2 D-brane interactions
2.1 Open string amplitudes
The perturbative scattering amplitudes involving open strings on a D-brane and closed strings in the bulk Minkowskian spacetime can be formulated by BRST quantization of the worldsheet path integral over surfaces with boundaries, generalizing the perturbation theory of closed strings. The open string asymptotic states are represented by insertions of boundary vertex operators that are BRST cohomology representatives on the strip. While 2 each handle comes with a factor of gs as before, each boundary component comes with a fac- tor of gs.Theworldsheetcorrelatorwiththeappropriateb ghost insertions will be integrated over the moduli space of Riemann surfaces with boundaries.
2.1.1 Disc amplitudes
Let us begin with tree level amplitudes of open string tachyons on a single Dp-brane in critical bosonic srtring theory, associated with worldsheet diagrams of disc topology. The open string tachyon vertex operator is of the form
ik X VT (k)=gpce · (2.1) inserted at a point y R on the boundary of the upper half plane, where c(y)isunderstood 2 ik X to be the limit of the bulk holomorphic c(z)ghostfieldinthelimitz y,ande · defined ! by boundary normal ordering. gp is the open string coupling on the Dp-brane. Its relation to the closed string coupling gs will be determined later. Using the conformal Killing group PSL(2, R)oftheupperhalfplane(ordisc),wecanfix the three tachyon vertex operators at three given points y1,y2,y3 in one or the other cyclic order. This leads to the 3-tachyon tree amplitude
3 o 3 ik X ATTT(k1,k2,k3)=gp ce · (yi) +(k2 k3) * + $ i=1 (2.2) Y 3 3 p+1 p+1 =2igpCp(2⇡) ( ki). i=1 X
Here Cp is a normalization constant associated with the disc topology, which a priori depends on p.
20 The 4-tachyon tree amplitude is evaluated as
3 1 o 4 ik X ik X(y4) A (k , ,k )=g ce · (y ) dy e · +(k k ) TTTT 1 ··· 4 p i 4 2 $ 3 *i=1 Z 1 + Y 4 1 4 p+1 p+1 2↵ k1 k4 2↵ k2 k4 = ig C (2⇡) ( k ) dy y 0 · 1 y 0 · +(k k ), p p i | | | | 2 $ 3 i=1 Z 1 X (2.3) where in deriving the second line we have have chosen y =0,y =1,y = ,andwritten 1 2 3 1 y4 = y.Theintegrationiny can be broken into three parts, y<0, 0
4 Ao (k , ,k )=2ig4C (2⇡)p+1 p+1( k )[I(s, t)+I(t, u)+I(s, u)] . (2.5) TTTT 1 ··· 4 p p i i=1 X The reduced amplitude (defined by stripping o↵ i(2⇡)p+1 p+1( k )from ) at fixed t A i i A has poles in s at b n P s = ,nZ 1, (2.6) ↵0 2 corresponding to exchange of open string states. In particular, near the tachyon pole s 1 ! ↵ ,wehave 0 2 Ao 2g4C . (2.7) TTTT ! p p ↵ s 1 0 o 2 1 1 Tree level unitarity relation demandsb that the RHS is equal to (A ) (s+ ) ,fromwhich TTT ↵0 we determine 1 b Cp = 2 . (2.8) ↵0gp One may write an e↵ective action that reproduces the tree level amplitudes of open string tachyons, of the form
