole charge is integer quantized, i.e. M F B ∈ Z. This can be demonstrated explicitly using (d 2 ν 2 0 ) i0 as a local representative of F B on an open set U i . Given a 3-cycle L (3) together with a polyhedral decomposition, we obtain indeed Note finally that given a 2-cycle L (2) together with a polyhedral decomposition, the (2-)holonomy So the space of q-form connections is equivalent to the space of equivalence classes in the q-th DB cohomology H q DB (M, Z), as illustrated above for the q = 1, 2 cases. We say a connection A (q) ∈ H q DB (M, Z) is flat if it lies in the kernel of the D (q,q+1) operator and thus we have the following isomorphism: This follows from the fact that a U(1) q-form connection needs to satisfy a single extra constraint in order to be in the kernel of D (q,q+1) that is Hence the curvature of the q-form connection vanishes locally on each open set.

Strict 2-group connections
Having described the space of gauge inequivalent configurations of higher form U(1) gauge theories in terms of Deligne-Beilinson cohomology, in this subsection we explore a scenario where the group bundle is a non-trivial product of bundles corresponding to 1-form U(1) connections and 2-form U(1) connections. Here by non-trivial product we mean that locally the data required on open sets, overlaps of open sets and so on is identical to that of a direct sum of some number of 1-form connections and 2-form connections. However, the gluing relations which were previously related to certain DB cocycle conditions are twisted in a way that we make precise below. Also, the redundancies or gauge transformations which were related to DB coboundaries are altered accordingly. This is the relevant situation when discussing the embedding of a finite group 2-form gauge theory into a toric gauge theory. That particular field theory (2.30) is the motivation for this subsection. By constructing the Gauß operators and the gauge transformations within this theory, we inferred that these transformations correspond to those of a toric strict 2-group bundle. Below we first briefly describe strict 2-groups and then carefully construct the corresponding strict 2-group bundles.
A strict toric 2-group [36,37,39] is defined by four pieces of data, namely G = U(1) P , U(1) Q , t, where This data needs to satisfy some consistency conditions which ensure that t and interact well with one another. 16 The consistency relations for some A ∈ U(1) Q and B ∈ U(1) P are t(A B) = t(B) and t(B) B = B . In the following, we choose = id. A homomorphism t may be written as 16 These consistency relations make G equivalent to a crossed module. For details please see [36] and references therein.
where I ∈ 1, . . . , P , J ∈ 1, . . . , Q and B ≡ (h 1 , . . . , h P ) ∈ U(1) P . In order to build a G-bundle, we require local data which corresponds to P 2-form U(1) connections and Q 1-form U(1) connections. Therefore, the local fields are Q DB 1-cochains A I and P DB 2-cochains B J : Henceforth, in order to keep the notation light, we specialize to the case P = Q = 1 which can be readily generalized to P, Q ∈ Z. Although the local data corresponds to a direct sum of an ordinary 1-form and 2-form U(1) gauge theory, the gluing (cocycle) conditions and gauge transformations are twisted by the homomorphism t ∈ Hom(U(1), U(1)) Z. We note that since U(1) R/Z fits in the canonical exact sequence 0 → Z → R → R/Z, the homomorphism lifts to t ∈ Hom(R, R) and t ∈ Hom(Z, Z). 17 Hence the homomorphism acts on all the local data of theČech-de Rham bicomplex. This is an essential ingredient in writing consistent gluing relations. The Let us look at the above gluing conditions a bit more closely. For example the 1-form connection A involves an assignment of (µ 1 0 ) i on open sets U i . On the overlap U i0i1 of two open sets U i0 and U i1 the local 1-form representatives are glued together by imposing Hence the gluing condition for the 1-form connection has been altered by the presence of the 2-form connection. Similarly, the gluing conditions on overlaps of all degrees are modified. In other words we need to impose that all the parenthesis in (3.29) vanish independently. Furthermore, this data is defined up to the following gauge transformations where Λ ∈ C 1 DB (M, Z) and Θ ∈ C 2 DB (M, Z). Note that (3.31) is nothing but (2.32) written more precisely in terms of the Deligne-Beilinson data. More explicitly, in terms of the local data, the former equivalence reads while the gauge transformations for B are the same as those for ordinary 2-form U(1) connections (3.21). We can readily check that D t (1,1) • D t (0,1) = 0 so that one may define an affine cohomology theory. The space of gauge inequivalent configurations of a strict 2-group G are isomorphic to this cohomology space that we denote by H 2,1 G (M). 17 We use 't' for the lifted homomorphisms as well in order to keep the notation light.
Having defined H 2,1 G (M), we then consider the subspace of flat connections. This will be important in what follows as it is the configuration space of topological G-gauge theories.
which, in terms of the local data, translates into It is easy to check that the flatness condition is preserved under the gauge transformations (3.31). Indeed, where we made use of the fact that . We now want to compute the integral of the curvature of the 2-group connection and reading off whether it satisfies any quantization conditions. First of all, we can immediately infer that since the gauge transformations of B are unaltered compared to the case of the 2-form gauge theory previously studied, the quantization condition also remains unaltered, i.e The situation is different as far as the curvature F A is concerned. Let us first try to construct a local representative of F A . The simplest possibility is d 1 dR µ 1 0 . Doing so, we realize that Using this representative, we may integrate the curvature over a closed 2-cycle L (2) Hence the field strength of a strict 2-group connection is not quantized but rather, as expected, the quantization is shifted by the holonomy of B.
The line operators are a bit more subtle since the naive guess (3.19) is not gauge invariant. Furthermore, a Wilson line can only be defined for homologically trivial 1-cycles in order to be (2-group) gauge invariant. Instead, the gauge invariant operator takes the form where ∂ −1 L (1) is a 2-chain whose boundary is L (1) . The corresponding polyhedral decomposition can be obtained by attaching a single disc-like region to the 1-cycle L (1) . Let us first focus on the l.h.s term of (3.42) whose integrand only depends on A. As mentioned earlier, the integral of A over L (1) is not invariant under gauge transformations by itself due to the modified gauge structure. Indeed, under gauge transformations one has exp 2πiQ The piece of data on the r.h.s that depends on B requires a bit more care. We attach a disc-like region to L We may now integrate B ∈ C 2 DB (M, Z) over ∂ −1 L (1) and write how it is modified under gauge transformations exp 2πiQ This confirms that (3.41) and (3.42) are the gauge invariant operators for a strict 2-group toric gauge theory. To conclude, we have shown above that the gauge transformations for the continuous topological gauge theory (2.30) correspond to a strict toric 2-group bundle. Furthermore, such a bundle can be defined rigorously using methods based on Deligne-Beilinson cohomology. Above, we constructed such a bundle, studied the quantization conditions for its topological sectors and constructed gauge invariant functions (operators in the quantum theory) in terms of local data.
Using the same technology, it is possible to write down rigorous actions for higher-form topological phases in terms of Deligne-Beilinson cocycles. Some explicit examples are provided in app. D. Note that this construction can also be adapted in order to describe flat connections for weak 2-group bundles and more generally for models built from Postnikov towers.
In the previous sections, we explored properties of higher-form topological theories with a special emphasis on 2-form topological gauge theories. More specifically, we explained how these models could be defined as sigma models with target space the classifying spaces B q G of a finite group G. Since these theories are defined in terms of finite groups, they are naturally defined in the discrete. However, it is also possible to embed them in the continuum. We explained that in the case of 2-form gauge theories, the corresponding continuous theory is based on a strict 2-group. We defined the space of gauge inequivalent q-form U(1) connections as the space of q-th Deligne-Beilinson cohomology classes. We then used this result in order to provide a rigorous definition of the strict 2-group continuous theory.
We shall now investigate the same theories but on the other end of the spectrum, namely in terms of Hamiltonian lattice realizations. More specifically, we are going to define the lattice Hamiltonian realization of a 2-form topological theory for a finite abelian group. To do so, we need a more explicit definition of the cohomology group H 4 (B 2 G, R/Z), which in turn requires a better understanding of the space B 2 G. But since the second classifying space B 2 G is an example of Eilenberg-MacLane space K(G, 2), we shall present the general theory of Eilenberg-MacLane spaces defined as abelian simplicial groups and present their so-called W -construction. This can in turn be used to define the so-called 2form cohomology that is identified with the cohomology of the classifying space B 2 G as provided by this W-construction. All the results presented in the following sections will then follow from the properties of the 2-form cohomology. Interestingly, the classification of the cohomology group H 4 (B 2 G, R/Z) in terms of quadratic forms also plays a very important role on the lattice.

Eilenberg-MacLane spaces
This section provides some review material for algebraic topology in order to motivate the construction of the 2-form cohomology presented in the next section. More specifically, we review the algebraic structure of Eilenberg-MacLane spaces that are defined as: Let q ∈ N and G a group (abelian if q ≥ 2), then an Eilenberg-MacLane space K(G, q) is a connected topological space such that π q (K(G, q)) G and π n (K(G, q)) 0 if n = q, where π n denotes the n-th homotopy group.
Eilenberg-MacLane spaces satisfy the following fundamental property: Therefore, we will often abusively refer to any Eilenberg-MacLane space K(G, q) as K(G, q) and, in particular, we identify thereafter the q-th classifying space B q (G), which is a space K(G, q), with K(G, q). There exists different constructions of Eilenberg-MacLane spaces [10,11,45,[63][64][65][66]. In this paper, we define them as simplicial abelian groups and we focus specifically on the so-called W -construction. This is the formulation we will use in sec. 5 in order to define the 2-form cohomology.

