4.1. Review of the Theory
Analogous to the fact that Chern–Simons theory is a topological theory based on a Lie group and a 3-dimensional manifold, the
theory is also a topological theory based on a notion of a three-group and a 4-dimensional manifold. The notion of a three-group represents a categorical generalization of the notion of a group, in the context of higher gauge theory (HGT); see [
15] for a review and motivation. For the purpose of defining the
theory, we are interested in particular in a strict Lie three-group, which is known to be isomorphic to a so-called Lie two-crossed module; see [
17,
18,
19] for details.
A Lie two-crossed module, denoted as
, is an algebraic structure specified by three Lie groups
G,
H, and
L, together with the homomorphisms
and
, an action ⊳ of the group
G on all three groups, and a
G-equivariant map, called the Peiffer lifting:
In order for this structure to form a two-crossed module, the structure constants of algebras
,
, and
(the Lie algebras corresponding to the Lie groups
G,
H, and
L, respectively), as well as the maps
∂ and
, the action ⊳, and the Peiffer lifting, must satisfy certain axioms; see [
20] for details.
Given a two-crossed module and a four-dimensional compact and orientable spacetime manifold
, one can introduce the notion of a trivial principal three-bundle, in analogy with the notion of a trivial principal bundle constructed from an ordinary Lie group and a manifold; see [
15]. Then, one can introduce the notion of a three-connection, an ordered triple
, where
,
, and
are algebra-valued differential forms,
,
, and
; see [
17,
18,
19]. The corresponding fake hree-curvature
is defined as:
Then, for a four-dimensional manifold
, one can define the gauge-invariant topological
action, based on the structure of a two-crossed module
, by the action
where
,
, and
are Lagrange multipliers and
,
, and
represent the fake three-curvature given by Equation (
28). The forms
,
, and
are
G-invariant symmetric nondegenerate bilinear forms on
,
, and
, respectively. The action (
29) is an example of the so-called higher gauge theory.
By choosing the three bases of generators
,
, and
of the three respective Lie algebras, one can expand all fields in the theory into components as
One can also make use of the following notation for the components of all maps present in the theory, in the same three bases:
The complete gauge symmetry of the
action was studied in [
8] using the techniques of Hamiltonian analysis. It consists of five types of gauge transformations,
G-,
H-,
L-,
M-, and
N-gauge transformations, determined with the independent parameters
,
,
,
, and
, respectively. The form variations of the fields
B,
C,
D,
,
, and
, obtained in [
8] are given as follows:
The gauge transformations (
30) form a group
:
where
denotes the group of
G-gauge transformations, the
H-gauge transformations together with the
L-gauge transformations form the group
, while
and
are the groups of
M- and
N-gauge transformations, respectively. All these groups are determined from the structure of the initial chosen two-crossed module that defines the theory; see [
8] for details.
However, as we have seen in the general theory in
Section 2 and in the example of the Chern–Simons theory in
Section 3, the symmetry group
determined by the Hamiltonian analysis does not include HT transformations, and therefore, the
total gauge group should in fact be
4.2. Explicit HT Transformations
Let us explicitly define the
transformations for the
action (
29). If we denote the dimensions of the Lie algebras
as
the number of independent field components in the theory can be counted according to the following table:
The total number of independent field components is, therefore,
Let
denote all field components, where
. We can write the fields schematically as a column-matrix with six blocks:
The HT transformation is then defined via the parameters
as
The requirement that the variation of the action vanishes enforces the antisymmetry restriction on the parameters,
, for all
. These transformations can be represented more explicitly as a tensorial
block-matrix equation, in the following form:
The coefficients multiplying the variations of the action in the column on the right-hand side are there to compensate the overcounting of the independent field components. Due to the antisymmetry of HT parameters, all
blocks (below the diagonal) are determined in terms of the
blocks (above the diagonal), as follows. For the first column of the parameter matrix in (
33), we have:
For the second column, we have:
The
parameters in the third column are determined via:
while the remaining
parameters in the fourth and fifth columns are determined as:
Finally, in addition to all these, the parameters in the blocks on the diagonal also have to satisfy certain antisymmetry relations, specifically:
Like in the example of the Chern–Simons theory from the previous section, these antisymmetry relations can be satisfied in various multiple ways. All those possibilities are allowed, as long as the identities (
38) are satisfied. The final ingredient in (
33) is the expressions for the variation of the action with respect to the fields, and these are given as follows:
4.3. Diffeomorphisms
As in the case of the Chern–Simons theory, it is instructive to discuss diffeomorphism symmetry. The
action (
29) obviously is diffeomorphism invariant, since it is formulated in a manifestly covariant way, using differential forms. However, one can check that the diffeomorphisms are not a subgroup of the gauge symmetry group
given by Equation (
31), but nevertheless can be obtained as a subgroup of the total gauge group (
32):
Let us demonstrate this. Like in the Chern–Simons case, we want to demonstrate that the form variation of all fields corresponding to diffeomorphisms can be obtained as a suitable combination of the form variations for the ordinary gauge transformations (
30) and the HT transformations (
33). In other words, for an arbitrary choice of the diffeomorphism parameters
from (
24), Equation (
25) should hold in the case of the
theory as well:
Indeed, this can be shown by a suitable choice of parameters. Regarding the parameters of the gauge transformations (
30), the appropriate choice is given as:
Regarding the parameters of the HT transformations (
33), we chose the following special case, with the majority of the parameters equated to zero:
Of course, due to antisymmetry, the nonzero
blocks take negative values of the corresponding
blocks, in accordance with (
34), (
35), and (
36). The three independent nonzero
blocks are chosen as
Finally, substituting (
42) and (
44) into (
30) and (
43), respectively, and then substituting all those results into (
41), after a certain amount of work, one obtains precisely the standard form variations corresponding to diffeomorphisms:
This establishes both relations (
40), as we set out to demonstrate. We note again that the HT transformations play a crucial role in obtaining the result, since we had to choose the parameters (
44) in a nontrivial manner.
