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HYPER VERSION Brane transfer operations and T-duality of non-BPS states

J H 1. Introduction 2. Type II theories 2.1 D = 10 2.2 General compacti cations 2.3 D = 9 2.4 D < 9 3. Type I Theory and its T-Duals 3.1 D=10 3.2 D = 9 3.3 D-brane decay

3.5 D < 9 4. Type eI Theories

1. Introduction
A closer look at the dynamics of various unstable D-brane systems (such as
braneantibrane systems) has recently opened a new perspective for understanding D-branes
and their conserved charges. Traditionally, D-branes are understood as RR-charged
stringy solitons on which strings can end; in the new framework [

A crucial role in this construction is played by our improved understanding of
the string theory tachyon [

Any such construction can be related through a hierarchy of embeddings to
bound states in the unstable system of a number of spacetime- lling D9-branes.
The worldvolume dynamics of this system contains U(N ) U(N ) Yang-Mills theory
and a Higgs eld (a.k.a. \tachyon") in the (N; N) representation in the case of
Type IIB theory [

The precise dynamics of these unstable D-brane systems is not known, but the
topological information needed for the complete classi cation of D-brane charges
can still be determined. This information is usefully encoded in K-theory [

One of the de ning qualities of D-branes, which in fact is how they were
discovered [

In section 2 we discuss the K-theory realization of T-duality in Type II string theory. As one of the central points of the paper, we show that in general string theory compacti cations, D-brane charges are classi ed by relative K-theory groups, such as K(Sp Y; Y ), where Y is the compacti cation manifold. In the case of Type II strings compacti ed on a circle, the relative K-theory group of D-brane charges splits into the sum of two groups, whose elements reflect the split of the D-brane charges between wrapped D-branes and unwrapped D-branes. The K-group for IIB on a circle, K(X S1; S1), is isomorphic to the K-group for IIA on a circle, K?1(X S1; S1), clearly in line with T-duality. Moreover, the isomorphism exchanges the subgroup associated with wrapped D-branes on one side with the subgroup for unwrapped D-branes on H E P 0 4 ( 1 9 9 9 ) 0 1 0 Z planes. Unlike in the Type II theory, there are now new non-BPS D-branes with 2 the other side. In the rest of this paper, we would like to see if this clear split between wrapped and unwrapped D-branes holds generically in K-theory.

Our rst test case, discussed in section 3, involves looking at the T-duality between Type I strings on a circle and the Type IA orientifold with two O8? orientifold valued charges. At rst, the situation looks quite similar to the Type II case: The K-group for Type I on a circle, KO(X S1; S1), is again found to be isomorphic to the Type IA K-group KR?1(X S1; S1), and it splits into two parts which one may naively interpret as corresponding to wrapped branes and unwrapped branes.

This interpretation raises two puzzles. First, some of the non-BPS D-branes that were stable in flat space are now only stable for a certain range of the circle's radius, even though they carry a conserved charge. Second, in Type IA we expect to nd unwrapped branes localized on each of the two orientifold planes, yet the corresponding charges seem to be missing from the subgroup we naively associate with unwrapped branes. Section 3 will demonstrate how these two puzzles are resolved. A key element in this resolution involves processes that we refer to as brane transfer operations.

Our second test case, presented in section 4, deals with the more exotic Type eI open string theory, which is T-dual to the Type IfA orientifold with both an O8? and an O8+ orientifold plane. The K-group associated with these theories, which turns out to be equivalent to a certain K-theory group known in the mathematical literature as KgSC(X), does not split naturally into the of sum of two sub-groups. The split between wrapped branes and unwrapped branes is totally defeated. We will show that the root of this problem is linked to the fact that some non-BPS brane con gurations locally stable at the O8? orientifold plane become unstable due to the presence of the O8+ plane and vice versa. This phenomenon can have important consequences for the piecewise analysis of the stable non-BPS D-brane spectra in various compacti cations, such as when one approximates singularities in

Many technical details required for our analysis have been relegated to an appendix, which also serves as a collection of basic facts in K-theory, and can therefore be of some independent interest.

