AHEP Advances in High Energy Physics 1687-7365 1687-7357 Hindawi Publishing Corporation 920486 10.1155/2012/920486 920486 Research Article The K-Theoretic Formulation of D-Brane Aharonov-Bohm Phases 0000-0001-6695-1210 Warren Aaron R. Geng C. Q. Department of Mathematics, Physics, and Statistics Purdue University North Central 1401 S. US-421 Westville IN 46391 USA pnc.edu 2012 30 12 2012 2012 30 09 2012 19 11 2012 24 11 2012 2012 Copyright © 2012 Aaron R. Warren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The topological calculation of Aharonov-Bohm phases associated with D-branes in the absence of a Neveu-Schwarz B-field is explored. The K-theoretic classification of Ramond-Ramond fields in Type II and Type I theories is used to produce formulae for the Aharonov-Bohm phase associated with a torsion flux. A topological construction shows that K-theoretic pairings to calculate such phases exist and are well defined. An analytic perspective is then taken, obtaining a means for determining Aharonov-Bohm phases by way of the reduced eta-invariant. This perspective is used to calculate the phase for an experiment involving the (−1) −8 system in Type I theory and compared with previous calculations performed using different methods.

1. Introduction

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 , and is shown in Figure 1.

The canonical Aharonov-Bohm setup, with electrically charged particles moving along each of two paths, C and C, before interfering on a screen.

First, note that the existence of the B-field induces a connection ω=(ie/)A, where A is the Lie-algebra valued one form determined by the field. Next, we let γ=C-CH1(X). Then it is found that the phase acquired by a particle traveling along γ is (1.1)Φ[A,γ]=ieγA. Imposing the reality condition Φ[A1,γ]~Φ[A2,γ] if A1-A2, the set of equivalence classes [Φ[A,γ]] are /U(1)-valued. So we may view [Φ[A,·]] as an element of H1(X;U(1)). Then we see that the Aharonov-Bohm phase is given by a pairing H1(X)×H1(X;U(1))U(1) defined as [Φ[A,γ]].

When we consider D-branes, however, things are not so simple. In [2, 3] it was shown that D-brane charges and Ramond-Ramond (RR) fields in Types IIA, IIB, and I theories are classified by K-theory. Therefore, the calculation of Aharonov-Bohm phases for D-branes will necessarily involve some sort of K-theoretic pairing.

In this paper, Sections 2.12.4 produce a number of details concerning the topological formulation of D-brane Aharonov-Bohm phases in Type IIA theory, building off of a brief speculative discussion in . It is shown that the pairing outlined in  exists and is well-defined. Section 2.5 provides adaptations of this pairing to the Type IIB and Type I settings. In Section 3, the focus shifts to the use of the reduced eta-invariant as a means for calculating the K-theoretic pairing. A brief overview of relevant mathematical technology is presented in Section 3.1 and then utilized in Section 3.2 for the (-1)-8 system in Type I theory. It is shown that our result agrees with a calculation performed in  using different methods.

2. The Topological Formulation

Let us begin by considering Type IIA theory on t×X9, and suppose we have a brane producing a torsion flux. This flux defines an element of Ktors0(X), the torsion subgroup of K0(X), where XX8=X9.

2.1. The Long Exact Sequence

We next wish to lift our element of Ktors0(X) to an element of K-1(X;U(1)). Before doing this, it is useful to consider the analogous cohomological situation. From the exact coefficient sequence (2.1)0U(1), we may obtain the long exact cohomological sequence (2.2)δk-1Hk(X;)ikHk(X;)jkHk(X;U(1))δkHk+1(X;)ik+1Hk+1(X;), where the maps δk:Hk(X;U(1))Hk+1(X;) are called the Bockstein homomorphisms.

Since /n0 for any n-{0}, the kernel of ik+1 is the set of torsion elements of Hk+1(X;), denoted Htorsk+1(X;). Therefore, we may write the following exact sequence, (2.3)Hk(X;U(1))δkHtorsk+1(X;)0. Thus for any torsion class there is a lift in Hk(X;U(1)). This lift is an integral cochain, that is, closed in U(1) but not in . Simple diagram-chasing shows that this lift is well defined .

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 (2.4)K-1(X)chK-1(X;)αK-1(X;U(1))βK0(X)chK0(X;) gives the exact sequence (2.5)K-1(X;U(1))βKtors0(X)ch0. Here, ch is the Chern character and β is the forgetful map. Lifting an element of Ktors0(X) via the Bockstein β then gives an element of K-1(X;U(1)). As in the cohomological case, diagram-chasing shows that this lift is well defined.

