A Relation between H^{*}(BG,\mathbb{Z}) and H^{*}(BG,\mathbb{R}/\mathbb{Z})

We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge “symmetry”) and the field strength fluctuates only weakly. We ask, what kind of topological terms (the terms that do not depend on space-time metrics) can be added to the Lagrangian of such a weakly coupled gauge theory. For example, for weakly coupled gauge theory in space-time dimensions, a Chern-Simons topological term can be added.

In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and terms, in any space-time dimensions and for any gauge groups (continuous or discrete). We can use each element of the topological cohomology classes on the classifying space of the gauge group to construct a quantized topological term in space-time dimensions.

In 3 or for finite gauge groups above 3, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories.

Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group are also classified by the same . (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.

## I Introduction

Quantum many-body systems can be described by Lagrangians in space-time. As we change the coupling constants in a Lagrangian a little, the new Lagrangian will describe a new quantum many-body system which is usually in the same phase as the original system. However, the Lagrangians may also contain the so called topological terms – the terms that do not depend on space-time metrics. Some of those topological terms are quantized. Adding quantized topological terms to a Lagrangian will generate new Lagrangians that usually describe different phases of quantized many-body systems.

So studying and classifying quantized topological terms is one way to understand and classify different possible phases of quantum many-body systems. For example, a non-linear -model

 L=1λ[∂g(xμ)]2,  g∈G (1)

with symmetry group can be in a disordered phase that does not break the symmetry when is large. By adding different quantized topological -terms to the Lagrangian , we can get different Lagrangians that describe different disordered phases that does not break the symmetry .Chen et al. (2011a) Those disordered phases are the symmetry protected topological (SPT) phases.Gu and Wen (2009); Pollmann et al. (2009) We find that the different quantized topological -terms in -dimensional space-time can be classified by the Borel group cohomology classes , which leads us to believe that the SPT phases in -dimensional space-time can be classified by .Chen et al. (2011b, a) So the possible topological terms in a non-linear -model help us to understand the possible phases of the non-linear -model.

This motivated us to study possible quantized topological terms in gauge theory, hoping to understand different phases of gauge theory in a more systematic way. But a gauge theory is a very complicated system whose low energy phases can be any thing. To make our problem better defined, we need to restrict ourselves to a special class of gauge theories that we will call “weakly coupled gauge theories” or “weakly coupled lattice gauge theories”. “Weakly coupled (lattice) gauge theories”, by definition, are lattice gauge theories in the weak coupling limit (where gauge flux through each plaquette is small) and all the gauge charged excitations have a large energy gap (ie no Higgs condensations).

We find that a quantized topological term in the weakly coupled gauge theories can be constructed from each element in topological cohomology class for the classifying space of the gauge group in space-time dimensions.

The result is obtained by applying the three approaches introduced by Baker and by Dijkgraaf and WittenBaker (1977); Dijkgraaf and Witten (1990) to lattice gauge theories.

In the first approach (see section II), we follow Dijkgraaf and WittenDijkgraaf and Witten (1990) to use cocycles in group cohomology class to construct topological terms in weakly coupled gauge theories in -dimensional space-time with a finite gauge group . Such weakly coupled gauge theories are gapped and describe topological phases. So we can use to describe different topological phases in weakly coupled gauge theories with finite gauge group.

In the second approach (see section III), we view a lattice gauge theory as a lattice non-linear -model whose target space is given by the classifying space of the gauge group . So the quantized topological terms in a weakly coupled lattice gauge theory can be described by the quantized topological terms in the non-linear -model of the classifying space . Such a quantized topological term is similar to the one used in the classification of SPT phases,Chen et al. (2011a); Gu and Wen (2012) where the target space is simply the symmetry group . Using this approach, we find that some quantized topological terms in weakly coupled gauge theories with gauge group can be constructed from the torsion elements in topological cohomology class of the classifying space . Since for finite groups,Chen et al. (2011a) the second approach contains the result of the first approach.

In the third approach (see section IV), we express -dimensional Chern-Simons terms (assuming = odd) in terms of the differential characters of Chern, Simons and Cheeger.Baker (1977); Dijkgraaf and Witten (1990) The different differential characters are classified by ,Baker (1977) thus in turn giving a classification of Chern-Simons terms with gauge group . We see that the quantized topological terms obtained from the third approach (classified by ) contain those obtained from the second approach (classified by ), in = odd space-time dimensions.

