The String Coffee Table

Next Tuesday, May 2, 2006 there is a talk

On the String 2-Group ($\to$)

at the Bonn Math Institute ( 17:00-18:00, seminar room E (room 3, Meckenheimer Allee 160)).

Abstract.
We try to convey the main idea for
$\bullet$ what the String-group $\mathrm{String}_G$ is
$\bullet$ and how it is the nerve of a 2-group $\mathrm{Str}_G$
as well as
$\bullet$ what a $\mathrm{Str}_G$-2-bundle is
$\bullet$ and how it is ‘the same’ as a $\mathrm{String}_G$-bundle.

The first point is due to [BCSS,Henriques], which will be reviewed in section 2. The second point has been discussed in [Jurčo] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of $\mathrm{Str}_G$.

This entry continues the talk transcript begun before.

After having given lots of well-known definitions and facts, we now want to describe the main new theorem.

The motivation is to study to what degree derived categories $D^b(X)$ of coherent sheaves on $X$, as well as their stable subcategories (with respect to some stability condition), can distinguish between different spaces $X$.

Using the dictionary between the study of these derived categories and that of physical and topological string theories with target space $X$, the question here is hence to what degree the string physics can distinguish between different “target spaces” $X$.

Yesterday we had Daniel Huybrechts from Bonn talking in the ZMP seminar about stability conditions on derived categories of coherent sheaves over K3 surfaces.

Daniel Huybrechts
Derived and Abelian Equivalence of K3 Surfaces
math.AG/0604150

which is based on work by Bridgeland, who defined and studied stability conditions on triangulated categories in

Tom Bridgeland
Stability Conditions on Triangulated Categories
math.AG/0212237

and applied that to the case of derived categories of coherent sheaves on K3 surfaces in

Tom Bridgeland
Stability conditions on K3 surfaces
math.AG/0307164 .

This is pure math, but there is a nice dictionary which maps it 1-1 to the physics of D-branes. For a good overview of how this works see

Paul S. Aspinwall
D-Branes on Calabi-Yau Manifolds
hep-th/0403166,

which I once tried to summarize here.

Roughly, this dictionary is as follows:

A “geometric brane” for a B-model topological string on $X$ (i.e. a brane which can be regarded as a submanifold of $X$ with a vector bundle on it) is encoded in a coherent sheaf on $X$.

The general brane for the B-model string is obtained by stacking a collection of geometric branes and anti-branes on top of each other and turning on tachyon condensates between them (which in part mutually annihilates them, the remaining piece being a general brane). In the formalism, this corresponds to a bounded complex of coherent sheaves on $X$ (with the differential of the complex encoding the tachyonic string condensates stretching between the branes).

But ultimately we are interested not in the boundary conditions (= branes) of the topological B-model string, but of the physical type II string (i.e. of a CFT instead of a TQFT).

It turns out that all the branes of the topological B-string correspond to branes of the physical string - but not all of them are “stable” for the physical string. Instead, every brane for the topological string is supposed to decay into a collection of stable branes - the physical BPS branes.

More precisely, the derived category of coherent sheaves is what is called “triangulated”, which means that it contains lots of certain triangle diagrams (i.e. collections of three of its objects with certain morphism between them, satisfying some conditions). These triangles precisely encode the “brane chemistry”. Roughly, a triangle

says (as familiar from exaxt sequences), that the brane $B$ is an “extension” of the brane $C$ by the brane $A$. In other words, the existence of this triangle encodes the potential brane reaction

This is very much like in chemistry. (And of course oversimplified, see section 6.2 of Aspinwall’s review for the details.)

Which branes are stable and which are not is encoded by a “stability condition”, which, on the physics side, is called $\Pi$-stability. As far as I am aware, the entire motivation for Bridgeland to define stability conditions on triangulated categories comes from the desire to axiomatize this piece of physical input. The abstract definition of a stability condition on a triangulated category may look completely ad hoc, its natural meaning becomes manifest once you think of distinguished triangles as describing reaction processes of fusion and decay of branes.

Hence, whether one is interested in D-branes or not, when reasoning about stability condtions on derived categories of coherent sheaves it helps a lot to keep the above dictionary in mind. It makes many of the constructions and results better memorizable. (For more on the physics side see Eric Sharpe’s encyclopedia entry $\to$).

