The Mathematics Department of the University of Hamburg has a postdoctoral position available in the area of Algebra, Conformal Field Theory and String Theory which is part of the Collaborative Research Centre 676 “Particles, Strings and the Early Universe: the Structure of Matter and Space-Time” funded by the German Science Foundation (DFG).

The position starts in the fall of 2007 and is for a period of 2 years with the possibility of an extension for an additional year. The candidate is expected to do research at the interface of Algebra, Conformal Field Theory and String Theory.

Applicants must have a PhD in Theoretical Physics or Mathematics.

See the full announcement.

Further links:

Prof. Ch. Schweigert’s homepage

Some entries discussing the group’s work:

The FRS Theorem on RCFT

FRS Reviews

Unoriented Strings and Gerbe Holonomy

some projected research

The same day as the “Lessons from the LQG string” appeared on hep-th, there was another paper by Corichi, Vukasniac and Zapata crosslisted from gr-qc discussing the loopy oscillator and coming to conclusions which at first sight comes just to the opposite conclusions. Their abstract starts out with “In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics.” while I derived that at high frequencies the absorption spectrum of the polymer oscillator is quite distinct from the usual Fock/Schrödinger version. How could this be?

Luckily, their paper is written in a very clear manner and free from the notational ballast which makes many LQG papers hard to read. With only a brief read one can find the resolution: The two papers are doing different things. That’s not too surprising. Let me spell this out in a bit more detail. In one sentence: I tried to take the polymer oscillator literally and work out the conclusions from what I am given while they apply some limiting/regularisation/renormalisation procedure to the system to finally end up with the usual Fock description.

As a warning I should say that what I am going to present here is probably some kind of caricature of their paper. I have had some email exchange with the authors from Mexico and they have been very helpful and responded to many questions and I am extremely thankful. What I write here is the result of my learning process but might not be the way they would summarise their paper. So: All the errors in this presentation are mine!

The first difference is that the two papers start from different polymer Hilbert spaces: In my case, I have a basis labelled by points in phase space and the Weyl operators by translations in the $x$ and $p$ directions. Because of the singular scalar product these actions are not continuous and neither $X$ nor $P$ exist as operators, only $e^{\mathrm{iaX}}$ and $e^{\mathrm{ibP}}$ (for real $a,b$). This has the advantage that the classical time evolution translates directly to a unitary operator in that Hilbert space: It just rotates the phase space by an angle proportional to $t$.

They start with a Hilbert space where a basis is labelled by points on the real line and there is an $X$ operator which even has normalisable eigenfunctions. However, there is still no $P$ operator but what would be $e^{\mathrm{ibP}}$ acts by translations by $b$. This Hilbert space does not come with a nice time evolution of the oscillator but we will see below what they do instead. However, this difference is I think only technical and does not really matter in the following.

Then they go through some mathematically involved (projective) limiting procedure and play the “go to the dual space”-game several times. The result is that they pick a squence of subspaces of countable dimension, namely at stage $n$ they consider only the span of vectors over points of the form $m/2^n$ for integer $m$. These are in one to one correspondance to characteristic functions of the interval $(m/2^n,(m+1)/2^n)$ in the usual Hilbert space $L^2(R)$. This mapping however is not in isometry: In $L^2(R)$ these characteristic functions have a norm given by their length, i.e. $2^{-n}$ while they have norm 1 in the polymer space. Now comes the trick: You redefine (“renormalise”) the norm on the polymer side by copying the norm form the $L^2(R)$ side. When doing this you should remind yourself that the norm/scalar product is where the choice of state showed up in the GNS construction. So, by redefining the norm you effectively revise your choice of state. And the two descriptions (Fock and polymer) only differed by the choice of state…

At each finite stage of this regularisation procedure, you have broken most translation operators $e^{\mathrm{ibP}}$, only the ones for which $2^{n}b$ is an integer survive. But you can use those to come up with finite difference versions $P_{\mathrm{fd}}$ for what would be the $P$ operator and use it to define a regularised oscillator Hamiltonian $H=\frac{1}{2}(P_{\mathrm{fd}}^2+X^2)$.

