The String Coffee Table

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 $\mathbf{A}$ 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 $\mathbf{P}_2(X)$.

Proceeding analogously for the transport category $\mathrm{Trans}(E^+)$, we obtain $\mathbf{Trans}(E^+,E^-)$, 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 $\mathbf{P}_2(X)$ 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 $\mathbf{Trans}(E^+,E^-)$.

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 $\mathbf{P}_2(X)$ 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 $\mathrm{tra}(t^n_x)$. 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 $\mathbf{P}_2(X)$ 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 $\mathrm{tra} : \mathbf{P}_2(X) \to \mathbf{Trans}(E^\bullet)$ on this quiver yields nothing but a quiver representation ($\to$).

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