Also, in even space-time dimensions, (see eqn. (87)). So the quantized topological terms are classified by in any dimensions. Since

 HdB(G,R/Z)≃Hd+1(BG,Z) (2)

[see eqn. (J32) in Ref. Chen et al., 2011a], this suggests that the quantized topological terms in weakly coupled gauge theories are described by . Here can be a continuous group. Thus the quantized topological terms is the dual of the SPT phases classified by the same .

We know that, for continuous gauge group , a kind of quantized topological terms – Chern-Simons terms – can be defined in any = odd dimensions:

 S3d =∫M2πK2!(2π)2Tr(AF−A33)+FμνFμνλ, S5d =∫M2πK3!(2π)3Tr(AF2−A3F2−A510)+FμνFμνλ, ...... (3)

where = integer, is the gauge potential one form and the gauge field strength two form. Also, for example, means the wedge product . We have also included the usual kinetic term in the above with coefficient . In space-time dimensions, a Chern-Simons gauge theory is gapped and describes a topologically ordered phase. So the topological phases of a weakly coupled gauge theory are described by . Beyond , the above Chern-Simons gauge theory is gapless for small , and the Chern-Simons term is irrelevant at low energies. However, the Chern-Simons term is not renormalized, and we believe that different Chern-Simons terms will describe different gapless phases of the weakly coupled gauge theory. Consider Chern-Simons theory, if the space-time has a topology of and the gauge field has a non-zero total flux through , the Chern-Simons theory on will becomes a non-trivial Chern-Simons theory on which describes a non-trivial topological phase.

The above discussion is for = odd dimensions. Similarly, in = even space-time dimensions, we can have the following weakly coupled gauge theories with topological term

 S4d =∫Mθ2!(2π)2Tr(F2)+FμνFμνλ, S6d =∫Mθ3!(2π)3Tr(F3)+FμνFμνλ, ...... (4)

where and is small.

The above topological terms are not quantized and do not give rise to new phases. In this paper, we show that quantized topological terms do exist for weakly coupled gauge theories in even space-time dimensions. Such quantized topological terms are described by Tor. Again, the quantized topological terms do not open a mass gap in dimensional space-time (for small ). Here we like to propose that the different quantized topological terms give rise to different gapless phases of the gauge theories. The situation is even more interesting (and unclear) in dimensional space-time, where some gauge theories are confined even in small limit. It is not clear if the different quantized topological terms give rise to different confined phases.

Since the introduction of topological order in 1989,Wen (1989, 1990) we have introduced many ways of constructing topologically ordered states (ie long range entangled states) on lattice: resonating-valence-bond state,Rokhsar and Kivelson (1988); Moessner and Sondhi (2003) projective construction,Baskaran et al. (1987); Affleck et al. (1988); Dagotto et al. (1988); Wen et al. (1989); Read and Sachdev (1991); Wen (1991, 1999) string-net condensation,Freedman et al. (2004); Levin and Wen (2005); Chen et al. (2010); Gu et al. (2010) weakly coupled gauge theory of finite gauge group, etc .

The usual construction of weakly coupled lattice gauge theories for a given finite gauge group only gives rise to one type of topological order. In this paper, we manage to “twist” the weakly coupled gauge theory with a given finite gauge group by adding topological terms to obtain more general topological orders. The added topological terms are constructed from (see table 1).

We also manage to “twist” the weakly coupled gauge theory of continuous gauge group by adding topological terms. The added topological terms in this case are constructed from . This leads to more general topological orders in (2+1)D than those described by the standard Chern-Simons terms. But in higher dimensions, this may lead to more general gapless phases of weakly coupled gauge theories (see table 1).

From Ref. Randal-Williams, 2011, we find that . So there are three non-trivial quantized topological terms that we can add to the weakly coupled gauge theory in -dimensional space-time. We think that one of the quantized topological term is given by

 SU(1)⋊Z2 =∫Mθ3!(2π)3F∧F∧F+(Fμν)2λ,  θ=π (5)

where is the field strength of the gauge field and the gauge transformation acts as .Bucher et al. (1992) Such a gauge theory can be viewed as the gauge theory with the gauged charge conjugation symmetry. Note that the theory has a charge conjugation symmetry only when .