All, right, below I reproduce a transcript of the talk.

In Vietri ($\to$) Branislav Jurčo gave a talk on

B. Jurčo
Nonabelian Gerbes, Differential Geometry and Stringy Applications.
$pdf$

The slides for the talk have kindly been made available now.

Among other things, the talk recalls Killinback’s old result on how the Green-Schwarz anomaly can be understood as the (image in $H^4(M,\mathbb{R})$ of the) obstruction to having a $\mathrm{String}(n)$-structure on target space $M$ ($\to$, $\to$) and how this can be understood ($\to$) as the obstruction to lifting a $\mathrm{Spin}(n)$ bundle on $M$ to a gerbe or 2-bundle on $M$, whose structure 2-group is $(\widehat{\Omega\mathrm{Spin}(n)}\to P\mathrm{Spin}(n))$ ($\to$, $\to$).

Closely related to that is the idea, promoted in

Paolo Aschieri, Branislav Jurco
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
hep-th/0409200,

on how the Diaconescu-Freed-Witten anomaly indicates that M5-branes support modules for abelian 2-gerbes - i.e. twisted 1-gerbes.

The reasoning here is precisely analogous, just one dimension higher, to how the Freed-Witten anomaly gives rise to D-branes supporting modules for abelian 1-gerbes - i.e. twisted bundles.

One day I should say more about modules for gerbes here. The concept was formally introduced in

P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, D. Stevenson
Twisted K-theory and K-theory of bundle gerbes
hep-th/0106194.

There is a nice description of the idea in terms of morphism of transport functors. Let $\mathrm{tra}$ be a transport $p$-functor that encodes parallel transport in a $(p-1)$-gerbe over a $p$-dimensional volume. There is an obvious notion for what it means for such a functor to be trivial (in the sense of how a bundle can be trivial). Let $\mathrm{tra}_0$ be a trivial $p$-transport. Now, a trivialization of $\mathrm{transport}$ (if it exists) is nothing but a choice of isomorphism

In general, of course, no such isomorphism will exist. It may exist however if we enlarge the ambient category sufficiently. If that is the case, we call the triviaization a module for $\mathrm{tra}$.

For instance a module for an abelian bundle gerbe as defined in the above paper is essentially nothing but a trivialization of that bundle gerbe (a “stable isomorphism to the trivial bundle gerbe”), but in the (2-)category of nonabelian bundle gerbes instead of in the (2-)category of abelian bundle gerbes.

There is a general pattern at work here, which says that

Branes are modules ($\to$).

[Update: I now have some pdf notes on this issue: Note on Lax Functors and RCFT .]

Recall some basics of quiver theory:

A quiver diagram is nothing but a finite directed graph ($\to$).

Mathematicians call such graphs “quivers” when they are interested in algebra, because quivers can be taken to encode algebras.

Field theorists call such graphs quivers (or “mooses”) when they are interested in susy gauge theory, because quivers can be taken to encode certain field content in such theories.

String theorists call such graphs quivers when they are interested in D-branes on spacetimes of the form $M^4 \times CY_6$ (where $CY_6$ is a global quotient $\mathbb{C}^3/G$ by a finite subgroup of $SU(3)$), because quivers can be taken to encode the available type of (fractional) D-branes and the sorts of strings stretching between these.

Michael R. Douglas, Gregory Moore,
D-branes, Quivers, and ALE Instantons
hep-th/9603167.

More precisely, every vertex of the quiver is identified with a type of D-brane, while every edge of the quiver is identified with a species of string (topological string, usually) stretching between the types of D-branes corresponding to the source and target vertex of the edge.

For an illlustration, pick any random string theory paper on quivers, for instance see figure 1 in

Marco Billo, Marialuisa Frau, Fabio Lonegro, Alberto Lerda
N=1/2 quiver gauge theories from open strings with R-R fluxes
hep-th/0502084.

More precisely, the configuration of these (topological) branes (and the string condensates between them) is not encoded by the quiver itself, but by a representation of the quiver ($\to$). This is essentially a functor from the quiver (regarded as a category) to vector spaces.

Now, and that’s the point of my note here, some generalization of the concept of a functor on a quiver secretly also plays a crucial role for determining the D-brane content in the FFRS description ($\to$) of rational conformal field theory. Maybe there is more to that.

