JHEP11(2009)067 es. Because August 24, 2009 October 27, 2009 : : November 16, 2009 ity : Received Accepted Published we discuss the (differential) 10.1088/1126-6708/2009/11/067 on of the A-roof genera in the doi: on to the coupling of D-branes to ifornia at Santa Barbara, Published by IOP Publishing for SISSA [email protected] , a,b 0908.2249 and Aaron Bergman a D-branes, Differential and Algebraic Geometry, String Dual We explore the differential geometry of T-duality and D-bran [email protected] SISSA 2009 Santa Barbara, CA 93106, U.S.A. E-mail: Department of Physics, Texas A&MCollege University, Station, TX 77843, U.S.A. Kavli Institute for Theoretical Physics, University of Cal b a c

ArXiv ePrint: Abstract: Katrin Becker Geometric aspects of D-branes and T-duality D-branes and RR-fields areK-theoretic properly generalization described of via T-duality K-theory, RR-fields. and its This applicati leads tocoupling. a puzzle involving the transformati Keywords: JHEP11(2009)067 1 3 6 8 10 13 16 18 and then ′ M (chosen to be B × M M l on ]), the gauge field can 1 s a torus bundle over the which is the T-dual field. sesses a compact Abelian , [ ], that these rules can be ′ mmetry, it can be gauged. ndles. Let us denote the T- dimensional conformal field 10 M form, it exchanges a circle of s a duality between two circle , 9 hat context, the transformation icular form on . Then, the fiber product is written as B – 1 – with base ]. One can also see T-duality from the target space ′ 4 – M . More generally, Buscher has derived transformation 2 /R ′ and α ]. ]. . This gives a RR-potential on 6 8 M the transformation of the Ramond-Ramond fields is described , , M 5 1 7 with a circle of radius . The Buscher rules are equivalent to taking the RR-potentia ′ R M B In the above references, For another derivation, see [ × 1 rules for nonlinear sigma models in [ M independent of the fiber direction), multiplying it by a part theory. When that fieldThrough theory a possesses a process compact known Abelian as sy Abelian duality (see, for example 1 Introduction A background for perturbative string theory is given by a two 6 The invariance of the couplings 7 The index pairing 8 T-duality of branes 3 Gauge invariance 4 Self-duality 5 K-theory and RR-fields Contents 1 Introduction 2 The setup point of view by compactifying supergravity on a circle. In t in components. It was soon recognized in, for example, [ be integrated out resultingsymmetry. in This a duality new isradius background called that T-duality. also In pos its simplest rules have been derived in [ written in a more geometricbundles over form. a In given particular, base.base T-duality We whose i can fiber form is the thedual fiber product pair product of of which the circle i fibers bundles of by the two circle bu integrating over the fiber of JHEP11(2009)067 . F (1.2) (1.1) + . B X that is a K-theory, a ] and is exploited ) depends on the ane. For a single 9 -connection or some 1.2 r IIB string theory to H differential they can be derived by f D-branes, RR-fields and This pairing is related to tain our T-dual fields, we field, and we need to add a structures and self-duality that cellation condition ensures rections. For example, one s in c n, an d normal bundles to T-duality transformation. We g the A-roof genus. Thus, as he K-theory class representing sses and is independent of the tized flux, but also the RR-field s appears in [ fferential K-theory used in this paper, uzzle. Instead, we will concentrate is not closed, ( g to spin -flux, we must use twisted differential C , C . B H B e X e ) ) ]. We will ignore these terms here, although C . by the gauge-invariant combination B ), recall that the correct quantization of D- 11 T X just on its cohomology class. Thus, it is an NX e B ( ( – 2 – b b X 1.2 A A Z is the NSNS-potential which is globally defined on not s B X Z ]. It is important to note that, while the total anomaly is trivializable on the D-brane. This formula is incomplete 13 ). In particular, these have a topological character and are , H ] is in terms of K-theory rather than ordinary cohomology. In 1.1 12 15 ]. However, life is not quite this simple. It is the RR-charge 2 10 corrections to ( ′ is the normal bundle to the D-brane, ]. α ] and RR-fluxes [ 16 14 is the total RR-potential, and In addition, because we have a nontrivial NX 3 C The topological nature of these corrections arises because To better understand the coupling ( The simplest version of the coupling of a D-brane in type IIA o There are some subtle contributions to this formula relatin For a nice introduction to the differential cohomology and di 2 3 anomaly inflow arguments [ where other connection on the bundles associated to the tangent an K-theory. The goal oftheir this couplings can paper all is be to viewed explore in how light T-duality of o the geometrical the brane with theindex K-theory theory and class naturally representingopposed incorporates the to the RR-field. integrating terms ashould involvin instead differential do form this integral over in the K-theory. fiber A version to ofclass thi ob in K-theory. The RR-potential is instead part of a clas this language, the brane coupling is given by the pairing of t is given by thechoice integral of over a forms product thatexplicit of represent form characteristic those cla of classes, the as A-roof genus and branes [ extensively in [ their incorporation into this formalism is an interesting p usually given as on certain background RR-fields can be written as follows Here generalization of K-theory which not onlyitself. includes the quan interesting question to ask the specific form of the above cor could form the A-roof genus out of the Levi-Civita connectio please see [ we are neglecting. the brane worldvolume becausethat the the Freed-Witten NSNS-field anomaly strength can in many ways.D-brane, For this example, is we accomplished by could replacing add a gauge field on the D-br For a stack of D-branes, however,more we complicated have set a of non-Abelian terms gauge due to Myers [ JHEP11(2009)067 , 2 ) is = 2 M useful N ( e nstraint , implying = λ e ∧ . As a general e λ heoretic version = is a circle bundle ing in K-theory is where , we recall how the dγ e M 5 = eometry of T-dual pairs re devoted to differential A he Buscher rules, we first d ise. The remainder of the cetime her rules and the integration section for the reader. In section n some examples with l-known, but we hope that a proper interpretation of these such that f D-branes and RR-fields. As e expected results. Combined mport of these conditions. In . brane coupling under T-duality he physics and may not even be B with Euler class c background has a free Abelian , and 2 that from two somewhat different ) B M A gauge invariance behaves under T- + over F λ , ′ dθ ( ) ↔ M A∧ b ( , we derive certain conditions under which φ + 6 B ) is a form pulled back from the base of the 2 e γ , we derive a condition such that the geometric – 3 – 8 ]. This obviously has consequences towards the 1.2 − + -flux given by is a three-form on 17 2 B = H γ ds H , this gives another demonstration of the invariance of = 8 , and 2 M B ds and 5 , we see how the 3 , these relations are physically mysterious. While they are 6 , we show that the Buscher rules preserve the self-duality co . We also have an 4 is a connection on the circle bundle M A + ) can be interpreted as a pairing in K-theory, and we give a K-t dθ 1.2 = , we show that, if the Buscher rules are uncorrected, the pair is a closed two-form on is closed. 