The gapless phases (in small limit) for and should be different.

Similarly, we can have a quantized topological term in weakly coupled gauge theory in -dimensional space-time:

 SU(1)×U(1)⋊Z2 =∫Mθ(2π)2~F∧F+(Fμν)2λ+(~Fμν)2λ, (6)

where , are the field strengthes of the two gauge fields and the gauge transformation acts as . In such a gauge theory, the magnetic monopole with monopole charge will carry electric charges .Witten (1979); Rosenberg and Franz (2010) We see that when , dyons with quantum numbers all appear, which is consistent with the gauge symmetry.

When , a dyon with quantum number has a statistics (where means Bose statistics and means Fermi statistics and note that , , and are all integers).Tamm (1931); Jackiw and Rebbi (1976); Wilczek (1982); Goldhaber (1982); Lechner and Marchetti (2000) It is shown in Ref. Goldhaber et al., 1989 that the statistics of a dyon in the presence of the term does not depend on . So for a non-zero , the satistics is given by

 (−)(q−~mθ2π)m+(~q−mθ2π)~m=(−)qm+~q~m−θπm~m (7)

where the electric charges are no longer integers: + integer and + integer. We see that when , the statistics is invariant under the gauge transformation . A dyon is a boson while a dyon is a fermion. Clearly, the and correspond to two different gapless phases of the gauge theory in space-time dimensions.

It is amazing to see that the -dimensional SPT phases and the -dimensional quantized topological terms in weakly coupled gauge theories are classified by the same thing . In this paper, we would like to clarify a duality relation between the -dimensional SPT phase and -dimensional topological gauge theory for finite groups (see section V). (Here a topological gauge theory with a finite gauge group is defined as a weakly coupled gauge theory with a finite gauge group and a quantized topological term.) Such a duality relation is exactly the duality relation first proposed by Levin and Gu in 3-dimensions.Levin and Gu (2012) We know that the SPT phases are described by topological non-linear -models with symmetry :

 Missing dimension or its units for \hskip (8)

where the -quantized topological term is classified by which describes different SPT phases. If we “gauge” the symmetry , the topological non-linear -model will become a gauge theory:

 L=1λs[(∂−iA)g(xμ)]2+iWtop(g,A)+(~Fμν)2λ. (9)

If we integrate out , we will get a pure gauge theory with a topological term

 L=(~Fμν)2λ+i~Wtop(A). (10)

This line of thinking suggests that the gauge-theory topological term is classified by . Since , the topological terms constructed this way agrees with the topological terms constructed through the classifying space, which realizes the duality relation.

We remark that the SPT phases are described by topological non-linear -models with symmetry . If we “gauge” the symmetry , the topological non-linear -model will become a gauge theory– a principal chiral model,Liu and Wen (2012); Chen and Wen (2012) which realizes the duality relation.

## Ii Lattice topological gauge theory

### ii.1 Discretize space-time

In this paper, we will consider gauge theories on discrete space-time which are well defined. We will discretize the space-time by considering its triangulation and define the -dimensional topological gauge theory on such a triangulation. We will call such a theory a lattice topological gauge theory. We will call the triangulation a space-time complex, and a cell in the complex a simplex.

In order to define a generic lattice theory on the space-time complex , it is important to give the vertices of each simplex a local order. A nice local scheme to order the vertices is given by a branching structure.Costantino (2005); Chen et al. (2011a) A branching structure is a choice of orientation of each edge in the -dimensional complex so that there is no oriented loop on any triangle (see Fig. 1).

The branching structure induces a local order of the vertices on each simplex. The first vertex of a simplex is the vertex with no incoming edges, and the second vertex is the vertex with only one incoming edge, etc . So the simplex in Fig. 1a has the following vertex ordering: .

The branching structure also gives the simplex (and its sub simplexes) an orientation. Fig. 1 illustrates two -simplices with opposite orientations. The red arrows indicate the orientations of the -simplices which are the subsimplices of the -simplices. The black arrows on the edges indicate the orientations of the -simplices.