Finially you take the $n\to \mathrm{\infty }$ limit everywhere. To nobodies’ surprise you end up with the usual Hamiltonian in the usual Fock space. Strictly speaking, you have only defined the operators $e^{\mathrm{ibP}}$ for those rational $b$ which have a denominator which is a power of two. But as you are taking limits anyway, you can use these and continuity to define them for all $b$. Of course, as Drs Stone and von Neumann have told you long ago, there is no other continious choice of representation of Weyl operators than the standart one.

So what do we learn? As Giuseppe put it to me (of course with his better manners in more polite words): Both papers agree that the polymer representation sucks. In my paper, I show that how much it sucks and in their paper they show how you can redefine it away and proceed to the usual Fock space.

But I should warn you, dear reader: The original motivation for considering polymer representations at all (not so much for the oscillator but for gauge theories and gravity) was that it gives an easy (trivial) implementation of diffeomorphism symmetries. This is a central part of all this “background independance” stuff.

But the procedure these people suggest is to introduce a regulator (the $1/2^n$ equal partitioning), do the calculation and then remove the regulator. This regulator is nothing but a background! And it breaks many of the nice symmetries you wanted to maintain.

So there are two obvious questions: 1) In systems more involved than the harmonic oscillator (which is just the free theory in 0+1 dimensions), is it possible to renormalise scalar products and operators in a way that the limit exists? This question is like the continuum limit for a lattice regularisation: In nice theories (like QCD) it exists, in other cases there is no good continuum limit like for example QED, because the theory is not asymptotically free. And in the case of gravity I would be worried that the well known non-renormalisability (in the usual treatment) shows up when you try to remove the regulator and find the whole thing exploding.

But let’s assume for a second this problem does not occur or you have found a way to solve it. Then there is still question 2), the anomalies: The regularisation has broken many essential symmetries. Thus it is non-trivial that these reappear in the continuum limit. And we know: In general they don’t. The polymer state didn’t have this problem as it preserved the symmetries. But now they are explicitly broken. So you are thrown back to the situation of the conventional treatment (with a UV cut-off say). If you don’t believe this, you are welcome to upgrade the content of the paper to the case of the bosonic string and show how Diff($S^1$) reappears in the continuum limit.

It’s now two years, that Giuseppe and I have put out out our paper comparing the usual quantisation of the bosonic string to Thiemann’s loop inspired version. A bit to my surprise, that paper was of interest to a number of people and the months afterwards I was lucky to tour half of Europe to give seminars about it (in that respect it was my most successful paper ever; the only talk I have given more often is my popular science talk “Phaser, Wurmloch, Warpantriebe” about physics with a Star Trek spin prepared for the Max Planck society public outreach).

That paper had quite a resonance in the blogosphere as well, but its results have not always been presented in a way we intended them. This might also be because the paper was in large parts quite technical and some of the main messages were burried in mathematical arguments.

So I thought it might be a good idea to put out a “mainly prose” version of the argument which leaves out the technicalities to bring home the main messages. This I did and you should be able to find it on hep-th as you read this.

Remember the philosophy of this investigation: The loopy people always insist that diffeomorphism invariance is so central to gravity that it is important to build it into a theory of quantum gravity right from the beginning and all the problems one has with perturbatively quantising GR are due to ignoring this important symmetry or at least not building it into the formalism but expanding around some background.

As GR is a complicated interacting theory it is easy to get lost in the technical difficulties and one should consider simpler examples to test such claims.

The world sheet theory of the bosonic string is such an example as it is extremely simple being a free theory but still has an infinite dimensional symmetry of diffeomorphisms of the lightcone coordinates. It is thus the ideal testbed for approaches to diffeomorphism invariant theories where one can compute everything and check if it makes sense.

The first part of today’s paper explains all this and shows that the difference in the treatments can be summarised by saying that the usual Fock space quantisation of the string uses a Hilbert space built upon a covariant state whereas the loopy approach insists on invariance of that state which is a much stronger requirement.

My point is that covariance is the property which is physically required (and in fact states in the classical field theory are covariant but not invariant) and thus statements like the LOST theorem have too strict assumtions.