7 with metric λ A H B The T-dual geometry is a new circle bundle This paper is organized as follows. The first three sections a where the Euler class of rule, we will use primes to denote T-dual quantities. To use t that and discuss how theywe relate to discuss the in K-theoretic section description o supersymmetry in the heterotic string [ will find various conditions that lead to invariance of the D- duality. In section Thus, section preserved by T-duality. Finally, in section over the fiber product described in the introduction. Our spa T-duality preserves the brane coupling, and we discuss the i the D-brane coupling. 2 The setup We will begin by making precise the relation between the Busc circle bundle. Something akin to this has been shown toambiguity hold discussed i in the previous paragraph.conditions as We leave a the puzzle. geometric aspects of T-duality.unified Much presentation of in this a materialwe geometric is rewrite language wel the will Buscherof be rules helpful D-branes. in a In geometric section form and giveon the the g RR-fields payingpaper special is attention devoted to to the the K-theoretic signs aspects that ofof ar T-duality. the In T-duality transformation. In section T-duality transformation can be appliedwith to the branes results to of give sections th coupling ( mathematically, they are surely not thecorrect complete in story that for context. t Itdirections, is we interesting arrive to at note, the however, conditionsymmetry, that, the when ratio a geometri of A-roof genera in ( over JHEP11(2009)067 and (2.1) (2.3) (2.5) (2.4) (2.2) , and be the m B C . Let B such that ′ γ . , we form the fiber M − ] are equivalent to and the circle bundle as expected. We also ′ × along the base, 11 λ H B ∧A M manner using differential = e ′ ∈ , the Buscher rules give the n in [ = ) A both α . θ , ′ ′ als. By assumption, it has no d , 2 ′ s act on the B-field and not just ) ′ α , ∧A , A e dβ . ) = ∧A m,m −A∧ ′ =  A + . + − β A , θ M ′ θ ∧A ( α α + ′ = d = e . ∧ A ( k ∧ ¯ ′ θ ) e e ∧A b B B e ( ∧ θθ µθ A the component of − φ = 2 C G = G . Thus, we have the (globally defined) T-dual k + , ∗ 1 − θ θ α, B dβ φ , k θ e π B = d – 4 – ∧ B  − B B ∧ = + µ A ∗ β, B = 2 A with θ A 2 B = = = ′ − π d ′ with ′ ′ = + ds A ∧ φ β A ′ B k = which satisfies β dα , γ dθ ′ = . This is a torus bundle over the base, B = dA α ′ = C ∧ B = k ′ dB + 2 M θ + B λ B B = θ ds dβ d + H = k = ′ ′ B . This tells us that the local version of the T-dual B-field is A dB = A + ∧ = B which is defined to be the set of all ( θ θ ′ ′ d B H ]. To summarize, the T-dual NSNS-fields are M = − ′ B 18 k A × B = M ′ k B Writing the coordinate for the T-dual circle bundle as In order to define a T-duality transformation on the RR-fields the component along the fiber. This implies that project to the same point in , it is possible to define this transformation in an invariant ′ dependence. The T-duality formulae for RR-potentials give θ cohomology [ It is important here thaton the its T-duality flux. transformation rule Even though we have worked locally, trivializing where the (locally defined) one-form M B and In components, we have note that locally where we have set T-dual metric as connection product m have that Similarly, we can locally trivialize since differential form representing theθ sum of all the RR-potenti JHEP11(2009)067 . 4 F . α M M ↔ (2.6) , and A∧ X e base θ B . Under . Then ∗ M 2 ′ ). This i ) ′ alone. See κ M + , M B + ) ( B ′ ∗ → α i A ) = ′ + + β → θ θ as M d d , and we have the respectively. This ( ( ( o that it wraps the can be replaced by the ∗ X C ′ X i A A . θ → : θ ∧ M 2 ′ to is a submanifold of C ) s . These are exactly the ′ A γ A ection on the brane − ∗ ∧ M . Now let M i ) θ and are now globally defined. + λ . X α C θ ∗ ordinate − ′ β i potential d be the embedding. We define + M + α M . = . . ′ ...... θ ...... 2 . ∧ . . . . 2 ...... d ...... e . A . . . to . and . . . π ρ ...... ′ . . ( ...... ∗ ...... ∧ . ]. We can derive the usual form of . . → B . k . . . ′ . . i . . dB ...... , and thus it represents a section of ...... α . . . . k ...... 9 . . . . ′ . . ∗ ...... hould really never talk about ...... M ...... C . . . . must be exact. This is indeed true as . . .. 1 C M = . More precisely, this means that there X B π M etric, i.e., ( λ + B dθ . : B + ∗ X × ...... ]. It is straightforward to verify that this ...... θ ...... ∧ ...... i . . 1 ...... = ...... i . . . . 1 ...... dβ be the embedding of ...... θ ...... θ . . × ...... π . . . . C . . . . ρ ...... 11 ...... It must obey the Freed-Witten anomaly ...... C . . . M . . . . . C . η ...... dα ...... = ...... M . X ...... i ...... represents an integral over the variable, . . = )) = + ′  ′ – 5 – ∗ k k , and 2 M )) M C B ...... π ...... ( ...... A . . . . over e = which can be exchanged for a gauge field under the which is the embedding of the T-dual D-brane. We ( . Then X + ) which we write as a map ∗ ...... ′ ...... ′ ...... i ...... M . . M . B C . . . µθ ...... i ...... ∗ . . . . Next, let . . dθ ...... B M . . . M 1 ...... given above, but since we restricting to the worldvolume . . . . ( . . . A, C . . . )( . . π ...... ∗ . . . dα . . . . to . . ∧ i α → B θ = M κ = + C X X α λ θ : − ∗ d . Thus, we have the following diagram of spaces: i ) to be the restriction of the bundle ], building on the work in [ s k M ◦ C 10 M . Let ( η ∗ ), the pushforward = → (1 + ( , such that the wrapping number over any nontrivial loop in th i restricted to the submanifold be a submanifold of dB ′ s θ = and the fiber of C are the projections from 2.5 M ′ =  C X θ 2 X M ) which means that