2.2. The K-Cup Product

Next, note that the test brane defines an element of K0(X). We will pair this with our element of K-1(X;U(1)) via the K-cup product . Again we begin by considering the analogous cohomological case. There, one starts with a mapping (2.6)Sp(X)×Sq(X;G)Sp+q(X;G), which assigns to each p-cochain cp and q-cochain cq a (p+q)-cochain cp+q by letting cp act on the front p-face and cq act on the q-back face, then multiplying the results by the usual product operation sending (n,g) to ng. It follows that gives a well-defined product operation  (2.7)Hp(X)×Hq(X;G)Hp+q(X;G).

In K-theory, we define a similar product operation via tensor products of bundles. The external K-cup product is a group morphism K(X)K(Y)K(X×Y) which assigns to each abK(X)K(Y) the element (K(px)(a))(K(py)(b))K(X×Y), where px:X×YX and py:X×YY are the projection operators.

When extended to higher K-groups, this product becomes K-i(X)×K-j(Y)K-i-j(X×Y). To see this, we start with a pairing (2.8)K~-i(X)×K~-j(Y)K~-i-j(XY) given by tensor product. Next we use the defined relationship K-i(X)K~-i(X+)K~(Σi(X+)), where X+X{pt.} is X with a disjoint basepoint and Σi(X)=SiX is the smash product of Si with X. Then we obtain a pairing K-i(X)×K-j(Y)K-i-j(X×Y).

Letting X=Y and then composing with the map from K-i(X×X) to K-i(X) induced by the diagonal map XX×X gives the product (2.9)K-i(X)×K-j(X)K-i-j(X). It follows that there is a pairing (2.10)K-i(X)×K-j(X;G)K-i-j(X;G), with (n,g)ng as in the cohomological case.

We see that we may therefore pair (2.11)K0(X)×K-1(X;U(1))K-1(X;U(1)) via the K-cup product with generalized coefficients. Thus, given a brane producing a torsion flux and a charged test brane, we obtain an element of K-1(X;U(1)).

2.3. K-Homology

To measure the Aharonov-Bohm phase at infinity, we must move the test brane on a closed path in X. This path defines an element of H1(X). In order to pair our path with K-1(X;U(1)), we must lift the path to an element of K1(X), the K-homology of X .

We may parametrize our path by a function f:S1X. Note that S1 is a compact Spin-manifold without boundary, and f is by definition a continuous map. To put a complex vector bundle on S1 is easy, since every complex vector bundle on S1 is trivial. It is natural, then, to let a K-cycle associated with our path be given by (S1,ϵn,f) where ϵn is the trivial complex vector bundle with fibre n. Since S1 is odd-dimensional, we have defined an element of K1(X).

To show that this lift is unique, we use the bordism and direct sum relations. Consider the K-cycle (S1,ϵ2,f2). Since 2= the direct sum relation gives (S1,ϵ2,f2)~(S1,ϵ1,f2)(S1,ϵ1,f2). Let W be the compact 2-dimensional Spin-manifold shown in Figure 2, and put the trivial rank 1 complex vector bundle on it.

Surface W carries the trivial rank 1 complex vector bundle on it and serves as a bordism between (-S1,ϵ1,f1) and two copies of (S1,ϵ1,f2).

Then for appropriate choice of continuous ϕ:WX, we have (2.12)(W,E|W,ϕ|W)(S1,ϵ1,f2)(S1,ϵ1,f2)(-S1,ϵ1,f1)~(S1,ϵ2,f2)(-S1,ϵ1,f1). Hence (2.13)(S1,ϵ2,f2)~(S1,ϵ1,f1). Clearly this generalizes to any K-cycle (S1,ϵN,fN). Thus our lift from H1(X) to K1(X) is unique.

2.4. The Intersection Form

In the cohomological case, there is an intersection pairing on a compact oriented n-dimensional manifold X(2.14)Hk(X;)×Hn-k(X;U(1))U(1) defined by (2.15)α×βα·βαβ,[X], that is, cup product followed by integration over an orientation class [X]Hn(X;).

We use this to define another pairing (2.16)Hk(X;)×Htorsn-k+1(X;)U(1) in the following way. Since any torsion class [α]Htorsn-k+1(X;) has a well-defined lift [α]Hn-k(X;U(1)), we may define the desired pairing as (2.17)βα,[X]U(1).

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 K1(X;U(1)) and K1(X). We would now like to pair these elements and get an element of U(1), the Aharonov-Bohm phase. This is achieved with the use of the so-called intersection form .

The intersection form is the nondegenerate pairing (2.18)Kcpti(T*X)×Ki(X;U(1))Kcpt0(T*X;U(1))p!U(1), (with cpt denoting compact support) which is induced by the K-cup product and the direct image mapping p!:K0(T*X,U(1))K0(pt,U(1))=U(1) corresponding to the map p:Xpt.