If you insists on invariance you end up with a Hilbert space representation which is not continuous as this is what LOST like theorems tell you. The question now is if this discontinuity makes your theory useless as a quantum theory. Well, everybody is free to set up the rules of the game they call “quantisation” and in the end only theories which do not disagree with experiments are good theories. But as we are all well aware, there are not too many experiments performed today which study properties of quantum gravity or bosonic string and thus this test is not available for the time being.

A weaker test would be to apply your rules of quantisation to other systems which are available for experimentation and see what they give there. Thus the second part (as in the original paper with Giuseppe) deals with a loop inspired quantisation of the harmonic oscillator. The old paper was criticised for providing a solid argument that it is observationally possible to distinguish the loopy oscillator from the Fock oscillator.

The second part of the new paper I think provides such an argument: It couples the oscillator to an electromagnetic radiation field and computes the absorption spectrum. Remember that usually the absorption for a transition between states $\mid m\rangle$ and $\mid m\prime \rangle$ goes like

$$\frac{1}{(\Omega -{\omega }_m+{\omega }_{m\prime }{)}^2}.$$

Here, $\Omega$ is the frequency of the radiation. Now, the loopy result is proportional to

$$\frac{1}{\sin ((\Omega -m+m\prime )/N{)}^2}$$

where $N$ is a large natural number characterising the states. Thus if $\Omega \ll N$ the two expressions agree but for large $\Omega$ they don’t (don’t worry about an overall constant).

Thus if I am only allowed to measure within a finite frequency band for $\Omega$ the states can be made similar by choosing $N$ large enough. But once that $N$ is chosen the experimenter can reveal the difference by studying the behaviour at large frequencies.

So are they the same or not? Well, that’s a long story for which you have to read the paper.

After you’ve done that, you can come back here and comment.

I’m a little reluctant to post much on the master constraint program because I haven’t read much on it. But I thought I’d post this if others want to comment on the subject.

My initial question is how does the master constraint program work in classical mechanics? In particular, say we are given some symplectic manifold and some set of constraints. The master constraint is$$M=K^{\mathrm{IJ}}{C}_I{C}_J.$$Using this, how does one obtain the constrained phase space?

Or is this the wrong question to ask?

I’ve been trying to understand Thomas Thiemann’s riposte to the papers of Nicolai, Peeters and Zamaklar, Nicolai and Peeters and Helling and Policastro. I’m fairly busy right now with a paper of my own and moving, so I’ll concentrate on the part where he describes how LQG-quantization replicates the usual quantization of the harmonic oscillator. Maybe later, I can get to trying to understand the master constraint program.

I’ve already posted some comments at Christine Dantas’s blog, but I thought I might also try to post them here. I really would love to see some sort of discussion on these points. One of the things I think any scientist should be able to do is to get in front of a chalkboard and be able to communicate a pretty good idea about what’s going on with their work. Think of this as a long distance chalkboard, and I’m the skeptical visitor.

Anyways, I will try in this post to summarize my understanding of the construction in Thiemann. I hope people will correct me if I get it wrong. (And I hope I get the algebra correct….)

In LQG-quantization, we start with a $C^{*}$-algebra, in this case the algebra given by the exponentiated position and momentum operators. This is often used because $p$ and $q$ are unbounded operators. In particular, we have

$$U=e^{\mathrm{iap}}V=e^{\mathrm{ibq}}$$

and

$$\mathrm{UV}=e^{iab}\mathrm{VU}$$

The Stone-von Neumann theorem tells us that ordinarily all unitary representations of this algebra are unitarily equivalent. In LQG, however, we have weakly continuous representations, so we can’t appeal to this any more. Unfortunately, the usual harmonic oscillator Hamiltonian

$$H=p^2+q^2$$

isn’t an element in this $C^{*}$-algebra. However, we can define a one parameter family of operators:

$$H_{ϵ}=({\sin }^2(ϵp)+{\sin }^2(ϵq))/{ϵ}^2$$

which are elements in the algebra. Classically, of course, this expression does converge to $H$ as $ϵ\to 0$. It would be nice now to determine the spectrum of this operator, but apparently for the usual LQG representation, we cannot. Thus, let us define the raising and lowing operators