X )= ′ i Z π : M →B ′ H X M = ( M ( i ′ M ∗ ∗ M . Let i and i C 1 M π The picture to keep in mind here is that the D-brane The T-dual brane does not wrap the fiber of Next, we wish to add a D-brane to this setup. We will choose it s One way to see this is that a wrapping of the brane around the co 4 gauge transformation. This gauge invariance means that we s This only has components along the base, and equation, the next section for more details. wrapping both pullbacks of the local form for thus we can write things as follows embedding is zero. This is dual to there being no gauge flux in the fiber dir the bundle of the brane, the Freed-Witten condition means that the D-brane expression appears in [ addition of a cohomologically non-trivial one-form to the m where choose the section, the Buscher rules as follows. First, we locally write the RR- transformation squares to the identity. fiber of These are precisely the same rules as given in [ or T-duality, this comes from adding exists a section of the bundle there is a map In the expression ( from above we have that trivializes JHEP11(2009)067 = s M γ ′ (3.2) (3.1) , with ∗ M i B . This is . We can − β F α and, hence, = ∧ In particular, ′ θ ) e B ( ′ ∗ M i and a T-dual brane uge flux, , are given by the same by the gauge invariant M to the worldvolumes of )= ′ B ′ acetime is denoted by B H ity. We will only consider λ , H ( ′ e of the Wilson loop around γ . sitate the inclusion of Myers = ater. In this section, we will ∗ M i ′ , , is at constant − and s ′ A Λ β . d H − = ∧A . . ′ e C , λ )= B ′ B  F = θ + ′ → M ) to the brane worldvolume as expected. . ( B and H e e k e dθA ) ) 2.3 and F , Z β = T X , – 6 – . Each is a circle bundle over a base, NX α ′ and ′ and ( , B ( which wraps the fiber of πi α b b F 2 A A M  e γ M where we restrict + Λ s −A∧ ′ = − X F exp ) is replaced by β H λ Z A = 0 on the brane. This is not a gauge invariant statement, d → = = 1.2 A∧ ′ F . Their components along the base, F B ′ )= = dB M . The rules for T-duality give that the T-dual field strength i M H ( e dθ . The B-fields on the worldvolumes of the branes are given by θ and embedded into , respectively. The Euler classes are given by B F ′ H M A + X k = , which means that the B-field on the brane is given by into F , then, is that it changes the position of the T-dual D-brane. . For example, ( dβ and θ X = dB F F = A . To see the Freed-Witten condition, note that embedded into F + α γ ∗ ]. B X i 11 )= Λ = 0. In particular, in the brane coupling, we replace − ′ d α To begin with, let us examine the effects of the addition of a ga Let us summarize the results of this section. Our original sp A ( ∧ ′ e ∗ M ∗ embedding of These satisfy the branes. This gives relations between given by 3 Gauge invariance In the previous section, we set The effect of Thus, we can choose coordinates such that the section, recall that the positionthe of circle the dual brane is given by the valu decompose however, as there exists a symmetry and our T-dual spacetime is given by and similarly for thesee other how forms this of symmetry theAbelian is coupling gauge implemented in discussed fields as the l terms non-Abelian context [ of degrees T-dual of freedom neces i We have a brane with connections quantity and the H-fluxes are given by i precisely the pullback of the local expression ( JHEP11(2009)067 , k 0) B dθ x, Λ − is equiv- B ) = ( x df → ( = B M i θ when restricted F ′ = 0, we can define , and we have that M θ f unchanged, while Λ θ d s a flux on it given by = given by F θ 1 A S . This function provides the variance on the T-dual side. . and Z × ) lets us define a new function / 2 θ e previous section that we have . ) R θ B X dφ 2 , then . ( ) Λ ∼ = ′ ∗ B k ). We make the decomposition Λ = f 1 − A ) into ′ S , F . In fact, we can write any closed one . . + = 3.1 A ). As we also have shifted 1 . ) θ θ X θ θ ) on + . As above, since S θ b + Λ dσ F Λ ( Λ f ( B → B dθ θ x,f was a gauge invariance, we should be able f f φ θ − ), we have that : d + 2 ′ ( mod 1 (where we have set the circumference . This is the location of the T-dual D-brane − ) = Λ B − f = ( φ A Z 3.2 e 2 f θ – 7 – k ) = ( / x,f dφ − ) ( x ′ + e R = → = 0. This leaves ( θ is shifted as and ′ F 2 B ∗ Λ ′ σ θ + is not exact, recall that the relation f M f A ) = ( i by F + A ds , the above discussions tells us that the embedding of θ 2 B ) and ( x ′ 1 ( F = dθ as a map from ds θ B S θ 2.2 Λ ( ′ 2 f = ′ M + is a coordinate on ds 2 i M f . This map trivializes the circle bundle valued function, + Λ ′ i φ ds F R M for some function . From ( satisfying → k → dφ we have that 1 F θ S X flux. The addition of a flux ) where B : θ for some ′ dφ = F ( B → ′ M ∗ i : df A f θ = Λ = f . We begin with the case Λ θ θ ), as F F dθ . are shifted by Λ θ k 2.4 k F . Thus, we can think of F To address the situation where Now let us consider the case where Λ = Λ We can now understand the gauge invariance ( + Λ = X ′ k from ( of the dual circle to be 1). alent to the dual position of the brane is given by form as the pullback of a function Since the original transformation of Thus, the coupling is invariant under this transformation. Thus, if position of the T-dual brane.an More embedding precisely, recall from th to recover the original T-dual brane and metric via a gauge in To do so, we make the following coordinate transformation Λ This changes the metric to Since we have shifted the T-dual brane is shifted to and to This means that the new T-dual metric is which is the embeddingF of the T-dual D-brane. This D-brane ha where, as above, we parametrize when there is no JHEP11(2009)067 . ) θ − d θ 3.1 over here ∧ (4.1) Λ ). In , but ν f means F M A B . Since ( s = θ + , we can X ∗ A ′ t s F . Thus, we B (1), and, as θ = = x,f F U t s θ ne embedding. ∧ F A ν to ). The Freed-Witten ) = ( U over some open set, M and x ) = ( ′ ( 3 ν Λ B M ˇ f H ove situation where the ∧ ′ X + s θ (∆ f M auge field’ on the D-brane One should only speak of sformation rule for such that ′ i F ever, recall the definition of ∗ M 5 Away from the sources, one i = M s ansformation of the form ( X k sformation. 