Generalized Higher Gauge Theory
Patricia Ritter, Christian Sämann and Lennart Schmidt
Dipartimento di Fisica ed Astronomia
Università di Bologna and INFN, Sezione di Bologna
Via Irnerio 46, I-40126 Bologna, Italy
[0.5cm] Maxwell Institute for Mathematical Sciences
Department of Mathematics, Heriot-Watt University
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.
[0.5cm] Email: email@example.com , C.Saemann@hw.ac.uk , firstname.lastname@example.org
We study a generalization of higher gauge theory which makes use of generalized geometry and seems to be closely related to double field theory. The local kinematical data of this theory is captured by morphisms of graded manifolds between the canonical exact Courant Lie 2-algebroid over some manifold and a semistrict gauge Lie 2-algebra. We discuss generalized curvatures and infinitesimal gauge transformations. Finite gauge transformation as well as global kinematical data are then obtained from principal 2-bundles over 2-spaces. As dynamical principle, we consider first the canonical Chern-Simons action for such a gauge theory. We then show that a previously proposed 3-Lie algebra model for the six-dimensional (2,0) theory is very naturally interpreted as a generalized higher gauge theory.
- 1 Introduction and results
- 2 Kinematical description
- 3 Dynamics
1 Introduction and results
Higher gauge theory describes the parallel transport of extended objects transforming under local internal symmetries. There are well-known no-go theorems stating that in a naive setting, the internal symmetry group has to be abelian for objects with positive dimension. To avoid these theorems, one has to categorify the ingredients of usual gauge theory, see e.g.  for details. This leads in particular to categorified structure groups, known as -groups, as well as categorified notions of principal bundles known as principal -bundles.
One severe open problem in higher gauge theory is the lack of non-trivial examples of non-abelian principal -bundles with connection. For example, one would expect categorified analogues of non-abelian monopoles and instantons to exist. Although higher analogues of the twistor descriptions of monopoles and instantons have been constructed [2, 3, 4], the known solutions, e.g. those of , do not quite fit the picture.111If higher gauge theory is to describe the parallel transport of some extended objects, then a condition needs to be imposed on the curvature of the principal -bundle to ensure that the parallel transport is invariant under reparameterizations. While the solutions of  do not directly satisfy this curvature condition, one could argue that at least in the case of the self-dual strings considered in , the underlying parallel transport of strings is trivial and the fake curvature condition becomes physically irrelevant.
This lack of examples presents an obstacle to both mathematical as well as physical progress in the study of higher gauge theory. It is therefore important to find generalized formulations which allow for interesting examples. In this paper, we study the case in which also the base manifold is categorified to what has been called a 2-space : a category internal to the category of smooth manifolds.
In recent developments in string theory, there are many pointers towards the necessity of using 2-spaces instead of ordinary space-time manifolds, in particular in relation with generalized geometry and double field theory. In both contexts, it is usually the exact Courant algebroid over some manifold , which is used to give expressions a coordinate-invariant meaning, see e.g. [7, 8, 9]. It is therefore only natural to ask whether a definition of gauge theory involving this algebroid has some interesting features.
Recall that the local kinematical data of ordinary gauge theory over some manifold can be described by a morphism of graded algebras between the Chevalley-Eilenberg algebra of the gauge Lie algebra and the Weil algebra of the manifold , which is the Chevalley-Eilenberg algebra of the tangent Lie algebroid . For higher gauge theory, the domain of this morphism is extended to the Chevalley-Eilenberg algebra of some -algebra. In this paper, we also generalize the range of this morphism to the Chevalley-Eilenberg algebra of the Courant algebroid, cf. figure 1. The latter should more properly be regarded as a symplectic Lie 2-algebroid, and thus we arrive at a notion of gauge theory over the 2-space .
We discuss in detail the case where the gauge -algebra consists of two terms, corresponding to a semistrict Lie 2-algebra. In particular, we derive the form of the gauge potential and its curvature, which are encoded in the morphism of graded algebras and its failure to be a morphism of differential graded algebras. We also give the relevant formulas for infinitesimal gauge transformations. As we show, these results can also be obtained from the homotopy Maurer-Cartan equations of an -algebra consisting of the tensor product of the gauge -algebra with the Weil algebra of .
To glue together local kinematical data to global ones, we need a generalized principal 2-bundle structure as well as finite gauge transformations. We find both by considering principal 2-bundles over the 2-space . We thus arrive at an explicit formulation of the first generalized higher Deligne cohomology class, encoding equivalence classes of these higher bundles with connection.
In a second part, we discuss two possible dynamical principles for the generalized higher connections. The first one is a Chern-Simons action, which is obtained via a straightforward generalization of the AKSZ procedure. The second one is a previously proposed set of equations for a 3-Lie algebra222not to be confused with a Lie 3-algebra-valued -tensor supermultiplet in six dimensions . We show that these equations find a very natural interpretation within generalized higher gauge theory. In particular, the 3-Lie algebra valued vector field featuring crucially in the equations is part of a generalized higher connection.