Abelian simplicial groups
Let us first present the general definition of an abelian simplicial group and then illustrate it by constructing the space K(G, 1), with G a finite group. An abelian simplicial group can be succinctly defined as a simplicial object in the category of abelian groups [65]. Nevertheless, we provide below a more explicit definition. Let us first introduce the notion of simplicial set: Definition 4.2 (Simplicial set). A simplicial set X is a collection {X n } n∈N of sets, together with homomorphisms subject to the identities The maps ∂ i and η i are referred to as face and degeneracy operators, respectively. The elements of X n are usually referred to as n-simplices and i, j in the equations above label the faces of these simplices. Given a simplicial set X, we define the boundary map ∂ = ∂ (n) : X n → X n−1 as ∂ (0) ≡ 0 for n = 0, and for n > 0 as From the identity (4.3) follows the usual rule ∂ • ∂ ≡ 0. The simplest example of simplicial set is provided by the standard n-simplex : Let us first define an n-simplex n as the smallest convex set in R n containing n + 1 points denoted by v 0 , . . . , v n such that they do not lie in a hyperplane of dimension less than n. The points v i are 0-simplices and are identified with the vertices of the n-simplex. In the following, we denote such n-simplex by (v 0 . . . v n ). Furthermore, the vertices are endowed with an ordering which induces an orientation of the edges (v i v j ), i < j, according to increasing subscripts. We then define a face of an n-simplex n as a subsimplex defined by the vertices which form a subset of {v 0 , . . . , v n }. The i-th face of the n-simplex can be defined as the image of the map ∂ i such that where the notation• indicates that the corresponding vertex is omitted from the list. The oriented boundary of an n-simplex is then obtained as the image of the operator ∂ defined according to (4.8) as Furthermore, the i-th degenerate simplex of an n-simplex is obtained as the image of the map η i defined as The set of n-tuples (v 0 . . . v n ) together with the face and degeneracy maps introduced above naturally form a simplicial set that is referred to as the standard n-simplex.
We can now straightforwardly define a simplicial group: A simplicial group is a simplicial set X such that each X n is a group and the degeneracy and face operators are homomorphisms between them. If all the X n are abelian, then X is an abelian simplicial group.
Given a simplicial group X, since the face and degeneracy maps are group homomorphisms, the boundary map ∂ defined as in (4.8) is also a homomorphism. Therefore, together with the property ∂ • ∂ ≡ 0, the simplicial group X defines a chain complex with chain groups {X n } n∈N [63]. This last remark is the main reason why the study of Eilenberg-MacLane spaces, which are examples of abelian simplicial groups, is relevant to group cohomology and its generalizations.

Classifying space BG
Let us now illustrate the concepts introduced above with the construction of the classifying space BG ≡ B(G) of a finite group G, which is an Eilenberg-MacLane space K(G, 1). We follow an admittedly minimal (but hopefully pedagogical) approach to define such classifying space, however this is enough for the purpose at hand. More details can be found in [10,11,45,[63][64][65][66].
The construction of the classifying space B(G) mimics the construction of the standard n-simplex such that the n-simplices are now abstract simplices whose vertices are labeled by group variables: Definition 4.4 (Classifying space). Let G be a finite group and E(G) the simplicial set such that E(G) n = G n+1 . The n-simplices of E(G) are therefore identified with the ordered (n+1)-tuples (g 0 , . . . , g n ), with g i ∈ G. The boundary of an n-simplex (g 0 , . . . , g n ) reads (4.12) ∼ 28 ∼ and the i-th degenerate simplex of an n-simplex reads η i (g 0 , . . . , g n ) := (g 0 , . . . , g i−1 , g i , g i , g i+1 , . . . , g q ) . (4.13) The group G has a left action on E(G) by left multiplication such that for all g ∈ G, g (g 0 , . . . , g n ) = (gg 0 , . . . , gg n ) . (4.14) The classifying space B(G) of G is finally defined as the quotient space By definition, the n-simplices of B(G) satisfy the equivalence relation (g 0 , . . . , g n ) ∼ (gg 0 , . . . , gg n ) which implicitly identifies all the 0-simplices (or vertices) of B(G) so that it only contains a single 0-simplex, namely (g). The presentation of B(G) as constructed above is sometimes referred to as the homogeneous one as opposed to the non-homogeneous one that we will now present. Let us consider n-tuples [g 1 , . . . , g n ] of elements g i ∈ G. To each such tuple, we associate an n-simplex of the simplicial group B(G) as follows where 1 denotes the group identity. Conversely, to each n-simplex (g 0 , . . . , g n ), we can assign an n-tuple according to which provides a one-to-one correspondence between the n-simplices (g 0 , . . . , g n ) and the n-tuples [g 1 , . . . , g n ]. In the following, we regard each n-simplex of B(G) as such an n-tuple so that ordered products of g i variables label the 1-simplices of B(G). It is straightforward to see that the action of the boundary map ∂ can now be rewritten . . , g n ] + (−1) n [g 1 , . . . , g n−1 ] .
(4.17) Since the definition of the classifying space mimics the one of the standard n-simplex, it is easy to see how we can represent geometrically the relations presented above by drawing simplices and labeling their edges with group variables and product of group variables (when working with the nonhomogeneous presentation). Note that given a 2-simplex, the oriented product of the group variables labeling its boundary 1-simplices is always equal to the identity, 18 hence the correspondence with (1form) flat connections on the lattice. The fact that we can represent n-simplices of B(G) graphically will turn out to be very useful in the following when dealing with more complex formulas.
As alluded earlier, a simplicial group together with a boundary homomorphism satisfying ∂ • ∂ ≡ 0 forms a chain complex whose chain groups are given by G n in the case of B(G). Consequently, we can 18 This follows directly from the definition of the non-homogeneous presentation. Let us consider the 2-simplex [g 1 , g 2 ]. By applying the boundary map (4.17), we obtain ∂ (2) [g 1 , g 2 ] = [g 2 ] − [g 1 g 2 ] + [g 1 ], which informs us that the 1-simplices bounding [g 1 , g 2 ] are labeled by g 2 , g 1 g 2 and g 1 , respectively. Note that the orientation of [g 1 g 2 ] is opposite to the one of [g 1 ] and [g 2 ] so that the oriented product is indeed g 2 · g −1 2 g −1 think of an n-tuple [g 1 , . . . , g n ] as an n-chain which we choose to be valued in a G-module A, defining the space of n-chains C n (B(G), A). We obtain the dual cohomology by defining an n-dimensional cochain over A as a function which associates to each n-simplex of the simplicial group an element of A, so that an n-cochain can be thought of as a function of n variables on G valued in A, and by dualizing the boundary operator. The resulting cohomology turns out to be identified with the so-called group cohomology whose definition is recalled below so that algebraic cocycles on G are equivalent to simplicial cocycles on B(G): Definition 4.5 (Group cohomology). Let G be a finite group and A a G-module which is an abelian group. The group G has an action on A which commutes with the multiplication rule of A. We call an n-cochain a function ω n : G n → A and we denote by C n (G, A) the space of n-cochains. We define the coboundary operator d (n) : where we chose to write the product rule in A multiplicatively. An n-cocycle is then defined as an n-cochain that satisfies d (n) ω n = 1 . We refer to (4.19) as the group n-cocycle condition and the subgroup of n-cocycles is denoted by Z n (G, A). Given an (n−1)-cochain ω n−1 , we define an n-coboundary as an n-cocycle of the form The subgroup of n-coboundaries is denoted by B n (G, A). We finally construct the n-th (group) cohomology group as the quotient space of n-cocycles defined up to n-coboundaries: It turns out that the construction of the classifying space B(G) as presented above is not confined to finite groups. More precisely it is possible to generalize it so as to assign to any abelian simplicial group X a classifying space B(X). There exist several such generalizations that yield different simplicial groups but whose cohomology groups are isomorphic. The most well-known generalizing procedure is referred to as the bar construction [10] of a simplicial group. In the case where X is chosen to be a finite abelian group, B(X) is itself an abelian simplicial group so that the procedure can be iterated. More precisely, starting from the canonical simplicial group constructed out of a finite abelian group G, and under this bar construction, it is possible to define a q-th classifying space B q (G) of G recursively as which is an Eilenberg-MacLane space K(G, q), i.e.
π n (B q (G)) = π n−1 (B q−1 (G)) = . . . = π n−q (G) = G if q = n 0 otherwise . (4.23) We do not expose the details of this bar construction here since it does not serve our purpose well. Instead, we will make use of an alternative construction which we now present.

W -construction
In this section, we introduce a construction where the Eilenberg-MacLane space B q+1 (G) (or K(G, q + 1)) is obtained recursively from B q (G) via a uniform process denoted by W . This construction is different from the bar construction but they both yield homologically equivalent complexes. It provides us with a specific presentation for the n-cycles of the homology group H n (B q (G), A) which, after dualization, will be used to define what we will call the q-form cohomology group H n (G [q] , A). This cohomology group is identified with the cohomology group H n (B q (G), A) the same way as the cohomology of a finite group is identified with the cohomology its classifying space. 19 The reason why this W -construction is more relevant than the bar construction is because the corresponding ncochains are naturally defined as functions of a certain number of variables that is equal to the number group variables necessary to define a flat q-form connections on an n-simplex. More specifically, under this construction a 2-form n-cochain depends on n(n−1) 2 group variables as expected from the study of 2-form flat connections. 20 In general, given a simplicial group X, we define a new simplicial group denoted by W (X) via the recursive formulas where denotes the single element of W (X) 0 and W (X) n the set of n-simplices of W (X). It is also possible to define W (X) n directly without recursion. Indeed, we have W (X) n = X n−1 ⊗ X n−2 ⊗ · · · ⊗ X 0 . (4.25) We denote the elements of an n-fold product by such that x i ∈ X i . Furthermore, the face and degeneracy operators W (X) n satisfy the following properties ∂ n x n−1 , . . . , x 0 = ∂ n−1 x n−1 , ∂ n−2 x n−2 , . . . , ∂ 1 x 1 η 0 x n−1 , . . . , x 0 = 1 n−1 , x n−2 , . . . , x 0 η i x n−1 , . . . , where 1 i denotes the identity element of X i . Regarding B 0 (G) = G as a simplicial group where the n-simplices are identified with the elements g i ∈ G, such that ∂ i g = g = η i g, then W (B 0 (G)) B(G) is the simplicial group whose n-simplices are the n-tuples g n−1 , . . . , g 0 ≡ [g n−1 , . . . , g 0 ] (4.28) 19 Note that the group does not need to be abelian as long as we are only interested in the (first) classifying space and, a fortiori, the group cohomology. 20 Recall that given a finite abelian group G and a manifold M equipped with a triangulation , we defined a flat 2-form connection by assigning to every 2-simplex (012) ⊂ , a group element g 012 = g, (012) such that for every 3-simplex (0123) ⊂ , the cocycle condition dg, (0123) = g 123 − g 023 + g 013 − g 012 = 0 is imposed. It follows that given an n-simplex of and a flat 2-form connection, only n(n−1) 2 variables are independent.

2-form (co)homology
In the previous section, we recalled the construction of the classifying space of a finite group G as a simplicial group, and explained how this becomes a chain complex when identifying the nsimplices with n-chains, from which we can define the cohomology H n (BG, A) of simplicial cocycles that is equivalent to the cohomology H n (G, A) of algebraic cocycles. In this section, we use the Wconstruction of the second classifying space B 2 (G) of a finite abelian group G in order to define the so-called 2-form cohomology H n (G [2] , A) that is the cohomology of algebraic cocycles identified with the cohomology H n (B 2 (G), A) of simplicial cocycles.