1 1 1 S = dp+1x (@ T )2 + m T 2 + gT 3 + , (2.9) T 2 µ 2 T 3! ··· Z 2 1 2 where mT = and g = gp. While the tachyon field indicates a classical instability of ↵0 ↵0 the D-brane in critical bosonic string theory, the cubic interaction suggests the possibility of a classical stable “tachyon vacuum” state in which the tachyon field acquires a positive expectation value. This is known as open string tachyon condensation. A rigorous analysis of the tachyon condensation, even at the classical level, requires going beyond leading orders
21 in perturbation theory. This will be treated in the framework of open string field theory in chapter 5. It is easy to generalize the above analysis to open strings with U(N)Chan-Patonfac- tor, describing excitations on N coincident D-branes. With the open string tachyon vertex operator a ik X a V = g ce · ij , (2.10) p ⌦ ij| i we have (using (2.8))
o gp a1 a2 a3 a1 a3 a2 ATTT( ai,ki )= Tr( + ). (2.11) { } ↵0 The 4-tachyon reducedb amplitude with Chan-Paton factor is
g2 o p a1 a2 a3 a4 (2.12) ATTTT( ai,ki )= [I(s, t)Tr( )+(permutationson2, 3, 4)] . { } ↵0
Theb canonically normalized open string gauge boson vertex operator is
2 µ ik X a (2.13) VA(a, e, k)=gp eµc@X e · ij ij , r↵0 ⌦ | i µ where both eµ and kµ are parallel to the world volume of the brane. Recall that @X (y)is 1 µ o the boundary limit of @X(z), which is equal to 2 @yX (y). The s =0poleofATTTT comes from the gauge boson exchange in the 12 34 channel. We can compute the tachyon- ! tachyon-gauge-boson 3-point disc amplitude as (fixing y1 o 3 2 µ ik1 X ik2 X ik3 X a1 a2 a3 AT T = gp ceµ@X e · (y1)ce · (y2)ce · (y3) Tr( )+(2 3) A r↵0 $ ⌦ 3 µ µ ↵ 2 p+1 p+1 k2 k3 2↵ k k +1 a1 a2 a3 = i g (2⇡) ( k )e + y 0 i· j Tr( )+(2 3). ↵ p i µ y y | ij| $ r 0 i=1 12 13 1 i The yi dependence then drops out of (2.14), giving the result 2 o a1 a2 a3 (2.16) AT T = gpe k23Tr( [ , ]). A r↵0 · b 22 The 3-point amplitudes considered so far can be captured by an open string field e↵ective action of the form 1 1 g S = dp+1xTr (D T )2 + m T 2 + p T 3 + , (2.17) T 2 µ 2 T 3↵ ··· Z 0 where D T = @ T i[A ,T]isthecovariantderivativeofthetachyonfieldwithrespectto µ µ µ the U(N)gaugefieldAµ(x). The disc amplitudes for open strings on D-branes in type II string theories can be com- puted analogously, with the additional ingredient that the total (left and right) picture number must be equal to 2. For instance, the gauge boson on the Dp-brane is represented by the boundary vertex operator 1 µ ik X VA (e, k)=gpce eµ e · (2.18) in the ( 1)-picture, or 0 eµ µ µ ik X VA(e, k)=gpc (i@X + ↵0k ) e · (2.19) p2↵0 · in the 0-picture. The disc 3-point amplitude of gauge bosons is o 3 1 1 0 a1 a2 a3 ( a ,e,k )=g cV (e ,k ; y )cV (e ,k ; y )cV (e ,k ; y ) Tr( )+(1 2), AAAA { i i i} p A 1 1 1 A 2 2 2 A 3 3 3 $ (2.20) ⌦ ↵ where the correlator is evaluated as 1 1 0 cVA (e1,k1; y1)cVA (e2,k2; y2)cVA(e3,k3; y3) 3 µ⌫ ⇢ ⇢ µ ⌫⇢ ⌫ µ⇢ ⌦ p+1 p+1 ↵0 ⌘ ↵ k1 k2 k3 ⌘ + k3 ⌘ = iCp(2⇡) ( ki)e1µe2⌫e3⇢ + + 2 y12 y13 y23 y13y23 (2.21) i=1 r ✓ ◆ X3 p+1 p+1 3 = iC (2⇡) ( k )2 2 p↵ [(e e )(e k )+(2cyclicperm)]. p i 0 1 · 2 3 · 12 i=1 X This is precisely the 3-point amplitude