Among the open questions we intend to study in future work are the following. First, an additional gauge algebra-valued vector field seems to be desirable in many open questions related to the six-dimensional (2,0)-theory. It would be interesting to see if such problems can be addressed within our framework. Second, the Courant algebroid appears in double field theory after imposing a section condition. One might therefore want to formulate a full double gauge theory, related to ours only after the section condition is imposed. Such a double gauge theory might have interesting applications in effectively describing string theory dualities. Third, it remains to be seen whether we can write down six-dimensional maximally superconformal gauge equations which are less restrictive than those obtained in , using generalized higher gauge theory. Finally, as stated above, it would be most interesting to extend the twistor descriptions of [2, 3, 4] to generalized higher gauge theory and to explore the possibility of genuinely non-trivial and non-abelian generalized principal 2-bundles with connection.
2 Kinematical description
We begin by reviewing the notion of N-manifolds and their relation to -algebras. We use this language to describe ordinary gauge theory in terms of morphisms of graded manifolds and show how this extends to higher gauge theory, following [11, 12, 13, 14, 15, 16]. This formulation naturally allows for a generalization to gauge theory involving the exact Courant algebroid .
Formally, an N-manifold is a locally ringed space , where is a manifold and is an -graded commutative ring replacing the ordinary structure sheaf over . More explicitly, we can think of an N-manifold as a tower of fibrations
where is a manifold and for are linear spaces with coordinates of degree , generating the structure sheaf. For more details on this, see e.g. . A morphism of N-manifolds is then a morphism of graded manifolds . In more detail, we have a map between the underlying manifolds and a degree-preserving map between the structure sheaves, which restricts to the pullback along on the sheaf of smooth function on , . Note that for higher degrees, is completely defined by its image on the local coordinates that generate .
An N-manifold is now an N-manifold together with a homological vector field , that is, a vector field of degree squaring to zero: . The algebra of functions on given by global sections of together with now forms a differential graded algebra. A morphism of N-manifolds is then a morphism between N-manifolds and that respects the derivation , i.e. .
Physicists may be familiar with N-manifolds from BRST quantization, where the coordinate degree and correspond to the ghost number and the BRST charge, respectively.
A basic example of an N-manifold is given by , where we always use to denote a shift of the degree of some linear space (often the fibers of a vector bundle) by . On , we have coordinates on the base and the fibers of degree and respectively, i.e. we have an N-manifold concentrated in the lowest two degrees. Note that the algebra of functions on can be identified with the differential forms . Moreover, endowing with the homological vector field promotes it to an N-manifold. In the identification , becomes the de Rham differential.
A more involved example is . Recall that the functor gives extra coordinates with opposite degree to the fibers in . Therefore, local coordinates , , on are of degree and , respectively. For convenience, we group the coordinates of degree into a single , where the index runs from 1 to . A canonical choice of homological vector field is now
which can be “twisted,” e.g., to
where is a closed 3-form on , cf. . We shall work mostly with the case . Altogether, we arrive at an N-manifold concentrated in degrees to . This N-manifold is the one underlying the exact Courant algebroid , and we will come back to this point later. Also, this example is part of a larger class of N-manifolds given by containing the Vinogradov algebroids . For more details, see e.g. .
Another important example of N-manifolds is that of a grade-shifted Lie algebra with basis of degree and coordinates of degree . The algebra of functions is given by and is necessarily of the form
where are the structure constants of the Lie algebra . The condition directly translates to the Jacobi identity. This alternative description of a (finite-dimensional) Lie algebra is the well-known Chevalley-Eilenberg algebra CE of and we can thus think of a Lie algebra as an N-manifold concentrated in degree 1. Analogously, we will refer to the differential graded algebra consisting of the algebra of functions on an N-manifold together with the differential given by the homological vector field as the Chevalley-Eilenberg algebra of .
We can readily extend the last example, replacing the shifted Lie algebra by some shifted graded vector space, which we also denote by . On the latter, we introduce a basis of degree and coordinates of degree in . The vector field is then of the form
where can be non-zero only if since is of degree . The minus signs and normalizations are chosen for convenience.
We now also introduce a basis on the unshifted , where we absorb all grading in the basis instead of the coordinates. Thus, has degree . The structure constants can then be used to define the following graded antisymmetric, -ary brackets on of degree :
For an N-manifold concentrated in degrees to , the condition amounts to the homotopy Jacobi relations of an -term -algebra with higher products , cf. [19, 20]. Such -term -algebras are expected to be categorically equivalent to semistrict Lie -algebras.
As a constructive example let us look at a -term -algebra originating from the N-manifold . In a basis and with corresponding coordinates and of degree and on and , respectively, the vector field reads as
We define corresponding -algebra products , and on via (2.6) and the condition leads to higher homotopy relations, which, in terms of the graded basis and on , are
More generally, if the N-algebra is concentrated in degrees to , one analogously obtains an -algebroid. In fact, and were particular examples of such -algebroids.
