The String Coffee Table

Holonomy and Deligne-Classes for Nonabelian Gerbes

Gerbes play a role in string theory mostly as gerbes with connecton, namely as structures that admit ‘parallel transport’ of strings and possibly of membranes. This allows physicists to write down globally well-defined (‘anomaly free’) action functionals for these objects.

According to an argument formalized by Aschieri & Jurčo [1] the endstrings of certain membranes are in particular expected to couple to a nonabelian gerbe with connection. While the concept of parallel transport as well as that of Deligne classification is well-known for abelian gerbes with connection, its generalization to the nonabelian case has only more recently emerged [2] within the context of ‘2-bundles’ [3], and has further been developed in [4].

In order to motivate this approach first reconsider ordinary $G$-bundles with connection from the point of view of parallel transport. The most immediate definition of an ordinary $G$-bundle with connection over a space $M$ in this sense is in terms of its holonomy-functor

which maps paths in $M$ (really morphisms of the groupoid $\mathcal{P}_1(M)$ of thin homotopy equivalence classes of paths) to morphisms of $G$-torsors in a suitable smooth way.

Locally, on contractible open subsets $U_i \subset M$, such a functor is naturally isomorphic to a smooth local holonomy functor

from paths to group elements. As is well known, these functors are specified by local connection 1-forms

On double intersections $U_{ij} = U_i \cap U_j$ such functors are found to be related by natural isomorphisms

which induce the familar cocycle conditions on $A_i$ and $g_{ij}$. Gauge transformations, coming from different choices of trivializations, correspond to natural isomorphisms of these fuctors.

This means that with respect to a good covering $\mathcal{U} = \{U_i\}_{i \in I}$ of $M$, the global holonomy functor $\hol : \mathcal{P}_1(M) \to G-\mathrm{Tor}$ defines an isomorphism class of functors from the Čech-groupoid of $\mathcal{U}$ (whose objects are open sets and whose morphisms are double intersections) to the category of local holonomy functors, which is best thought of, equivalently, as a homotopy class of simplicial maps between the nerve of the covering and that of the holonomy functor-category [5].

Now categorify this situation. Fix a smooth category $G_2$ with strict group structure, called a strict 2-group [6], and consider 2-holonomy 2-functors:

that assign 2-morphisms of $G_2$-2-torsors to surfaces in $M$. Given a piece of worldsheet $\Sigma$ of a ‘nonabelian string’, $\mathrm{hol}(\Sigma)$ is supposed to be the parallel transport of a string across $\Sigma$.

What is the information encoded in the specification of such a 2-functor $\mathrm{hol}$? A combination of abstract diagrammatic reasoning together with some path space analysis shows that such 2-functors specify nonabelian gerbes with connection and curving whose fake curvature vanishes and which have a notion of surface holonomy given by $\mathrm{hol}$.

More in detail, the 2-functor $\mathrm{hol}$ is locally isomorphic to 2-functors

that are specified by pairs $A_i \in \Omega^1(U_i, \mathrm{Lie}(G))$ and $B_i \in \Omega^2(U_i,\mathrm{Lie}(H))$ satisfying the fake flatness contion [7]

(Here $H \overset{t}{\to} G$ is a crossed module of groups associated with $G_2$.)

These 2-functors are related on double overlaps by pseudonatural transformations

which themselves are related on triple overlaps by ‘modifications’ of pseudonatural transformations

These modifications finally satify a tetrahedron coherence law on quadruple overlaps.

In terms of the pairs $(A_i,B_i)$ all these transformations and coherence laws translate precisely into the cocycle conditions for a nonabelian gerbe with connection and curving (as displayed in [8,9]).

As before, we can think of this situation as a 2-functor from the Čech 2-groupoid of the covering $\mathcal{U}$ to the 2-category of local 2-holonomy 2-functors. One checks that natural isomorphism of such an assignment correspond to gauge transformations of the above cocycle data. Again [5], it is useful to think of this, equivalently, as a homotopy class of simplicial maps between the respective nerves (now using the notion of nerves of 2-categories as in [10]).