6 d strengths. In this section, we F even though we mean the gauge . logy group → − , the fact that it involves , and B corrections to the Buscher rules. However,  θ ). In a new trivialization t k M X of a self-dual field. For a discussion of the t trivialization of this class. F . Thus, it appears that the change in F F ∧A : X ( ′ ν θ . ∗ defines a map from this element to the brane is trivial. The correct s d A ∧ 1 . It is straightforward to verify that the e ) = θ t ρ ∧ ) C θ -flux is trivial. F depends on a local trivialization of A d − ∧ F = ( H ( ′ ) is mapped under T-duality to a coordinate = s be a local section of + ∗ 1 H s dθ θ ′ ∗ M π – 8 – θ ) s β i F 3.1 t +  F . Thus, the T-dual B-field shifts by ∆ ∗ = F ν 2 + ( ′ π dC k and + B − F is not gauge invariant. Of course, neither is s = ′ = F ) ). Then we have ′ A s ∗ F F F s dφ F = ( t ∗ = ) A = ( t s k ′ . The difference t F − θ F . F s F ) ]. + = ( 19 and 2.3 ν B , we have t k t F ). But these are precisely the original T-dual metric and bra x,f , we have that ∆ . Then we define k F X The moral of this story is that the traditional concept of a ‘g The careful reader will have noticed, however, that the tran Formally, the B-field is an element in the differential cohomo One might expect that this restricts the form of the possible ) = ( = ⊂ 5 6 θ . To make this dependence precise, let ′ Λ It follows from the above discussion that Thus, we see that the gauge invariance ( the trivialization see that the change in trivialization is precisely a gauge tr In these new coordinates, the brane embedding is given by on the T-dual side.choice In of a fact, trivialization this masqueraded is as very a much coordinate the tran sameis as not the an ab invariantgauge notion invariant even quantities, in when and the can define this field strength as transformation, and everything is invariant. Not only is this expression only defined locally on invariant quantity notion of an Abelian ‘gauge field’ on the brane is as an explici anomaly cancellation condition states that the pullback of one should be veryrelevant subtleties, careful see when [ discussing the quantization Buscher rules give 4 Self-duality Self-duality imposes a significant constraint on the RR-fiel will show that the Buscher rules preserve this constraint. is not well-defined because the form to save space, for the rest of the paper we will write f that it also depends on an explicit trivialization X U similarly define F above, a one-form trivialization has led usthe to T-dual a B-field different ( T-dual gauge flux. How JHEP11(2009)067 to here . with A (4.4) (4.3) (4.2) q ]. In φ φ ω e isfy e M q 19 1) )= − A , + + ( p ) θ dθ ω )) ( 1 for IIA and IIB β p θ , φ rcle bundle ) B β e 1) . ⋆ A · presented in [ − ∧ φ = θ ,  = 0 θ e ≀ ∧A α ϑ 0 α ) ∧ e κ  θ ≀ . , ) + ) = ( ⋆F ) ⋆β ⋆β . q ≀ ( n θ ) + ( ity to act on arbitrary forms. d ∧ β ω + ⋆β ≀ , ⋆β α ) ( B β α ...µ 1 2 . d + 1 − ⋆ · = ) + ( ( µ ≀ ≀ 2 p M ∧ ⋆F ⋆F ≀ ≀ ) ϑ β ∧ − and add Z α ω ) 1 ( θ n α ( A − − , − (( ( d ∧ α B = n ⋆β 3 4 ) θ φ φ ...µ + A n 1 e e + (( α − ⋆F , φ ...a µ on ⋆F ⋆F ≀ d θ e 1 ∧ B B dθ mod 2. Thus, ...µ ) ) + a , ( a − ( α ∧ Z Z 1 + + q | κ e 2 α θ φ θ µ A ) d 5 4 2 ) ω π π ⋆β φ α β B ( e F F 10+ e g n ∧ = ⋆β ) + ≀ ∧ occurs in an expression such as the above, it is – 9 – + ( = 2 = 2 + + ≀ θ − ...µ α κ ∧ and 2 B 3 2 d 1 n form α ≀ α  det( µ | F F α q ds φ = ( + α = ≀ ...µ  e | 1 p + + ≀ α ) = µ φ α F B 1 0 α g e β ( Z = F F 2 M n )= M M ≀ π and a n = = Z Z det( ds β ...µ | p · 1 ...a ω 1 µ ≀ = 2 = = F F p a : : α α | α ( M ) ) can be written . Note that, despite the regrettable collision of notation, ⋆β g ). Z ∧ θ form M 4.2 IIA α IIB β p · det( M | θ Z and not a one-form. Recall that the Hodge star is defined to sat α p in IIB. These fluxes obey the self-duality relation B M 5 Z F − = is the degree of the form and 5 d ⋆F We want to write a formula for the Hodge star on an arbitrary ci Now, let us consider the self-duality constraint in the form The right side of ( with respectively. In what follows, whenever the metric the degree of theFor form example, following given the a sign, extended by linear we have that, in orthonormal components, where Thus, choosing a mostly plus signature, We choose an orthonormal frame of one-forms where Lorentzian signature, one first defines the total RR-flux If we write and similarly for ( complete the frame on is a function on JHEP11(2009)067 . ) of M 4.5 (4.5) (4.6) (5.1) (4.7) (4.8) to ial is by the . From , C ′ X θ M F B ⋆ d constraint ( + ′ 2 κ wisted differential 2+ ) . − We will briefly depart ( A ) 7 φ d e RR-potential by + ′ ∧  ≀ are the same as the degrees κ ) ( tten condition. The twist is β ′ θ − spects of T-duality. First, we B F ) . ∧A ⋆ ′ B − ≀ ⋆ + A F θ φ . ion for the relation of D-brane charges to , = ( , e β embedded into a manifold ≀ θ ′ θ ], and we write the twisted differential ) B F F (1 + = ⋆ C F X B 18 d B ′ . ∧ B θ ) ⋆ ) ⋆ ⋆ ) F ) [ ′ − d d A d )ch( − − φ + A M 2 2 κ κ 2 κ e ( φ M = ( 3 ( ∧ 1+ − 1+ 2+ θ b θ e ˇ ) ) and ) A H A – 10 – F F ∧ φ − − − φ ≀ q ∈ e e θ + F ∧ ≀ F = = ( = ( = ( θ + H + φ F ) ≀ ′ ≀ θ ( e θ C . Since the degrees of F F F  ⋆β κ F is Lorentzian and ten dimensional, the self-duality con- B = ( ∗ ∧A ⋆ ≀ 2 ′ φ − ≀ ) for the brane coupling as follows. e F π + ( and discuss a brane φ F M ≀ e is that 2 ) = = = d 1.2 = 1 − ′ (1) for the push-forward of the fundamental class of ′ A 2 κ ⋆β ) . Physically, this is just the fact that we have to keep track ∗ ), we obtain φ F κ e 1+ ]. The derivation here is well-known. M M ) i = ( ( 4.4 ∧ ], we have that the differential form version of the RR-potent 20 θ is − • H F 15 ˇ ⋆β F K = ( + ≀ θ ) and ( F F φ mod 2 and the first sign is 1. Thus, we recover the self-duality = 4.3 e d F = ′ ≀ = . Since we know that D-branes and RR-fields naturally live in t ′ F κ M i , The T-dual of ′ A variant of this argument was, in fact, the original motivat 7 F This is possible inprovided twisted by an K-theory element in because differential of the Freed-Wi K-theory and appears in [ where we have used that Comparing ( Returning to the casestraint where on the B-field and not just its flux. We will denote the class of the K-theory, let us write for the T-dual field strength. 5 K-theory and RR-fields In this section, wewill begin ‘derive’ our the discussion expression of ( the K-theoretic a K-theory group as from the notation of section map and, equivalently, of Hence To verify self-duality, we compute Moore and Witten [ JHEP11(2009)067 ttle = 0 (5.2) (5.3) (5.5) (5.6) (5.4) ection B in differ- structures. c . . 8 ) 0