Definition and 2-form cocycle conditions
Let G be a finite abelian group and A an abelian group that is a G-module whose product rule is written multiplicatively. We assume for notational convenience that G has a trivial action on the abelian group A. 21 We call a 2-form n-cochain a function ω n : G ( n 2 ) → A. We denote by C n (G [2] , A) the space of 2-form n-cochains. Given a 2-form n-cochain ω ∈ C n (G [2] , A), we make use of the following notation ω(g ( n 2 ) , g ( n 2 )−1 , . . . |g ( n−1 2 ) , . . . | . . . |g 3 , g 2 |g 1 ) ∈ A where the presence of bars | is there to remind of the underlying tensor product structure of the corresponding n-simplices according to (4.25). We find this notation convenient in order to make the algebraic structure more manifest but this can easily be omitted as well.

∼ 33 ∼
In particular, it follows from ex. 4.2, 4.3 and 4.4 the action of the 2-form coboundary operator d (2) : the action of the 2-form coboundary operator d (3) : 5) and the action of the 2-form coboundary operator d (4) : (5.6) respectively, where a, b, . . . , i, j ∈ G. Using the general recursive definition (4.27) together with the correspondence spelled out above, we can then find the defining formula of any 2-form coboundary operator d (n) . Nevertheless, we will only make use of the previous three formulas in this work. A 2-form n-cocycle is then defined as a 2-form n-cochain that satisfies The subgroup of 2-form n-cocycles is denoted by Z n (G [2] , A). Given a 2-form (n−1)-cochain ω n−1 , we define an n-coboundary as a 2-form n-cocycle of the form The subgroup of 2-form n-coboundaries is denoted by B q (G [2] , A). We finally construct the 2-form q-th cohomology group as the quotient space of 2-form n-cocycles defined up to 2-form n-coboundaries: H n (G [2] , A) := Z n (G [2] , A) B n (G [2] , A) = ker(d (n) ) im(d (n−1) ) . (5.9)

Geometric realization
We mentioned above how we can think of a given evaluation of the cocycle ω as a pairing between ω and the corresponding n-simplex identified with the relevant n-chain so that a 2-form n-cocycle assigns to each n-simplex of B 2 (G) an element of the group A. This can be used in order to provide a geometric interpretation to the cocycle conditions presented above. In order to do so, we need the geometric realization of B 2 (G) which is, loosely speaking, obtained by identifying a given n-simplex of B 2 (G) with a standard n-simplex together with a G [2] -coloring, i.e. an assignment of group elements g ∈ G to every 2-simplex such that for every 3-simplex the 2-cocycle constraint dg = 0 is satisfied. Let us for instance consider the 3-simplex g 3 , g 2 , g 1 , ∈ (B 2 (G)) 3 . Recall that its boundary reads so that g 1 , , g 20 + g 1 , , g 21 + g 20 , and g 21 , are 2-simplices. Let us now think about the 3-simplex g 3 , g 2 , g 1 , ∈ (B 2 (G)) 3 as a standard 3-simplex (0123) so that 0, 1, 2 and 3 ∼ 34 ∼ label its vertices, together with a G [2] -coloring which assigns to every face (xyz) the group element g xyz = g, (xyz) . From the expression of the boundary of the 3-simplex g 3 , g 2 , g 1 , , we read-off the correspondence g 123 = g 1 , g 023 = g 2 + g 1 , g 013 = g 3 + g 2 , g 012 = g 3 . (5.11) We can check that this coloring automatically satisfies the cocycle constraint dg = 0 since dg, (0123) = g 123 − g 023 + g 013 − g 012 = g 1 − (g 2 + g 1 ) + (g 3 + g 2 ) − g 3 = 0. So given a 3-simplex of B 2 (G), we can assign to it a standard 3-simplex whose G [2] -coloring is provided by (5.11). Conversely, given a standard 3-simplex (0123) with a given G [2] -coloring, we assign to it a 3-simplex of B 2 (G) which reads g 012 , g 013 − g 012 , g 123 , . Since a 2-form 3-cocycle assigns to a 3-simplex of B 2 (G) an element of the group A, we can use the previous correspondence to further assign to the standard 3-simplex (0123) the evaluation α, (0123) ≡ α(g 012 , g 013 − g 012 |g 123 ) . (5.12) Note that we use a slightly abusive notation since we treat the standard 3-simplex (0123) as g 012 , g 013 − g 012 , g 123 , in light of the identification we have just made. The same procedure can be iterated so as to assign to a given standard n-simplex together with a G [2] -coloring the evaluation of a 2-form n-cocycle. For instance, we will make extensive use of the following correspondence between a 2-form 4-cocycle and the standard 4-simplex (01234): ω, (01234) = ω(g 012 , g 013 − g 012 , g 014 − g 013 |g 123 , g 124 − g 123 |g 234 ) . = π(g 012 , g 013 − g 012 , g 014 − g 013 , g 015 − g 014 |g 123 , g 124 − g 123 , g 125 − g 124 |g 234 , g 235 − g 234 |g 345 ) from which we can easily read-off the structure underlying this construction and therefore 'guess' the subsequent formulas.
Let us now use this correspondence in order to provide a geometric interpretation to the cocycle conditions. It turns out that an n-cocycle condition is associated with a so-called n-dimensional Pachner move [43]. Given a piecewise linear manifold M and its triangulation , a Pachner move replaces by another triangulation associated with a manifold M homeomorphic to M. Given two triangulations of a given manifold, it is always possible to relate one to the other by a finite sequence of Pachner moves. In three dimensions, we distinguish two Pachner moves, namely the 2-3 and the 1-4 move denoted by P 2 →3 and P 1 →4 , respectively. The 1-4 move subdivides a 3-simplex into four 3-simplices by introducing an additional vertex inside, while the 2-3 move decomposes the gluing of two 3-simplices into three 3-simplices. Graphically, this latter move can be represented as where ⊗ represent the fourth vertex as seen from behind the triangle.

Normalization conditions
In the following section, we construct a lattice Hamiltonian model whose input data is a finite abelian group G and a 2-form cohomology class [ω] ∈ H 4 (G [2] , U(1)). Before doing so, it is convenient to derive some normalization conditions for 2-form cocycles. This will allow us to greatly simplify some formulas but also to provide useful information regarding the algebraic structure of the cocycles. The list of normalization conditions presented here may not be exhaustive but these are all the ones we need for our purpose. By definition, [ω] ∈ H n (G [2] , U(1)) is an equivalence class of 2-form n-cocycles defined up to 2form n-coboundaries and in order to represent the class [ω], we can choose any cocycle ω ∈ [ω]. The purpose of this section is, given an equivalence class [ω], to find a representative ω ∈ [ω] which satisfies as many normalization conditions as possible. Since the data entering the definition of our lattice model is a 2-form 4-cocycle, we will focus our attention on the cohomology group H 4 (G [2] , U(1)) but the strategy presented here is very general and can be applied to any 2-form cocycle.

Hamiltonian realization of 2-form TQFTs
In this section, we use the results of the previous section in order to construct and study an exactly solvable model whose ground state is described by a topological lattice 2-form gauge theory. In the next section, we will study how this model turns out to be related to the Walker-Wang model for abelian braided fusion categories.