While this paper was being written, two papers [

2. Type II theories As a warm-up exercise, we set the stage for our later analysis of stable D-branes in various orientifold models by rst analyzing the case of Type II theories compacti ed on a circle to nine dimensions. (As we will see, this procedure can be easily iterated to understand Tn compacti cations). It turns out that all stable D-brane states preH E P 0 4 ( 1 9 9 9 ) 0 1 0

Ke (X)

K?1(X) 2.1 D = 10 D6 S3 0 D7 S2 D8

S1 D9 S0 0 K?1(Sn) = n?1 (U(2N )=U(N )

Ke (Sn) = n?1(U(N )):

D4 S5 0 D5

S4 dicted by K-theory in these compacti cations carry conventional Ramond-Ramond charges. Therefore, we do not expect any surprises; the main goal in this brief section is to see that T-duality of Type II theories is indeed a manifest symmetry in

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R Stable Dp-brane charges in Type IIA and Type IIB theory on 10 are classi ed
by the K-theory groups K?1(S9?p) and Ke (S9?p) respectively, where the sphere S9?p
represents the dimensions transverse to the worldvolume of the p-brane, compacti ed
by adding a point at in nity. This result can be derived by realizing supersymmetric
Type II D-branes as stable topological defects in the Higgs eld on the worldvolume of
a system of spacetime- lling D9-branes [

R entire compacti cation manifold Y at in nity. Therefore n is e ectively replaced R to the vacuum asymptotically in the transverse space n, i.e. along a copy of the R 9?d. Such charges will arise both from D-branes located at particular points in Y , R pacti cations on 9?d Y , at least when no RR backgrounds or non-trivial B The next step would be to consider Type II theory compacti ed on S1. Before we discuss this case in detail, it seems worthwhile to rst study the classi cation of D-brane charges for a more general compacti cation space Y of dimension d. We are interested in nding all D-brane charges of codimension n in the non-compact space and from D-branes which wrap non-trivial cycles in Y . Since we are only interested in objects of nite energy (or action), we consider only con gurations that are equivalent with Sn by adding a point at in nity, which corresponds in the full theory to adding a copy of the compacti cation manifold Y at in nity. In mathematical terms, this requires us to consider bundles which are trivialized on the compacti cation manifold Y at in nity; such bundles de ne groups known in the mathematical literature as relative K-theory groups (cf. the appendix). Thus, we conclude that the proper way of understanding the spectrum of D-brane charges is in terms of relative K-theory. In Type IIB and Type IIA theory on Y , the relative groups that classify D-brane charges are denoted by K(Sn Y; Y ) and K?1(Sn Y; Y ), respectively.

The argument leading to the appearance of relative K-theory groups is essentially independent of the type of string theory considered. It suggests the following prescription for identifying stable D-brane charges in general string theory combackgrounds are excited: Stable charges carried by D-branes of codimension n in the non-compact dimensions are classi ed by the relative K-theory groups K?q(Sn Y; Y ). The value of q and the type K of K-theory depends on the type of string theory and the compacti cation manifold Y . 2.3 D = 9 Having clari ed the appearance of relative K-theory groups in the classi cation of D-brane charges in general compacti cations, we can now return to Type II theory on a circle. Using (A.9) and arguments presented in the appendix, the relative groups that classify D-brane charges in Type IIB and Type IIA theory on S1 can be shown to decompose as follows: and

K(X

S1; S1) = K?1(X)

Ke (X) ; K?1(X

S1; S1) = Ke ?2(X)

K?1(X) ; (2.3) (2.4) where in both cases the rst term is the contribution of unwrapped branes to the nine-dimensional D-brane charge, and the second term is the contribution of wrapped

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 K(X

K(X

S1; S1) = K?1(X m ! n 2.4 D < 9

m Tm; Tm) = M

n=0 branes. Since by Bott periodicity Ke ?2(X) = Ke (X), the above groups are isomorphic We can iterate the steps of the previous subsection, and extend our results to higher toroidal compacti cations. Thus, the relative group of D-brane charges in Type IIB theory is Z group of D-brane charges is | the RR charge of the corresponding Dp-brane. and we recover the result that the spectrum of D-brane charges in nine dimensions is identical for Type IIA and Type IIB. In fact, for each X = Sn, the relative K-theory

Furthermore, since the above isomorphism maps the rst (second) term in (2.3) to the second ( rst) term in (2.4), and therefore exchanges unwrapped and wrapped D-branes, it corresponds precisely to T-duality. More rigorously, this follows from a derivation of (2.5) that keeps track of the multiplicative structure of K-theory (see (A.12) of the appendix and the discussion therein).