I am grateful for the comments to the recent entry “Generalized Worldsheets?”, but I feel they indicate that I did not manage to get my main point across. I now tried again, providing more details, in the latest comment to that entry ($\to$), which, for that reason, I thought I should lift to blog top-level. It is reproduced below.

I’ll very quickly mention a few aspects of some further talks that we heard ($\to$) on Thursday. A collection of slides for all talks should be available online soon.

This morning we had a talk ($\to$) by Pietro Fré on issues of M-theory compactifications using the tool of “free differential algebras”. Before I say something about this talk, consider the following general question.

What on earth IS an RR-form, really ($\to$)?

It’s not quite something living in twisted K-theory ($\to$). Jarah Evslin today kindly tried to enlighten me a little about the latest ideas, but I guess I am being dense. His talk tomorrow ($\to$) might help.

Somehow, in the end, something like the following should be true, at least to my mind: over eleven dimensional spacetime there should be nothing but a plain 2-gerbe with connection, which is locally given by the supergravity 3-form. Upon compactifying down to IIA or IIB, that 2-gerbe with connection should decompose into the Kalb-Ramond 1-gerbe with connection and some other strucure in which the RR fields live.

It sounds like a straightforward task to examine the structure obtained by “compactifying a $p$-gerbe with connection” this way, which should completely resolve the issue of the nature of the RR-forms (or shouldn’t it?). But I don’t see that anyone has tried to do this.

(However the approach by Sati and Kriz, trying to identify elliptic cohomology as the correct refinement of K-theory should be closely related

Igor Kriz, Hisham Sati
Type IIB String Theory, S-Duality, and Generalized Cohomology
hep-th/0410293.

After all, it is expected on general grounds that ellitptic cohomology is to 2-gerbes as K-theory is to 1-gerbes ($\to$)).

But there are some tantalizing hints coming from the description of the field content of supergravity in terms of (free) (graded) differential algebras.

Leonardo Castellani, Alberto Perotto
Free Differential Algebras: Their Use in Field Theory and Dual Formulation
hep-th/9509031.

Recall ($\to$) that a graded differential algebra concentrated in the $p$ lowest degrees is nothing but a $p$-term $L_\infty$ algebra (-algebroid), which in turn is nothing but a semistrict Lie $p$-algebra. There is a way to encode connections on $p$-gerbes in terms of morphisms of such dg-algebras ($\to$, section 14), which is governed by the double complex obtained from deRham and the dg-algebra itself.

Now, precisely these structures have been found useful in organizing the field content of supergravity theories. This goes back to old seminal work by Fré and Auria on supergravity, and has recently again attracted increased attention, see for instance

A. Bandos, J.A. de Azcarraga, J.M. Izquierdo, M. Picon, O. Varela
On the underlying gauge group structure of D=11 supergravity
hep-th/0406020

Or see the recent article that Pietro Fré talked about today in Vietri sul Mare:

Pietro Fré
M-theory FDA, Twisted Tori and Chevalley Cohomology
hep-th/0510068.

The various $p$-form fields appearing in supergravity theories can nicely be regarded as components of a free graded differential algebra, with the differential being induced from the various generalized Maurer-Cartan equations. Field strengths and gauge transformations follow precisely the same logic as in the dg-algebra description of gerbes with connection.

For some reason I had not payed attention to this fact before. I am grateful to B. Jurco for addressing this point. To me it suggests that the 2-gerbe interpretation of RR-fields might be simply obtained by staring at this SUGRA dg-algebra long enough, comparing it with the algebroid description of gerbe connections. As soon as I find a working printer and some time (this may not be very soon), I should take a closer look at this.

I am curently at the Amalfi coast, attending the annual IIASS conference ($\to$). The talks of most interest to me are yet to come, but here is a quick note on a talk we heard yesterday.

Fedele Lizzi reported on recent attemts

Fedele Lizzi, Sachindeo Vaidya, Patrizia Vitale
Twisted Conformal Symmetry in Noncommutative Two-Dimensional Quantum Field Theory
hep-th/0601056

to define conformal field theory on a noncommutative parameter space, more precisely, to construct something that would deserve to be called a “Virasoro algebra living on the noncommutative Moyal plane”.

Essentially, what is done in the above paper is an application of the general prescription described in

Paolo Aschieri, Christian Blohmann, Marija Dimitrijevic, Frank Meyer, Peter Schupp, Julius Wess
A Gravity Theory on Noncommutative Spaces
hep-th/0504183.