Fixed point wave functions
One can in general define gapped phases of matter in terms of equivalence classes of many body wave functions under local unitary transformations [32,67]. Given a graph or a lattice, these local transformations can be used so as to implement a wave function renormalization group flow. Defining equivalence classes of wave functions under such transformations then boils down to finding so-called fixed point wave functions. By definition, these fixed point wave functions are expected to capture the long-range entanglement pattern which is the defining feature of intrinsic topological order.
Levin and Wen introduced in [32] so-called string net models as a way to systematically construct fixed point wave functions in two dimensions. A string net is essentially defined in terms of a graph labeled by objects satisfying compatibility conditions such that each graph with a given consistent labeling represents a state (or many-body wave function). The linear superposition of spatial configurations of string nets define the Hilbert space of the model. Local unitary transformations are defined at the level of the graph and uniquely specify the fixed point wave functions. These fixed point wave functions are then found to be ground states of given Hamiltonians.
In this section we follow an approach similar to Levin and Wen to construct an exactly solvable model whose fixed point wave functions define the ground states of a lattice Hamiltonian which has a 2-form gauge theory interpretation. Our setup is the following: Let M be a compact oriented four-manifold and Σ a closed three-dimensional hypersurface equipped with a triangulation . Each 2-simplex 2 = (xyz) ⊂ of this triangulation is decorated by a group element g xyz ∈ G with G a finite abelian group such that the group identity 0 ∈ G is the vacuum sector. Furthermore, we define compatibility conditions referred to as branching rules at every 3-simplex of the triangulation: Given a 3-simplex 3 = (wxyz) ⊂ , the branching rules impose that the oriented product of the super-selection sectors labeling the 2-simplices vanishes, i.e. g xyz − g wyz + g wxz − g wxy = 0. Using the differential on cochains and the canonical pairing between simplices and cochains, this can be rewritten: ∀ (wxyz) ⊂ , dg, (wxyz) = 0. A labeling of the 2-simplices of such that the branching rules are everywhere satisfied is said to be consistent and we refer to it as a G [2] -coloring. The set of G [2] -colorings is denoted by Col(Σ, G [2] ) and defines a local description of the set of flat 2-form connections.
Since the branching rules effectively reduce to the group multiplication, the input data of our lattice Hamiltonian model is particularly simple, namely a tuple {G, [ω]} where G is a finite abelian group and [ω] ∈ H 4 (G [2] , U(1)) a class of 2-form 4-cocycles valued in U(1). Since the model is defined in terms of a 2-form 4-cocycle up to 4-coboundaries, we can choose whichever representative of the class we want to carry out our calculations: We choose it so as to satisfy the normalization conditions where the G [2] -coloring is left implicit. These two equations dictate how a fixed point wave function is modified under a 2-3 Pachner move P 2 →3 and a 4-1 Pachner move P 4 →1 , respectively. The equations associated with the opposite moves, namely P 3 →2 and P 1 →4 , are obtained in an obvious way. Both equation (6.1) and (6.2) depends on a factor ω, (01234) ± (01234) that is the pairing between the 2form 4-cocycle ω and the 2-form chain identified with the standard 4-simplex (01234) ⊂ that is the 4-simplex bounded by the 3-simplices appearing in the corresponding Pachner moves. The sign factor (01234) depends on the orientation of the 4-simplex. Since the convention for the P 4 →1 follows from the one of the P 2 →3 , we only explain the latter one in detail: Convention 6.1 (2-3 Pachner move). Pick one of the 3-simplices in the source complex and assume that it is labeled by (wxyz) such that w < x < y < z. The remaining vertex of the source complex is labeled by o. We denote by (wxyzo) the sign factor associated with the corresponding 2-3 Pachner move. First, determine the orientation (wxyz) of the 3-simplex (wxyz) according to convention 5.1, then consider the list of vertices {o, w, x, y, z}. If it takes an even number of permutations to bring this list to the ascending ordered one, then (wxyzo) = + (wxyz), otherwise (wxyzo) = − (wxyz). The same convention applies to find the orientation of a 4-simplex.
The constraints (6.1) and (6.2) satisfied by the wave functions under local unitary transformations are only valid together with the corresponding consistency conditions. We will show later that these are guaranteed by the fact that ω is a 2-form 4-cocycle, but before doing so we are going to investigate in more detail these local unitary transformations. First of all, let us present an alternative presentation for these mutations which is closer related to the original formulation of Levin and Wen for 2d string net models as well as the one of the 3d Walker-Wang model which we review in sec. 7. Instead of working with the triangulation , we consider the one-skeleton of its dual polyhedral decomposition denoted by Υ such that the 2-simplices 2 ⊂ are dual to links l ⊂ Υ and the 3-simplices 3 ⊂ are dual to nodes n ⊂ Υ. The branching rules or (compatibility conditions or 2-flatness constraints) are now encoded at every node. Because we now work on the dual graph, it is inconvenient to keep the labeling completely implicit, thus we label each link by its dual 2-simplex. Note, however, that it is not strictly necessary to specify explicitly the orientation of each edge since it can be easily deduced from the definition of the constraint dg, (wxyz) = g xyz − g wyz + g wxz − g wxy = 0. In terms of the .
Now is a good time to recall that the pairing between the 2-form 4-cocycle and the standard 4-simplex identified with the corresponding 4-chain (of the simplicial group B 2 (G)) is explicitly given by ω, (01234) = ω(g 012 , g 013 − g 012 , g 014 − g 013 |g 123 , g 124 − g 123 |g 234 ) . (6.5) In the following, we are interested in special cases of the equations above which correspond to setting some of the group variables appearing in (6.5) to the identity. In particular, we study what these special cases reveal about the algebraic structure of 2-form 4-cocycles using arguments similar to the ones presented at the end of sec. 5.3. Let us for instance consider the P 3 →2 move dual to the P 2 →3 move depicted in (6.3) but with a different distribution of vertices and such that g 013 = g 134 = g 234 = g 012 = 0: where the dashed line represents links (or 2-simplices) labeled by the vacuum sector. Furthermore, we set g 014 = g 034 = b and g 023 = g 123 = g 124 = a so that ω, (01234) = ω(0, 0, b|a, 0|0) .
It should be obvious from this presentation that when setting g 013 = g 134 = g 234 = g 012 = 0, the P 2 →3 move effectively reduces to a braiding move. The 2-form 4-cocycle correspondingly reduces to a group 2-cochain R : (a, b) → ω(0, 0, b|a, 0|0) such that where the r.h.s corresponds to the situation where the left strand undercrosses the right one. Note that instead of starting from the P 2 →3 move, we could have considered a special case of the P 4 →1 move instead and it would have led exactly to the same result.
There is another interesting special case of the same P 3 →2 move that is obtained by setting g 124 = g 234 = g 134 = g 123 = 0: 22 .
Let us furthermore set g 012 = a, g 023 = b and g 034 = c so that ω, (01234) = ω(a, b, c|0, 0|0) . (6.9) It should be obvious from this presentation that when setting g 124 = g 234 = g 134 = g 123 = 0, the P 2 →3 move effectively reduces to a 2-2 Pachner move P 2 →2 if we were to consider the two-dimensional triangulation dual to the three-valent graph defined by the bold edges. Indeed, the 2-form 4-cocycle reduces to a group 3-cochain α : (a, b, c) → ω −1 (a, b, c|0, 0|0) such that (6.10) What we have just shown, from a graphical point of view, is how the local unitary transformation associated with a 2-form 4-cochain reduces to a braiding move or a 2-2 Pachner move. In the following subsection, we study the coherence relations of these transformations which allows us to make this correspondence more precise. 22 Note that we use yet another distribution of vertices compared to the one in (6.3). Nevertheless, it obviously does not matter how we choose such labeling and we could have performed the same analysis with any other. However, we always choose the one which makes the evaluation ω, (01234) as simple as possible. In general, because of the inherent redundancy of the algebraic structure underlying the 2-form 4-cocycle, there are many ways to write the same thing. We tried to choose our examples so as to make the results as manifest as possible.

Consistency conditions
We presented earlier the equations that fixed wave functions must satisfy under local unitary transformations. However, for these equations to be self-consistent, some coherence relations must be satisfied. We are now going to study these coherence relations with an emphasis on the correspondence put forward at the end of the previous subsection.
Let us consider the union of three 3-simplices. There are two different sequences of three P 2 →3 moves which lead to the same complex that is the union of six 3-simplices. This situation is represented in fig. 3 where the direction of the arrows is decided according to convention 6.1. According to equation (6.3), the amplitude of the map performing each P 2 →3 move is given in terms of the 2-form 4-cochain ω or its inverse. Applying this definition to the two sequences of P 2 →3 and requiring that any composition of maps yielding the same final configuration must be identified, the coherence relation implies the following equality: ω, (02345) ω, (01245) ω, (01234) = ω, (01235) ω, (01345) ω, (12345) . (6.11) It turns out that this equality is nothing else than the 2-form 4-cocycle condition and fig. 3 its graphical interpretation. The coherence of the local unitary transformation is therefore ensured by the fact that ω ∈ Z 4 (G [2] , U(1)). To check explicitly that this is indeed the cocycle condition d (4) ω(a, b, c, d|e, f, g|h, i|j) = 1 as written in (5.6), we just need to use (5.13) and label the face variables as follows: g 012 = a, g 013 = a + b, g 014 = a + b + c, g 015 = a + b + c + d g 123 = e, g 124 = e + f , g 125 = e + f + g, g 234 = h, g 235 = h + i and g 345 = j. Note that there is another way to interpret the 2-form 4-cocycle condition following the lines of sec. 5.2. Indeed, considering a 3-3 Pachner move and identifying each one of the oriented six 4-simplices with a 2-form 4-cocycle, we would obtain (6.11) as well. This relies on the fact that there is a canonical way to assign a cyclic sequence of five P 2 →2 moves to a P 2 →3 move, a cyclic sequence of six P 2 →3 moves to a P 3 →3 move, and so on and so forth, as exploited in [20].
In the same way we investigated earlier special cases of the P 2 →3 move, we will now study special cases of the cocycle condition d (4) ω(a, b, c, d|e, f, g|h, i|j) = 1. In particular, we are interested in the graphical interpretation of these special cases in light of the correspondence between (6.11) and fig. 3.
Let us now consider the 4-coboundary d (4) ω(0, 0, c, d|e, 0, 0|0, 0|0) which yields the cocycle condition ω(e, 0, 0|0, 0|0) ω(0, c, d|0, 0|0) ω(0, 0, c + d|e, 0|0) ω(e, c, d|0, 0|0) ω(0, c, d|e, 0|0) ω(0, 0, c|e, 0|0) = 1 . In order to obtain this figure, we proceed as follows: (i) Reproduce fig. 3 but for the one-skeleton of the polyhedral decomposition Υ dual to the triangulation and for a different (judicious) numbering of the vertices. (ii) Identify the 2-simplex variables from (6.15) using the correspondence (5.13) and draw the dual links with a dashed line when the corresponding labeling vanishes. Focusing on the bold edges appearing in fig. 6.2, we can draw several remarks: Firstly, as expected, the two P 2 →3 moves associated with the 4-cocycle evaluations normalized to one do not modify the combinatorics of the diagram. Secondly, two P 2 →3 effectively reduce to a braiding move which is consistent with the fact two R-matrices appear in equation (6.16). Thirdly, one P 2 →3 effectively reduces to a P 2 →2 which is We will now repeat the previous analysis starting from a different cocycle condition in order to provide an alternative decomposition for the map c. Let us consider the cocycle condition , ω, (01345) = α(c, h, g) and ω, (01235) = α(c, g, h) −1 . We made use between the first and the second line of the normalization conditions as well as (6.14). As before, this cocycle condition can be represented graphically (see lower panel of fig. 6.2) by identifying all the 2-simplex variables and make a judicious choice of numbering of the vertices which dictates, among other things, which P 2 →3 move each term of (6.18) corresponds to. In terms of string diagrams, this effectively boils down to: 23 c(c,g,h) (6.19) 23 The correspondence between fig. 6.2 and the effective string diagrams is not as obvious as earlier where it was directly provided by the bold links of the dual complex. Indeed, because of the presence of additional links labeled by non-trivial group variables, the correspondence is not quite as transparent. However, by looking carefully at the value of each 2-simplex (or dual link) variables, the reader should be able to convince itself that it does reduce to the equation in terms of string diagrams. First of all, it should be clear that the links labeled by (014) and (024) are irrelevant from a combinatorial point of view in the top-left, top-right and right complexes (of the lower panel). Furthermore, from ω, (01245) = ω(g 012 , g 014 − g 012 , g 015 − g 014 |g 124 , g 125 − g 124 |g 245 ) = ω(0, c, 0|0, g|0), we read off in particular that g 014 = g 015 = c and g 045 = g 145 = g so that we can effectively 'forget' about the links labeled by (014)  where we omitted as before the trivial maps.
Let us summarize what we have shown so far: Using two special cases of the 2-form 4-cocycle condition, together with their geometrical interpretation that relies on the fact that the map performing a 2-3 Pachner move evaluates to a 2-form 4-cocycle, we have obtained two different decompositions for the map c : (a, b, c) → ω(0, a, b|c, 0|0) in terms of the group 2-cochain R and the group 3-cocycle α, both algebraically and geometrically. Equating these two decompositions, we obtain α (a, b, c) R(c, a + b) α(c, a, b) = R(c, a) α(a, c, b) R(c, b) (6.20) which is nothing else than one of the hexagon relations appearing in the definition of a (abelian) braided monoidal category. This equation is the consistency condition for the braiding move whose amplitude is provided by the 2-cochain R, the same way (6.13) is the one for the P 2 →2 move. We can deduce very easily the corresponding graphical interpretation in terms of string diagrams and it reads: It turns out, there is another hexagon equation which can be obtained similarly starting from two others special cocycle conditions: whose graphical interpretation can be derived by proceeding as before.
Thinking of ω, R and α as isomorphisms, what we have essentially shown in this part is that the input data of our model, namely {G, ω}, reduces to the input data of an abelian braided monoidal category, namely {G, R, α}, and that the consistency conditions of the constraints satisfied by the fixed point wave functions under P 2 →3 moves reduce to the consistency conditions of R and α. Note that a similar analysis has been carried out for instance in [68][69][70] but using the bar construction of B 2 (G) instead of the W -construction as we did. The computations are in this case more straightforward. However, because the bar presentation does not yield a geometrical interpretation in terms of 2-form flat connections defined on a triangulation, it is neither possible to show this correspondence from an intuitive simplicial point of view, nor to define the corresponding lattice Hamiltonian model as we are doing. We will provide more category theoretical details in sec. 7 and exploit this result to show to which extent our model is related to the Walker-Wang model. id Figure 4. Graphical depiction of the 2-form cocycle condition d (4) ω(0, 0, c, d|e, 0, 0|0, 0|0) = 1 and d (4) ω(0, 0, c, 0|0, 0, g|h, 0|0) = 1. The dashed line are labeled by the identity group element 0 ∈ G. Each arrow of the diagram is labeled by a 4-simplex (abcde) such that ω, (abcde) is the evaluation of the 2-form 4-cocycle ω that is the amplitude of the corresponding 2-3 Pachner move, as well as a symbol α, R or id depending on whether the 2-3 Pachner move effectively reduces to a 2-2 Pachner move, a braiding move or a trivial move, respectively. Together, these two consistency conditions effectively reduce to a so-called hexagon relation.