) ) C 0 ( 4 ) C F ∗ M 4 ( i C ∧ ∗ M ( e a connection on the i 2tr(  ∗ M ] to obtain ∧ ) i − . 2 s as follows (we set  . ∧ e a twist, however, we will 21 ) 2 and requires a Riemannian l bundle gives rise to certain )  ))  2 ) should be interpreted as a  B ) )) e F T M . 1 ( 5.4 T M ) M 1 − ( tr( tial K-theory. For ordinary twisted K- ), and the integral takes this to ( al, this formula is meant to be impres- p C 2 T M ( b )) C . A p ( 4 M ( 1 ∧ ∗ M ( ch( π i q p 1 . ∗ M 1 ( ∧ H 1 4 i on of the pushforward in differential K-theory. C ˇ K X/M ality constraint and the issue of Spin 2 1 ∗ M 128 + (  ·C i + , we have (1)) ∗ M 2 1 − 2 N F i ∗ ( dC (1) )) )= b − )) M ∗ A B i e V  = M )) def ( – 11 – i ∗ 2 )) C X/M )ch( M X/M M ( i ( H Z M X/M d N ( N ( ( 1 X/M ( b 1 ( A = 2 , p N p p ) ( N 3 1 M 2 F 1 (1)) = (  ]. As always, we are neglecting many issues here and making li Z p ∗ b  F roughly gives that A 22  M 9 i 1 X tr( 1 1 Z 2 24 1440 640 1 ch( π . Inserting this into the above expression and using the proj 8 + + )+ X − 8 C ( )= ∗ M . i V which we can exponentiate and add to the action. The integral ( 7 1 ] for more details. D Z TM p / Z 21 R ∼ = ) ] and [ To write this in cohomology, we use the index theorem of [ pt 16 Like all the formulae in this paper involving the RR-potenti To our knowledge, this remains unproven for twisted differen ( 8 9 1 ˇ K pretense towards rigor, leaving that for other work. ential K-theory is relatedstructure to on the index of the Dirac operator The integrand lives in the differential K-group The coupling is given by the K-theoretic integral The Riemann-Roch theorem sionistic. It can beHowever, we made are sense neglecting, of among by other things, unravelling the the self-du definiti For a D7 brane, we can write this in terms of Pontryagin classe for convenience): See [ theory, the theorem is proven in [ To write a Pontryaginrelevant bundle. class If in we terms denote its of curvature forms, as we have to choos and In the untwisted case, theconnections Riemannian structure that on determine the the norma leave A-roof the genus. A-roof Since genus we do unspecified. hav The right side of ( current supported on formula, we obtain JHEP11(2009)067 ]. ]. ). 10 18 5.3 (5.7) (5.8) ), on ) [ (5.10) ′ logous, d issues M ))]. If we ) X/M B ( × 1.2 N X/M M This bundle is ( ( ⊕ 3 F . N way the Buscher ˇ ( H . B  10 b T X e A ]. [ ) is most natural, but ) given by a canonical ∧A ! f genera. This is the ′ ′ ∪ 21 [ A C CS 5.5 M ∗ )] e 1 ) X ng for any representative π B ). To obtain an expression eads to a difference in the C Z rpreted along the lines of ( ))) , so we can write the above × T X C ∧A elated to the ‘extra’ terms in ( ′ = B ∗ 1 ∧A ch( b ′ M A π A ∗ 1 ( e C sted differential K-theory, but we are ′ A π to as elements of ( H H ′ d ) into the expression ( (Θ( d ˇ  ∗ M K dinary (not differential) twisted K-theory  B ′ M 2 = ′ e 5.5 π ′ )) (5.9) C . M M C ′ B H ]. ′ ∗ and B 1 e CS B ) and )ch( π ) M ′ − ) ′ × 23 M × X , we can suggest the K-theoretic formula M )] factors as [ 2 Z M M H M (Θ( CS, and we have that ( T X M B NX ∗ M d = b ( (  A 2 ( ×  b – 12 – b -genera. The difference will always amount to a A b π A T A C T q b A decomposes as the direct sum M  B =  s ( e ). The second is the fact that we have not specified b ′ A b H ) are equal. Since the difference between the ratio of X A X )= C ) ′ ) ′ ˇ Z 5.7  K CS C to 5.7 ∗ d M M 2 ( ( π X b b ). This formula appears for non-differential K-theory in [ A )ch( ) A Z ′ ′ T M 2.5 ) and ( s M M ) and ( ( (

)= b b 5.5 A A ∗ 2 5.5 5.7 q q π ( is not closed, even though the two ratios of genera are cohomo − = = = ). We should emphasize that there are two separate but relate ) ′ C C 5.8 B 5.5 e ( ]. Again, this formula is impressionistic and should be inte ). For the first issue, K-theory suggests that the form ( 24 ). Returning to the setup of section 5.8 2.5 Because these objects live in K-theory we must extend in some Because the pullback of As with the Riemann-Roch theorem, this theorem is for nontwi 10 isomorphism of the pullbacks of the B-fields on as an obvious refinement of ( in terms of cohomology, we take Chern characters giving assuming it holds foris the given in twisted [ case. The pushforward in or choice of CS in ( take this to hold on the level of forms, we can turn ( However, it becomes precise when applied to field strengths. Here Θ is an isomorphism between This is related to the canonical trivialization here. The firstdifference is between, say, the ( formhow of to form the the couplingform relevant of in A-roof ( terms genera. ofit Each the tells of us A-roo these less issues aboutthe l the brane second coupling issue. found These by choices Craps may and be Roose r [ rules ( canonically isomorphic to that of the projection from genera is an exact form, we can write it as However, since We can, in principle,in write the down cohomology class a of version the of ratio the of brane coupli Here, we have used the index theorem in differential K-theory it does not mean that ( the level of cohomology, the pullback of [ JHEP11(2009)067 ): . 5.5 (6.3) (6.2) (6.1) ). To   (5.11) ) given C 6.1 ∗ 1 π 1.2 . But since ∧A X ′ is precisely the A ) below: it is a e he T-dual side in- M )) 6.4 ) X B M ( as . M/ b ( A X T ! correction to the Buscher ( q d the following calculation. C b . ∗ , the integral in the coupling 1 A do is to compare the explicit π ) ves and not just cohomology om the expression (   M ume of the brane, ly evaluate this expression. For te them in the manner of ( . C ) so that it is equal to ( ed in the next section. ∗ ( ∧A ; that is the meaning of the push- 2 ′ ) ′ . The T-dual brane, on the other ot attempt to do so here. π ∗ M A 6.2 ′ i M e C X ( . Now, our expression for the T-dual ∗ M ′ ′ )) i ∧A ) ∗ M α β B i M e e β over β e )) e 5.11 M/ ′ ) ( ) ′ )) M ′ )) ′ T ) ( M M ( ( b to verify that the coupling is invariant, it suffices M /M A – 13 – b X/M ( b A ( A ) ′ ) b M ′ X/M ′ just done in two parts: first over the fiber to obtain 11 A ( N ∗ M X ∗ M ( i M M ∗ M to obtain the coupling. However, ( i M N ( i ( b ( A b b X A N q b A X q A ( X b A Z s X Z

