The String Coffee Table

Title: From Loop Space Mechanics to Nonabelian Strings (pdf, ps.zip)

Abstract: Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged under a fiber bundle thereby turns into a string charged under a 2-bundle, or gerbe. This stringification is nothing but categorification. We look at supersymmetric quantum mechanics on loop space and demonstrate how deformations here give rise to superstring background fields and boundary states, and, when generalized, to local nonabelian connections on loop space. In order to get a global description of these connections we introduce and study categorified global holonomy in the form of 2-bundles with 2-holonomy. We show how these relate to nonabelian gerbes and go beyond by obtaining global nonabelian surface holonomy, thus providing a class of action functionals for nonabelian strings. The examination of the differential formulation, which is adapted to the study of nonabelian p-form gauge theories, gives rise to generalized nonabelian Deligne hypercohomology. The (possible) relation of this to strings in Kalb-Ramond backgrounds, to M2/M5 brane systems, to spinning strings and to the derived category description of D-branes is discussed. In particular, there is a 2-group related to the String-group which should be the right structure 2-group for the global description of spinning strings.

Part I: Overview (pdf, ps.zip)

Part II: SQM on Loop Space (pdf)

Part III: Nonabelian Strings (pdf)

(in the pdf version some figures in part I are broken)

These are the main results:

From deformations of the superstring’s supercharges one can obtain local connection 1-forms on loop space that give rise to a notion of possibly nonabelian surface holonomy in target space:

(Many thanks once again to Eric Forgy for this figure.)

These 1-forms can be shown to be precisely those that give rise to a 2-functor

from the 2-path 2-groupoid $\mathcal{P}_2(U_i)$ of a patch $U_i$ of target space to a strict structure 2-group $G_2$, called the 2-holonomy 2-functor.

A locally trivialized principal 2-bundle with 2-connection and 2-holonomy is a map from Čech-2-simplices to the 2-functor 2-category of local 2-holonomy 2-functors as indicated by the left arrow $\Omega$ in the following figure:

Here the

are pseudonatural transformations

between the local 2-holonomy 2-functors and the

are quasi-modifications between these

Such a map is characterized by the same cocycle conditions as a nonabelian gerbe, but subject to the constraint of ‘vanishing fake curvature’ which encodes the 2-functoriality of $\mathrm{hol}_i$.

This is joint work with John Baez. It will be presented at the ‘Streetfest’.

Given this data, it is possible to glue the local $\mathrm{hol}_i$ 2-functors to obtain a global 2-holonomy 2-functor. Its action is as follows:

This is the diagrammatic version and generalization to nonabelian (and weak) 2-groups of the well-known formula for the coupling of strings to the Kalb-Ramond field as described for instance in equation (2.14) of

K. Gawedzki & N. Reis
WZW branes and Gerbes
hep-th/0205233

and in equation (3.11) of

A. Carey, S. Johnson & M. Murray
Holonomy on D-branes
hep-th/0204199.

One can elegantly summarize all these structures by saying that a smooth principal $G_p$-$p$-bundle $E \to M$ with $p$-connection and $p$-holonomy over a categorically trivial base space $M$ is a single global smooth $p$-functor

where $\mathcal{P}_p(M)$ is the $p$-category of $p$-paths in $M$ and $\mathrm{Trans}_p(E)$ is the smooth $p$-category whose objects are the fibers $E_x$ of $E$ for each point $x\in M$, regarded as $G_p$-$p$-torsors, and whose $n$-morphisms are the $Gp$-$p$-torsor $n$-morphisms between these. This is demonstrated for $p=1$ and $p=2$.

There is a differential version of all these structures, where the 2-path 2-groupoid is replaced by a 2-algebroid $\mathfrak{p}_2(U_i)$ (compare I, II, III, IV, V), the structure 2-group by a 2-algebra or 2-algebroid $\mathfrak{g}_2$ and the 2-holonomy functor by a 2-connection morphism

This is indicated on the right of the above figures. The assignment, $\omega$, of transition $n$-morphisms of the 2-connection to Čech-$n$-simplices can be shown to be encoded in an equation

where $\delta$ is the Čech coboundary operator acting on complexes of sheaves of 2-connection $n$-morphisms and $Q$ is the natural coboundary operator acting on these which is induced from the dual description of 2-algebroids in terms of chain complexes.

The operator

generalizes the well-known strict and abelian Deligne coboundary operator known from the study of abelian gerbes. Gauge transformations correspond to shifts by $D$-exact terms, $\omega \to \omega + D\lambda$.

I discuss possible applications of this formalism to stringy physics, like the derived category description of D-branes( I, II, III, IV). In particular, the semistrict Lie 2-algebra $\mathfrak{spin}(n)_1$ is equivalent to a Frechét Lie 2-algebra $\mathcal{P}_1 \mathfrak{spin}(n)$ whose Lie 2-group $\mathcal{P}_1 \mathrm{Spin}(n)$ is related to the group $\mathrm{String}(n)$ and seems to have all the right properties to be the structure 2-group for 2-bundles describing parallel transport of spinning strings (superstrings).

This is joint work with John Baez, Alissa Crans and Danny Stevenson. Alissa and Danny will talk about that at the Streetfest, too: I, II.