Poincaré duality and the Thom isomorphism give [10, 12] (2.19)Kcpt1(T*X)K1(X). To see this isomorphism topologically, first let S(T*X) denote the unit sphere bundle of T*X. Also let π:S(T*X)X be the projection. It was shown in  that elements of K1(T*X) are in one-to-one correspondence with stable homotopy classes of self-adjoint symbols on X. Then an element of K1(T*X) is a pair (E,σ) where E is a Hermitian vector bundle on X and σ:π*(E)π*(E) is a self-adjoint automorphism of π*(E). Then σ gives the decomposition π*(E)=E+E- where E± is spanned by the eigenvectors with ± eigenvalues of σ.

Setting X^=S(T*X), note that dim(X^)=2(dim(X))-1 so that it is odd-dimensional. Furthermore, X^ is a Spin-manifold since TX^ has a Spin structure from TXT*XTX. Then the triple (X^,E+,π) is an element of K1(X). If we define c(E,σ)=(X^,E+,π), then (2.20)c:K1(T*X)K1(X) is an isomorphism.

Thus the intersection form is a nondegenerate pairing between K-homology and K-theory, (2.21)K1(X)×K1(X;U(1))U(1). This is precisely what we need to give the Aharonov-Bohm phase.

To summarize our formulation for the Type IIA case, the torsion flux defined an element of Ktors0(X) which we lifted to K-1(X;U(1)) by the long exact sequence of K-groups associated with the exact coefficient sequence. The test brane defined an element of K0(X) which we paired with our element of K-1(X;U(1)) via the K-cup product to again get an element of K-1(X;U(1)). The path of our test brane defined an element of H1(X;) which we lifted to an element of K1(X;). The intersection form then took K1(X;)×K1(X;U(1))U(1), which we call the Aharonov-Bohm phase.

2.5. The Type IIB and Type I Cases

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 Ktors1(X), with X=X9 as before. Then we may again use the exact coefficient sequence to give a long exact sequence of K-groups and lift our element of Ktors1(X) to K0(X;U(1)). Now our test brane defines an element of K1(X), and we again use the K-cup product to pair these elements as (2.22)K1(X)×K0(X;U(1))K1(X;U(1)). Again we lift the path of the test brane from H1(X) to K1(X) and use the intersection form to pair (2.23)K1(X)×K1(X;U(1))U(1).

Finally, we may make a similar proposal in the Type I scenario. Here, the torsion flux is valued in KOtors-1(X), and the test brane defines an element of KO-1(X). All the properties of the complex K-theory that we employed carry over to the KO-groups. The only real difference between these theories is the form of Bott Periodicity, but that does not seriously affect our discussion. So we lift the torsion flux to KO(X;U(1)), then pair the test brane charge to it, (2.24)KO-1(X)×KO(X;U(1))KO-1(X;U(1)), by a KO-cup product which is completely analogous to the K-cup product. We can complexify to obtain an element of K-1(X;U(1)). Now we lift the path of the test brane from H1(X) to K1(X). Then we use the intersection form to pair (2.25)K1(X)×K-1(X;U(1))U(1), giving us the Aharonov-Bohm phase.

Some explanation is required to justify labelling this U(1) as an Aharonov-Bohm phase. The question is whether this phase occurs within the partition function for a D-brane that participates in an Aharonov-Bohm experiment. The interaction between the pair of D-branes involved in such an experiment will be mediated by open strings connecting the two branes, and to produce an Aharonov-Bohm phase they must be sensitive to their relative orientations. Only fermions which become massless when the branes coincide are capable of detecting their relative orientations. While the Neveu-Schwarz sector open string zero point energy is sometimes greater than zero , Ramond-sector open strings always have a zero point energy equal to zero. Therefore, there will be massless fermions whose sensitivity to relative orientation will affect the partition function, generating an Aharonov-Bohm phase.

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 3.2.

3. Analytical Aspects

Here we describe the pairing (3.1)K1(X;)×K1(X;U(1))U(1) from an analytic point of view. We begin by reviewing relevant material from  to define the reduced eta-invariant, and relate it to the topological pairing from Section 2. We then use the eta-invariant to calculate the phase for an Aharonov-Bohm experiment involving a (-1)-8 brane system in Type I theory.

3.1. The Analytic Formulation

First we define a 2-graded cocycle in K1(X;U(1)) to be a quadruple 𝒱=(V±,hV±,E±,ω), where

V=V+V- is a 2-graded vector bundle on X,

hV=hV+hV- is a Hermitian metric on V,

V=V+V- is a Hermitian connection on V,

ωΩodd/im(d) satisfies dω=ch(V).

We may define a Z2-graded cocycle in KO1(X;U(1)) analogously, by replacing the adjectives “complex” and “Hermitian” with “real” and “symmetric,” respectively.