M ∗ X )= 2 ′ Z and write the coupling for the brane wraps the fiber of π C ′ ( ′ = M C ′ ∗ M X i C . In particular, that is the content of the formula ( β e M ) )) ′ X ′ M ( b A X/M ′ . On the other hand, the expression for the brane coupling on t ( ∗ ∗ M 2 i N π ( q b A Before proceeding, it is worth describing the picture behin Because T-duality is its own inverse, X This is nontrivial to prove in the K-theory case, and we will n Z 11 forward can be expressed as an integral over volves the integral of the T-dual RR-fields over the worldvol RR-fields expresses them as an integral over the fiber of the T-dual RR-fields are given by integrals over the fiberthe of T-dual RR-field, and then over hand, does not wrap the fiber of the T-dual space worldvolume of our original brane.expressions Thus, over all that remains to Recall that our brane expression as As noted in the previous section, these expressions differ fr in the introduction. to check only one direction. Thus, we will manipulate ( begin with, we plug in the T-duality equation ( Again, since we areclasses, we concerned must still with determine particularan the A-roof representati arbitrary genera choice to of proper rules the for A-roof T-duality. The genera, significance this of appears this to will be be discuss a 6 The invariance of theTo verify couplings invariance of the brane couplings, we will first wri Similarly, we can define JHEP11(2009)067 ′ . . to m X ∧A ′ ′ (6.4) (6.5) (6.6) A M . The e )= B x M ( and then which we ′ × to denote X θ ′ M 1 i . This then M ρ first restricts to M except M ∗ 1 . τ X ρ M ∗ . 1 . .   ρ over the image of ∧A !   ∧A ′ α , . )) A , ′ ∧A M ) ) such that 0 e ′ ′ ′ 0 0 ) C e A . e M s about the A-roof genera. C integrates along ck the fiber of ) ( ! ( −→ ∗ ∗ 1 −→ b X/M β M t sequences: −→ A C . Explicitly, this is given by 2 ) ( e ′ π ( ) τ X π lled back from ∗ 1 ′ ′ x,m,m N ∗ M )) )) π ′ i ( ) /M T X ) = ′ ∗ M B b giving ∗ 1 s ( i A )) M ρ M ) M is a circle bundle over B M

( X/M M ( X which appears above. It is equivalent M/ ′ ( b ∗ 1 ( b X/M ( B M A A X ∗ ( ρ ′ ( . On the other hand ), and we wish to compare it to the cou- N 2 −→ M/ T M N × ′ b ( ( q A N π ∗ M )) ′ B i ( , so we really do obtain M b T M 6.2 X i A i B b ( q ∗ M A × −→ i −→ b   ∼ = ′ A =

T X M/ ∗ ) M ∗ 1 ′ (   s τ – 14 – ρ which we denote T ∗ T M T M ◦ M ( . This map is the restriction of ! ′ τ −→ ′   β ∗ M ∗ η b B B ) given by A ∗ M i 1 e )) i M ◦ ′ ρ × B . This is precisely the fiber of ) ∗ M )) B i to 2 ′ β B ) M ′ X e ρ −→ M M , giving the same answer. × M/ , we have −→ = ( ( θ ( M/ )) M M in ′ b α is redundant, and we obtain the space on the right. Note ′ ( = ) A M ( T ′ )) M ′ B b X/M M T ′ over A X corrects the B-field on the brane, and the two brane couplings ( ( q ′ ∗ M T X ′ × M ) m × i M )= T X b ( ′ i → N ∗ M /M A . The resulting space is b X/M ( i M ∧A M A ′ X X ( . Here we have abused notation slightly and used ′ A ◦ ′ b ( M   A ( −→ ∗ M A ′ b M −→ 2 −→ N A ∗ M τ X i e = 0 ρ (

i ∗ M X ( ∗ ∗ M 0 0 i ∗ M 1 b i ∗ 1 A i N ρ M )). ρ by ). Thus M ( ′ x X X b X ( M A ′ Z X m and Z X ( ′ M 2 Z ρ M i m, i ) = ). The factor lifts m and then integrates over ( 6.1 ) = ( τ 1 M ρ To make this more precise, recall that Now, examine the push-pull combination We can now write the coupling as X x,m ( We apply the projection formula along the fiber of rewriting of the coupling for the T-dual brane, ( Notice that everything in the second set of parentheses is pu This can be easily seen, for example, from the following exac pling ( On the level of cohomology, we have that agree provided me make a number of compatibility assumption Since can pull back to the base by the section of This can be seen intuitively by nothing that we are pulling ba Explicitly, the space on the left is the collection of triple the projection of and restricts to the embedding of that the composition τ gives us a map from to to the composition equivalence of these two operations can be seen because JHEP11(2009)067 . . ′ B sses, (6.7) T M ) becomes ) is exactly 5.11 5.11 robably not how , . ) B )) aragraph provide the / ′ B can be written as / ), the fact that SO(1) is ′ M hese splittings, then the ( he structure group to the X e. There are a number of , M 6.7 t, there does as we can see T ey may not be induced from er rules are uncorrected and ( al component is also trivial. ) ounds whether or not there nnection in the A-roof genus pulled back from the base ection on the principal circle ivated from the point of view xplore their consequences. In ) represents the U(1) action ng invariant for any choice of T ). B ′ , in fact, corrected. However, ( there exists a set of connections / ′ B⊕ b ons ( ) is induced from that on 6.1 A . ′ ). Still, it is not too hard to add C , ly postulated in ( T M ) ∗ 2 )) ( the Buscher rules in ( X/M ρ ∧A 6.5 B ( B T α ′ e ∼ = X/M N X/ ( β ∗ M ′ M/ ( i e ( N N )) ⊕ T given by ( ∗ 1 ) ( ) ρ b ) will not be true on the level of forms for B A M ) is not sufficient to ensure the invariance of M /M ( ∼ = X q b M ) 6.5 A X/ – 15 – , T M ( 5.11 X ∗ M ) ( i N /M B )) = N q ( M ∼ = B ) subbundle of b A ) M/ X ′ B ( ( / M/ M ′ T ( N X T M Z ( ( X/M ( B⊕ b T A T N ) ) ′ ∗ 1 ρ M M ( ) holds on the level of forms as opposed to just cohomology cla ( ∼ = b b A A . The ′ 6.5 s M T M and ) holds. Nonetheless, as we will discuss in a moment, this is p 6.5 M acting on the brane, and the splitting of ′ While we are not claiming that the relations of the previous p Of course, as we have emphasized, ( While these connections exist mathematically (although th M we have that the coupling of the T-dual RR-fields to the brane The first two splittings arebundles equivalent to the choice of a conn on If we form therelation ( A-roof genera out of connections that respect t A Riemannian structure andspecial orientation orthogonal on group. a If bundle that reduces respects the t decompositi If we assume that ( in correction terms by hand that render the full brane coupli things are realized in physics. physically correct answer, it isparticular, worth these spending relations a imply moment that to the e ‘correction’ to trivial means that the inducedIn connection particular, on this a implies one that dimension the correction we original the coupling; we need to further assume the relation ( equal to one. Thus, wethat are every led connection to the which conclusion appears that in the the Busch brane coupling is arbitrary connections on the various bundles. One can ask if on the five bundlesfrom above the such following that bundle this isomorphims: relation holds. In fac which is exactly the coupling to the brane the needed structures in theof index the theorems), they physics. are unmot exists D-brane an couplings Abelian should symmetry, and, applyshould as in such, be all the universal. choice backgr ofpossible the Their resolutions. co exact The formas first remains, we is then, have that a seen, the puzzl the Buscher simple rules correction are ( JHEP11(2009)067 ) (6.8) (6.9) 5.11 nd derive ) must be s topological 5.9 on ( ted (as we have seen ainder of the paper to erential K-theory classes erstand the local form of . le D-brane (in contrast to rection, it is possible that erhaps. Another possibility ee in what follows that one st contain additional terms ation of the string scattering . This may not be a problem   ric such that the equations of formula that seems difficult to erved by T-duality. In other f A-roof genera