August 24, 2005

CFT, Gerbes and K-Theory in Oberwolfach, VII

Posted by Urs Schreiber

You’ll have noticed that I failed to report on the talks of the last two days in Oberwolfach, either because I was too tired, or too occupied or both. Here are some symposium-postprocessing notes.

Update: Peter Woit has followed the Oberwolfach discussion here and has written a related entry.

While searching for literature on what I now know is called ‘differential’ or ‘smooth’ K-theory I came across the report of another recent Oberwolfach workshop which is very closely related to the things I mentioned here recently:

P. Teichner & St. Stolz (Eds.)
Geometric Topology and Connections with Quantum Field Theory
OW Report No. 27/2005

This report is valuable for the references it assembles.

In the above fashion, every participant of an Oberwolfach workshop is supposed to provide a summary of his talk for the OW-Reports. A draft for my summary is the one below:

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

(1)$\mathrm{hol}:{𝒫}_{1}\left(M\right)\to G-\mathrm{Tor}$

which maps paths in $M$ (really morphisms of the groupoid ${𝒫}_{1}\left(M\right)$ 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

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

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

(3)${A}_{i}\in {\Omega }^{1}\left({U}_{i},\mathrm{Lie}\left(G\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

On double intersections ${U}_{\mathrm{ij}}={U}_{i}\cap {U}_{j}$ such functors are found to be related by natural isomorphisms

(4)${\mathrm{hol}}_{i}\stackrel{{g}_{\mathrm{ij}}}{\to }{\mathrm{hol}}_{j}$

which induce the familar cocycle conditions on ${A}_{i}$ and ${g}_{\mathrm{ij}}$. Gauge transformations, coming from different choices of trivializations, correspond to natural isomorphisms of these fuctors.

This means that with respect to a good covering $𝒰=\left\{{U}_{i}{\right\}}_{i\in I}$ of $M$, the global holonomy functor $hol:{𝒫}_{1}\left(M\right)\to G-\mathrm{Tor}$ defines an isomorphism class of functors from the Čech-groupoid of $𝒰$ (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:

(5)$\mathrm{hol}:{𝒫}_{2}\left(M\right)\to {G}_{2}-2\mathrm{Tor}$

that assign 2-morphisms of ${G}_{2}$-2-torsors to surfaces in $M$. Given a piece of worldsheet $\Sigma$ of a ‘nonabelian string’, $\mathrm{hol}\left(\Sigma \right)$ 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

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

that are specified by pairs ${A}_{i}\in {\Omega }^{1}\left({U}_{i},\mathrm{Lie}\left(G\right)\right)$ and ${B}_{i}\in {\Omega }^{2}\left({U}_{i},\mathrm{Lie}\left(H\right)\right)$ satisfying the fake flatness contion [7]

(7)${F}_{{A}_{i}}+\mathrm{dt}\left({B}_{i}\right)=0\phantom{\rule{thinmathspace}{0ex}}.$

(Here $H\stackrel{t}{\to }G$ is a crossed module of groups associated with ${G}_{2}$.)

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

(8)${\mathrm{hol}}_{i}\stackrel{{g}_{\mathrm{ij}}}{\to }{\mathrm{hol}}_{j}$

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

(9)${g}_{\mathrm{ik}}\stackrel{{f}_{\mathrm{ijk}}}{\to }{g}_{\mathrm{ij}}\circ {g}_{\mathrm{jk}}\phantom{\rule{thinmathspace}{0ex}}.$

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

In terms of the pairs $\left({A}_{i},{B}_{i}\right)$ 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 $𝒰$ 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 $\left\{{B}_{i}\right\}$ 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 ${𝒫}_{2}\left({U}_{i}\right)$ should be replaced by ‘2-path 2-algebroids’ ${𝔭}_{2}\left({U}_{i}\right)$, the structure 2-group ${G}_{2}$ similarly by a Lie 2-algebra ${𝔭}_{2}\left({U}_{i}\right)$, the 2-holonomy 2-functor by a morphism

(10)${\mathrm{con}}_{i}:{𝔭}_{2}\left({U}_{i}\right)\to {𝔤}_{2}$

of 2-algebroids, the transition transformation ${g}_{\mathrm{ij}}$ by a respective 2-algebroid 2-morphism ${𝔤}_{\mathrm{ij}}$ 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 ${\mathrm{con}}_{i}:{𝔭}_{2}\left({U}_{i}\right)\to {𝔤}_{2}$ of dg-algebras. Using the differentials of ${𝔭}_{2}\left({U}_{i}\right)$ and ${𝔤}_{2}$ one naturally obtains a nilpotent operator $Q$ which makes the sheaves ${L}^{n}$ of dg-algebra $n$-morphisms into a complex of sheaves