corresponding to the Yang-Mills Lagrangian. Note ⌫ ⇢ µ that a coupling of the form Tr(Fµ F⌫ F⇢ ) is absent. This is in fact required by the 16 supersymmetries preserved by the Dp-brane. Next, let us consider the disc 4-point amplitude of gauge bosons, o ( a ,e,k ) AAAAA { i i i} 1 4 1 1 0 0 a1 a4 a2 a3 = gp cVA (e1,k1; y1)cVA (e2,k2; y2)cVA(e3,k3; y3) dy4VA(e4,k4; y4) Tr( ) ⌧ Z0 +(permson 1, 2, 3 ) { } (2.22) 23 Setting y =0,y =1,y = ,thecorrelatorisevaluatedas 1 2 3 1 1 1 0 0 cV (e ,k ;0)cV (e ,k ;1)(cV )0(e ,k ; )V (e ,k ; y) A 1 1 A 2 2 A 3 3 1 A 4 4 4 ⌦ p+1 p+1 1 2↵ k1 k4 2↵ k2 k4 ↵ 2 = iC (2⇡) ( k ) y 0 · (1 y) 0 · e e ↵0 ( k k e e + k e k e ) p i 2↵ 1 · 2 3 · 4 3 · 4 3 · 4 4 · 3 i=1 0 ( X 2 k1 e4 k2 e4 ↵0 + ↵0 (k e + yk e ) · + · + e e 2 · 3 4 · 3 y y 1 2 3 · 4 ✓ ◆ 2 k1 e4 k2 e4 + ↵0 (e k e e e k e e ) · + · 2 · 3 1 · 3 1 · 3 2 · 3 y y 1 ✓ ◆ +(e k e e e k e e )(k e + yk e ) 2 · 4 1 · 4 1 · 4 2 · 4 2 · 3 4 · 3 ↵ 2 + 0 e k e k e e + e e e e k k e e e k e k e k e e e k y 1 1 · 3 2 · 4 3 · 4 1 · 3 2 · 4 3 · 4 1 · 3 2 · 4 4 · 3 1 · 3 2 · 4 3 · 4 ✓ ◆ ↵ 2 0 e k e k e e + e e e e k k e e e k e k e k e e e k y 1 · 4 2 · 3 3 · 4 1 · 4 2 · 3 3 · 4 1 · 4 2 · 3 3 · 4 1 · 4 2 · 3 4 · 3 ✓ ◆) (2.23) Performing the y-integral gives the result o ( a ,e,k ) AAAAA { i i i} 2 4 ( ↵0s) ( ↵0u) a1 a2 a3 a4 =b ↵0 gpCpK( ei,ki ) Tr( )+(permutatiomson2, 3, 4) . { } (1 ↵0s ↵0u) (2.24) Here 4 µ1⌫1 µ4⌫4 K( e ,k )=t ··· e k , (2.25) { i i} 8 iµi i⌫i i=1 Y where the tensor t8 is defined as in the sphere 4-point amplitude of massless (NS, NS) closed string states. Let us examine the low-momentum expansion 2 ( ↵0s) ( ↵0u) 1 ⇡ 2 = + (↵0 ). (2.26) (1 ↵ s ↵ u) ↵ 2su 6 O 0 0 0 The leading term gives rise to the tree-level 4-point amplitude of Yang-Mills theory. We can compare the part of the residue at s =0proportionalto(e e )(e e ) 1 · 2 3 · 4 g4 (e e )(e e )tu + o ( a ,e,k ) p C 1 · 2 3 · 4 ···Tr( a1 a2 a3 a4 ) (2.27) AAAAA { i i i} 4 p su with theb factorization into a pair of 3-point amplitudes, summed over intermediate gauge boson states. This leads to the identificaftion 1 Cp = 2 . (2.28) ↵0gp 24 2.1.2 Cylinder amplitudes and tadpole cancelation The simplest open string loop amplitude is the cylinder diagram that contributes to the ground state energy density of the D-brane. As a toy example, let us consider the cylinder with the two boundary components on two di↵erent Dp-branes extended in X0, ,Xp ··· directions in critical bosonic string theory, separated by xi, i = p +1, , 25. We can take ··· the cylinder to be of circumference 2⇡t and length ⇡. The corresponding amplitude is 1 dt Nbc 2⇡tL0 cyl =2 Tr o ( ) b0c0e , (2.29) A 2t H Z0 where the overall factor of 2 comes from the summation of two orientations of the open string, or equivalently two ways of assigning the boundary conditions on the two ends of the 1 cylinder. The factor 2t in the integration measure comes from the volume of the conformal Killing group, namely the translation symmetry along the circumference of the cylinder, and the orientation-preserving Z2 reflection. The trace over the strip Hilbert space in (2.29) can be evaluated as 2 p+1 2 ( x) d k 2⇡t↵0 k + 2 Nbc 2⇡tL0 (2⇡↵0) 24 Tr o ( ) b0c0e = iVp+1 e ⌘(it) H (2⇡)p+1 Z (2.30) 23 p ( x)2 iVp+1 t 24 2 2⇡↵ = p+1 t e 0 ⌘(i/t) . 