September 23, 2005

A. Gustavsson on Surface Holonomy

Posted by urs

A while ago there appeared a new paper on surface holonomy, which I had missed at that time, being busy with other things, and which I came across by chance yesterday:

Andreas Gustavsson
A reparameterization invariant surface ordering
hep-th/0508243 .

Here is a summary and some discussion.

Update 28 Sept. 2005: As a reaction to the following entry the paper has been replaced by a revised version. For instance the metric factor discussed below no longer appears there.

According to the introduction, the author is motivated by the desire to find a nonablian surface holonomy that does not involve an auxiliary 1-form. I assume this is referring to approaches like in hep-th/9903168, hep-th/0309173, hep-th/0412325 and hep-th/0509163, which realize 2-functors from surfaces to strict 2-groups as pairs of a 2-form $B$ and a 1-form $A$ with values in a differential crossed module of Lie algebras.

Another motivation for A. Gustavsson is to make contact with the ideas recently published by E. Akhmedov (hep-th/0503234), which we had some discussion about here and here.

There are three main aspects in A. Gustavsson’s paper:

1) First of all he defines a certain 1-form on loop space constructed from a globally defined $\mathrm{Lie}\left(G\right)$-valued 2-form on target space and later justifies this definition by claiming that its holonomy in loop space gives a well-defined surface holonmy.

2) Supplementary to this construction there is a discussion how this is motivated from ideas presented by E. Akhmedov.

3) Finally there are some comments on how one might construct action functionals for the nonabelian 2-form, following ideas by T. Larsson (math-ph/0205017).

I will concentrate in the following on the first of these aspects. As before with the discussion of Akhmedov’s work, my interest is in seeing if we can get a 2-group valued 2-functor from the proposed connection 1-form on loop space. The reason for this is that in that case (and currently only in that case) would we know immediately, by the general nonsense discussed here, how to make this notion of surface holonomy globally well defined.

Namely, if we have 2-holonomy 2-functors on each contractible patch, we can glue them on double overlaps by pseudonatural transformations and glue these on triple overlaps by modifications of pseudonatural transformations which themselves are glued on quadruple overlaps by an identity 3-isomorphism.

The point of this is that one can show that given such data glued in such a way, it follows that we can glue the local surface holonomy on each patch consistently, gauge invariantly and independent of choices of covering etc. (think ‘global anomaly free’ if you are a physicist) and hence get well-defined action functionals for strings coupled to that loop space connection.

Below I will sketch some observations on how it might be possible to indeed get a 2-functor with values in a weak 2-group from (a slight modification of) A. Gustavsson’s construction.

(1)$\phantom{\rule{thinmathspace}{0ex}}$

So, let’s fix some base manifold $M$, which we’d maybe better assume to be contractible for the moment, and fix global coordinates $\left\{{x}^{\mu }\right\}$ on $M$. Also fix some Lie algebra $\mathrm{Lie}\left(G\right)$ and choose a basis $\left\{{t}_{a}{\right\}}_{a\in A}$ for it in terms of which the structure constants are

(2)$\left[{t}_{a},{t}_{b}\right]={C}_{\mathrm{ab}}{}^{c}{t}_{c}\phantom{\rule{thinmathspace}{0ex}}.$

Let $\mathrm{LM}$ be the free loop space over $M$ (nothing of the following would crucially change if we’d considered based loop space or pinned path space instead).

A. Gustavsson now defines an infinite-dimensional algebra with generators called

(3)${t}_{a}\left(\sigma ,\gamma \right)$

for all $a\in A$, $\sigma \in \left[0,1\right]$ and $\gamma \in \mathrm{LM}$.

The antisymmetric product on the generators which come from the same $\gamma$ he defines to be

(4)$\left[{t}_{a}\left(\sigma ,\gamma \right),{t}_{b}\left(\sigma \prime ,\gamma \right)\right]={C}_{\mathrm{ab}}{}^{c}\frac{\delta \left(\sigma -\sigma \prime \right)}{\sqrt{\mid \gamma \prime {\mid }^{2}\left(\sigma \right)}}{t}_{c}\left(\sigma ,\gamma \right)\phantom{\rule{thinmathspace}{0ex}}.$

The square root term is apparently supposed to be necessary to ensure reparameterization invariance of some surface holonomy later on, though I must say I am not sure that we really need it at this point. Maybe I am missing something. More on that below.