(11)$\dots \to {L}^{n}\stackrel{Q}{\to }{L}^{n-1}\stackrel{Q}{\to }{L}^{n-2}\to \dots \phantom{\rule{thinmathspace}{0ex}}.$

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

(12)$\dots \to {ℋ}^{0}\left(𝒰,{L}^{•}\right)\stackrel{D=\delta ±Q}{\to }{ℋ}^{1}\left(𝒰,{L}^{•}\right)\to \dots \phantom{\rule{thinmathspace}{0ex}},$

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

(13)${ℋ}^{•}\left(𝒰,{L}^{•}\right)\to {ℋ}^{•}\left(𝒰,{\Omega }^{•}\right)$

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 ${\mathrm{𝔰𝔬}}_{k}\left(n\right)$ 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

Posted at August 24, 2005 1:32 PM UTC

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Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

Ah yes, you’ve found it at last. ;) Anyway, there are plenty of examples/reports where that came from.

An update on the notes I promised: they are almost typed up now, but I need to look over it to check that everything makes sense. If anyone wants an extremely rough first draft, let me know.

Posted by: Rongmin on August 25, 2005 6:12 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

Ah yes, you’ve found it at last. ;)

Right, you told me! Sometimes I forget what I wanted to search for five minutes ago, being occupied by searching for something else already. :-)

Anyway, there are plenty of examples/reports where that came from.

I have to admit that I still haven’t made much progrees with really understanding the details of differential K-theory from either that report of Hopkins&Singer. To my embarrassment I have to admit that I am apparently confused by their basic notation. For instance, what precisely is the differential $\delta$ in the very last equation on the bottom of p. 8?

Posted by: Urs on August 25, 2005 10:24 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

what precisely is

Ah, never mind, it’s just the coboundary operator on smooth singular cubic cochains! They are using the notation as in

J Cheeger & J. Simons:
Differential Characters and Geometric Invariants, in Geometry and Topology, J. Alexander & J. Harer (Eds.) pp. 50, Lecture Notes in Mathematics, Springer (1980)

Before reading that physicists might want to look at

N. Kalogeropoulos: The Diarc monopole and differential characters, math-ph/0504005.

Posted by: Urs on August 25, 2005 11:43 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

I have found that a very good motivation of and introduction to (generalized) differential cohomology (and hence differential K-theory) is in section 1 and 2 of

D. Freed: Dirac Charge Quantization and Generalized Differential Cohomology, hep-th/0011220

This clarifies in the text leading to its example 1.10 the otherwise somewhat mysterious definition 2.5. in Hopkins&Singer’s math.AT/0211216

With a little luck I manage to understand enough of this stuff to write a short summary before I leave for Jena next monday.

Posted by: Urs on August 25, 2005 6:48 PM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

I still don’t understand the relation between gerbes/2-bundles and 1-gauge theory in string space. A general 1-connection in string space is certainly not flat, even if the gauge group is semisimple. Thus the condition of fake flatness must correspond to some very peculiar connections in string space.

I recently realized that my own construction can be considered as 1-gauge theory on the space of straight strings. It is easy to see that a straight string in n dimensions is coordinatized by a pair of n-vectors (x,s), modulo equivalence relations

x ~ x + as, s ~ bs

for scalars a,b. The configuration space of straight strings strings is thus a manifold of dimension 2n-2, and the 1-gauge theory on such a finite-dimensional manifold is manifestly well-defined; there is no need to prove consistency. The 1-form connection is related to a 2-form, but not quite the usual 2-form of the abelian case. The functions of (x,s) that I wrote down before are simply straight string fields - functions of the straight string coordinates.

As you said, a gerbe should be a structure that admits parallel transport of strings, in particular straight strings. So how do gerbe connections act on straight strings, and how does one recover the non-zero curvature for semisimple gauge groups?

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

Thus the condition of fake flatness must correspond to some very peculiar connections in string space.

Yes. It describes precisely those connections on path space which locally take values in strict 2-groups and are reparameterization invariant.