2 (8⇡ ↵0) 2 The cylinder amplitude can be written as 21 p ( x)2 iVp+1 1 t 24 2 2⇡↵ cyl = p+1 dt t e 0 ⌘(i/t) A (8⇡2↵ ) 2 0 0 Z (2.31) 21 p ( x)2 iVp+1 1 t 2⇡/t 2⇡/t 2 2⇡↵ = p+1 dt t e 0 e +24+ (e ) , 2 2 O (8⇡ ↵0) Z0 ⇥ ⇤ 2⇡/t where in the last line we have expanded the eta function in powers of e .Thecontribution 2⇡(n 1)/t of order e comes from exchange of closed string modes at oscillator level n between the pair of D-branes, whose mass squared is proportional to n 1. For instance, the leading term e2⇡/t corresponds to the closed string tachyon exchange, the next-to-leading term 24 corresponds to graviton and dilaton exchange (there is no B-field exchange before the latter does not couple to the D-brane at linear order), and so forth. /(iV )istheenergy Acyl p+1 density of the pair of D-branes due to closed string exchange (at leading order in gs). Its derivative with respect to the distance x gives the force between the D-branes. Of course, the tachyon exchange gives a divergent contribution. Nonetheless, the massless closed string exchanges give a finite contribution, and can be compared to the corresponding amplitude computed from an e↵ective action that couples the D-brane to the massless closed string fields. In the next section, we will use this relation to determine the tension of the Dp-brane. 25 Next let us consider the cylinder amplitude between a pair of parallel Dp-branes in type II string theory separated by xi in transverse directions, 1 dt =2 Z (t), (2.32) Acyl 2t cyl Z0 where Zcyl(t) is the partition function of the worldsheet CFT on a cylinder of modulus t with appropriate ghost insertion and GSO projection, 1+( )F Nbc+N 2⇡tL0 Zcyl(t)= Tr NS Tr R ( ) b0c0e . (2.33) Ho Ho 2 The relative minus sign of R sector contribution is due to the R sector open string states being fermionic excitations on the D-brane. Strictly speaking, similarly to the torus vacuum amplitude in type II string theory, the trace over R with ( )F +N insertion is ill-defined Ho due to the infinite degeneracy of the R sector ground state of system. The role of the latter will be to cancel the contribution to the partition function from matter CFT oscillators in the lightcone direction, so that (2.33) reduces to a trace over transverse oscillator excitations in the lightcone gauge, F 1 1+( ) 2⇡t(L0 ) Zcyl(t)= Tr NS Tr R e 2 . (2.34) Ho,l.c. Ho,l.c. 2 ⇣ ⌘ Let us first consider the contribution from the trace without ( )F insertion, corresponding to the cylinder partition function with anti-periodic boundary condition for fermions along the Euclidean time circle, 1 1 2⇡t(L0 2 ) Zcyl (t)= Tr NS Tr R e Ho,l.c. Ho,l.c. 2 ⇣ ⌘ 2 4 4 (2.35) iVp+1 2 p+1 t ( x) 8 ✓3(0 it) ✓2(0 it) = (8⇡ ↵0t) 2 e 2⇡↵0 (⌘(it)) | | , 2 ⌘(it) ⌘(it) ( ) 1 n+ 1 2 1 where we have used ✓ (0 it)/⌘(it)=q 24 1 (1+q 2 ) and ✓ (0 it)/⌘(it)=2q 12 1 (1+ 3 | n=0 2 | n=1 qn)2. The contribution of (2.35) to (2.32) can be interpreted as the amplitude of (NS, NS) Q Q closed string exchange between the D-branes, NSNS 1 dt cyl = Zcyl (t) A 0 t Z 4 4 p+1 5 p ( x)2 iVp+1 2 1 t 8 ✓3(0 i/t) ✓4(0 i/t) = (8⇡ ↵0) 2 dt t 2 e 2⇡↵0 (⌘(i/t)) | | 2 ⌘(i/t) ⌘(i/t) Z0 ( ) 2 p+1 5 p ( x) 1 2⇡ 1 1 2⇡ 1 1 2⇡ iVp+1 2 1 t 8 (n+ ) 8 (n+ ) 8 = (8⇡ ↵0) 2 dt t 2 e 2⇡↵0 (⌘(i/t)) e 6 t (1 + e 2 t ) (1 e 2 t ) 2 Z0 "n=0 n=0 # 2 Y Y p+1 5 p ( x) 2⇡ iVp+1 2 1 t = (8⇡ ↵0) 2 dt t 2 e 2⇡↵0 16 + (e t ) . 2 0 O Z h i (2.36) 26 2⇡ In the last line, terms of order e t in the large t limit are due to exchange of massive (NS, NS) closed string modes. The exchange of massless (NS, NS) closed string modes, namely that of the graviton and the dilaton, gives rise to an attractive potential between the pair p 7 of parallel D-branes that falls o↵like x .Thisattractivepotentialwillbecanceledby | | arepulsivepotentialduetotheexchangeofmassless(R,R)closedstringstates.Indeed, + RR = 1 dt Z (t)isequalto NSNS and the total cylinder amplitude vanishes. In the Acyl 0 t cyl Acyl next section, we will compare NSNS and RR to the exchange amplitude computed from R Acyl Acyl the D-brane e↵ective action to determine the tension and the charge of the Dp-brane in type II string theory. Now we turn to the question of massless tadpole in type I string theory. Just as how aconformalboundaryconditioncanberepresentedbyaboundarystate B ,acrosscrap | ii can be represented by a “cross cap state” .Letusinvestigatetheexpectationvalueof |⌦ii the Euclidean CFT propagator in , |⌦ii ⇡ c c (L0+L0 ) 2⇡t(L0+L0 ) e 2t 12 =Tr⌦e 12 , (2.37) hh⌦| |⌦ii e e where on the RHS we represented the same amplitude as the partition function of the Klein bottle, computed by a trace over the CFT Hilbert space on the circle twisted by the parity transformation ⌦. In a CFT with fermions, one must also specify how the fermion fields are identified along the cross cap, as there can be a sign ambiguity. We will adopt a definition of the cross cap state that is compatible with GSO projection, as follows. The cross cap state consists of its (NS, NS) and (R, R) components, = + . (2.38) |⌦ii |⌦iiNSNS |⌦iiRR The Klein bottle amplitudes with di↵erent fermion periodicity conditions are given by F F ⇡ c c (L0+L0 ) ( ) +( ) 2⇡t(L0+L0 ) NSNS e 2t 12 NSNS =Tr ⌦ e 12 (2.39) hh⌦| |⌦ii 4 e e e and F +F ⇡ c c (L0+L0 ) 1+( ) 2⇡t(L0+L0 ) RR e 2t 12 RR =Tr ⌦ e 12 . (2.40) hh⌦| |⌦ii 4 e e e Note that the trace on the RHS only receives contribution from (NS, NS) and (R, R) sectors, and not from (NS, R) nor (R,NS) sectors. Altogether, we can write F F ⇡ c c (L0+L0 ) 1+( ) 1+( ) 2⇡t(L0+L0 ) e 2t 12 =Tr ⌦ e 12 . (2.41) hh⌦| |⌦ii 2 2 e e e It will also be useful to consider the overlap between an ordinary boundary state B and | ii the