This establishes the notion of nonabelian gerbes/2-bundles with connection and with holonomy. A little exercise in diagram-gluing produces an explicit formula for computing nonabelian surface holonomy from local 2-forms $\{B_i\}$ which generalizes a similar formula well-known for abelian gerbes [11,12].

While the classification of these objects in terms of classes of maps between the Čech 2-groupoid and a 2-functor 2-category is available, it does not seem to lend itself to computations. An efficient nonabelian generalization of the cocycle description of abelian gerbes with connection in terms Deligne hypercohomology would therefore be desireable.

It turns out that, at least at the linearized level, such a description is obtainable by suitably ‘differentiating’ all elements of the above discussion. In order to do so the 2-path 2-groupoids $\mathcal{P}_2(U_i)$ should be replaced by ‘2-path 2-algebroids’ $\mathfrak{p}_2(U_i)$, the structure 2-group $G_2$ similarly by a Lie 2-algebra $\mathfrak{p}_2(U_i)$, the 2-holonomy 2-functor by a morphism

of 2-algebroids, the transition transformation $g_{ij}$ by a respective 2-algebroid 2-morphism $\mathfrak{g}_{ij}$ and so on.

One technical complication for this program is that $p$-algebroids have yet to be formulated in a convenient category-theoretic framework that would allow to extract them by mere application of some $p$-functor from the above disucssion. But for all semistrict Lie $p$-algebras as well as for 1-algebroids and certain 2-algebroids it is known that they have a dual description in terms of $p$-term differential graded algebras. These again fit naturally in $p$-categories whose 1-morphsims are given by chain maps, 2-morphism by chain homotopies, and so on.

Using this dictionary, the entire above discussion translates into the study of simplicial maps from Čech-simplices to categories of morphisms $\mathrm{con}_i : \mathfrak{p}_2(U_i) \to \mathfrak{g}_2$ of dg-algebras. Using the differentials of $\mathfrak{p}_2(U_i)$ and $\mathfrak{g}_2$ one naturally obtains a nilpotent operator $Q$ which makes the sheaves $L^n$ of dg-algebra $n$-morphisms into a complex of sheaves

One finds that a simplicial map from the Čech-groupoid to the category of algebroid morphisms is equivalent to a cochain in the hypercohomology complex

where $\delta$ is the Čech coboundary operator. Homotopies of such maps correspond to shifts by $D$-exact terms.

One checks that in the abelian case this reduces to ordinary Deligne hypercohomology

classifying abelian ($p$-)gerbes with ($p+1$)connection. In the nonabelian case the equation $D\omega = 0$ provides an efficient tool for computing the linearized cocycle conditions of these objects.

The algebroid formalism allows to handle objects with (weak) structure Lie $p$-algebras that are not integrable to Lie $p$-groups and hence have no proper $p$-gerbe analog. An interesting example for this is the semistrict Lie-2-algebra $\mathfrak{so}_k(n)$ which is related to the $\mathrm{String}$-group [13].

[1] P. Aschieri, B. Jurčo, Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory, hep-th/0409200
[2] J, Baez, U. Schreiber, Higher Gauge Theory: 2-Connections on 2-Bundles, hep-th/0412325
[3] T. Bartels, Higher gauge theory: 2-Bundles, math.CT/0410328
[4] U. Schreiber, From Loop Space Mechanics to Nonabelian Strings, PhD thesis
[5] see the contribution by B. Jurčo to this report.
[6] J. Baez, A. Lauda, Higher-Dimensional Algebra V: 2-Groups, math.QA/0307200
[7] H. Pfeiffer, F. Girelli, Higher gauge theory – differential versus integral formulation, hep-th/0309173
[8] L. Breen, Messing, Differential Geometry of Gerbes, math.AG/0106083
[9] P. Aschieri, L. Cantini, Jurčo, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory, hep-th/0312154
[10] J. Duskin Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, TAC 9 (2001)
[11] K. Gawedzki & N. Reis, WZW branes and Gerbes , hep-th/0205233
[12] A. Carey, S. Johnson & M. Murray, Holonomy on D-branes, hep-th/0204199
[13] J. Baez, A. Crans, D. Stevenson, U. Schreiber, From Loop Groups to 2-Groups, math.QA/0504123