Note that in order to make sense of the above expression we need a metric on $M$. A metric structure is not what one would expect to need in order to construct surface holonomy. I’d believe one can do without.

In order to define the product between generators ${t}_{a}\left(\sigma ,{\gamma }_{1}\right)$ and ${t}_{b}\left(\sigma \prime ,{\gamma }_{2}\right)$ that are not associated to the same loop, A. Gustavsson tells us to pick any third loop ${\gamma }_{3}$ and parameters $\kappa$ and $\kappa \prime$ such that

(5)${\gamma }_{3}\left(\kappa \right)={\gamma }_{1}\left(\sigma \right)$
(6)${\gamma }_{3}\left(\kappa \prime \right)={\gamma }_{2}\left(\sigma \prime \right)\phantom{\rule{thinmathspace}{0ex}}.$

He gives a calculation that shows that the definition

(7)$\left[{t}_{a}\left(\sigma ,{\gamma }_{1}\right),{t}_{b}\left(\sigma \prime ,{\gamma }_{2}\right)\right]:=\left[{t}_{a}\left(\kappa ,{\gamma }_{3}\right),{t}_{b}\left(\kappa \prime ,{\gamma }_{3}\right)\right]$

in terms of the above defined bracket is independent of these choices.

So this defines an infinite-dimensional algebra. Consider a 1-form $𝒜$ on loop space with values in this algebra of the form

(8)$𝒜\left(\gamma \right)={\int }_{0}^{1}d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }^{a}\left(\sigma ,\gamma \right){t}_{a}\left(\sigma ,\gamma \right)\phantom{\rule{thinmathspace}{0ex}}d{\gamma }^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}},$

where $d$ is the exterior derivative on loop space. The holonomy of this 1-form over a path $\Sigma :\tau ↦{\gamma }_{\tau }$ of loops would be

(9)$\mathrm{hol}\left(\Sigma \right)={\mathrm{P}}_{\tau }\int d\tau {\int }_{0}^{1}d\sigma \phantom{\rule{thinmathspace}{0ex}}{\stackrel{˙}{\gamma }}_{\tau }^{\mu }\left(\sigma \right){A}_{\mu }^{a}\left(\sigma ,\gamma \right){t}_{a}\left(\sigma ,\gamma \right)\phantom{\rule{thinmathspace}{0ex}},$

where ${\stackrel{˙}{\gamma }}_{\tau }=\frac{\partial {\gamma }_{\tau }}{\partial \tau }$ and ${\mathrm{P}}_{\tau }$ denotes path ordering with respect to $\tau$. This is the surface holonomy proposed by Gustavsson. It takes values in the group obtained by formally exponentiating the algebra of the ${t}_{a}\left(\sigma ,\gamma \right)$.

One should think of $A\left(\sigma ,\gamma \right)$ as coming from the pull-back of a $\mathrm{Lie}\left(G\right)$-valued 2-form $B$ on $M$:

(10)${A}_{\mu }\left(\sigma ,\gamma \right)={B}_{\mu \nu }\left(\gamma \left(\sigma \right)\right)\gamma {\prime }^{\nu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

With reference to the old paper by Teitelboim Gustavsson claims that this holonomy is reparameterization invariant since ${t}_{a}\left(\sigma ,\gamma \right)$ and ${t}_{b}\left(\sigma \prime ,\gamma \right)$ commute for $\sigma \ne \sigma \prime$.

(11)$\phantom{\rule{thinmathspace}{0ex}}$

I am wondering if we cannot slightly simplify this construction as follows:

Instead of defining the ${t}_{a}\left(\sigma ,\gamma \right)$, why not simply use the loop algebra of $\mathrm{Lie}\left(G\right)$?