This proves T-duality of D-brane charges in Type II theory on Tm, and gives the expected degeneracy of Dp-brane charges arising from wrapping all higher supersymmetric branes on various cycles of the torus. All in all, this shows that in the case of Type II theory on Tm, D-brane charges predicted by the more precise Ktheory arguments coincide with those predicted by the somewhat cruder argument that relates D-brane charges to RR charges (and therefore to the cohomology of the compacti cation manifold). where the second equality follows by Bott periodicity. An analogous calculation on the Type IIA side (cf. the appendix) reveals

Ke ?n(X) = Ke(X) 2m?1 3. Type I Theory and its T-Duals

K?1(X) 2m?1 ; Tm; Tm) = K(X

Tm; Tm) : K?1(X

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 S1; S1) ; (2.5) (2.6) (2.7)

D9

S0 3.1 D=10

D8 S1

D7

S2

D4 S5 0

D3 S6 0

D2 S7 0

In Type I theory, the full spectrum of D-brane charges can be determined from
the dynamics of unstable systems of multiple D9-brane D9-brane pairs. Since the
action of the orientifold group is antilinear on Chan-Paton bundles, the K-theory that
arises in such systems is the KO-theory of real virtual bundles [

The values for KgO(Sn) reproduce the known BPS Dp-branes of ten-dimensional
charged non-BPS Dp-branes with p = ?1; 0; 7 and 8. The former carry RR charge
and correspond to boundary states of the form
and are their own antibranes. All the properties of the non-BPS D-branes can be
obtained from these boundary states via tree-level overlaps with other boundary
states [

The most useful description of the non-BPS D-branes is often in terms of bound
states of a single BPS D-brane D-antibrane pair with lowest possible dimension. In
this approach, the D0-brane and D8-brane simply correspond to topologically stable
kinks in the tachyon (Higgs) eld living on the worldvolume of the D1-D1 and D9-D9
systems, respectively [

1 jDpi = p2 (jBpiNSNS jDpi = jBpiNSNS ; jBpiRR) ;

D5 USp(2)

D0 S9 (3.1) (3.2) D5 S4

D1

S8 D7

D8

D9

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 D3 S5 J H E P 0 4 ( 1 9 9 9 ) 0 1 0

S1; S1) = KgO?1(X)

Thus, we obtain the nine-dimensional stable D-brane charge spectrum as table 4: As in the Type II case, D-brane charges of Type I compacti ed on a circle are classi ed by the relative K-theory group KO(X S1; S1). This group is evaluated in the appendix, giving: 3.2 D = 9

KO(X

KgO(X)

KgO?1(X) R of the form 9 S1=? I, known as Type IA (or Type I0); here ? acts as a reflection

Note that the relative K-theory groups correctly include the nine-dimensional Dpbrane charges that correspond to unwrapped Dp-branes as well as wrapped D(p + 1)branes in the ten-dimensional theory.

Under T-duality, Type I string theory is mapped to an orientifold of Type IIA
on the worldsheet, and I acts as a reflection on the compact direction. The compact
direction is therefore an interval, rather than a circle. The associated relative
Ktheory group is given by KR?1(X S1;1; S1;1) [

whose elements correspond to D-brane charges of the orientifold compacti cation.

The isomorphism (3.5) again maps the rst term of the relative KO-group in (3.3) to the second term of the relative KR-group in (3.4), and vice versa. It is therefore tempting to identify the respective terms as the contributions to nine-dimensional D-brane charges coming from unwrapped and wrapped ten-dimensional D-branes. Therefore, as in the Type II case, T-duality between Type I and Type IA theory manifests itself as an isomorphism between the relative K-theory groups,

S1;1; S1;1) ;

Consider for example the non-BPS D0-brane in Type I. The spectrum of open strings beginning and ending on the D0-brane is tachyon-free in ten dimensions. Once we compactify on a circle however, the ground state at winding number 1 will have a classical mass squared given by ( 0 = 1) Z Z con gurations, i.e. 2 from the wrapped non-BPS D1-brane, and from unwrapped Z For example, nine-dimensional 0-brane charge in Type I receives a 2 contribution Z from the unwrapped non-BPS D0-brane, and a contribution from wrapped BPS Z responsible for the 2 charges when the non-BPS D-branes are unstable? To answer D1-branes. In the Type IA description, it receives contributions from the T-dual BPS D0-branes. However, on the face of it, there seems to be a problem with this interpretation: the non-BPS D-branes are not stable for all radii, and therefore cannot contribute conserved charges everywhere in moduli space. What is then this question, let us rst recall how non-BPS D-branes decay.

m2 = ?12 + R2 ;

(3.6)

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 R?

R ?R?

Z2 Dp-brane

D(p+1)-brane

T-dual picture:

R = 1 Z to the other orientifold plane. Thus we see that the puzzle of missing 2 charges in Z R0 < p2, there seem to be two distinct sources of 2 charge associated with D0Z D0-brane 2 charge.