### ii.2 Lattice gauge theory on a branched space-time complex – finite gauge group

Now, we are ready to define lattice gauge theory on a branched space-time complex. We first choose a gauge group . For the time being, let us assume that is finite. We then assign to each link in the space-time complex , where is the orientation of the link. For each simplex (with vertices ), we assign a complex action amplitude .

The imaginary-time path integral of the lattice gauge theory is given by

 Z=∑{Gij}′∏[ij...k]αs(i,j,...,k)({Gij}), (11)

where is the product over all the simplices in the space-time complex , and or depending on the orientation of the simplex defined by the branching structure. Two sets and are said to be gauge equivalent if there exists a set , such that

 G′ij=giGijg−1j. (12)

We require the above lattice theory to be gauge invariant:

 ∏[ij...k]αs(i,j,...,k)({giGijg−1j}) =∏[ij...k]αs(i,j,...,k)({Gij}), ∀gi ∈G, (13)

for any closed space-time complex : . In eqn. (11), sums over the gauge equivalent classes of . This way, we define a lattice gauge theory with gauge group . We may rewrite the path integral as

 Z=|G|−Nv∑{Gij}∏[ij...k]αs(i,j,...,k)({Gij}), (14)

where is the number of elements in , the number of vertices. The overall normalization factor can be understood as modding out the theory by its overall redundant phase volume.

### ii.3 Lattice topological gauge theory – finite gauge group

What is the low energy fixed-point theory of the lattice gauge theory defined above, if the theory is in a gapped phase? In a study of SPT phases, we have discussed the gapped phases of non-linear -model and the related fixed-point theories (or topological theories).Chen et al. (2011a); Gu and Wen (2012) There, the fixed-point theories have the following defining properties, that the action amplitudes for any paths are always equal to 1 if the space-time manifold has a spherical topology. Here we will use the similar idea to study the fixed-point theory of the gapped phases of a lattice gauge theory.

First, one possible low energy fixed-point theory is given by the following action amplitude

 α({Gij})=1. (15)

Such an action amplitude does not change under the renormalization of the coarse graining and describes a fixed-point theory. However, such a fixed-point theory describes a confined phase with trivial topological order. In this paper, we will regard such a fixed-point theory to have a trivial gauge group.

We would like to ask, is this the only way for a gauge theory to become gapped? Can gauge theory become gapped without confinement and the reduction of gauge group? The answer to the above questions is yes: a gauge theory with a finite gauge group can be gapped even without confinement. So in this section, we will consider gauge theories with a finite gauge group.

The fixed-point action (15) describes a confined phase since the Wilson loop operator, such as , can fluctuate strongly. So to obtain a fixed-point theory with the original gauge group , we require that

 |α({Gij})|=1  only when GijGjkGki=1, (16)

on all the triangles of the simplex on which is defined. In other words, the amplitude of a path is zero if there is a non-zero flux on some trangles. We will call this condition a flat connection condition since it corresponds to requiring the “field strength ”. In order for to describe a fixed-point topological theory, we also require that

 ∏[ij...k]αs(i,j,...,k)({Gij})=1 (17)

on all the complex that have a spherical topology. (It would be too strong to require on any closed complex .)

In (1+1) dimensions, the simplest sphere is a tetrahedron. Due to the flat connection condition, we can use , , and to label all the ’s (see Fig. 2a). For example, . On a tetrahedron, the condition (17) becomes

 α(G1,G2)α(G1G2,G3)α(G2,G3)α(G1,G2G3)=1. (18)

We note that if is a solution of the above equation, then defined below is also a solution of the above equation:

 α′(G1,G2)=α(G1,G2)β(G1)β(G2)β(G1G2). (19)

We regard the two solutions to be equivalent. The equivalent classes of the solutions correspond to different lattice topological gauge theories in (1+1) dimensions.

In (2+1) dimensions, the simplest 3-sphere is a pentachoron. Due to the flat connection condition, we can use , , , and to label all the ’s (see Fig. 2b). For example, . On a tetrahedron, the condition (17) becomes

 α(G1,G2,G3)α(G1,G2G3,G4)α(G2,G3,G4)α(G1G2,G3,G4)α(G1,G2,G3G4)=1 (20)

We note that if is a solution of the above equation, then defined below is also a solution of the above equation:

 α′(G1,G2,G3)=α(G1,G2,G3)β(G1,G2)β(G1G2,G3)β(G2,G3)β(G1,G2G3). (21)

Again, the above defines an equivalence relation between the solutions. The distinct equivalence classes of the solutions correspond to different lattice topological gauge theories in (2+1) dimensions.