in the D-brane ) such correction (including that . Nonetheless, one might hope )) t expect T-duality to have in a ′ hat one important consequence ′ . but it is somewhat disturbing. B M ]. If we further assume that ( ( b 25 A )). It was shown that there are no X/M ′ ( ∗ M )) = 1 6.5 i N B ( ]. q b A 30 M/   ]. One incomplete attempt in that direction ( , ∗ 1 T 26 ρ 29 ( – 16 – , b A = ) 23 ) ′ )) ) M M ( ( b M /M b A A ( ) hold. This would imply that T-duality only holds on- b M A s . This is important because it is not difficult to show that X 6.8 ∗ M ( B i N q ( b A ) and ( 6.5 ) and the assumption that the Buscher rules are uncorrected a ) is simply incomplete, and there exist further terms of a les ) is that 6.9 ] extending the work of [ 6.1 6.8 ) is incorrect, and that the K-theoretic T-duality expressi 28 5.11 )) almost inevitable spoils the self-duality derived above ) and ( ]. More generally, this can be resolved by an explicit calcul We do not have a solution to this puzzle and will devote the rem We can instead postulate that the Buscher rules are uncorrec 6.5 5.11 27 the invariance of the D-brane coupling. 7 The index pairing The goal of thisinvariant section under is the to circle show action, that, the when index evaluated on pairing diff is pres that there exist connectionsmotion involving fields imply beyond that the ( met A close examination will reveal thatsatisfy this unless is all an extremely the odd relevant forms are pulled back from if it is notwhich true, couple then a T-duality D-branethe implies to Myers that transverse terms RR-fields the which even coupling arecan for start inherently mu a with non-Abelian). ( sing We will s As mentioned in the introduction, thiscoupling means is that the pulled ratio back o from in ( forms representing the ratio of A-roof genera. However, any is [ because of the subtleties associated with a self-dual field, follows from one attempt to satisfy theis relation the ( correct expression, we have that character. One way tothe address anomaly these and questions demand might its be cancellation to [ und amplitudes [ understanding how it relates toK-theoretic various context. properties Before one proceeding, migh of however, we ( note t corrections in the NSNS sector by direct calculation in [ shell. One might go evenis further that and ( require supersymmetrymodified p to keep thethe Buscher coupling rules ( uncorrected. In another di JHEP11(2009)067 not (7.3) (7.1) (7.2) (7.4) . ), but ′ 6.8 · D ′ C ′ . ). ) and their primed M  .  Z ) D M
5.9 C 1 ∗ 1 ′ ( ∗ 1 d π • H 2 C · D π )) = c ˇ ∧A K ∧A 1 1 M ′ ∧A D d ′ , d Z In other words, we have A A 1) e ession is chosen so that the ivalent to the statement that re uncorrected ( + e  D∈ − . ∧A  . Then, for the right hand side 2 )= . ∗ (CH( 2 d 2 ∗ ), ) . B d D 2 T , + ( π  . ) D π ∗ ∗ 2 2 ) ) M ∧ 2 1 ( + , ′ ( C  d d ) )) D 1 T )ch( ′ 1 2 H ( C d C c c · D ∧ C C C π C π ∗ • 1 − ∗ 1 ( 1 ) ′ ∗ ∗ 1 1 π )ch( ˇ d π = B K C π π ′ C ′ ′ Z ′ ( 1) )ch( M ∧A (CH( ′ ( T M − b A ′ T C ∈ M Z A∧A A ) is given by A∧A A∧A )= e ′ ( e M e e b ),  + ( q =  A )) = CH( M ′ ′ Z 7.4 – 17 – ∗ , D 2 ∗ C Z ∧A 2 M M M 2 d M 2 = = ), let us define ( def π B B 1 π d Z 3 c ) ) by × × ′ ∧A ( )= C · D ˇ D 5.1 2 C = + H def B M M M (CH( c ∧ D 1 7.4 )= M Z Z ) Z Z ∈ T d C Z + D C CH( = = = )( ( 1 H M CH( c ) = T Z ∧ CH( ∧A ) = D ) and ( 2 ( C ∧ ) are invariant under the circle action, we can define c ) C T 7.3 , let us denote the Buscher T-dual by C ) + 7.4 C C CH( 1 are all implicitly pulled back from ( c ( 2 CH( T M d ′ Z M M Z M Z ) holds. Following ( = Z and 6.5 1 C · D , d 2 ) follows from ( is some K-theoretic version of T-duality along the lines of ( M , c Z ′ 1 7.1 C c ), we have: Now, given a form Since the forms in ( We will show that 7.4 where The assumption that the Buscher rules are uncorrected is equ T-duals, we want that For this calculation, we will assume that the Buscher rules a and similarly for anypairing in differential cohomology K-theory is class. the same as This the expr pairing in K-theory. that relation ( On the other hand, the left hand side of ( of ( words, given invariant elements where Then ( Thus, we see that they are equal, and we are done. JHEP11(2009)067 the ). That ) is that From the σ ( 5.9 , then they ∗ ′ ω 12 M i ) that T-duality , so the currents   X   η 7.3 B e 2 vate the answer, recall uality only holds if the . and )) . n ) rential forms.  M estions we want to answer  M /M ′ B ′ X ( e B η s. They are the analogue of other words, we will only be ned similarly to distributions b M he T-duality of branes should sible ‘test forms’ 1 A e manner. Given that RR-fields 1 − X ∗ M ( − nder that action. Let us choose elements in currential K-theory. Since i )) )) equal? Unfortunately, the answer N ′ q ( b -theory is the same as that in cohomology, /M X A η ad hoc M   , ∗ X/M ω , X ) ( ( ) M ω and i N N σ ( ′ ∗ 1 ( ( ( . We will assume as in ( M b π ∗ b ∗ M A ′ . The definition of the current ) or its K-theoretic generalization ( ′ A ′ ′ X H   M η ∗ – 18 – ∗ i σ i ′ − 2.5 M ′ A∧A M X M e i i M Z ) ) Z ′   ∗ M 2 ( M π ( b A b A q a form on )= q (1). After taking (modified) Chern characters, we obtain the ω M = ∗ def = def X M η M i ( that the differential K-theory class of a D-brane is defined by X and X η T 5 η X = def M ′ X η , we treated the T-dual D-brane in an 2 is a form on σ We saw in section As stated above, it is easily verified that this is false for Even more, we should really view these as Chern characters of 12 differential form Recall that the D-branes are twisted by is no. How,that then, these do are we notdelta reconcile differential functions this forms in the with but worldas are T-duality? of elements instead differential in To current forms the moti and dual are vector defi space to some nice space of diffe we are choosing thehowever, modified and Chern we character, can the ignore pairing this in point. K 8 T-duality of branes In section We are now faced with the question: are is the goal of this section. K-theoretic pushforward point of view ofinvolve the the physics, integrals what of this the means form is that all the qu and the modified Chern character commute. Thus, we have On the other hand, we have the T-dual brane as defined in sectio where measure the charges of D-branes,follow however, one from expects a that geometric t formula such as ( this expression is equivalent to which we can easily evaluate. If two currents agree on all pos are unequal. We canRR-fields now are understand invariant with thetesting respect problem, our to however. currents the T-d against circle differential action. forms In invariant u are equal. JHEP11(2009)067 ), 6.5 (8.2) (8.1) , ) α . ∧ ) 2 α ω ∗ ) ∧ i 2  ω . To begin with, ) ( ω ′ ω + . ∗ ω ∗ 2 i 1 (   π  , . Ignoring, for the ω ∗ ∧A 2 ) + ′ 2   ∗ π   i X ′ ω 1 ω B ( ( ·C B e ω β e ∗ 2 + ∗ e . i )) n now rederive the invari- π 1 A∧A ( )) ′ ) (1) e ω ) = ∗ β ′    e M /M