The picture of D-brane decay described above o ers insight for the resolution of another puzzle which is most clearly illustrated in the Type IA picture. When branes in the nine-dimensional theory. The rst one is due to the possibility of locating a single D0-brane at either orientifold plane, while the second is due to the stretched non-BPS D1-brane. However, K-theory indicates that there is only one

As we saw above, when R0 > p2 the D1-brane decays into a D0-brane at one O8plane, and a D0-brane at the other O8-plane. This is a crucial clue for the resolution of our puzzle. Consider a con guration consisting of a stuck D0-brane (half D0-brane) at one orientifold plane and a wrapped non-BPS D1-brane. As far as conserved Dbrane charges are concerned, this con guration is completely equivalent to a stuck D0-brane at the other orientifold plane, and in fact unstable to decay into it. The same is true for the unwrapped D7 and D8-brane. In each case, a D-brane stuck at one orientifold plane is \transferred" by a wrapped D-brane of one higher dimension

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 12 Dp-brane Brane Transfer

Operation 21 Dp-brane Our analysis can be straightforwardly extended to T-duality of Type I theory on higher tori. D-brane charges in Type I theory on Tm correspond to the relative

KO(X

m Tm; Tm) = M p=0 m ! p

KgO?p(X) ; (3.7) to be compared to the corresponding group on the T-dual side.

The T-dual orientifold theory is IIA/? Im if m is odd, and IIB/? Im if m is
even, with Im the involution (times appropriate factors of (?1)FL) that reflects m
compact dimensions. D-brane charges in these theories will therefore be classi ed
by the relative KR-group KR?n(X Tm; Tm) (for some n), where the involution of
KR-theory acts trivially on X, and reflects all the dimensions of Tm. An interesting
subtlety arises when we try to determine the value of n that corresponds to Tm. In
Type IA theory, i.e. for m = 1, one could use the alternative string theory de nition
of KR?1(X) to demonstrate that the appropriate value of n is in fact n = 1 [

Agreement with the known spectrum of supersymmetric D-branes determines that n = m mod 4. Since the Bott periodicity of KO-theory is eight, this leaves an uncertainty as to whether n equals m or m + 4. We claim that the correct value is n = m, and the D-brane charges in the Type I T-dual models are classi ed by the relative group

KR?m(X Indeed, iterating (A.23), and using the (1; 1) Bott periodicity of KR-theory, one can show that

KR?n(X

m Tm; Tm) = M p=0 m ! p

m KgRn;p(X) = M p=0 m ! p

KgOp?n(X) ;

(3.9)

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 4. Type eI Theories this coincides with (3.7) for n = m.3 Thus, we again get a T-duality isomorphism

R the relative group KR(X S0;2; S0;2).4 Since S0;2 is just a circle in 2 with both
Z another type of orientifold plane and annihilating there. Also, stretched 2 charged
Z worked out in [

To classify the possible D-brane charges of Type IfA we will once again analyze possible tachyon backgrounds of unstable D9-branes using K-theory. For this purpose, it is easier to start with the T-dual of Type IfA, Type eI theory. This T-dual was is realized by composing worldsheet parity reversal (?) with a half-circumference shift along the circle. The natural K-group of D-brane charges for Type eI is then dimensions reflected, the involution of KR-theory indeed acts on S0;2 by the required shift.

The Type eI K-group, KR(X S0;2), is known in the mathematics literature to
be isomorphic to the K-group KSC(X) of self-conjugate bundles on X (see [

4We are again using the Sp;q notation reviewed in the appendix.

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 D5

D4

D3 D8

D7

D1

D0

KR(X

D(?1)

S0;2; S0;2) :

S0;2; S0;2) = KR?4(X

First of all, using the relation to KSC-theory, we can show for the relative groups that D2 0 The appearance of KR?4(X S0;2) is very interesting here, as this group associates a symplectic projection to ?. Thus, the period of four indicated by (4.1) ts nicely with having both an O8? plane and O8+ plane in Type IfA | indeed, it means that orthogonal and symplectic groups appear on the same footing in this model.

Second, KSC groups have been calculated for all spheres Sn [

Unlike the relative K-theory groups that appeared in the previous sections, KSC(X) does not naturally split into subgroups related to wrapped and unwrapped branes. Undeterred, we will try to analyze the physical spectrum listed above in terms of the wrapped and unwrapped Dp-branes of Type IfA string theory, hoping to learn an interesting lesson when this strategy becomes inadequate.