cross cap state, ⇡ c c (L0+L0 ) 2⇡t(L0 ) B e 4t 12 =Tr ⌦ e 24 , (2.42) hh | |⌦ii HBB e 27 where is the Hilbert space of the CFT on the interval of length ⇡,subjecttoboundary HBB condition B on both ends. The RHS can be interpreted as the partition function of the M¨obius strip. In a CFT with fermions, writing B = B + B , we have | ii | iiNSNS | iiRR F ⇡ c c (L0+L0 ) 1+( ) 2⇡t(L0 ) NSNS B e 4t 12 NSNS =Tr NS ⌦ e 24 , hh | |⌦ii HBB 2 e F (2.43) ⇡ c c (L0+L0 ) 1+( ) 2⇡t(L0 ) RR B e 4t 12 RR =Tr R ⌦ e 24 . hh | |⌦ii HBB 2 e 0 In type I string theory, the vacuum energy density at order gs receives contribution from both open and closed string 1-loop diagrams, including the topologies of cylinder, M¨obius strip, torus, and Klein bottle. The normalization of these vacuum amplitudes can be fixed as follows. The cylinder vacuum amplitude in type IIB string theory is now replaced by the ⌦-projected cylinder amplitude, F 1 dt 1+( ) 1+⌦ Nbc+N 2⇡tL0 STr o ( ) b0c0e , (2.44) t H 2 2 Z0 where STr o =Tr NS Tr R is the trace with an extra sign for spacetime fermions. Here o H Ho Ho H is the strip Hilbert space with U(N)Chan-Patonfactor,onwhich⌦actsaccordingto(1.66), leading to either SO(N)orSp(N) Chan-Panton factor for the unoriented open string. The torus amplitude of type IIB string theory is now replaced by 1 F F 2 d⌧ 1 1 d⌧ ⌦ 1+( ) 1+( ) 2 2 Nbc+N 2⇡i⌧L0 2⇡i⌧¯L0 d⌧1 STr b0c0b0c0 + STr b0c0 ( ) e . 1 e ⌧2>0, ⌧ >1 2⌧2 2 0 2⌧2 2 2 2 Z 2 Z | | Z e (2.45) e e We have written the Klein bottle amplitude with a ⌧1-integral, for the sake of comparison with the torus amplitude. While the torus partition function is integrated over the fundamental domain : ⌧ > 0, ⌧ < 1 , ⌧ > 1, the Klein bottle amplitude is integrated over the entire F 2 | 1| 2 | | positive ⌧2-axis. The trace with ⌦insertion only receives contribution from (NS, NS) and (R, R) states with L0 = L0,andhencethe⌧1-integral drops out. Note that the ⌦-projection is only implemented at the level of physical string states, as seen from the ⌧ limit of 2 !1 the integrand of (2.45), note at the level of worldsheet CFT. The (NS, NS) tadpole can be extracted from ⇡ 1 (L0+L0) 1 F F N +N 2⇡t(L0+L0) e 2t bc = Tr ( ) +( ) ⌦ b c ( ) bc e 2t NSNShh⌦| |⌦iiNSNS 16 0 0 e h ei e(2.46) and F 1 ⇡ N 1+( ) (L0+L0) Nbc+N 2⇡tL0 NSNS B e 4t bc NSNS = ± Tr NS ⌦ b0c0( ) e , (2.47) 4t hh | |⌦ii 2 Ho 2 e 28 where represents the strip Hilbert space without Chan-Paton factor, and the factor +N Ho applies to the case of SO(N)Chan-Patonfactorand N applies to Sp(N). On the LHS, we have replaced the average of b and c by the insertion of a pair of bc ghosts at an arbitrary position (as the result does not depend on the location of their positions), and perform a conformal rescaling so that the circumference is 2⇡. To proceed, we compute the (NS, NS) part of the Klein bottle partition function ⇡ ⇡s(L0+L0) 1 F F N +N (L0+L0) e bc = Tr ( ) +( ) ⌦ b c ( ) bc e s NSNShh⌦| |⌦iiNSNS 16s 0 0 e h 4 ei 4 e iV 2 5 5 8 ✓3(0 i/s) ