June 23, 2005

From Loop Space Mechanics to Nonabelian Strings

Posted by Urs Schreiber

I spent June with finishing my thesis (see the links below). A doughnut and an acknowledgement for every helpful comment that makes it into the last changes.

Update 22 Sept. 2005: It is now available on the arXiv as hep-th/0509163.

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

(1)${\mathrm{hol}}_{i}:{𝒫}_{2}\left({U}_{i}\right)\to {G}_{2}$

from the 2-path 2-groupoid ${𝒫}_{2}\left({U}_{i}\right)$ 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

(2)${g}_{\mathrm{ij}}:{\mathrm{hol}}_{i}\to {\mathrm{hol}}_{j}$

are pseudonatural transformations

between the local 2-holonomy 2-functors and the

(3)${f}_{\mathrm{ijk}}:{g}_{\mathrm{ik}}\to {g}_{\mathrm{ij}}\circ {g}_{\mathrm{jk}}$

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

(4)$\mathrm{hol}:{𝒫}_{p}\left(M\right)\to {\mathrm{Trans}}_{p}\left(E\right)\phantom{\rule{thinmathspace}{0ex}},$

where ${𝒫}_{p}\left(M\right)$ is the $p$-category of $p$-paths in $M$ and ${\mathrm{Trans}}_{p}\left(E\right)$ 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 $\mathrm{Gp}$-$p$-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 ${𝔭}_{2}\left({U}_{i}\right)$ (compare I, II, III, IV, V), the structure 2-group by a 2-algebra or 2-algebroid ${𝔤}_{2}$ and the 2-holonomy functor by a 2-connection morphism

(5)${\mathrm{con}}_{i}:𝔭\left({U}_{i}\right)\to {𝔤}_{2}\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)$\left(\delta +Q\right)\omega =0\phantom{\rule{thinmathspace}{0ex}},$

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

(7)$D=\delta +Q$

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 $\mathrm{𝔰𝔭𝔦𝔫}\left(n{\right)}_{1}$ is equivalent to a Frechét Lie 2-algebra ${𝒫}_{1}\mathrm{𝔰𝔭𝔦𝔫}\left(n\right)$ whose Lie 2-group ${𝒫}_{1}\mathrm{Spin}\left(n\right)$ is related to the group $\mathrm{String}\left(n\right)$ 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.

Posted at June 23, 2005 8:57 AM UTC

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Re: Almost Done

Congratulations, Urs. I look forward to hearing John’s talk at the Streetfest! Would you really send a doughnut to the other side of the world? I like chocolate…

All the best, Kea

Posted by: Kea on June 23, 2005 9:56 PM | Permalink | Reply to this

Doughnuts to Australia

Would you really send a doughnut to the other side of the world?

I would!

Maybe I could even give it to you personally. Since it seems that I can submit my thesis earlier than originally planned, I am now playing with the crazy thought of coming to the Streetfest after all. I see that application deadline is June 30, so that would be fine.

Only thing I don’t know is: how long does it take for the visa or ETA to be issued after application? Do I still have a chance?

What makes this plan a little crazy is that I’d have to be back on June 15 afternoon to give a talk in Bad Honnef.

Posted by: Urs Schreiber on June 24, 2005 11:45 AM | Permalink | Reply to this

Re: Doughnuts to Australia

Wow - it would be great to see you! Unfortunately I’m not sure how easy it would be to get a visa. It depends what passport you have. I suspect that if you have a European passport it shouldn’t be a problem.

Posted by: Kea on June 24, 2005 10:05 PM | Permalink | Reply to this

Re: Doughnuts to Australia

Coincidentally, my new supervisor will be at Streetfest as well. When I heard he was going, I asked if I could go too, but alas :) It sounds like a great time. Hope you can make it.

Cheers,
Eric

Posted by: Eric on June 24, 2005 10:19 PM | Permalink | Reply to this

Re: Doughnuts to Australia

My manager now says she disapproves of a hasty and expensive journey to Australia which would make me miss my father’s 60th birthday party. I guess she is right – too bad.

Posted by: Urs on June 26, 2005 1:22 PM | Permalink | Reply to this

Re: Almost Done

Excellent news!

I will do my very best to try to give some feedback (and maybe a figure or two), but I wouldn’t hold your breath. I accepted an offer in California and am leaving this weekend. Tomorrow is my last day in NY and I’m taking care of loose ends. It is madness! I’m sure you can relate :)

Best wishes,
Eric

Posted by: Eric on June 23, 2005 10:01 PM | Permalink | Reply to this

(and maybe a figure or two)

That would be fantastic. However, deadline now is July 1st, unfortunately.

(By the way, I still haven’t solved that problem with your figures: they still produce an error when I try to convert my dvi into pdf and then they are missing in the resulting pdf. That’s why I provided also a ps version, where this problem does not occur.)

I accepted an offer in California and am leaving this weekend.

Cool! Congratulations!

You sure have had a rather exciting life lately, it seems. All my best wishes.

I hope that next time that I am in the US we can manage to meet, finally.

Posted by: Urs Schreiber on June 24, 2005 12:03 PM | Permalink | Reply to this
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