$$a_{ϵ}^{†}=(\sin (qϵ)+i\sin (pϵ))/ϵ$$

and similarly for $a_{ϵ}$. It is not true $a_{ϵ}^{†}{a}_{ϵ}+1/2$ is $H_{ϵ}$, but it is true to order ${ϵ}^2$ if I haven’t screwed up my algebra. Now, define the operators$$b_{n,ϵ}=\frac{1}{n!}(a_{ϵ}{)}^n{a}_{ϵ}^{†}{a}_{ϵ}(a_{ϵ}^{†}{)}^n$$For the usual harmonic oscillator, we have that $b_n\mid 0\rangle =n\mid 0\rangle$ and so the vev of the $b_n$ is $n$. By weak continuity, we can ensure that, for $n<N$ and for any $\delta >0$, there exists an ${ϵ}_0$ such that for all $ϵ<{ϵ}_0$:$$\mid \langle 0\mid b_{n,ϵ}\mid 0\rangle -n\mid <\delta$$

We’re still nowhere near the LQG harmonic oscillator, though. To summarize what we’ve done so far, we have defined a one parameter family of operators that exist in the algebra generated by $U$ and $V$ that, such that if we interpret them as in the algebra with the $p$ and $q$, they would go to the usual Hamiltonian and raising and lowering operators. There, the vevs of the $b_n$ give exactly the spectrum of the Hamiltonian. We would have ensired that the vevs of a finite number of the $b_{n,ϵ}$ are close to the integers, but ut’s not at all clear to me what these vevs have to do with the spectrum of $H_{ϵ}$.

Regardless, we don’t necessarily have the harmonic oscillator vacuum floating around. It is a theorem (apparently) that the space of traceclass operators on our (GNS-)Hilbert space is dense in the space of states on the $C^{*}$-algebra. Thus, we can find a traceclass operator, $\rho$, such $\mathrm{Tr}(\rho b_{n,ϵ})$ is as close to $\langle b_{n,ϵ}\rangle$ in the Harmonic oscillator vaccum as we desire. Since that’s as close to the integers as we specify, we’ve produced a state (actually, an infinite number of them) such that, for $n<N$,$$\mid \mathrm{Tr}(\rho b_{n,ϵ})-n\mid <\delta$$

So, what have we proven? We’ve produced an infinite number of mixed states for the GNS-representation provided by LQG-quantization. In each of these mixed states, the vevs of the $b_{n,ϵ}$ are within a specified closeness to the integers.

Let’s grant for the moment that the vevs of these $b_{n,ϵ}$ operators are really related to what we measure for the Harmonic oscillator energy states. Given a particular choice of mixed state, there is a prediction that at some accuracy, these values will differ from the standard prediction. Presumably, something similar will hold for any comparison of LQG-quantization and standard quantization.

What we have not furnished at this point is a first principles method for determining the correct mixed state. I don’t see how we have any predictivity without this.

What’s more, there are plenty of other traceclass operators. For example, take the density matrix for the harmonic oscillator:$$\rho =\frac{1}{2}(\mid 0\rangle \langle 0\mid +\mid 1\rangle \langle 1\mid )$$Then,$$\mathrm{Tr}(\rho b_n)=\frac{1}{2}(n+(n+1{)}^2)$$I can just as well produce a density matrix such that the vevs of the $b_{n,ϵ}$ give these numbers to the specified accuracy. Can I distinguish from first principles why I should choose one of the above density matrices that give us ‘good’ answers rather than one of the ones that give these values?

Given this wide variety of ‘energy levels’ for the Harmonic oscillator, it seems to me that LQG has not predicted (or retrodicted in this case) anything. Worse, no matter how accurate traditional quantization appears to be, we can always find a state that reproduces our measurement to the needed accuracy, so LQG does not predict when we should see deviations from standard quantization.

This isn’t completely awful in situations like the Harmonic oscillator where traditional quantum mechanics tells us the answer to look for. But what happens in quantum gravity, then, when we don’t have a traditional quantization to guide us? Does this surfeit of mixed states cease to exist? How do we get any predictions at all?

Review of Not Even Wrong

Any comments and corrections are greatly appreciated. The first paragraph follows after the jump.