This algebra is defined to consist of all (suitably well behaved) maps

(12)$f:\left[0,1\right]\to \mathrm{Lie}\left(G\right)$

with the bracket defined pointwise:

(13)$\left[f,h\right]=\left(s↦\left[f\left(s\right),h\left(s\right)\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$

In terms of Fourier modes

(14)${f}_{a}^{m}:=\left(s↦{e}^{\mathrm{im}s}{t}_{a}\right)$

this gives the bracket

(15)$\left[{f}_{a}^{m},{f}_{b}^{n}\right]={C}_{\mathrm{ab}}{}^{c}{f}_{c}^{m+n}$

familiar from current algebras (at level $k=0$).

Alternatively, we might pass to a ‘delta-basis’

(16)${t}_{a}^{\sigma }$

with bracket

(17)$\left[{t}_{a}^{\sigma },{t}_{b}^{\sigma \prime }\right]=\delta \left(\sigma -\sigma \prime \right){C}_{\mathrm{ab}}{}^{c}{t}_{c}^{\sigma }$

so that we can write each $f\in L\mathrm{Lie}\left(G\right)$ as

(18)$f={\int }_{0}^{1}d\sigma \phantom{\rule{thinmathspace}{0ex}}{f}^{a}\left(\sigma \right){t}_{a}^{\sigma }\phantom{\rule{thinmathspace}{0ex}}.$

Given any 2-form $B$ on $M$ we get an $L\mathrm{Lie}\left(G\right)$-valued 1-form $𝒜$ on $\mathrm{LM}$ by

(19)$𝒜\left(\gamma \right)={\int }_{0}^{1}d\sigma \phantom{\rule{thickmathspace}{0ex}}{t}_{a}^{\sigma }{B}_{\mu \nu }^{a}\left(\gamma \left(\sigma \right)\right)\gamma {\prime }^{\nu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}d{\gamma }^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

The curvature of this 1-form is something like

(20)$ℱ\left(\gamma \right)={c}_{1}{\int }_{0}^{1}d\sigma \phantom{\rule{thickmathspace}{0ex}}{t}_{a}^{\sigma }\left(\mathrm{dB}{\right)}_{\mu \nu \rho }^{a}\left(\gamma \left(\sigma \right)\right)\gamma {\prime }^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}d{\gamma }^{\mu }\left(\sigma \right)\wedge d{\gamma }^{\rho }\left(\sigma \right)+{c}_{2}{\int }_{0}^{1}d\sigma {\int }_{0}^{\sigma }d\sigma \prime {B}_{\mu \nu }^{a}\left(\gamma \left(\sigma \right)\right){B}_{\mu \prime \nu \prime }^{b}\left(\gamma \left(\sigma \prime \right)\right)\gamma {\prime }^{\mu }\left(\sigma \right)\gamma {\prime }^{\mu \prime }\left(\sigma \prime \right)\phantom{\rule{thinmathspace}{0ex}}d{\gamma }^{\nu }\left(\sigma \right)\wedge d{\gamma }^{\nu \prime }\left(\sigma \prime \right)\left[{t}_{a}^{\sigma },{t}_{b}^{\sigma \prime }\right]\phantom{\rule{thinmathspace}{0ex}}.$

$𝒜$ gives a reparameterization invariant surface holonomy if its holonomy is invariant under deformations tangential to a given surface $\tau ↦{\gamma }_{\tau }$. This is the case if the evaluation of $ℱ$ on respective loop space tangent vectors vanishes:

(21)$ℱ\left(\sigma ↦{\stackrel{˙}{\gamma }}_{\tau }\left(\sigma \right),\sigma \prime ↦a\left(\sigma \prime \right){\stackrel{˙}{\gamma }}_{\tau }\left(\sigma \prime \right)+b\left(\sigma \prime \right)\gamma \prime \left(\sigma \prime \right)\right)=0\phantom{\rule{thinmathspace}{0ex}}.$

And this indeed holds for the above $𝒜$.

So we do get a well defined surface holonomy here with values in the loop group $\Omega G$ of $G$. I have made before some vague comments that such a loop group should arise from E. Akhmedov’s approach.

This has already been a long entry. Hence I’ll postpone a discussion of the relation of this to 2-groups.

Posted at September 23, 2005 11:27 AM UTC

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