Lattice Hamiltonian
In this subsection, we introduce our lattice model which is an Hamiltonian realization of the 2-form topological invariant (2.16) whose ground states are described by the fixed point wave functions satisfying (6.1) and (6.2). These ground states correspond to the physical states (as defined in app. C) that span the Hilbert space H Σ obtained upon quantization of the 2-form TQFT on a manifold of the form M = Σ × R.
We introduced in sec. 2.3 a general formula (2.16) for the partition function of topological q-form lattice gauge theories. We reproduce below this formula for the case q = 2 in a slightly different form: where ω ∈ Z 4 (G [2] , U(1)) is the 2-form 4-cocycle, ω, 4 ( 4 ) ≡ S ω [ 4 ] is the topological action such that ( 4 ) = ±1 is determined according to conv. 6.1, and Col(M, G [2] ) is the set of G [2] -colorings of M, i.e. an assignment of group elements g ∈ G to every 2-simplex which satisfy the 2-cocycle condition dg = 0 where dg, (wxyz) = g xyz − g wyz + g wxz − g wxy . Note that we are now considering cocycles valued in U(1) instead of cocycles valued in R/Z, this is merely a choice of convention. Furthermore, we now pick a cocycle in the cohomology H 4 (G [2] , U(1)) instead of H 4 (B 2 (G), U(1)), which is the same by construction. The 2-form 4-cocycle condition ensures the topological invariance of the partition function.
Let us now define an Hamiltonian realization of this topological field theory on a space-like threedimensional hypersurface Σ endowed with a triangulation whose 2-simplices are labeled by group variables in the finite abelian group G. To every 3-simplex of , we associate a projector B ( 3 ) which enforces the zero-flux condition. To every 1-simplex of , we associate a projector A ( 1 ) which enforces the twisted 1-form gauge invariance. The zero-flux condition is particularly simple for our model since it boils down to the branching rules. In other words, it enforces the fact that the labeling of the 2-simplices define a G [2] -coloring, i.e. a local description of a flat 2-form connection. Given a state | (0123) that is the state of a labeled 3-simplex whose vertices are numbered 0, 1, 2 and 3, the action of the operator B ( 3 ) explicitly reads B (0123) | (0123) = δ g123−g023+g013−g012,0 | (0123) . (6.23) The action of the operator A ( 1 ) is a little more subtle but there is a particularly convenient way of defining it via a so-called tent move in terms of the state-sum invariant. Given a 2-simplex (xy) ⊂ , the operator A (xy) acting on (xy) can be written succinctly as 24 is the topological action, cl(xy) is the minimal subcomplex of that contains all the simplices such that (xy) is one of their subsimplices, and j denotes the join operation [71].
Let us illustrate the definitions of cl(•) and j with a simple example in one lower dimension. Let (0) be a 0-simplex shared by only three 2-simplices, namely (012), (023) and (013). The operation (0 ) j cl(0) then reads where g 245 = g 345 = g 145 = 0. 26 It is not obvious from the drawing but the complex (5) j cl(04) does not contain the 2-simplex (123) and as such it only contains three 4-simplices, namely (02345), (01245) and (01345). Let us further suppose w.l.o.g that we have the following initial G [2] -coloring: g 012 = a, g 013 = a+b, g 014 = a + b + c, g 123 = e, g 124 = e + f and g 234 = h and we denote the 1-form gauge parameter by g 045 = d. Denoting the initial state in (6.25) by |ψ init. and the final state in (6.26) by |ψ fin. , the amplitude of the operator A d (04) for such coloring explicitly reads where we used the correspondence (5.13). As it turns out, we could have anticipated this result. Indeed, if we embed the initial complex made of four 3-simplices meeting at (4) in a four-dimensional 25 We focus our definition of the Hamiltonian model on this special example in order to show later on how it is related to the Walker-Wang model. We postpone a more through study of this lattice Hamiltonian to another paper. 26 The operator A acts on the 1-simplex (04) only so that we must have g 025 = g 024 + g 045 , g 035 = g 034 + g 045 and g 015 = g 014 + g 045 . It then follows from the 2-cocycle condition dg = 0 that g 245 = g 345 = g 145 = 0. manifold, we can think of it as a 4-simplex (01234). But the topological action assigns to this 4-simplex an amplitude ω, (01234) = ω(g 012 , g 013 − g 012 , g 014 − g 013 |g 123 , g 124 − g 123 |g 234 ) = ω(a, b, c|e, f |h) (6.27) together with the G [2] -coloring defined above. Upon 1-form gauge transformation at the edge (04), one has g 014 → g 014 − θ 04 , g 024 → g 024 − θ 04 and g 034 → g 034 − θ 04 so that the topological action transforms as ω(a, b, c|e, f |h) → ω(a, b, c + d|e, f |h) (6.28) where θ 04 = −d. We can then deduce from the 2-form 4-cocycle condition d (4) ω(a, b, c, d|e, f, 0|h, 0|0) = 1 that the topological action is modified under this transformation by the following factor which is exactly the amplitude of the operator as obtained above from the tent move. It follows from the 2-form 4-cocycle condition that the operators A ( 1 ) and B ( 3 ) as defined above commute 27 and the lattice Hamiltonian projector finally reads The fact that the ground states of this Hamiltonian satisfy equations (6.1) and (6.2) under local unitary transformations follows directly from the topological invariance of (6.22), or more precisely from the 2-form 4-cocycle condition.