String theory, the enormously ambitious and speculative endeavor that has, for the past thirty years, attempted to unify our understanding of quantum mechanics and gravity has failed to live up to its initial promise. Its relative domination of the field of fundamental theoretical physics has long led to criticism within the scientific community. In the last few years, however, a number of popular books and television shows have made the case for string theory to the public. There is certainly a place, then, for these criticisms to also be presented to the public. More so, the sociology of modern theoretical physics could provide a fascinating context in which to present a reasonably disinterested discussion of the pros and cons of both string theory as a research program and the way in which modern theoretical physics is pursued. Dr. Woit has instead chosen to write a tendentious account providing little guidance as to why, even in the face of such criticism, so many have chosen to work on string theory. After reading this book and some of the unfortunate innuendo it contains, one might conclude not that string theorists are honest researchers doing the best they can to understand the nature of the universe, but rather are misguided devotees of a failed cult mired in self-delusion.

…

A new group blog has been created.

The $n$-Category Café.

It’s hosted by John Baez, David Corfield and myself.

The café is supposed to be the right place for the sort of discussion of mathematical physics that you know from John’s This Week’s Finds in Mathematical Physics, maybe from some of the stuff that I have been posting here, hopefully close to the constructive style that is practised on David’s blog.

We are very glad to be able to use Jacques Distler’s sophisticated blog technology.

I will probably move much of my activity from the coffee table to the café.

On the occasion of the availability of the new edition of Anders Kock’s book on synthetic differential geometry ($\to$) I want to go through an exercise which I wanted to type long time ago already.

I’ll redo the derivation of the transition laws for 2-connections ($\to$) using synthetic language. This greatly simplifies the derivation, to the extent that the equations in terms of differential forms become almost identical to the diagrammatic equations that we derive them from.

The main two facts of synthetic differential geometry that I’ll need are the following.

1) A function from $n$-simplices to some group, which sends degenerate $n$-simplices to the identity element is a (group-valued) $n$-form. For the additive group $ℝ$ this is an ordiary $n$-form. More generally, this function is to be thought of as the infinitesimal exponential of a Lie algebra-valued $p$-form.

2) Given a group-valued 1-form

its gauge covariant curvature is the 2-form

So fix some space $X$ together with a good covering by open sets $U_i$. On each $U_i$ we have a transport 2-functor which sends infinitesimal 2-simplices

to elements of the 2-group $H\to G$:

From the source-target relation in 2-groups we read off that the 1-form $A$ and 2-form $B$ satisfy the fake flatness condition

On double intersections, two such 2-functors are related by a pseudonatural transformation. This is a 1-form

Its values are composed horizontally using whiskering in $(H\to G)$. We may hence think of this as a 1-form taking values in the semidirect product group $G⋉H$. I’ll denote the $H$-part of its synthetic curvature by $\exp F_{a_{\mathrm{ij}},A_i}$ below.

The mere existence of the 2-cell on the right above means that

which is equivalent to the transition law for $A_i$.

In order to qualify as a pseudonatural transformation, this 1-form must satisfy the equation

But using the two SDG facts stated above, together with the rules for composition in the 2-group $(H\to G)$, this immediately says that

Next, on triple intersections the pseudonatural transformations $\exp a_{\mathrm{ij}}$ are related by an isomodification. This says that there is a 0-form

which satisfies

Here we just need to collect exponents to get the expected transition law.

Finally, there is the tetrahedron law on quadruple intersections. This only involves 0-forms and SDG does not tell us anything here that we did no know before.

In conclusion, SDG here mainly helps handling the otherwise somewhat subtle curvature $F_{a_{\mathrm{ij}},A_i}$ in the transition on double intersections, and makes reading off the law on triple intersections a little more systematic.

As explained for instance in

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

a special case of Quillen’s concept of superconnections can be used to elegantly subsume both the gauge connection as well as the tachyon field on non-BPS D-branes into a single entity.

Assuming that this is not just a coincidence, one might ask what it really means. What notion of functorial parallel transport ($\to$) is encoded in these superconnections?

I’ll give an interpretation below. With hindsight, it is absolutely obvious. But I haven’t seen it discussed before, and - trivial as it may be - it deserves to be stated.

Superconnections.

Let $E\to X$ be some ${\mathbb{Z}}_2$-graded vector bundle. Let $\Omega (X,E)=\Omega (X)\otimes \Gamma (E)$ be the space of differential forms taking values in sections of this bundle. This is an $\Omega (X)$-module in the obvious way. It inherits a ${\mathbb{Z}}_2$-grading from the combined ${\mathbb{Z}}_2$-grading of $\Omega (X)$ and $\Gamma (E)$.