The above discussion can be generalized to any dimensions. We also note that the equivalence class defined above is nothing but the group cohomology class . Therefore, the lattice topological gauge theory with a finite gauge group is classified by in dimensions, a result first obtained by Dijkgraaf and Witten.Dijkgraaf and Witten (1990) From the above discussions, we see that the result can also be phrase in a more physical way: the gapped phases of a lattice gauge theory with a finite gauge group is classified by in space-time dimensions, provided that there is no confinement and the reduction of gauge group (ie the “field strength ” fluctuate weakly).

We see that a lattice gauge theory can have many different gapped phases. One kind of gapped phases have no confinement nor reduction of gauge group (say due to the Higgs mechanism). This kind of gapped phases are classified by for finite group and in space-time dimensions. Other kind of gapped phases may have confinement or reduction of gauge group. Those gapped phases may be described by where is the unbroken gauge group.

## Iii Topological gauge theory as a non-linear σ-model with classifying space as the target space

### iii.1 Classification of G-bundles on a d-manifold M via classifying space Bg and universal bundles Eg of group G

In order to define topological gauge theory for continuous group (as well as for finite group), Dijkgraaf and Witten pointed out that all the gauge configurations on can be understood through classifying space and universal bundles (with a connection): all -bundles on with all the possible connections can be obtained by choosing a suitable map of into , .Dijkgraaf and Witten (1990) is a very large space, often infinite dimensional.If we pick a connection in the universal bundle , even the different connections in the same -bundle on can be obtained by different maps . Therefore, we can express the imaginary-time path integral of a gauge theory as

 Z=∑γeiS[γ] (22)

where sum over all the maps : , and is the action for the map . The dynamics of the gauge theory is controlled by the action and the connection on . In other words, once we specify a connection on , every map will define a connection on . Thus we can view the action as a function of the connection, (plus, possibly other gauge invariant degrees of freedom). We see that, in some sense, a gauge theory can be viewed as a non-linear -model with classifying space as the target space.

We like to remark that when we study gauge theory in a fixed space-time dimension , we can choose a truncated classifying space which has a finite dimension and a finite volume. We can view a gauge theory as a non-linear -model with the truncated classifying space as the target space.

In the following, we will use this point of view to study topological gauge theory. We have to say that such an approach is quite indirect compared to the discussion in section II. But, as we will see later, the two approaches give rise to the same classification of topological gauge theories for finite gauge groups.

### iii.2 Topological gauge theory from the non-linear σ-model of Bg

Viewing a gauge theory as a non-linear -model with classifying space as the target space, we can study topological terms in the gauge theory by studying the topological terms in the corresponding non-linear -model. Here, we write as . The term is independent of space-time metrics and is called the topological term. We are mainly concerned about the question whether the systems described by and are in the same phase or not. In general, a quantized topological term may make and to describe different phases. So we may gain some understanding of quantum phases by studying quantized topological terms.

In Ref. Chen et al., 2011b, a, we studied the quantized topological -terms in lattice non-linear -model with the symmetry group as the target space. We find that such quantized topological terms are classified by Borel cohomology classes . In this case, the different quantized topological terms do give rise to different quantum phases. Here, we can use a similar approach to construct/classify topological terms in non-linear -model with classifying space as the target space.

To use the above idea to study lattice gauge theories, we need to put the above discussion on a lattice by trianglating the space-time manifold into a complex . The mapping from to now becomes a mapping from to . However, the mapping from to can be defined differently, with extra structures and information in some definitions as oppose to others.

We may define the map from to as a map from the vertices of to . We have chosen such kind of map when we use lattice topological non-linear -model with the symmetry group as the target space to classify the SPT phases. However, such maps are not adequate to define lattice gauge theory, since the maps of the vertices do not allow us to obtain a connection on by pulling back the connection on .

