The topological calculation of Aharonov-Bohm phases associated with D-branes in the absence of a Neveu-Schwarz

It is well known that the existence of a magnetic field will affect the phase of electrically charged particles, even when the particles do not pass through the region containing the magnetic field. The canonical example was formulated by Aharonov and Bohm [

The canonical Aharonov-Bohm setup, with electrically charged particles moving along each of two paths,

First, note that the existence of the

When we consider D-branes, however, things are not so simple. In [

In this paper, Sections

Let us begin by considering Type IIA theory on

We next wish to lift our element of

Since

A completely analogous argument goes through in K-theory. Indeed, one may write a long exact sequence similar to that above, and all subsequent statements and actions carry over. Specifically, the long exact sequence

Next, note that the test brane defines an element of

In K-theory, we define a similar product operation via tensor products of bundles. The external K-cup product is a group morphism

When extended to higher K-groups, this product becomes

Letting

We see that we may therefore pair

To measure the Aharonov-Bohm phase at infinity, we must move the test brane on a closed path in

We may parametrize our path by a function

To show that this lift is unique, we use the bordism and direct sum relations. Consider the K-cycle

Surface

Then for appropriate choice of continuous

In the cohomological case, there is an intersection pairing on a compact oriented

We use this to define another pairing

Returning to the K-theoretic case, recall that from our torsion flux, test brane, and path of the test brane we have defined elements of

The intersection form is the nondegenerate pairing

Poincaré duality and the Thom isomorphism give [

Setting

Thus the intersection form is a nondegenerate pairing between K-homology and K-theory,

To summarize our formulation for the Type IIA case, the torsion flux defined an element of

Now that we have given the topological details of the K-theoretic formula for Aharonov-Bohm phase in the Type IIA case, we would like to develop similar statements for the Type IIB and Type I cases.

In the Type IIB situation, the torsion flux takes values in

Finally, we may make a similar proposal in the Type I scenario. Here, the torsion flux is valued in

Some explanation is required to justify labelling this

In the Type I case, these open string interactions can be viewed from the perspective of the effective gauge theory defined on the worldvolume of 9-branes used to construct the D-brane system. The two D-branes correspond to topological defects in the gauge bundle defined on the 9-brane system, and the K-theoretic pairing specified above measures the topological phase induced by the relative motion of the defects. We will return to this gauge bundle perspective later, in Section

Here we describe the pairing

First we define a

We may define a

Next, recall that a K-cycle in

Since

Let

Then given a

Finally, given a cycle

Note that we can use the D-brane charges instead of the RR fields in our prescriptions. The K-theory class associated with an RR-field may be mapped in a well-defined way to the K-theory class associated with the D-brane charge via the isomorphism

We now consider an Aharonov-Bohm experiment for a

The 8-brane determines the nontrivial element of

Next, the (−1)-brane determines an element of

We also have the associated Dirac operator along

Our analytic pairing can be evaluated by using the pullback via

Then we find that our pairing gives

This result agrees with the calculation performed in [

In this paper we have developed formulae to calculate the Aharonov-Bohm phase of torsion Ramond-Ramond fluxes in the Type II and Type I string theories based upon the K-theoretic classification of Ramond-Ramond fields and D-brane charges. These formulae were constructed in two different but equivalent fashions, one being purely topological and the other employing the reduced eta-invariant. The topological pairing was shown to exist and be well defined. The analytic perspective was used to calculate the phase for the

The author would like to thank G. W. Moore for the helpful discussions.