A superconnection on $E$ is defined to be any odd graded endomorphism $A$ of $\Omega (X,E)$ which satisfies the Leibnitz rule

for all $\omega \in \Omega (X)$.

Such superconnections arise in the form of ordinary connections plus an odd element $A$ of $\Omega (X,\mathrm{End}(E))$:

Hence, in particular, they may contain a 0-form contribution, taking values in odd-graded endomorphisms of $E$.

Superconnections on D-branes.

For the applications to D-branes, all we need of superconnections is this additional 0-form degree of freedom.

The graded bundle $E=E^{+}\oplus E^{-}$ is interpreted as the Chan-Paton bundle of some D-branes plus that of some anti D-branes. The tachyon field arises from strings stretching beween branes and anti-branes, and is hence a 0-form taking values in odd endomorphisms of $E$. Combined with the ordinary connections on $E^{+}$ and $E^{-}$ we get a superconnection.

While this way of looking at tachyon fields is very fruitful, it is noteworthy that we have severely restricted the full freedom of superconnections. This might indicate that the “super”-point of view is not precisely the most natural one describing this situation. I shall now argue for what I feel is a more natural way of looking at the situation.

Functorial reformulation.

Consider first just a stack of D-branes, without any anti-D-branes. They carry a gerbe module (a twisted vector bundle) and parallel transport of open strings ending on the brane and coupled to a possibly non-vanishing Kalb-Ramond fields is described by a 2-functor which takes endpoints of strings to fibers of an algebra bundle of compact operators, which takes open strings to bimodules for the algebras associated to the endpoints, and which takes pieces of worldsheet to homomorphisms between the bimodule of the incoming and the outgoing string.

This in particular encodes a parallel transport

in the (twisted) bundle on the D-brane which takes paths

in the base manifold (the D-brane’s worldvoume) to morphisms of fibers

If we conveniently take the point of view of synthetic differential geometry ($\to$) and assume $x$, $y$ and $z$ to be infinitesimal neighbours, then this assignment is an endomorphism-valued connection 1-form.

The above paths live in some sort of (2-)path (2-)groupoid $P_2(X)$, roughly encoding open strings ending on the stack of D-branes.

Now, quite obviously, if we want to incorporate in addition a stack of anti D-branes with a bundle $E^{-}$ over them, we need to enrich $P_2(X)$ by additional morphisms encoding the strings stretching between $E^{+}$ and $E^{-}$.

Let’s take two copies of $P_2(X)$, and throw in precisely one 1-morphism going between every ordered pair of two copies of the same object in $P_2(X)$. These will encode paths with no spatial extension, but whose endpoints lie on different copies of stacks of branes. Call the 2-category freely generated this way $P_2(X)$.

Proceeding analogously for the transport category $\mathrm{Trans}(E^{+})$, we obtain $\mathrm{Trans}(E^{+},E^{-})$, which is generated from $\mathrm{Trans}(E^{+})$, $\mathrm{Trans}(E^{-})$ and all morphisms between fibers over the same point.

The connection data encoded by the functor.

Now a connection of the brane/anti-brane stack is a functor

A 1-morphism diagram in $P_2(X)$ would look for instance like this

where ${\gamma }^{+}$ is a path with endpoints on the stack of branes, and ${\gamma }^{-}$ is the same path (as a path in $X$) but with the endpoints taken to sit on the stack of anti-branes. The morphism $t_x$ are constant paths (as paths in $X$) with one endpoint on the branes, the other on the anti-branes.

Hitting this with our functor produces a diagram

in $\mathrm{Trans}(E^{+},E^{-})$.

We read off what data the new functor encodes: it contains an ordinary connection ${\mathrm{tra}}^{+}$ on $E^{+}$, as well as an ordinary connection ${\mathrm{tra}}^{-}$ on $E^{-}$. Both are given by ordinary 1-forms.

In addition, there is now an assignment of morphisms $\mathrm{tra}(t_x):E_x^{+}\to E_x^{-}$ for every point $x$. Since $x^{+}$ and $x^{-}$ belong to the same point in $X$, this is a morphism-valued 0-form.