Excitations
Given a closed three-dimensional hypersurface Σ endowed with a triangulation , the lattice Hamiltonian is provided by (6.30), the states are defined as superpositions of labeled graph states, and the ground states of the Hamiltonian are defined as states |ψ satisfying A ( 1 ) |ψ = |ψ and B ( 3 ) |ψ = |ψ for each 1 , 3 ⊂ . The Hilbert space of ground states on Σ endowed with the triangulation is denoted by H . Recall that the two conditions enforced by the operators B ( 3 ) and A ( 1 ) are the 2-form flatness condition and the twisted 1-form gauge invariance, respectively. But, flat 2-form connections on Σ can be defined as homomorphisms from the second homotopy group π 2 (Σ) to G, so that non-trivial 2-holonomies can only be found along non-contractible 2-paths. This means that by imposing the 2form flatness condition at every 3 ⊂ , we make the implicit assumption that each 3 is associated to a contractible 2-path. We define an excitation as a local neighborhood of the triangulation where the energy density is higher than that of the ground state, i.e. a state for which the conditions A ( 1 ) |ψ = |ψ and B ( 3 ) |ψ = |ψ are violated in a local neighborhood. We refer to a state for which one constraint A ( 1 ) |ψ = |ψ is violated as an electric charge excitation and a state for which one constraint B ( 3 ) |ψ = |ψ is violated as a magnetic flux excitation.
Let us first focus on magnetic excitations. By definition, these excitations occur when a given state violates the 2-form flatness condition at one or several 3-simplices. But, if we want the 2-form connection interpretation to persist, this violation must be associated with a non-contractible closed 2-path. Given a closed three-dimensional manifold Σ, such a non-contractible 2-path can be produced by removing an appropriate three-manifold B from Σ, hence turning Σ into an open manifold Σ\B whose boundary is given by the boundary ∂B of the three-manifold B. For instance, this can be done by removing a solid two-torus or a solid two-sphere from Σ. The resulting manifold would then have a torus boundary or a sphere boundary, respectively, that can support point-like magnetic flux excitations.
So we constrain the magnetic excitations to occur at boundary components of the three-manifold. Similarly, we restrict the electric charge excitations to occur at the boundary. More specifically, we endow each component of ∂Σ that has at least one non-contractible 1-cycle with a marked link for each non-contractible 1-cycle and allow for the 1-form twisted gauge invariance at a 1-simplex to be violated if and only if it coincides with a marked link. So the lattice Hamiltonian yields point-like magnetic flux excitations and string-like electric charge excitations, both located at the boundary of the manifold. Given a closed three-manifold Σ equipped with a triangulation, we can remove threemanifolds from it and we think of the states defined on the resulting manifold as being excited with respect to the ground states defined on Σ.
It is well-known that in such context manifolds of the form N × [0, 1] play a special role [39,[72][73][74][75][76]. For instance, in two dimensions, the Hamiltonian realization of the 3d Dijkgraaf-Witten TQFT yields point-like electric and magnetic excitations located at punctures. The twice-punctured two-sphere (or cylinder), i.e. T[S 1 ] ≡ S 1 × [0, 1] is then the simplest topology supporting both type of excitations. Moreover, the gluing of two cylinders results in a manifold homeomorphic to a cylinder, hence defining an algebra on the Hilbert space of states that is referred to as Ocneanu's tube algebra [77,78]. By defining specific excited states on the cylinder, we can confirm explicitly that this algebra is equivalent to the twisted Drinfeld double of the gauge group G It turns out that the number of independent excited states on a manifold of the form N × [0, 1] corresponds to the ground state degeneracy on the manifold N × S 1 . Therefore, there is a systematic way to compute the ground state degeneracy of a given lattice Hamiltonian on a manifold of the form N × S 1 : Consider the tube algebra of T[N ] ≡ N × [0, 1], derive its irreducible representations, and find the ground states degeneracy as the number of such irreducible representations.
The strategy outlined above has been extensively employed to study gauge models of topological phases. But it can also be used in the context of 2-form topological models. We briefly sketch such strategy here and postpone to another paper a more thorough treatment. Let us first consider the manifold T[S 2 ] ≡ S 2 × [0, 1]. The two-sphere S 2 has the following Betti numbers: b 0 = 1, b 1 = 0 and b 2 = 1. In other words, S 2 has a single connected component, zero non-contractible 1-cycle and one non-contractible 2-cycle. In our context, this means that the tube T[S 2 ] can only support a single point-like magnetic excitation. In order to have a manifold that supports both electric and magnetic excitations, it is necessary to introduce non-contractible 1-cycles. The natural choice would be to consider the manifold T[T 2 ] ≡ T 2 × [0, 1] obtained by cutting open the three-torus. This manifold would actually support one type of magnetic excitation and two types of electric excitations. However, the structure of the excitations associated with this manifold is rather involved. Therefore, as an intermediary step, we could focus instead on the three-pseudo-manifold T[S 2 no. ] ≡ S 2 no. × [0, 1] where S 2 no. is the nodal sphere obtained after identifying two points of the two-sphere S 2 . This pseudomanifold is homeomorphic to a pinched two-torus and can be graphically depicted as The manifold S 2 no. that possesses one non-contractible 2-cycle and one non-contractible 1-cycle (as opposed to the manifold T[T 2 ] that possesses two non-contractible 1-cycles) supports one point-like flux and string-like charge. As a matter of fact, we can think of T[S 2 no. ] as a special case of T[T 2 ] in the sense that some of the excitations supported by T[T 2 ] are condensed. As such, the study of the algebra associated to this tube cannot reveal as much information about the excitation content of the theory. It is however considerably simpler and constitutes an interesting intermediary case. We will compute in a follow-up work such tube algebras for several boundary manifolds and derive the ground state degeneracy on the corresponding closed manifolds. But let us conclude this section by computing explicitly the tube algebra for the nodal sphere in the case where the 2-form cocycle is trivial.
So we are interested in the algebraic structure underlying the states defined on the manifold T[S 2 no. ] ≡ S 2 no. × [0, 1]. Let us first find a basis for these states. To do so, we need to introduce a presentation of this pseudo-manifold. Since the two-sphere can be discretized by a 2-gon whose edges are identified, we can present the nodal sphere as a 2-gon whose edges and vertices are identified from which a discretization of T[S 2 no. ] can easily be obtained. Representing identified edges and identified vertices by an identical arrow and an identical dot, respectively, we have the following presentation: Each nodal two-sphere bounding T[S 2 no. ] corresponds to a non-contractible 2-cycle, hence supporting a flux excitation. Furthermore, each nodal sphere is equipped with a marked closed link which coincides ∼ 52 ∼ with the edges of the discretization so that the 1-form gauge invariance is there relaxed. The electric excitation is captured by the face variable b ∈ G, 28 while the magnetic excitation on the upper nodal sphere is captured by the face variable a ∈ G. It follows from the 2-flatness condition imposed on the 2-cycle 'between the two spheres' that the face variable labeling the bottom sphere is also a ∈ G. This defines a basis of states associated with the manifold T[S 2 no. ] that is labeled by two group elements. Let us denote by H T[S 2 no. ] the Hilbert space spanned by these states. We can decompose this Hilbert space in terms of boundary colorings as follows no. ] =: a∈Col(S 2 no. ×{0},G [2] ) where we implicitly made use of the fact that the 2-form flatness condition implies Col(S 2 no. ×{0}, G [2] ) = Col(S 2 no. × {1}, G [2] ). The Hilbert space H S 2 no. ×[0,1] [a, a] is therefore spanned by states labeled by one group variable and is denoted by no. ] [a, a]. In general, the tube algebra for a manifold T[N ] is defined by gluing two copies of T[N ] along one of the boundary components. This gluing operation must be performed such that the marked links on the boundary are identified. The result is a manifold homeomorphic to the initial one. This yields an algebra product denoted by which consists of two operations: (i) A gluing map G which identifies the boundary configurations, (ii) the projection via A ( 1 ) onto the subspace of states satisfying the 1-form gauge invariance everywhere but at the marked links on the boundary. 29 The projection step (ii) is required since after gluing there are new bulk 1-simplices, namely the ones that coincide with the marked links that are identified, which are not located at the boundary anymore and thus at which the 1-form gauge invariance must be enforced. Once the gauge invariance is enforced, the constraints are everywhere satisfied in the bulk of the resulting manifold so that the corresponding states satisfy the equations (6.1) and (6.
where H ext.
no. ] is the Hilbert space of states defined on the pseudo-manifold resulting from the gluing such that the 1-form gauge invariance is not yet enforced at the new bulk 1-simplex. Since the cocycle is taken to be trivial, both steps in the definition of the algebra product are particularly simple and the algebra simply reads Finding the irreducible representations is immediate and the ground state degeneracy on the manifold S 2 no. × S 1 is thus |G| 2 as expected. Naturally, the situation is considerably more complicated when the 2-form 4-cocycle is not trivial and this will be the subject of a follow-up work.
We exposed in the previous section how the local unitary transformations whose amplitudes are given in terms of a 2-form 4-cocycle can be reduced to a 2-2 Pachner move or a braiding move. Furthermore, we showed, algebraically and geometrically, how the 2-form 4-cocycle condition yields the so-called pentagon and hexagon relations which are the defining equations of a certain braided monoidal category. But it turns out that this braided monoidal category is the input data of another lattice Hamiltonian model, namely the Walker-Wang model [17]. 30 In this section, we first provide further detail regarding the interplay between the 2-form cohomology group H 4 (G [2] , U(1)) and abelian braided monoidal categories, then we study to which extent our 2-form gauge model is related to the Walker-Wang model.

Braided monoidal categories
First let us provide some basic definitions of category theory. More details can be found for instance in [86]: Definition 7.1 (Monoidal category). A monoidal category is a sextuple (C, ⊗, 1, , r, α) where: • C is a category whose collection of objects is denoted by Ob(C) and for x, y ∈ Ob(C), the collection of morphisms between them is denoted by Hom C (x, y).
• ⊗ is a bifunctor ⊗ : C × C → C referred to as the tensor product.
• α, and r are natural isomorphisms: referred to as the associator, the left unitor and the right unitor, respectively. These natural isomorphisms are subject to some coherence relations that we omit for now, namely the pentagon relation and the triangle relation.
In this article, we are only interested in a specific monoidal category, namely the category C-Vec G of G-graded vector spaces over the field of complex numbers, where G is a finite abelian group. 31 We define a G-graded vector space as a vector space V which satisfies V = g∈G V g and the tensor product of two G-graded vector spaces reads 30 More precisely, the input data of the Walker-Wang model is a unitary fusion braided category. It is possible to endow the monoidal category we are interested in, namely the category of G-graded vector spaces, with the structures necessary to turn it into a unitary fusion braided category. However, for this specific example, it is not required to do so as far as the definition of the Hamiltonian is concerned. 31 As mentioned earlier, the category C-Vec G is actually an example of fusion category but we do not need the corresponding additional structures in order to define the Walker-Wang model and show the correspondence with our 2-form gauge model.
The category C-Vec G has finitely many simple objects provided by the 1-dimensional G-graded vector spaces which are in one-to-one correspondence with group elements g ∈ G. We denote these simple objects by δ g∈G and they satisfy by definition End(δ g ) = C, Naturally, the tensor product of simple objects boils down to the group multiplication: δ g ⊗ δ h ∼ = δ g+h . Since it is enough to define the associator of the category on the simple objects, we are looking for an isomorphism determined by a function α : The pentagon relation then implies that α is a group 3-cocyle in H 3 (G, C × ) (which is the same as H 3 (G, U(1)) ). The triangle relation implies that if the left and the right unitors are trivial, then the 3-cocycle α is normalized, i.e. α(g, 1, h) = 1, ∀g, h ∈ G. We will now turn the category C-Vec G into a braided monoidal category: Definition 7.2 (Braided monoidal category). Given a monoidal category C, a braiding on C is a natural isomorphism R x,y : x ⊗ y ∼ − → y ⊗ x that is subject to the so-called hexagon relations. A braided monoidal category is then defined as a pair {C, R}.
In order to turn C-Vec G into a braided monoidal category, we only need to add a braiding, i.e a group 2-cochain R ∈ C 2 (G, C × ), satisfying the hexagon equations which are exactly (6.20) and (6.21). Interestingly, the set of associators and braidings as defined above enters the definition of the following cohomology: The set of all abelian coboundaries is denoted by B 3 ab (G, C × ). We finally define the abelian cohomology group as the quotient space: It results from the definitions above that isomorphism classes of braided monoidal categories whose simple objects form an abelian group G are classified by H 3 ab (G, C × ).
Using these definitions, we can rephrase our previous result: Given a 2-form 4-cocycle ω, the group cochains α : (a, b, c) → ω(a, b, c|0, 0|0) −1 and R : (a, b) → ω(0, 0, b|a, 0|0) form an abelian cocycle. Furthermore, it follows from (5.32) and (5.34) that the pair {d (3) α(a, b, c|0, 0|0), d (3) α(0, 0, a|b, 0|0)} forms an abelian coboundary. Putting everything together, it should not surprise the reader that there is a bijection between H 4 (G [2] , U(1)) and H 3 ab (G, U(1)). 32 If the relationship between H 4 (G [2] , U(1)) and H 3 ab (G, U(1)) is natural in light of our derivations in the previous section, a complete proof of this bijection would require more care. Since our work does not strictly rely on this bijection, we refer the reader to [68] instead. Nonetheless, let us assume this result until the end of this subsection and let us pursue our analysis. Recall that we defined earlier quadratic forms on a finite abelian group G valued in C × as a function q : G → C × such that q(g) = q(−g) and is bilinear. We denote the group of quadratic forms on G by Quad(G). Given a braided monoidal category whose simple objects form the abelian group G (such as C-Vec G ), we can construct easily a quadratic form q : G → C × such that for all g ∈ G, q(g) = R(g, g) ∈ Aut C (g ⊗ g) = C × . It follows directly that is a homomorphism. But, and this is a result by Eilenberg and MacLane presented in a succinct way in [86], this homomorphism turns out to be an isomorphism. This means that abelian cocycles are classified by quadratic forms. Since we assumed that there was a bijection between H 4 (G [2] , U(1)) and H 3 ab (G, U(1)), this also proves that H 4 (G [2] , U(1)) is classified by quadratic forms on G. Despite the numerous gaps we left, we hope this brief review provides some intuition as to why this is the case. This analysis thus completes the study initiated in sec. 2 where we made use of the same bijection in order to write down explicitly the action of a 2-form gauge theory in terms of a quadratic function q and the Pontrjagin square P. In any case, we do not need this result to display how our 2-form gauge model is related to the Walker-Wang model for the braided monoidal category of G-graded vector spaces.