Our strategy for determining which nine-dimensional Dp-branes come from
unwrapped D-branes of Type IfA (which are point-like along the interval) is to use our
knowledge of Type IA theory to list the stable D-brane spectrum near an O8? plane,
and to use a simple period shift to list the stable D-brane spectrum near an O8+
plane (cf. [

4.1 Unwrapped D-branes of Type IfA D(?1) D2 0 D5 0

D3 0 D8

D7

D6

D4

D0

J H E P 0 4 ( 1 9 9 9 ) 0 1 0

D4 D8

D3

D2

D0 D7 0

The O8+ plane di ers from O8? by interchanging SO and Sp projections. Due to Bott periodicity between KO- and KSp-theory, the switch from O8? to O8+ corresponds to swapping Dp-branes with D(p + 4)-branes, and leads to the spectrum near the O8+ plane:

Z puzzle reveals an interesting new e ect. Take, for example, the stable 2-charged Z no need of extra 2 charges for brane transfer operations.

Z run into an interesting puzzle: while these tables correctly account for the 2-charged Z D(?1)-brane, D3-brane and D7-brane of table 5, they also predict a 2-charged D2Z Looking rst at the {valued D-brane charges, we see that we can correctly Z charges, we now turn to the 2-charged non-BPS D-branes of tables 6 and 7. Here we

account for the BPS D0-branes and D4-branes of Type IfA. The fact that these appear in both table 6 and table 7 reflects the fact that two half-D0-branes on the O8? plane can combine to make a single D0-brane in the bulk which then becomes a D0-brane on the O8+ plane, and similarly for the half-D4-branes on the O8+ plane. Since half-D-branes are now limited to living on only one of the O8 planes, there is

Having been successful with the BPS D-branes that carry conventional RR brane and D6-brane, which are however absent in table 5. The resolution of this non-BPS D6-brane identi ed in table 6 near the O8? plane. It consists of a D6-brane and its mirror D6-brane, where the usual tachyon between the two has been removed by the orientifold projection. Just like the Type I non-BPS D7-brane in section 3, this system carries a U(1) gauge group, and can separate in a symmetric fashion and transfer over to the other O8 plane. In Type IfA, however, the orientifold projection is di erent at the other O8 plane, and therefore the tachyon is no longer removed. This implies that the non-BPS D6-brane locally stable near the O8? plane is no longer stable in the global theory.

This e ect has important consequences for the analysis of non-BPS D-branes on compact manifolds. Typically, when looking at the BPS spectrum of D-branes near singularities of a compact manifold such as K3 orientifolds, one can look piecewise at the singularities (i.e. approximate them with ALE spaces) and add the corresponding spectra (being careful to match the bulk D-branes). We now see that for stable nonBPS D-branes this is a dangerous procedure, as D-branes locally stable near one kind of singularity can become unstable due to other singularities in the complete space.

Now that we have successfully accounted for the unwrapped D-branes, we can move on to look at the charges in table 5 which come from wrapped D-branes of

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 5. Conclusions Z is no contradiction with the fact that K-theory predicts only a single 2 (table 5) Z cluded a half-shift along a circle. Requiring that we respect the 2 symmetry means Z which are now unwrapped. The IIB 2 symmetry we gauged to get Type eI strings inZ sources of 2 p-brane charge in Type eI for p = ?1; 3; 7; the rst is due to wrapped Z guration is BPS, and correspondingly yields the stable -charged D1-brane and D5 Z non-BPS D-branes. It is clear that these states carry a 2 charge, since when two To examine how the wrapped D-branes of Type IfA contribute to the nine-dimensional spectrum listed in table 5, it is more convenient to shift to the T-dual Type eI point of view. We can now use a simple construction to build their dual Type eI D-branes, we match each unwrapped Dp-brane with another Dp-brane at the opposite position along the circle for p = 1 or 5, and with a Dp-brane for p = ?1, 3, or 7. The rst conbrane in table 5. Note that these D-branes will carry a U(N ) gauge group, and will correspond to doubly wrapped BPS D2-branes and D6-branes in the Type IfA theory.

The second kind of con guration above is more interesting, as it yields stable of them are present there is an allowed motion which enables the D-branes and Dantibranes to annihilate, as can be seen in g. 3. Consequently, there appear to be two (p + 1)-branes (the T-duals of the unwrapped Type IfA D-branes discussed in the previous subsection), and the second is due to the above p ? p combinations. There since the above states are stable in complementary regions of moduli space ( g. 4). Z of some 2 charges in the string theory construction is resolved by brane transfer In this paper, we used the picture of stable D-branes as topological defects in unstable brane systems to study the charges of stable non-BPS D-branes in string compacti cations. K-theory turns out to be a useful tool in this pursuit.