It’s precisely the tachyon field 0-form that we expect to see.

And of course there is a similar 0-form with morphisms going the other way, not depicted in the above diagram.

In conclusion, the transport functor on the path category which allows paths to end on points colored by two different colors encodes precisely the information contained in the superconnections which appear on brane/anti-brane pairs. In fact, as I vaguely indicated, the construction seamlessly generlizes to KR-twisted bundles and the NS-NS surface transport associated with that.

Curvatures functorially.

One can also nicely deduce the $n$-form curvatures from that, by transporting around infinitesimal loops.

First, there is a 0-form curvature obtained by transport along

Next, there is a 1-form curvature obtained by transport around

for $x$ and $y$ first order infinitesimal neighbours.

Finally, there are two ordinary curvatures obtained, as usual, by transporting around infinitesimal loops

The 0-curvature is just the square of the tachyon field. The 1-form curvature is a gauge covariant derivative of the tachyon field, with respect to the two gauge connections on $E^{+}$ and $E^{-}$.

I believe it is easy to see that this does reproduce the three curvature equations (2.12), (2.13) and (2.14) in Richard Szabo’s text.

Gauge tranformations functorially.

Gauge transformations are discussed similarly. For us, a gauge transformation is a natural isomorphism

If we assume, as usual, the base space to be fixed, then this is an assignment of an isomorphism

to every point $x^{+}$ on the stack of D-branes, and an isoomorphism

to every point on the stack of anti D-branes, such that all diagrams of the form

and

commute.

Clearly, the first reproduces the ordinary gauge transformations of the two ordinary connections, while the second gives the correct transformation of the tachyon field (compare Szabo’s equation (2.22)).

Example.

An especially important form of tachyon fields are those discussed for instance on p. 29 and p. 38 of Szabo’s text. Here the bundle $E=E^{+}\oplus E^{-}$ contains in particular a spinor bundle $S=S^{+}\oplus S^{-}$ with the usual ${\mathbb{Z}}_2$ spinor grading.

We may hence consider tachyon fields which on a local coordinate patch $\{x\}$ look like

where ${\gamma }_i$ are some generators of a representation of the relevant Clifford algebra on $E$. If the tachyon field going the other way looks similar, the corresponding 0-form curvature

looks like

This indicates that all the $n$ D-branes in the rank-$n$ bundle $E$ decy into a single one, following the tachyon profile $f^2$.

Notice that $\mathrm{tra}(t_x)$ here is like a Dirac operator expressed in a plane-wave basis, i.e. Fourier-transformed/T-dualized. Compare this with the general relationship between tachyon fields and operators appearing in spectral triples and Fredholm modules ($\to$).

Complexes of D-branes and derived categories.

While I am just giving a rather obvious reformulation of superconnections in terms of certain functors, I might add that the above seamlessly generalizes to the setup usually considered in detail for instance for topological strings, where we don’t just have branes/anti-branes, but an entire $\mathbb{Z}$-grading of branes.

This $\mathbb{Z}$-grading plays a major role in deriving that D-branes are described by derived categories ($\to$), since it is responsible for the fact that there is not just one tachyon field

but an entire complex

of them.

There is nothing more natural than generalizing the above setup to this situation. Now we take $P_2(X)$ to contain diagrams of the form

These encode strings stretching between stacks of D-branes of arbitrary ghost charge - or whatever you call the $\mathbb{Z}$-grading here.

All of the above discussion generalizes straightforwardly to this setup in the obvious way.

While everything is encoded in the single functor

we may again manifestly write this in terms of its components ${\mathrm{tra}}^n$ encoding ordinary connections on $E^n$, together with almost-morphisms

endoced by the $\mathrm{tra}(t_x^n)$. Notice that the failure of these to be natural transformations in measured precisely by the 1-form curvature of $\mathrm{tra}$, i.e. by the gauge-covariant derivative of the tachyon.

Because, as I vaguely indicated, all the above 1-functorial discussion secretly sits inside a 2-functorial description of surface transport, there is the possibility that we realize that what looks like the failure of a natural transformation of 1-functors is actually a pseudonatural transformation of 2-functors. But I won’t go into that at the moment.