The natural notion of inner product on an -algebra arises from an additional symplectic structure on the underlying N-manifold. A symplectic N-manifold of degree is an N-manifold endowed with a closed, non-degenerate 2-form of degree333This is the degree in , not the form degree of . satisfying . If the degree of is odd, such symplectic N-manifolds are also known as P-manifolds  or P-manifolds . In the general case, symplectic N-manifolds of degree are also called -manifolds .
A simple example of a symplectic N-manifold of degree is with coordinates , homological vector field for some anti-symmetric bivector and symplectic form . Indeed, .
We are mostly interested in the N-manifold with coordinates as defined above. With
becomes a symplectic N-manifold of degree 2: we have
As shown in , the data specifying a symplectic N-structure on are equivalent to the data specifying a Courant algebroid structure on the bundle . In particular, sections of the Courant algebroid are functions in which are linear in the coordinates . Moreover, a metric on sections of originates from the Poisson bracket induced by the symplectic structure (2.9). With the symplectic structure in coordinates , we have
where is the inverse matrix to . With our choice of symplectic structure (2.9), we have
For simplicity, we will refer to both the symplectic N-manifolds and the vector bundle with Courant algebroid structure as Courant algebroid.
Note that the exact Courant algebroid features prominently in generalized geometry and double field theory. We therefore expect our following constructions to be relevant in this context.
2.2 Gauge connections as morphisms of N-manifolds
In ordinary gauge theory, we consider connections on principal -bundles over some manifold and encode them locally as 1-forms taking values in a Lie algebra . The curvature of is and gauge transformations are parameterized by -valued functions and act on via . At infinitesimal level, these are given by , where with a -valued function.
Let us now reformulate the local description of gauge theory using morphisms of N-manifolds. As discussed in section 2.1, differential forms can be encoded as functions on the N-manifold . In terms of coordinates of degree and , respectively, the de Rham differential corresponds to the homological vector field . We also regard as an N-manifold with coordinates of degree and , cf. again section 2.1.
A local connection 1-form is then encoded in a morphism of N-manifolds between these two N-manifolds
Recall that it suffices to define the action of on the local coordinates of , so we define
where is a basis on . The curvature of then describes the failure of to be a morphism of N-manifolds:
Indeed, we have
where denotes the Lie bracket on .
Gauge transformations between and are encoded in flat homotopies between these, that is, morphisms which are flat along the additional direction . More precisely, given coordinates along and on , we have
Note that defines a gauge potential
and a curvature
where we used the amended on . For the homotopy to be flat, we require , which implies that
Restricting to yields the usual formula for gauge transformations with infinitesimal gauge parameter . Integrating (2.19) with the boundary condition , we obtain the finite form
where is the path-ordered exponential of along .
2.3 Higher gauge connections
We can readily extend the picture of the previous section to the case of higher gauge theory. Here, we simply replace the gauge Lie algebra by a general -algebra .444In principle, we could also allow for -algebroids, which would lead us to higher gauged sigma models. The morphism of N-manifolds now also contains forms of higher degree. Similarly, the curvature, which is again given by the failure of to be a morphism of N-manifolds, leads to higher curvature forms.
As an instructive example let us look at the 2-term -algebra introduced before in section 2.1. The image of the pullback morphism on the coordinates and of degree and on the shifted vector space is given by
where in addition to the -valued -form potential we now also have a -valued -form potential . We combine both into the 2-connection
With from (2.7), we compute the curvature components
which we combine into the 2-curvature
Again, the infinitesimal gauge transformations between and are encoded in homotopies that are flat in the extra homotopy direction. We use coordinates on and we have as well as . Then defines a gauge potentials as before, that is,
Using the extended vector field , we calculate the curvature defined by along the additional direction to be
As before, the infinitesimal gauge transformations are encoded in the flat homotopies for which the above curvature in the directions including vanishes. This leads to the transformations
which are parameterized by two infinitesimal gauge parameters: the -valued function and a -valued -form . We thus obtain the infinitesimal gauge transformations of semistrict higher gauge theory as found e.g. in .555An alternative approach to finite gauge transformations of semistrict higher gauge theory is found in . Putting , we obtain the infinitesimal gauge transformations of strict higher gauge theory, which can be integrated as done in . Setting and to zero reduces the transformation back to the case of ordinary gauge theory.
2.4 Local description of generalized higher gauge theory
We now come to our extension of higher gauge theory to generalized higher gauge theory. To this end, we replace the domain of the morphism of N-manifolds, which has been so far, by the Courant algebroid with coordinates of degree and , respectively, and homological vector field , see section 2.1.
Generalized higher gauge theory is thus given by a morphism of N-manifolds , where is an arbitrary -algebra. We again focus on the example where is a -term -algebra and we introduce a basis and coordinates of degree and , respectively, on . The homological vector field is given in (2.7). The images of the coordinates of under the morphism are
where can now be regarded as the sum of a 1-form and a vector field, which are both -valued. Similarly, consists of a 2-form, a bivector, a tensor of degree (1,1) and a vector field, all taking values in . We combine all these into the generalized 2-connection
The generalized 2-curvature is again obtained from the failure of to be a morphism of N-manifolds, and splits into components according to
The components of are computed to be