To define a lattice gauge theory where gauge degrees of freedom reside on the edges of the triangulation , the map therefore need at least to specify how the set of 1-simplices, in is mapped into . In principle, no further detail is necessary to define the gauge theory. However, we will take a less general route and instead regard the map as an embedding of into . This means that information about the mapping of all the higher simplices, such as 2d faces that connect the edges are also completely specified. As will be evident in more detailed discussion of the Mathematics of the construction in section IV.3, such a choice of map requires that the lattice gauge theory is in the semiclassical limit where the fluctuations in the field strength are weak. In this case, the connection on naturally becomes a connection on . Different embeddings correspond to different gauge field configurations on . In order to write down an action for the lattice topological gauge theory on a -dimensional complex , we assign a phase mod to each -dimensional simplex in the triangulated classifying space . Such an assignment correspond to a -cochain in . Then, the action is the sum of the phases on the simplices in . The resulting total phase corresponds to evaluating the cochain on the complex :

 Stop=2π⟨μd,γ(Mtri)⟩  mod 2π. (23)

Such an action amplitude depends on the embedding and defines a dynamical gauge theory. This way, we write a lattice gauge theory as a lattice non-linear -model with as target space, through the embedding map .

To define a lattice topological term, we may choose mod for any maps as long as has no boundary. This is the action that we choose to classify the SPT phase using lattice topological non-linear -model.

But here, we like to choose a more general topological term . As a topological term, should not depend on the “metrics” of the complex (ie the size and the shape of the ). We would also like to consider restricting such that it has no dependence on the connection on , as long as has no boundary. But may depend on the topology of , or more precisely on the homological class of the embedding in . Those considerations suggest that we can define a topological action by choosing a cocycle :

 Stop[γ]=2π⟨αd,γ(Mtri)⟩  mod 2π. (24)

Note that the -cocycle are special -cochains whose evaluation on any -cycles [ie -dimensional closed complexes] are equal to mod 1 if the -cycles are boundaries of some -dimensional complex. So, each -cocycle in defines a lattice topological gauge theory in -dimensions.

If two -cocycles, , differ by a coboundary: , , then, the corresponding action amplitudes, and , can smoothly deform into each other without phase transition. So and , or and , describe the same quantum phase. Therefore, we regard and to be equivalent. The equivalent classes of the -cocycles form the cohomology class . We conclude that the topological terms in weakly coupled lattice gauge theories are described by in space-time dimensions.

For finite gauge group, we can choose a flat connection for the -bundle . Given that, the connection on is always flat regardless of the embedding . In this case, the topological gauge theory defined via the classifying space is closely related to the lattice topological gauge theory defined in section II. On the other hand, we can also choose a non-flat connection for the -bundle . In this case, the different embeding will give rise to different connections on . So the gapped phases of the gauge theory classified by can appear even when there are weak fluctuations of the “field strength ”. Certainly, those gapped phases can also appear when the “field strength ” are zero, as discussed in section II. For finite group , we have (see eqn. (76)).

For continuous gauge group, the connection for the -bundle is always non-flat. In this case, the different embeddings always give rise to different connections on . So the the gauge theory in general contain fluctuations of the “field strength ”.

In appendix B, we show that has a form . So for continuous groups, may not be discrete and the corresponding topological terms are also not quantized. So the quantized topological terms are described by the discrete part of :

 Dis[Hd(BG,R/Z)] =Zn1⊕Zn2⊕... =Tor[Hd+1(BG,Z)] (25)

(see eqn. (85)). (Note that for finite group .) We can use the torsion of the cohomology class of the classifying space to construct the quantized topological terms.

### iii.3 The relation between the first and the second constructions

For finite gauge group , its classifying space has a property . So, each non-trivial loop in can be associated with a non-trivial element in , while the trivial loop (or a point) is associated with the identity element in . For continuous group, we can choose a one-to-one mapping between the non-trivial elements in and a set of loops in that all go through the base point in . As an element approaches the identity, its loop shrinks to the base point. Using such a property, we can understand the relation between the first and the second constructions discussed above.

The lattice gauge theory in the first construction is defined on a space-time complex . A lattice gauge configuration is given by a set of group elements, , on each link . So a lattice gauge configuration corresponds to a 1-skeleton in . The 1-skeleton is formed by the loops that correspond to .