In any case, we may consider the situation where $T^{n+1}\circ T^n=0$, in wich case we get a complex of vector bundles with connection, with morphisms being tachyon fields - as known from the derived category description of D-branes.

Without any additional effort, we actually have the ability here to describe a complex of twisted bundles with connection, even a complex of twisted bundles together with their associated Kalb-Ramond gerbes with connection. But I won’t go into that at the moment.

Quivers

Assume all our D-branes are pointlike in some sense (“fractional”, maybe). The vector bundles over them are then just vector spaces, and all ordinary connections ${\mathrm{tra}}^n$ in our superconnection $\mathrm{tra}$ disappear. We are left only with the tachyon component of the superconnections.

In this case, any diagram in $P_2(X)$ consists only of point-like D-branes and strings stretching between these. This is usually called a quiver diagram describing the D-brane configuration ($\to$).

Applying our superconnection transport functor $\mathrm{tra}:P_2(X)\to \mathrm{Trans}(E^{•})$ on this quiver yields nothing but a quiver representation ($\to$).

That’s not supposed to be deep. But it’s true.

Before finishing the last entry I should review some basic facts about K-theory and D-branes, beyond of what I had in my previous notes ($\to$).

Apart from the Brodzki-Mathai-Rosenberg-Szabo paper ($\to$) I’ll mainly follow

T. Asakawa, S. Sugimoto, S. Terashima
D-branes, Matrix Theory and K-homology
hep-th/0108085

which is based in part on

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

and

Jeffrey A. Harvey, Gregory Moore
Noncommutative Tachyons and K-Theory
hep-th/0009030.

Since Witten’s original claim that D-branes are described by K-theory

Edward Witten
Overview Of K-Theory Applied To Strings
hep-th/0007175

several refinements of the precise relationship have been discussed. Usually, the decategorification and Grothedieck group completion performed in forming K-theory from the category of vector bundles is identified with the physical process of partial mutual annihilation of space-filling D9 brane and anti-brane pairs, thereby realizing all lower-dimensional branes as decay products of D9-brane configurations.

T. Asakawa, S. Sugimoto and S. Terashima in their paper promote an alternative point of view, which, as they aim to demonstrate, makes more direct contact with the conception of K-theory in terms of $C^{*}$-algebras. Namely, they use the fact that one can go the other way around, and realize all higher-dimensional branes as composites of non-BPS D(-1) branes, using a certain flavor of what is called Matrix Theory.

From this point of view the world is described by $N\to \mathrm{\infty }$ objects called non-BPS D-instantons, whose geometric configuration is encoded by ten operators

called the scalar fields, as well as an operator

called the tachyon field, all represented on some seperable Hilbert space

The mathematically inclined reader not familiar with this might (or might not) get an impression of what is going on here by looking at

Alain Connes, Michael R. Douglas, Albert Schwarz
Noncommutative Geometry and Matrix Theory
hep-th/9711162.

The dynamics of these funny objects is encoded by a functional which contains terms like

This is something one derives from boundary string field theory.

One expects to be able to make full sense of this only if at least

is trace class, hence $T$ is an unbounded operator, and

are in

the space of bounded operators on $\mathscr{H}$.

Anyone who has run across a spectral triple before should now have a déjà vu. If we like Fredholm modules better than spectral triples, we may instead consider the normalized tachyon operator

This is now bounded and $e^{-T^2}$ being trace class implies that $T_b^2-1$ is compact. Similarly the $[T,{\Phi }^{\mu }]$ are now required to be compact operators - and we have obtained a Fredholm module from our spectral triple describing D-instanton dynamics.

This story is thought to extend to an entire dictionary, which should look roughly like this.

Asakawa, Sugimoto and Terashima are mainly interested in identifying the very last entry on the right. They argue in

T. Asakawa, S. Sugimoto, S. Terashima
D-branes and KK-theory in Type I String Theory
hep-th/0202165

that the KK-theory $\mathrm{KK}(A,B)$ describes D-brane configurations on product spaces $A\otimes B$ (with $A$ and $B$ the corresponding $C^{*}$-algebras.) I am not sure that I precisely follow this, but it is certainly compelling to associate KK-theory to pairs of D-brane configurations. Given that D-branes “are modules” (