Next, recall that a K-cycle in K1(X) is a triple (M,E,f) with M a closed odd-dimensional Spin-manifold, E a complex vector bundle on M, and f:MX a continuous map. We will in fact let f be smooth here. Again note that there is an analogous real formulation. For the rest of this subsection, however, we will restrict our attention to the complex case.

Since M is Spin, the principle GL(dim(M))-bundle on M may be reduced to a principle Spin-bundle, call it P. We may associate to P a Hermitian line bundle L on M . Choose a Hermitian connection L on L, a Hermitian metric hE on E, and a Hermitian connection E on E.

Let A^(TM)Ωeven(M) be the closed form representing A^(TM)Heven(M;). Also, let exp[c1(L)/2]Ωeven(M) be the closed form representing exp[c1(L)/2]Heven(M;). Finally, let the spinor bundle of M be SM.

Then given a 2-graded cocycle 𝒱K1(X;U(1)), we let Df*V± denote the Dirac-type operator acting on sections of SMEf*V±. Its reduced eta-invariant is  (3.2)η-(Df*V±)=12[η(Df*V±)+dim(Ker(Df*V±))]mod. Then the reduced eta-invariant of f*𝒱 is the /-valued function (3.3)η-(f*𝒱)=η-(Df*V+)-η-(Df*V-)-MA^(TM)exp[c1(L)2]ch(E)f*ω.

Finally, given a cycle 𝒦=(M,E,f) in K1(X) and a 2-graded cocycle 𝒱 for K1(X;U(1)), their /-valued pairing is (3.4)[𝒦],[𝒱]=η-(f*𝒱). The proof that this is in fact the correct pairing is given in , and is based largely on the corresponding proof in . We claim that this yields the Aharonov-Bohm phase as (3.5)2πiη-(f*𝒱).

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 (3.6)Kcpti(M)Ki-1(M)j(Ki-1(M)), where j restricts a K-theory class from M to M. The analogous isomorphism holds for the KO-theory as well. Therefore, if desired we may change our prescriptions to begin with the K-theory classes associated with the charges of the D-branes.

3.2. Calculation in the Type I Case

We now consider an Aharonov-Bohm experiment for a (-1)-8 system of Type I D-branes. The path of the instanton defines the K-cycle (M,ϵ1,f) as discussed above. We use the 32 nine-branes required for tadpole cancellation to construct our system without adding extra branes/antibranes. It will be convenient to work from the K-theory classes of the D-brane charges instead of those of the RR fields.

The 8-brane determines the nontrivial element of KOtors0(S1)=2, which we view as KOtors0(1) with compact support. Such an element is given by the pair (E8,pt) where E8 is a rank 1 bundle and pt is the trivial rank 0 bundle. We lift this via the Bockstein to an element (E8,pt)KO-1(;U(1)). Note that E8 is also a rank 1 bundle.

Next, the (−1)-brane determines an element of KO0(S10)=2. This is the pair (E-1,pt). Taking the KO-cup product we get (E-1E8,pt)KO-1(11;U(1)). After complexifying these bundles, we then obtain a 2-graded cocycle V=(E-1E8)pt in K-1(11;U(1)).

We also have the associated Dirac operator along M for the E-1E8 component of the cocycle, (3.7)Da+=D(A-1)+D(A8)+Γ9a, where a parametrizes M as the distance between the 8-brane and the (−1)-brane as above, and the + superscript indicates that it corresponds to the V+E-1E8 component. Since E8 is a rank 1 bundle, the index theorem [17, 18] says that D(A-1)+D(A8) has one zero mode of definite chirality with respect to Πμ=09Γμ and Γ9 (in the nontrivial instanton number sector). Thus Da+ has eigenvalue a.

Our analytic pairing can be evaluated by using the pullback via f of V=(E-1E8)pt from X to M. Since the eigenvalue of Da+ is equal to a, we get η-(Df*V+)=1/2. Also, since E-1 and E8 are each SO(n)-bundles, their connection and curvature forms are so(n)-valued, hence have zero trace. Complexification does not change this, so that the Chern character of V is zero. Then since dω=ch(V)=0 and ωΩodd/im(d), we get ω=0 and f*ω=0. Finally, note that η-(Df*V-)=0 since V-=pt.

Then we find that our pairing gives (3.8)η-(f*𝒱)=η-(Df*V+)-η-(Df*V-)-MA^(TM)e(c1(L)/2)ch(ϵ1)f*ω=12. Consequently, we get an Aharonov-Bohm phase of 2πi(1/2)=iπ, and the monodromy is exp(iπ)=-1.

This result agrees with the calculation performed in  for the (-1)-8 system, which was performed by examining changes in massless fermionic contributions to the amplitude as the instanton is moved.

4. Summary

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 (-1)-8 system in Type I theory, allowing us to test our formulae by comparison with independent calculations.

Acknowledgment

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

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