So how do gerbe connections act on straight strings

The concept of a connection on an ablian gere is generally regarded as a procedure that maps closed surfaces to unimodular complex numbers. So this would be applicable to ‘straight strings’ in your sense only when the target is compactified, somehow, I guess.

We claim that, more generally, a connection on a (non)-abelian gerbe is something that associates 2-torsor 2-morphisms to possibly open compact surfaces. But I haven’t thought about how this would generalize to non-compact surfaces, as would arise as worldsheets of ‘straight strings’.

There is a precise statement here, with precise assumptions: Any suitably smooth 2-functor from the 2-path 2-groupoid to 2-torsors describes a fake flat gerbe with connection and curving. That’s a matter of going through a couple of calculations.

Of course you can say you want something different than a 2-functor on the 2-path 2-groupoid. There might be other concepts which might also deserve to be addressed as a sort of ‘parallel transort of string’ and which do not lead to a condition like fake flatness.

Note however, that even with something as simple as ‘straight strings’ (in the sense that you have drastically reduced the number of parameters needed to describe a certain configuration) you need to take care of reparameterization invariance. Let $E\subset {ℝ}^{m}$ be a plane whose holonomy you wish to compute (or a half-plane, maybe). There are different ways to folitae it in terms of ‘straight stings’. That is, different paths in the space of straight strings will map to the same surface. If you want to have a notion of surface holonomy you will have to restrict to connections on the space of straight strings which respect this freedom. This is the sort of reparameterization invariance condition which in the case I am talking about leads to the requirement of fake flatness.

Posted by: Urs on August 29, 2005 8:39 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

Re reparametrization invariance: To describe straight strings in curvilinear coordinates is no problem. This is why my fields carry a representation of the diffemorphism group (diffeos in x, not in (x,s)).

Re foliations: I have never claimed that surface holonomy is independent on the split of the boundary into an in and out side. On the contrary, this would be very unnatural from a string space point of view: vertical transport of a horizontal string is not the same as horizontal transport of a vertical string. However, once a split on the boundary is done, the surface holonomy is unique - we can define it as the unique 1-holonomy in straight string space.

Note that I am talking about the continuum theory here, which we may define as 1-gauge theory in straight string space. The lattice theory may suffer from lattice artefacts, just as 1-LGT may, but it would be rather surprising if the 1-holonomy over a finite-dimensional manifold were not parametrization dependent, wouldn’t it?

I suspect that the difference is this: I consider general connections in string space, whereas you restrict attention to those for which the surface holonomy is independent of the split of the boundary. This condition is very restrictive, and rules out everything of interest in the semisimple case.

Re: CFT, Gerbes and K-Theory in Oberwolfach, VII

You wrote:

To describe straight strings in curvilinear coordinates is no problem.

Before you said:

a straight string in n dimensions is coordinatized by a pair of n-vectors $\left(x,s\right)$, modulo equivalence relations

Maybe I don’t see what you mean by a ‘straight string’. It seemed to me you were thinking of something given by a basepoint $x\in {ℝ}^{n}$ and a vector $s\in {T}_{x}{ℝ}^{n}$, namely the line through $x$ with tangent $s$. By ‘equivalence relations’ I guessed you meant to identify tuples of $\left(x,s\right)$ if they describe the same line.

I thought that you imagined a 1-parameter family of such tuples

(1)$\tau ↦\Sigma \left(\tau \right)=\left(x\left(\tau \right),s\left(\tau \right)\right)$

to describe the ‘worldsheet’ of such a ‘straight string’ and were thinking of associating a holonomy with this worldsheet by integrating a 1-form

(2)$A\in {\Omega }^{1}\left(T{ℝ}^{n},\mathrm{Lie}\left(G\right)\right)$

on the space of the above tuples over the curve $\Sigma \left(\tau \right)$.

I was pointing out that different curves $\Sigma$ as curves in $T{ℝ}^{n}$ may have the same image as surfaces in ${ℝ}^{n}$. If we want to associate the holonomy with the surface and not with its parameterization, then that 1-form $A$ has to satisfy a suitable invariance condition.

I have never claimed that surface holonomy is independent on the split of the boundary

Let us concentrate on the case of holonomy of closed surfaces. (In the case of ‘straight strings’ we can imagin putting them not on ${ℝ}^{n}$ but on the $n$-torus, for instance.)

Posted by: Urs Schreiber on September 2, 2005 8:12 AM | Permalink | Reply to this

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