Walker-Wang model for the category of G-graded vector spaces
The Walker-Wang model was first introduced in [17] as a generalization of Levin-Wen models to 3+1 dimensions. In general, the input data for the Walker-Wang model is a unitary braided fusion category. Crucially, the properties of the corresponding topological phase depends on whether the category is modular. Indeed, if the category is modular, then the model is trivial in the sense that it displays neither ground state degeneracy nor fractionalized excitations. In this section, we are only interested in the Walker-Wang model based upon the braided (fusion) monoidal category of G-graded vector spaces whose input data is a finite abelian group G, a group 3-cocycle α and a group 2-cochain R which together satisfy the pentagon and the hexagon relations. In light of the correspondence between abelian braided monoidal categories and the cohomology class of 2-form 4-cocycles, we want to emphasize how our 2-form gauge model is related to this Walker-Wang model. The lattice Hamiltonian introduced by Walker and Wang was originally defined on a cubic lattice such that all the nodes are six-valent. Crucially, in order to define the action of the commuting operators, it is necessary to split the six-valent nodes into three-valent ones so that the action of the plaquette operator can be expressed in terms of 2-2 Pachner moves (or F-moves) and braiding moves. The Hilbert space of the model is then spanned by all graph states obtained by labeling the edges of the graph obtained after such splitting. Different splittings must lead to equivalent models as they all match in the continuum limit, but a specific choice needs to be made nonetheless and it is referred to as a choice of resolution of the vertices. Note that this model can be generalized to richer input data such as G-crossed braided fusion categories, see [71].
In this section, we study the Walker-Wang model based upon the monoidal braided category of G-graded vector spaces. However, instead of working with a cubic discretization, we define the model on the one-skeleton of the 2-complex Υ dual to the triangulation . Naturally, since is obtained as a gluing of 3-simplices all the nodes of the one-skeleteon of Υ are four-valent. Therefore, it is still necessary to perform a (single) splitting of the nodes in order to obtain a graph whose nodes are all three-valent. The lattice Hamiltonian is given by such that to each three-valent node, we assign an operator A n which enforces the oriented product of the group variables labeling the edges meeting at the node n to vanish, and to each plaquette, we assign an operator B p which modifies the group configuration of the edges adjacent to p by 'fusing' a loop of defect into the boundary of p. We can define more precisely the action of B p using some graphical calculus in a way which is reminiscent of (2+1)d string net models. To do so, we consider a special example, namely the triangular plaquette that is the one-skeleton of the dual graph of the union of the three 3-simplices (0134), (0124) and (0234) as depicted in (6.25) so that we have the correspondence: where in the second drawing we kept some labeling implicit as they can be deduced from the branching rules implemented at each node by the operator A n . Furthermore, the orientation of the edges is also kept implicit, however it is always such that the group variable associated with an unlabeled link is obtained as the sum of the group variables labeling the other two links meeting at this node. We write the plaquette operator B p = 1 ∼ 57 ∼ where the last state is obtained by fusing the loop defect labeled by d into the plaquette via trivial P 2 →2 moves.
It now remains to use local unitary transformation so as to obtain a state whose underlying graph is identical to the initial one. To do so, we first perform three R-moves in order to move aside the links labeled by e + f , f + h and h, so that P 2 →2 moves can be performed (as in 2d) without worrying about non-trivial braidings. Once all the P 2 →2 moves are performed, the links labeled by e + f , f + h and h are brought back to their original positions using three R-moves. Putting everything together, these transformations read R Denoting the initial state in (7.8) as |ψ init. and the final state in (7.13) as |ψ fin. , the plaquette term for this configuration is ψ fin. | B d p |ψ init. and reads The example we chose in order to illustrate the definition of the plaquette operator is admittedly very special but it can be generalized easily to any other situation (in particular with a different distribution of legs pointing inward or outward the plaquette). But, since this special configuration is the one corresponding to the situation chosen to illustrate the definition of the operator A ( 1 ) of the 2form gauge model in sec. 6.3, we are now able to draw a correspondence between the two Hamiltonian models.

From the 2-form gauge model to the Walker-Wang model
In this section, we sketch the correspondence between the 2-form gauge model whose input data is {G, ω ∈ Z 4 (G [2] , U(1))} and the Walker-Wang model for the braided monoidal category of G-graded vector spaces whose input data is {G, (α, R) ∈ Z 3 ab (G, U(1))}. We will not prove this correspondence in its full generality but merely focus on the specific example used above to define the two models.
First of all, notice that the operator A n of the Walker-Wang model and the operator B ( 3 ) of the 2form gauge model are essentially the same, both implement the branching rules. Therefore, our focus is on the action of the operator B p and A ( 1 ) , More specifically, we want to compare their amplitude in the case of the configuration (7.4). It is clear that both of them enforce a twisted 1-form gauge invariance. Furthermore, it follows from the duality relation between and Υ that a twisted 1-form gauge transformation at the 1-simplex (04) as performed by A ( 1 ) acts on the same 2-simplex variables as the plaquette operator B p via a loop of defect, namely (014), (024) and (034). However, it is not clear how the amplitudes of these two operators match, especially in light of the fact that the Walker-Wang model requires a splitting of nodes into three-valent ones.
Graphical depiction of the 2-form cocycle condition d (4) ω(a, 0, c, d|0, 0, 0|h, 0|0) ≡ d (4) ω, (012345) = 1. This illustrates how the term ω(a, c, d|h, 0|0) = ω, (01345) encodes all the defining steps of the plaquette operator in the Walker-Wang model: the splitting of the four-valent node into three valent ones, the combination of P2 →2 moves and braiding moves as well as the recombination of three-valent nodes into a single four-valent one. sequences of P 2 →3 moves but by equating two sequences composed of two P 1 →4 moves and one P 2 →3 move so that each term appearing in d (4) ω, (012345) = 1 corresponds to a P 2 →3 move or a P 1 →4 move. The 2-simplex variables are identified using the correspondence (5.13) and we represent by a dashed line vanishing variables. The result is represented fig. 5. We recognize that the different P 2 →3 and P 1 →4 moves reduce to: a move which splits the 4-valent node into two 3-valent ones whose amplitude is given by the function s, a trivial move which does not change the combinatorics of the graph built out of the bold links, a combination of braiding moves and P 2 →2 whose amplitude is given by the function c as represented in (6.17), a P 2 →2 move, and finally a move which puts together two 3-valent nodes into a single 4-valent one whose amplitude is given by s −1 . We deduce that the effective action of s can be graphically interpreted in terms of string diagrams as The presence of the s-terms in (7.17) is therefore explained by the fact that the definition of the operator B p in the Walker-Wang model requires an ad hoc splitting of the nodes into 3-valent nodes while our model is defined directly in terms of the 4-valent initial ones. So to summarize, the analysis carried out in this part confirms two things: (i) The correspondence between our model based on a 2-form 4-cocycle ω ∈ Z 4 (G [2] , U(1)) and the Walker-Wang model for the category of G-graded vector spaces, (ii) The fact that the ad hoc resolution of the vertices required to define the plaquette term in the Walker-Wang model is directly included in the definition of the 2-form 4-cocycle. Furthermore, our approach makes transparent the fact that the plaquette operator of the Walker-Wang model for the case of the category C-Vec G actually implements the invariance under twisted 1-form gauge transformations at the 1-simplex dual to the plaquette, as it is obvious from the definition of the operator A ( 1 ) of our 2-form gauge model.

Conclusion
Gauge and higher gauge models of topological phases of matter have been under intense investigation in the past years, one reason being that they seem to encapsulate most of the known models displaying non-trivial topological order in (3+1)d. In this paper, we studied in detail models that have a 2form gauge theory interpretation. The goal of this paper was two-fold: Study properties of 2-form topological gauge theories in the continuum and present an 3d exactly solvable model that is the Hamiltonian realization of a (3+1)d 2-form topological field theory.
Firstly, we defined 2-form topological theories as sigma models whose target space is provided by the second classifying space B 2 G of a finite abelian group G. These are classified by cohomology classes [ω] ∈ H 4 (B 2 G, R/Z). It turns out that this cohomology group is isomorphic to the group of (possibly degenerate) quadratic functions on R/Z allowing for a more explicit expression of the partition function. Furthermore, such discrete 2-form gauge theories can be embedded into continuous strict 2-group gauge theories. We described the construction of U(1) strict 2-group connections using the language of Deligne-Beilinson cohomology. More specifically, we showed that the space of gauge inequivalent q-form U(1) connections was isomorphic to the q-th Deligne-Beilinson cohomology and, within the same framework, how U(1) 1-form and 2-form connections could be combined so as to obtain a strict 2-group connection.
Secondly, we defined a lattice Hamiltonian realization of a 2-form gauge theory. To do so, we defined the 2-form cohomology of an abelian group that is isomorphic to the cohomology of its second classifying space as provided by the W -construction, and derived its properties. We showed in particular how a 2-form 4-cocycle reduces to an abelian 3-cocycle that is the input data of an abelian braided monoidal category. But these monoidal categories are classified by quadratic functions on the group, hence closing the loop with the results obtained in the first part. Correspondingly, we explained how our 2-form gauge model is related to the Walker-Wang model. Interestingly, we displayed how the ad hoc splitting into three-valent vertices required for the definition of the Walker-Wang Hamiltonian is now directly encoded in the 2-form cocycle itself.
The tools developed in this manuscript can be generalized and used for other purposes. For instance, the Deligne-Beilinson cohomology can be used to define weak 2-group connections, while the strategy followed to define the 2-form cohomology can be extended to define a weak 2-group cohomology. As a matter of fact the study of weak 2-group gauge models of topological phases as initiated in [20] was one of the motivations for the present work and we believe that this work is useful to study more systematically these higher group gauge theories. Furthermore, in light of the ∼ 61 ∼ correspondence between 2-form 4-cocycles and abelian braided monoidal categories, we believe that the tools developed in this manuscript could be used to study the braiding of higher-dimensional excitations from a cohomological point of view. More specifically, we anticipate 2-form 5-cocycles to be related to the braiding statistics of loop-like excitations [57,[87][88][89][90][91].
Apart from phases displaying intrinsic topological order as studied in this manuscript, it is possible to define symmetry protected topological phases of matter (SPTs). In general, SPTs are gapped phases of matter that are short-range entangled and have a global symmetry acting locally so that the phase can be adiabatically connected to the trivial one upon breaking the symmetry. It is possible for SPTs to contain operators that are localized on (q−1)-dimensional submanifolds (see e.g. [92,93]), in which case the global symmetry is referred to as a (q−1)-form global symmetry [14]. Gauging such a (q−1)form global symmetry requires the introduction of q-form flat connection and the resulting theory is a q-form topological gauge theory. This gauging process was studied in [20] both at the level of the action and in terms of its lattice realization, and could be reformulated in light of the constructions presented in this paper.