= M /M ∗ ( ) M ) ) ) A∧A )) ( ′ 1 ′ ′ i b M ′ e A ω M b ′ π ality. However, this proof gives a . (1) when evaluated on invariant A  X M M ∗ 2 ∧A ∗ M X M ( ′ ( ( ∗ i ∗ M   1 ( Z 1 i b b M A A because rather than assuming ( N B ω X/M π X/M ′ i ′ q N ( ( e ( q = ( 6 b + . ∗ M ∗ M ∗ M A ′ b N )) i i i A N 2 ( ) ( B   b ω ·C q ·C q b   A ∗ e ( A ′ M ∗ /M (1) is ( M ∗ )) X   M i ∗ M M b ) (1) A i Z i ∗ 1 M ∗ 1 ∗ (1)) ∗ 1 i X from section B π ρ ∗ M ∗ M /M ( ′ π = M e i – 19 – . ( ′ i α M 2 = M b N ω i A q ∗ ( . Recall that the coupling can be expressed as the ′ ( M 1 X ′ b A∧A 6 ∗ M )) A ( Z A∧A ρ ∗ M e i ) i M e ′ N   −A∧ ′ Z   = q ( B M /M ∗ ∗ b β e M ( 2 A = ) B b M M ) π A ′ ′  i = ′ × ) M X ) above. ∗ M ·C ), we see that they are equal when ( ω M M X M M i B ( ( Z Z Z Z from section N ∗ 2 b 6.9 8.2 q A ( X/M (1) π ′ ( = = = = b ′ ∗ β A ) ∗ M M N ω i = i ( ∧ ′ b A∧A 5.2 q A M e M ) and ( B Z  ′ X X ∗ η Z 8.1 1 ′ π = M . Because of its invariance we can write it as ∗ M Z ω , we evaluate i ω ) implies that ∧ B M e X ′ X 7.1  η η ′ ∗ we have 1 M ρ X We want to integrate over the fiber to obtain something on In light of this result and that of the previous section, we ca Comparing ( For Z η making use of the definition moment, the terms involving the A-roof genus, we have This is precisely equation ( As we have established that the T-dual of for such a form Here, we have used which establishes that the coupling issomewhat invariant under different T-du perspective than that in section ance of the coupling shown in section K-theoretic integral ( classes, ( JHEP11(2009)067 , , ]. vity , , (1998) 69 , ]. , , SPIRES (1999) 281 -models time spinors -models B 517 σ ] [ 3 ounds σ ) and that the A- ]. SPIRES ] [ 7.3 , ]. ussions. AB also thanks ]. ed ( r Kahle, Ruben Minasian, Center for Physics for their SPIRES HY05-05757, PHY05-51164 sions and e-mails. We thank Nucl. Phys. ] [ ). Understanding the correct , SPIRES SPIRES ]. ]. ]. 6.9 spitality and support while most ] [ hep-th/9810188 ( ] [ [ ]. B hep-th/9910053 [ SPIRES SPIRES SPIRES ] [ ] [ ] [ Adv. Theor. Math. Phys. SPIRES , (1998) 019 hep-th/9605033 [ ] [ I-brane inflow and anomalous couplings on Target space duality in string theory 12 (1999) 022 T-duality: Topology change from H-flux hep-th/9806120 – 20 – [ hep-th/0306062 Duality in the type-II superstring effective action [ ]. ]. ]. 12 JHEP (1997) 47 , hep-th/9504081 hep-th/9907152 hep-th/9912236 14 [ [ [ SPIRES SPIRES JHEP [ [ SPIRES , [ Anomalies, branes and currents (1999) 195 (2004) 383 hep-th/9401139 ]. ]. [ An Sl(2,Z) multiplet of nine-dimensional type-II supergra 249 B 541 (1995) 547 (2000) 145 (2000) 431 SPIRES SPIRES (1988) 466 (1987) 59 (1985) 127 ] [ ] [ (1994) 77 Quantum corrections and extended supersymmetry in new Path Integral Derivation of Quantum Duality in Nonlinear A Symmetry of the String Background Field Equations T-duality, space-time spinors and RR fields in curved backgr SO(d,d) transformations of Ramond-Ramond fields and space- Dielectric-branes B 451 B 568 B 583 244 B 201 B 194 B 159 D-branes and k-theory Class. Quant. Grav. Nucl. Phys. D-branes, T-duality and index theory , , hep-th/9710206 hep-th/9902102 theories [ Commun. Math. Phys. Nucl. Phys. Nucl. Phys. Nucl. Phys. Phys. Lett. Phys. Rept. Phys. Lett. Phys. Lett. D-branes [ [1] A. Giveon, M. Porrati and E. Rabinovici, [6] E. Bergshoeff, C.M. Hull and T. Ort´ın, [5] P. Meessen and T. Ort´ın, [4] T.H. Buscher, [7] S.F. Hassan, [8] S.F. Hassan, [9] K. Hori, [3] T.H. Buscher, [2] T.H. Buscher, [11] R.C. Myers, [13] Y.-K.E. Cheung and Z. Yin, [14] E. Witten, [10] P. Bouwknegt, J. Evslin and V. Mathai, [12] M.B. Green, J.A. Harvey and G.W. Moore, we have instead assumed that the Buscher rules are uncorrect perspective will have to be the subject of futureAcknowledgments work. We thank Mohammed Garousi and SavBen Sethi Craps, for many Jacques helpful Distler, disc Greg Dan Moore, Rob Freed, Myers, Jeff and Harvey,the Alessandro Alexande the Tomasiello for Kavli Institute discus forof Theoretical this Physics work for was their being ho accomplished.hospitality. KB This also work thanks is the supported Aspen and by PHY05-55575 the and NSF by under Texas grants A&M P University. References roof corrections in the coupling are pulled back from JHEP11(2009)067 , , , , ary . (1997) 002 Surveys in (1997) 7940 le, MA U.S.A. , 11 , in D 56 ]. JHEP , omology , arXiv:0907.3508 ]. , ]. SPIRES Phys. Rev. ] [ , ]. SPIRES ]. SPIRES ] [ ] [ Supersymmetry breaking, heterotic strings SPIRES ]. SPIRES ] [ ]. ] [ Differential twisted k-theory and applications arXiv:0904.2932 [ SPIRES – 21 – SPIRES ] [ hep-th/9812149 [ ] [ hep-th/9808074 [ Duality beyond the first loop hep-th/9812088 K-theory and Ramond-Ramond charge (2009) 428 [ Type II actions from 11-dimensional Chern-Simons theories hep-th/0010022 A note on the torsion dependence of D-brane RR couplings Thom isomorphism and Push-forward map in twisted k-theory [ Self-duality, Ramond-Ramond fields and k-theory ]. ]. (1999) 358 ]. (1998) 150 Anomalous D-brane and orientifold couplings from the bound (Non-)anomalous D-brane and O-plane couplings: The normal An index theorem in differential k-theory B 823 hep-th/9912279 [ , Surveys in differential geometry VII, Int. Press, Somervil hep-th/0011220 (1999) 275 B 450 SPIRES SPIRES (2001) 47 SPIRES . B 445 [ ] [ ] [ Gravitational couplings of D-branes and O-planes . B 548 Dirac charge quantization and generalized differential coh B 504 Nucl. Phys. (2000) 032 , Phys. Lett. 05 , Phys. Lett. , hep-th/9705193 hep-th/9710230 state (2000) pg. 129–194 [ differential geometry Nucl. Phys. Phys. Lett. and fluxes math/0507414 [ [ arXiv:0708.3114 hep-th/0611020 JHEP bundle [22] A.L. Carey, J. Mickelsson and B.-L. Wang, [24] A.L. Carey and B.-L. Wang, [26] K. Becker et. al,[27] work in C.A. progress. Scrucca and M. Serone, [28] K. Becker et. al,[29] work in B. progress. Craps and F. Roose, [30] B. Stefanski Jr., [16] D.S. Freed, [17] K. Becker, C. Bertinato, Y.-C. Chung and G. Guo, [20] R. Minasian and G.W. Moore, [21] D.S. Freed and J. Lott, [18] D.S. Freed, private communication. [19] D.M. Belov and G.W. Moore, [23] B. Craps and F. Roose, [15] G.W. Moore and E. Witten, [25] N. Kaloper and K.A. Meissner,