4.4. Symmetry Breaking in Theory
Let us now turn to the topic of symmetry breaking and the way it influences HT transformations. To that end, we studied the topological
action, which is a special case of the
action (
29) without the last term:
In order to be even more concrete, let us fix a two-crossed module structure with the following choice of groups:
In other words, we interpret group
G as the Lorentz group, group
H as the spacetime translations group, while group
L is trivial, for simplicity. This choice corresponds to the so-called Poincaré two-group; see [
16] for details. Since the generators of the Lorentz group can be conveniently counted using the antisymmetric combinations of indices from the group of translations, instead of the
G-group indices
, we shall systematically write
, where
are
H-group indices, and the brackets denote antisymmetrization. With a further change in notation from the connection 1-form
to the spin-connection 1-form
, the curvature 2-form
to
, and interpreting the Lagrange multiplier 1-form
C as the tetrad 1-form
e, the
action can be rewritten in new notation as
The ordinary gauge symmetry group for this action has a form similar to (
31):
while the total group of gauge symmetries is extended by the HT transformations, so that
The explicit
transformations are written as a tensorial
block-matrix equation, in the form
where the usual antisymmetry rules apply. Here, we have
The
action (
46) is topological, in the sense that it has no local propagating degrees of freedom. In this sense, it does not represent a theory of any realistic physics. In order to construct a more realistic theory, one proceeds by introducing the so-called
simplicity constraint term into the action, which changes the equations of motion of the theory so that it does have nontrivial degrees of freedom. An example is the action
where the new constraint term features another Lagrange multiplier two-form
. By virtue of the simplicity constraint, the theory becomes equivalent to general relativity, in the sense that the corresponding equations of motion reduce to vacuum Einstein field equations (see [
16] for the analysis and proof). In this sense, constraint terms of various types are important when building more realistic theories; see [
20] for more examples.
However, adding the simplicity constraint term also changes the gauge symmetry of the theory. In particular, it breaks the gauge group
from (
48) down to one of its subgroups, so that the symmetry group of the action
is
This is expected and unsurprising. What is less obvious, however, is that the group of HT transformations
of the action
is not a subgroup of the HT group
of the original action
:
which implies that
despite (
53).
Let us demonstrate this. Since the action (
52) features an additional field
, the HT transformations (
50) have to be modified to take this into account and obtain the following
block-matrix form:
where
We can now investigate the differences in the form of HT transformations for the topological and constrained theory. First, comparing (
56) to (
50), we see that the HT transformations in the constrained theory feature
more gauge parameters than are present in the topological theory. Namely, compared to
, the action
features an extra Lagrange multiplier two-form
, which extends the matrix of HT parameters from
blocks to
blocks, and, therefore, introduces the new parameters
and
(and
, which are the negative of
due to antisymmetry). This means that the group
for the constrained theory is
larger than the group
for the topological theory. On the one hand, this immediately proves (
54) and, consequently, (
55). On the other hand, one can ask the opposite question—given that
is larger than
, is the latter maybe a subgroup of the former?
The answer to this question is negative:
which together with (
54) implies our final conclusion:
In order to demonstrate (
58), we can try to set all extra parameters
,
, and
to zero in (
56), reducing it to the same form as (
50). This would naively suggest that
indeed is a subgroup of
. However, upon closer inspection, we can observe that this is not true, since the functional derivatives (
57) are different from (
51). Namely, even taking into account that the choice
eliminates the fifth equation from (
57), the first four equations are still different from their counterparts (
51) because of the presence of the Lagrange multiplier
in the action. The Lagrange multiplier is a field in the theory, and generically, it is not zero, since it is determined by the equation of motion:
Therefore, the HT transformations (
56) in fact cannot be reduced to the HT transformations (
50) by setting the extra parameters equal to zero, which proves (
58) and (
59).
The overall consequences from the above analysis are as follows. The topological action
has a large ordinary gauge group
and a small HT symmetry group
. When one changes the action to
by adding a simplicity constraint term, two things happen—the ordinary gauge group breaks down to its subgroup
, so that it becomes smaller, while the HT symmetry group
grows larger to a completely different group
. In effect, the
total gauge groups for the two actions are intrinsically different:
in the sense that neither is a subgroup of the other. This conclusion is often overlooked in the literature, which mostly puts emphasis on the symmetry breaking of the ordinary gauge group down to its subgroup.
Let us state here, without proof, that the action (
52) represents an example of a non-topological action, for which one can also demonstrate a property analogous to (
40), that diffeomorphisms are not a subgroup of its ordinary gauge group, but are a subgroup of the total gauge group. Simply put, given that the simplicity constraint term in (
52) breaks the ordinary gauge symmetry group
into its subgroup
(see (
53)), one can expect that diffeomorphisms are not a subgroup of
, since they are not a subgroup of the larger group
of the topological action (
46). Nevertheless, since the action (
52) is written in a manifestly covariant form, diffeomorphisms are certainly a symmetry of the action and, thus, must be a subgroup of the total gauge group
, in line with the statement analogous to (
40). We leave the details of the proof as an exercise for the reader. The point of this analysis was to demonstrate that the interplay (
40) between diffeomorphisms and the HT symmetry is a generic property of a large class of actions, including the physically relevant ones, and not limited to examples of topological theories such as the Chern–Simons or
models.
As the last comment, let us remark that, in fact, almost all conclusions discussed for the cases of the Chern–Simons, , and theories are not really specific to these concrete cases. One can easily generalize our analysis to any other theory, and the conclusions should remain unchanged, except maybe in some corner cases.