We have seen that T-duality appears to be a manifest symmetry in K-theory. In
Type II and Type I on Tm, we have identi ed the relative K-theory groups on both
sides of T-duality, and have demonstrated that they are isomorphic. In the slightly
more exotic Type eI Type IfA T-duality, we have identi ed the K-theory group of
Dbrane charges only on the Type eI side, and then demonstrated that it exactly
corresponds to D-brane charges on the Type IfA side. Of course, one should be able to
identify the K-groups on both sides of T-duality from rst principles, by studying how the
orientifold group acts on unstable systems of spacetime- lling branes. We certainly
expect that such a direct analysis will con rm our ndings, and will provide an extra
check that T-duality is a manifest symmetry in K-theory. It would also be instructive
to extend our analysis of Type eI to all Type I models without vector structure [

In the process of identifying the D-branes which carry the charges predicted by K-theory, we have come across several interesting e ects. An apparent abundance H E P 0 4 ( 1 9 9 9 ) 0 1 0 R

R

Dp-brane

Stable

Configuration Z operations. Other 2 charges, apparently conserved locally near an orientifold plane, dissipate in the full theory due to the presence of another O-plane. We believe that these phenomena occur in a more general class of compacti cations, indicating that the piecewise analysis of stable non-BPS D-brane spectra should only be trusted when these phenomena are taken into account.

T-duality is to be contrasted with other string theory dualities, such as S-duality
of Type IIB string theory; whether or not there is an extension of K-theory that
incorporates Type IIB S-duality | and in particular explains NS states | remains
one of the many intriguing open questions of this framework. (For a possible step in
this direction, see [

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In this somewhat extensive appendix, we present a summary of some basic notions of
K-theory (and its connection to string theory), as well as technical details of various
K-theory calculations needed for some of our arguments in the body of the paper.
For a more comprehensive introduction to K-theory, the reader should consult [

Z x0 ! X. The full K-theory group splits canonically as K(X) = Ke (X) . In Z K(x0) to the K-theory group K(x0) = of a point x0 in X, induced from the map Unitary K-theory K?n(X) The unitary K-theory group K(X) is de ned, for a given compact manifold X,5 as the group of equivalence classes of pairs of unitary bundles (E; F ) on X, where two pairs are declared equivalent if they can be made isomorphic to each other by adding pairs of isomorphic bundles (H; H).

The reduced group Ke (X) is de ned as the kernel of the natural map K(X) ! Type IIB string theory on X, E and F are the Chan-Paton bundles of branes and anti-branes wrapping X, and Ke (X) classi es invariant charges that can be carried by general tadpole-cancelling con gurations of brane-antibrane pairs, modulo pair creation and annihilation.

Higher (reduced) K-theory groups Ke ?n(X) are de ned by

(For example, S1 ^ S1 = S2, Sn ^ S1 = Sn+1.) The higher reduced groups are related to the unreduced groups by K?n(X) = Ke?n(X) K?n(x0), where the K-theory groups of a point are given by Here the wedge product is de ned for two manifolds X and Y with a marked point x0 in X and y0 in Y , as the (topological) coset6

Ke ?n(X) = Ke (X ^ Sn) :

K?2p?1(x0) = 0 : X ^ Y = X y0) [ (Y (A.1) (A.2) (A.3) x0) :

Y =(X 5Unless explicitly stated otherwise, X in this paper is always a compact connected manifold. K-theory can be extended to non-compact manifolds, as K-theory with compact support; one essentially de nes K(Z) = Ke(Ze), where Z is the one-point compacti cation of Z. Thus, K-theory with compact support can be related to K-theory of compact manifolds, and we will not use it explicitly in this paper.

6Given a closed submanifold Y in a compact manifold X, the topological coset X=Y is de ned by shrinking Y | as a subset in X | into a point.

J H E P 0 4 ( 1 9 9 9 ) 0 1 0

The de nition of the higher K-theory group K?1(X) (which classi es Type IIA
theory D-branes) in terms of the p + 1-dimensional extension X ^ S1 of the
pdimensional manifold X is rather awkward for string theory purposes, as it invokes
an extra spacetime dimension S1. There is an alternative de nition of K?1(X), in
terms of pairs (E; ) where E is a bundle on X and is an automorphism on E
(see, e.g., [

K-theory is intimately related to homotopy theory. K-theory groups of any compact manifold X can be understood as the groups of homotopy classes of maps from X to certain classifying spaces, R where Bn is the unit ball in n, and Sn?1 is its boundary sphere. The relative groups (A.5) represent a generalization of the reduced groups Ke ?n(X), since one can write Ke ?n(X) = K?n(X; x0) with x0 a point in X. Also, relative groups are related to the reduced groups by K?n(X; Y ) = Ke ?n(X=Y ) whenever both sides of this equation make sense.

The relative groups K?n(X; Y ) are important because they connect the groups of X and Y via the exact sequence Here U and BU are the N ! 1 limits of the unitary group U(N ) and the (group) coset U(2N )=U(N ) U(N ), respectively.7

Given a closed submanifold Y in a compact manifold X, one de nes the relative
K-theory group K(X; Y ) as follows. Just like in the de nition of K(X), we start with
a pair of bundles (E; F ) on X. In addition, we choose a \trivialization" along Y , i.e.
an isomorphism : EjY ! F jY between the restrictions of E and F to the
submanifold Y . One de nes a certain equivalence relation on such triples (E; F; ), declaring
two such triples equivalent if they can be made isomorphic by creation or annihilation
of triples (H; H; idH ) (see [

Ke (X) = [X; BU ] ;

K?1(X) = [X; U ] : K?n(X; Y ) = K(X

Sn?1 [ Y (A.4) (A.5)

Bn) ;

Bn; X K?n?1(Y ) ! K?n?1(X) ! K?n(X; Y ) ! K?n(Y ) ! K?n(X) (A.6) (valid for any n 0), which is reminiscent of similar exact sequences from cohomology theory. In fact, K-theory is a generalized cohomology theory | it satis es all the axioms of cohomology theory except for the dimension axiom.

7More generally, Ke?n(X) = [X; ?nBU ], where ?nY is the n-th iterated loop space of Y . One can prove that ?BU is homotopically equivalent to U , and ?2BU is homotopically equivalent to BU . In conjunction with (A.4), this fact leads to Bott periodicity, K?n?2(X) = K?n(X).

J H E P 0 4 ( 1 9 9 9 ) 0 1 0

Y ) = Ke ?n(X ^ Y )

Ke (X K?1(X

Y ) ! K?n(Y ) ! 0 ;

In the case of relative K-theory groups K(W; Y ) that appear in this paper, the
pairs W , Y are of a very special type, with W = X Y for some manifold Y . For
such pairs (or more generally, whenever W is a \retract" of Y ), (A.6) can be reduced
to the following split exact sequence (cf. [

Y; Y ) to the calculation of the K-theory groups of W Y . The latter can be expressed
through K-theory groups of X and Y with the use of the following formula ([

Y; Y ) ! K?n(X Z a graded ring, with the obvious 2 graded structure. We have a K-theory analog of This allows us to determine K(X S1; S1) and K?1(X S1; S1) using (A.8), leading to (2.3) and (2.4).

Alternatively, we can calculate K?n(X S1) in a manner that keeps track of the
multiplicative structure of the theory. De ne K#(X) = K(X) K?1(X). K#(X) is
the Kušnneth formula,
which is valid if either K#(X) or K#(Y ) is freely generated (see, e.g., [

The case of our primary interest in section 2 is Y = S1. Using (A.9) together with Ke ?n(X ^ S1) = Ke ?n?1(X) and Bott periodicity, we get

S1) S1) = Ke (X)

= K?1(X) Y ) = K#(X) ? K#(Y ) ; (K?1(X) ? K?1(S1));

(K?1(X) ? K(S1)): S1) = (K(X) ? K?1(S1)) S1) = (K(X) ? K(S1))

Ke (X);

J H E P 0 4 ( 1 9 9 9 ) 0 1 0 thus leading to (A.7) (A.8) (A.9) (A.10) Ke ?n(X)

Ke ?n(Y ) : (A.11) (A.12) K?n(X

Y ) = K?n(X Just like in the unitary case, the full KO-groups are related to the reduced groups KgO?m(X) by Orthogonal K-theory KO?n(X) and symplectic K-theory KSp?n(X) KO(X) is the group of virtual real bundles, de ned by replacing complex bundles with real bundles in the de nition of K(X) groups. Higher KO groups are again de ned via Z that K-theory groups of S1 have shortened periodicity, K?m(S1) = K?m?1(S1) = .

With the insight from (A.12), Type II T-duality can thus be traced back to the fact Also, using (A.12), the fact that T-duality swaps wrapped and unwrapped branes corresponds to the fact that under the isomorphism (2.5) of the K-groups, Ke(X) ? K(S1) maps to Ke (X) ? K?1(S1) (and similarly for K?1(X)), with K(S1) factors and

with x0 a point in X. The key to the appearance of KO-theory in the bound-state construction of Type I D-branes is again its relation to homotopy theory. Just as in the unitary case, we have KO?n(X) = [X; ?nBO],8 where BO is de ned as the large-N limit of O(2N )=O(N ) O(N ), and ?BO = O can be similarly approximated by O(N ). In this case, the statement of Bott periodicity KO?m(X) = KO?m?8(X) follows from the fact that ?m+8BO is homotopically equivalent to ?mBO.

By replacing O(N ) with Sp(N ), and real bundles with symplectic bundles, one can similarly de ne the symplectic K-theory groups KSp?n(X). Bott periodicity can be re ned to show that KO?n(X) = KSp?n?4(X) for any n, which means that any calculation in KSp-theory can be done in KO-theory; therefore, we will not discuss

Relative K-theory groups KO?n(Z; Y ) are de ned by replacing complex bundles with real bundles in the de nition of K?n(Z; Y ) reviewed above. For our purposes, we will again be interested in relative groups for a special class of pairs, KO?n(X Y; Y ); for such pairs, one can relate the relative group to KO?n(X Y ) and KO?n(Y ) via the following split exact sequence,

Y ) ! KO?n(Y ) ! 0 ;

KO?m(x0) ; 0 ! KO?n(X

Y; Y ) ! KO?n(X

Y ) = KgO(X ^ Y )

KO?n(Y ) : Y ) = KO?n(X (A.14) leading to J H E P 0 4 ( 1 9 9 9 ) 0 1 0 (A.13) (A.17) (A.15) (A.16) KO?n(X KgO(X

Y; Y ) KgO(X)

KgO(Y ) :

KgO(X This formula is used in section 3 on the Type I side of the proof of T-duality between

Z In the case of our main interest, Y = S1, we obtain (using KgO(S1) = 2, and KgO(X ^ S1) = KgO?1(X))

S1) = KgO?1(X)
(A.18)
Real K-theory KRp;q(X) KR-theory (introduced, under the name of \Real
Ktheory," by Atiyah [

Just like in KO-theory, one can de ne higher groups KR?m(X), by R (x; ?y). Similarly, one de nes Sp;q as the unit sphere (of dimension p + q ? 1) in p;q R trivial actions of the involution. Consider p;q, as a real manifold of dimension p + q the involution of X is extended to the involution of X ^ Sm that acts trivially on Sm.

More generally, we can consider replacing Sm in (A.19) by spheres with nonwith coordinates (x1; : : : ; xp; y1; : : : ; yq), and with involution that takes (x; y) ! with respect to the flat euclidean metric.9

Now, we can de ne a two-parameter set KRp;q(X) of higher KR-theory groups, by R R R where e p;q is the one-point compacti cation of p;q (i.e. e p;q is topologically a p + qsphere). By de nition, (A.19) are related to (A.20) by KR?m(X) = KRm;0(X). One can de ne relative K-theory groups KR?n(Z; Y ), again by repeating steps used in the de nition of relative K-groups in K-theory and KO-theory.

Bott periodicity in KR-theory states that KRp;q(X) = KRp+1;q+1(X), and that KR?m(X) = KR?m?8(X). Due to the rst relation, KRp;q(X) depends only on the di erence p?q, and one has KRp;q(X) = KRq?p(X). It is interesting to notice that in KR-theory, spheres with antipodal involutions play the role of negative-dimensional spheres.

KgR?m(X) = KgR(X ^ Sm); (A.19) (A.20) H E P 0 4 ( 1 9 9 9 ) 0 1 0 KgR?1(X Z classi es D-brane charges in Type IA theory. (The orientifold 2 acts as a reflection In particular, one can derive Bott periodicity in K(X), KO(X) and KSp(X) from the periodicities of KR-theory.

Now we are equipped to calculate the relative group KR?1(X S1;1; S1;1) that on the circle S1;1, and trivially on X.) This relative group is again related to KR(X S1;1) via

K?m(X) = KR?m(X Z KO(pt) = , our result (3.9) follows from (A.22) and (A.23).

where we have used the (1; 1) periodicity of KR-theory, and the fact that KR (X) = KO (X) for the trivial orientifold action on X. Since KgR?1(S1;1) = KR?1+1(pt) =

S1;1) = KgR?1(X ^ S1;1)

KR?1(S1;1) :

S1;1; S1;1) = KgR1;1(X)

S1;1) = KR?1(X
(A.21)
(A.22)
(A.23)
S0;1) ;
Self-conjugate K-theory KSC?n(X) Given a compact manifold X, one de nes a
self-conjugate bundle on X as a bundle E equipped with an antilinear automorphism
: E ! E. Self-conjugate K-theory KSC(X) (see [

One can prove that Bott periodicity of the self-conjugate K-theory is four. This
can be shown either by a direct analysis of the homotopy properties of the classifying
space [

S0;2) ;
(A.24)
and using Bott periodicity of KR-theory (see [

The relation (A.24) between KSC and KR groups plays a central role in our analysis of T-duality between Type eI and Type IfA in section 4. H E P 0 4 ( 1 9 9 9 ) 0 1 0 H E P 0 4 ( 1 9 9 9 ) 0 1 0 H P 1