A triangle in is mapped to a loop in using the above correspondence. If the gauge configuration is flat: , the loop is contractible. If is finite for . So there is a unique way to extend the above contractible loop to a disk in . This way, we extend the 1-skeleton to a 2-skeleton. Since , we can extend the 2-skeleton to 3-skeleton, etc . Therefore, for a finite group, we can obtain a canonical map from a lattice gauge configuration to an embedding map . Such an embedding map relate the group cohomology cocycle for the group to the topological cocycle in . So there is a clear one-to-one relation between the first and the second construction for finite gauge groups.

For continuous groups, are non-trivial. So the relation between the second construction and lattice gauge theory is less clear. For a lattice gauge configuration with , there is a unique way to extend the 1-skeleton to an embedding map . For example, even when , we can still uniquely extend a small triangle to a disk with the smallest area.

We can use this idea to find a map from a lattice gauge configuration to an embedding map by choosing the extension with the minimal area/volume. The topological action obtained this way is topological at least when . We can extend to any values of far from and still keep its topological properties. The resulting may not be a continuous function of the lattice gauge configuration . But it is a measurable function (ie the discontinuity happens only on a measure-zero set).

## Iv Differential character and topological gauge theory in d = odd space-time dimensions

In the last section, we constructed topological terms in a weakly coupled gauge theory assuming that the action does not depend on the connection on the space-time complex , as long as has no boundary. In this section we are going to relax such a restriction and allow the action to depend on the gauge connection for = odd space-time dimensions. However, we will still assume that the action is independent of the “metrics” of , which ensure the constructed term to be topological. Such a generalized topological term corresponds to a Chern-Simons term.Dijkgraaf and Witten (1990) For simplicity, the the rest of this section, we will concentrate on = 3 space-time dimensions. However, the results and approaches can be easily generalized to any odd dimensions.

### iv.1 3d Chern-Simons theory

First, let us define the Chern-Simons theory carefully. Naively, a Chern-Simons theory of gauge group on a closed 3 space-time manifold is defined by the action

 SCS=∫MK4πTr(AF−13A3). (26)

However, such a definition is incomplete, since for some smooth gauge configurations , the gauge potential cannot be well defined smooth functions on . To fix this problem, we may try to view as the boundary of : , and try to define the Chern-Simons theory action asDijkgraaf and Witten (1990)

 SCS=∫BK4πTr(F2). (27)

But it may not be always possible to extend the gauge configuration on to . Let us assume that the boundary of is copies of : , and let us assume that for a proper , the gauge configuration on can be extended to . In this case, we can define the Chern-Simons theory action asDijkgraaf and Witten (1990)

 SCS=1n∫BK4πTr(F2). (28)

In the following, we will implement the above idea more rigorously, which allow us to define a generalized Chern-Simons in any odd space-time dimensions and for any gauge group .

### iv.2 3d Chern-Simons theory of gauge group G

In our brief discussion of constructing Chern-Simons terms above, we have introduced the need for a four dimensional manifold in which embeds. A most natural choice, given our task to classify these terms that depends on gauge connections, would be to choose some inside the classifying space , such that .

To understand how the integer emerges, let us consider -homology class of the classifying space . This classifies the obstruction for a given closed three manifold to be the boundary of some four manifold in . For a finite group however, contains only torsion.1 For continuous group, also contains only torsion if is odd. Thus contains only torsion.

Let be 3-dimensional and let be the integer such that . So for any embedding , is a boundary of -dimensional complex : inside . Following the idea in section IV.1, a suitable action of the Chern-Simons theory is given byDijkgraaf and Witten (1990)

 SCS2π=−1n(∫BK4π2F2−⟨ω,B⟩)mod 1, (29)

for some . This definition works both for finite and continuous compact groups. One can see that Eqn. (29) is basically the Chern-Simons action (28). We note that the choice of the pair and defines the theory. However, they are not independent. In fact, has to be chosen such that for all closed manifolds .2 This implies that the action is in fact exact, and the theory is truly three dimensional, which contrasts with WZW theories. In other words, there must be some analogue of Chern-Simons forms depending on the connection ,3 such that

 dθ(A)=K4π2F2−ω, (30)

and that the action can be rewritten as

 SCS2π=⟨~θ(A),γ(M)⟩, (31)

where . The connection evaluated on is determined by the embedding . Therefore is a function of . i.e. We write . It turns out that indeed exists, and the corresponding , is called the differential characters, which is uniquely determined for given for compact groups.