Postnikov towers and sigma models
In this appendix, we present further generalizations of the sigma models introduced in sec. 2 where the target space is provided by a k-stage Postnikov tower.
As the name suggests, the k-stage Postnikov tower E k can be built in a sequence of k steps. The first stage is provided by a classifying space E 1 = B q1 G 1 . The second stage is provided by a fibration over E 1 with the fiber being isomorphic to B q2 G 2 . This step is captured in the sequence 0 → B q2 G 2 → E 2 → E 1 → 0 (A.1) whose extension class is [α 2 ] ∈ H q2+1 (E 1 , G 2 ). At the third stage, we build a space E 3 as a fibration over E 2 so that whose extension class is [α 3 ] ∈ H q3+1 (E 2 , G 3 ). 33 This sequence proceeds iteratively until whose extension class is [α k ] ∈ H q k +1 (E k−1 , G k ). A homotopy class of map from M to E k is provided by a k-tuple A k Furthermore, we require that A k ∈ ker(D E k ) which amounts to imposing the following cocycle conditions dA 1 = 0 . . . dA k = α k (A 1 , A 2 , . . . , A k−1 ) .

(A.5)
There is a gauge redundancy A k ∼ A k + D E k Φ k generated by the null homotopy D E k Φ k where Φ k is the k-tuple The definition of D E k is such that . . .
One can check that D E k • D E k = 0 so that we can define the cohomology where q = (q 1 , q 2 , . . . , q k ), so that cohomology classes label isomorphism classes of data A k . Finally, we may build a generalized topological gauge theory by constructing a topological action from a cohomology class [ω] ∈ H d+1 (E k , R/Z) whose partition function reads Following the examples provided in sec. 2, we know that we can obtain a lattice realization of a topological model whose target space is given by a k-stage Postnikov tower by reproducing the construction above, except that we now work with a triangulation of M and that instead of summing over cohomology classes in a generalized cohomology group H q E k (M), we sum over colorings g ∈ Col(M, E k ). An element g = {g 1 , g 2 , . . . , g k } ∈ Col(M, E k ) is such that g i is a coloring of the q i -simplices of . These colorings are such that the corresponding group variables satisfy local constraints which are the analogue of the (twisted) cocycle conditions presented earlier that depend on the cohomology classes [α p ] ∈ H qp+1 (E p−1 , G p ). Using the differential on cochains, these local constraints read dg 1 , q1+1 = 0 dg 2 − α 2 (g 1 ), q2+1 = 0 . . . dg k − α k (g 1 , g 2 , . . . , g k ), q k +1 = 0 . (A.10) Finally, the partition function is provided by

Pontrjagin square
In this appendix, we collect some important properties of the Pontrjagin square P [94]: Property B.1. If f ∈ Z 2 (M, Z n ), then P(f ) ∈ Z 4 (M, Z 2n ) with n even, while with n odd we have P(f ) ∈ Z 4 (M, Z n ). An explicit expression for P(f ) can then be written in terms of ∼ 64 ∼ Steenrod's higher cup products 34 as wheref is the integer lift of f , i.ef ∈ C 2 (M, Z) such that df = nu for u ∈ B 3 (M, Z) and f =f mod n.
We can check that P(f ) as defined above is indeed closed. We consider the two cases separately. When n is odd, dP(f ) = df f + f df = 0, and when n is even Property B.2. The Pontrjagin square refines the bilinear form 2f g. Indeed, let f, g ∈ Z 2 (M, Z n ). If n is odd, one has where d = is an equality up to exact terms. If n is even, we write df = nu and dg = nv, and we get P(f + g) − P(f ) − P(g) =f g +g f + n(f 1 v +g 1 u) = 2f g + d(f 1g ) + n [−u 1 g − f 1 v + f 1 v + g 1 u] = 2f g + d(f 1g ) − 2nu 1g − nd (u 2 g) − n 2 u 2 v d = 2f g (mod 2n) .

Operators, quantization and invertibility of 2-form topological theories
In this appendix, we review some of the properties of the 2-form topological theory introduced in sec. 2. More precisely, we consider the partition function of the 2-form gauge theory formulated as a continuous topological field theory, construct its gauge invariant operators, quantize it, and study its invertibility. We follow closely the analysis of [14,59].
In order to keep the notations lighter and focus on the physical aspects, we consider the simpler case of a 2-form gauge theory with gauge group Z n . Let us consider the topological action which is exactly (2.30) for G = Z n , where p prescribes a choice of homomorphism in Hom(Γ(Z n ), R/Z). Instead of differentiating the cases for n being an odd or even integer, we work with general n and restrict the values of p. More precisely, one takes p ∈ Z when n is even, and p ∈ 2Z when n is odd. Since p ∼ p + 2n, 35 there are n distinct topological gauge theories for n odd and 2n distinct topological gauge theories for n even. This agrees with the order of the universal quadratic group for Z n . Usually gauge invariant operators in (3+1)d topological gauge theories are defined on closed lines and surfaces. Such operators assign topological data in the form of correlation functions to certain linked configurations for the corresponding lines and surfaces. For example, the (3+1)d BF theory assigns a non-trivial phase to a linked line and surface embedded in the (3+1)-dimensional spacetime manifold. More interestingly, non-trivial discrete gauge theories, namely Dijkgraaf-Witten theories, may have topological correlation functions associated to linked configurations of three or four surface operators (cf. for example [55,57,88,96]). Gauge invariance ( ) is an inherent line operator if p · Q ∈ nZ. The reason for this is that exp 2πin ∂ −1 L (1) B is the identity operator so that the operator W Q does not have a surface attached to it (or equivalently has a transparent surface attached to it) and is therefore a genuine line operator. From the above constraint on genuine line operators, one may read off that W := W n/gcd(n,p) is the simplest non-trivial line operator and since W n is trivial, there are gcd(n, p) such non-trivial operators. Similarly, some surface operators can end on closed lines and are therefore topologically trivial. The number of surface operators that cannot end on lines match the number of line operators, namely gcd(n, p). As a quick illustration of this last point, let us have a look at two examples: Example C.1 (n = 12 and p = 4). Naively, one would say that the surface operators are U M (L (2) ) with M = 0, . . . , 12, however, when M /4 ∈ Z, such a surface operator can end on a line. For instance, if M = 4, one could have the operator exp 2πi ∂L (2) A + 8πi L (2) B . Therefore, the number of surface operators modulo the number of trivial surface operators is gcd(n, p).
for an untwisted Dijkgraaf-Witten theory with gauge group G = Z gcd(n,p) (equivalently a Z gcd(n,p) BF theory) up to a geometric counterterm χ(M) ln(gcd(n, p)), or alternatively is the partition function of a 2-form topological gauge theory with gauge group G = Z gcd(n,p) and trivial cohomology class.
By quantizing the theory on manifolds of the form M = Σ × R, we obtain Hilbert spaces H Σ of physical states. The dimension of these Hilbert spaces is an interesting class of objects. Indeed, given a surface Σ, the dimensions of H Σ corresponds to the ground state degeneracy of the lattice Hamiltonian realization of the theory on Σ. By computing explicitly the ground state degeneracy, we can then confirm for which values of n and p the theory is invertible. By definition, an invertible (3+1)d TQFT assigns a single physical state to any 3-manifold so that the dimension of the Hilbert space H Σ obtained upon quantization of the theory on Σ × R incurs a contribution only from the non-invertible part of the theory. Since the non-invertible part of the partition function can be mapped (dualized) to an untwisted Z gcd(n,p) Dijkgraaf-Witten theory, one has dim H Σ = Z[Σ × S 1 ] = gcd(n, p) b1(Σ)=b2(Σ) . where the surface operators U are defined according to (C.2). Such a basis can be explicitly constructed as where the vacuum is normalized to have unit eigenvalue for all the non-trivial surface operators. A similar basis can be constructed that diagonalizes the line operators W defined in (C.3). Denoting this basis by | Q , one has the following overlap Q | M = exp 2πi Q · M gcd(n, p) (C.10) that can be viewed as the partition function on a four-sphere S 4 with the line operators W Q i ([L (1) ] i ) and U M j ([L (2) ] j ) inserted such that the linking number link([L (1) ] i , [L (2) ] j ) = δ ij . In order to visualize this, it is possible to start with a four-sphere and hollow out a four manifold B Σ whose boundary is Σ. Then using standard surgery where the surgery involves carving out B Σ from S 4 and inserting line operators in the carved out B Σ so as to create the state | Q on Σ. Similarly inserting surface operators in Z\B Σ such as to create the state | M on ∂(Z\B Σ ) = Σ, then filling in B Σ into Z\B Σ , which amounts to the overlap Q | M . This is nothing but the S 4 partition function with linked configuration of lines and surfaces as described above.
Since it is known that invertible topological theories are short-range entangled it is illustrative to compute the topological entanglement entropy [100][101][102][103][104] to confirm this. The computation is rather straightforward. We are interested in the situation where Σ = S 3 and want to compute the topological piece in the topological entanglement entropy. We bipartition S 3 into subregions Σ 1 and Σ 2 such that for ∂Σ 1 = ∂Σ 2 = D 2 . Following a well-known recipe [104][105][106] to compute the topological entanglement entropy, we first need to compute the n-th Renyi entropy S where we have implicitly absorbed the terms that depends on the Euler characteristic χ(M) into local geometric counterterms. So as expected the 2-form TQFT is short-range entangled (or invertible) when gcd(n, p) = 1, and not otherwise. It is known [17,18,59] that when gcd(n, p) = 1, i.e when the quadratic form defining the topological action is non-degenerate, the theory admits a gapped boundary condition with non-trivial line operators which form a modular tensor category. On the other hand when gcd(n, p) = 1, there also exist gapped boundaries with non-trivial operators however these do not form a modular category anymore but a premodular one.

Topological actions in terms of Deligne-Beilinson cocycles
In this appendix we derive expressions for various topological actions built from Deligne-Beilinson cohomological data.

D.1 (2+1)d BF theory
First, let us consider BF theory in (2+1)d. The BF topological action is commonly written as where n ∈ Z is a parameter of the theory, B and A are 1-form U(1) connections and M is an oriented 3-manifold. But this expression does not make sense when we include topological sectors of A and B.
To give a more precise definition of the BF topological action, we consider A, B ∈ H 1 DB (M, Z) together with the following pairing: Let the local data that defines A and B as DB 1-cochains be denoted by A = µ 1 0 , µ 0 1 , n A