The main point is that, when products of functions are deformed by the Moyal star, one can introduce a related twist on the Leibnitz rule for derivations on the original function algebra such that symmetries represented on the original algebra remain unbroken when sent to the deformed algebra. This is intended to allow one to have, for instance, ordinary Poincaré or conformal symmetry implemented on noncommutative spaces.

I must say I haven’t looked closely enough at these constructions yet to say anything of value about the technical details. (But Fedele Lizzi himself emphasized that their construction is little more than a first idea at the moment.) I am wondering, though, what the big picture is that is lurking in the background here, that, which does not depend, in particular, on restricting attention to the Moyal product.

I think the question is this:

Can one sensibly define a generalization of 2-dimensional conformal field theory on some sort of generalized Riemann surfaces, which are not ordinary manifolds?

(Since it lead to some discussion after the above mentioned talk, I should maybe emphasize that the question here is not about noncommutativity in target space. I hear that people have defined and investigated CFTs with the target being a quantum group, for instance. But this is not what should be the issue here, I think. These are still CFTs defined on the Riemann sphere, the complex torus, etc.

But is there anything known about how to generalize the concept of a conformal 2-dimensional field theory to something whose parameter spaces are, say, … general 2-dimensional schemes? Or, maybe better, (since we want something like a conformal structure on our generalized parameter space), where the parameter spaces are noncommutative Riemann surfaces defined following Connes’ NCG?

If anybody knows relevant references, plase drop me a note!

Of course one thing that immediately comes to mind are constructions like Matrix Strings or other worldsheet discretizations that make an appearance here and there. But I am not sure I have ever seen a definition of CFT in these context in a way that deserves this name.

If I were to make a guess myself, I would maybe note the following. It seems like one can capture 2-dimensional CFT by a notion of 2-transport ($\to$), i.e. by 2-functors which assign CFT propagators to Riemann surface elements. In the general formalism governing this construction, there is nothing which forces one to take the domain 2-category of these 2-functors to really be one whose 2-morphisms are surface elements. The general construction works for much more general 2-categories. Hence, from that point of view, it would be sort of straightforward to define a 2d-CFT on a generalized parameter space to be a certain 2-transport 2-functor on suitably generalized domain 2-categories. Maybe.

Picard and Brauer groups of representation categories of vertex algebras encode symmetries of full CFTs ($\to$, $\to$). In the CFT context the elements of the Picard group are called simple currents (e.g. Int. J.Mod.Phys. A 5 (1990) 2903).

It is rarely explicitly admitted that the Picard group is really a 2-group ($\to$), and so is the Brauer group.

Even better, both fit into a certain 3-group. All this has been well known to a handful of experts, apparently ($\to$) - which of course does not stop me from enjoying rediscovering this from my personal point of view.

Namely, this 3-group is closely related to 2-dimensional field theory, and in particular to CFT. The fact that it is an $n$-group for precisely $n=3$ is closely related to the holography-like phenomenon that 2D CFT can be described in terms of 3D TFT.

A while ago, Toby Bartels had a paper on the arXiv in which a notion of a categorified bundle, a 2-bundle, was defined ($\to$)

Toby Bartels
Categorified gauge theory: 2-bundles
math.CT/0410328

Meanwhile this material has evolved. A draft of a refined version is now available:

Toby Bartels
Higher gauge theory: 2-Bundles (draft)
ps.

A collaboration of field and string theorists, mathematicians and experimentalists at Hamburg University and at the DESY accelerator is applying for a grant for a Collaborative Research Centre to be called

Particles, Strings and the Early Universe: the Structure of Matter and Space-Time.

Here is a list of the participating group’s research (taken from the funding proposal) and in particular a description of the research that I am involved in.

The university of Chicago hosts a conference, from April 7 to April 11, in memory of Saunders MacLane ($\to$, $\to$, $\to$) (who died April last year): Category Theory and its Applications in conjunction with the Unni Namboodiri Lectures, given by John Baez on Higher Gauge Theory ($\to$, $\to$, $\to$). Other speakers include Alissa Crans and Danny Stevenson ($\to$) on the $\mathrm{String}(n)$ ($\to$) 2-group ($\to$) , higher Schreier theory ($\to$) and related stuff.