Therefore the Chern-Simons action is classified by . Having defined the action, the path-integral is given by a sum over embedding , corresponding to a sum over different bundles and connections on

 Z=∑γeiSCS[γ]−S% dyn. (32)

We can see that in this formulation of the Chern-Simons theory, its connection with the non-linear sigma model discussed in section III is very explicit, where space-time manifold is embedded in the target space with the embedding . Eqn. (29) however, is sensitive to the connection, therefore relaxing the requirement in section III. Consider several limiting cases. For simply connected compact groups, such as , there is no non-trivial torsion, and . The term involving in Eqn. (29) contributes only to an integer and thus becomes trivial, and the action exactly reduces to (28). i.e. The differential character reduces to the Chern-Simons form. On the other hand, by comparing with Eqn. (24), we realize that when , the differential character reduces to a cocycle in , and thus coincide with the non-linear sigma model. In other words, the non-linear sigma model in forms only a subset of the Chern-Simons theory. In the case of a finite group however , and is isomorphic to . Thus in these cases the non-linear sigma models is in one-to-one correspondence with the Chern-Simons theories. As we will discuss in the next subsection, this is in fact precisely the topological lattice gauge theory.

### iv.3 3d topological lattice gauge theory and Chern-Simons theory

In this section, we would like to make connection between the Chern-Simons theory for finite group defined in the previous section and the topological gauge theory in section II. The discussion here closely parallels that in Ref. Dijkgraaf and Witten, 1990.

In the remaining part of this paper, we will only consider the case of finite gauge groups. In the case of finite groups, we can choose such that there is no non-trivial field strength, by setting for any configurations with non-trivial field strength. In this case, finite field strength gives rise to gapped excitations. So the low energy physics below the gap is controlled by configurations. We choose for configurations with zero field strength. Since in the following, we will limit ourselves to zero-field-strength configurations only, we will drop .

Those field configurations can be characterized by Wilson loops, corresponding to maps from the fundamental group to . This assignment of group element on each loop in depends on which loop in it is mapped to. In other words, the assignment depends entirely on the embedding , since each homotopy class of loops in is assigned a unique element .4 Also, as already noted in the previous section, in this case the differential character also reduces to a 3-cocycle in . The path-integral can then be understood as

 Z = Missing dimension or its units for \hskip (33) = ∑{gi},i∈π1(M)⟨γ∗α({gi}),M⟩,

where is the set of group elements in assigned to each homotopy class of loops in , and we have rewritten the dependence of the Chern-Simons action on the embedding as a dependence on the set . An admissible set is not arbitrary, as we will explain in more detail in some simple examples later. In fact, they form a representation of the fundamental group . This action is already very suggestive that we are dealing with a topological lattice gauge theory. To make precise the connection with the lattice theory, one needs to triangulate the space-time manifold for simplices each with some orientation , and demonstrate that Eqn. (33) can be broken down into local contributions from each simplex. This can indeed be achieved in two steps.

#### Path-Integral on a single simplex

First, one needs to define the path-integral for a single simplex . A path-integral for a simplex is one for which the manifold concerned has boundaries. Let us comment briefly on the physical meaning of a path integral on a manifold with a boundary. Consider a space-time manifold with a single boundary , on which we need to specify boundary conditions. i.e. we fix the boundary value of the embedding map , and we only sum over all maps which reduce to the boundary value in the path “integral”. The boundary has thus led to some physical degrees of freedom that reside at the boundary, to which one can associate with it a Hilbert space and the path integral with specific boundary condition can then be understood as the wavefunction that describes a particular state defined on a given dimensional fixed-time slice. Here we are identifying the direction orthogonal to as time. Note that if has multiple boundaries, a Hilbert space would be associated to each boundary, and the path-integral is a multi-linear map that maps to a phase. Let us now return to the path-integral of a single simplex. For concreteness, consider and the simplex is a tetrahedron. The surface of a tetrahedron is bounded by four triangles with six edges connecting four vertices. This provides a convenient way to obtain a basis for the Hilbert space on the surface of the tetrahedron. The idea is that a base point is chosen in , such that for each embedding they are deformed to make sure that all the vertices in the tetrahedron (and ultimately the entire triangulation of space-time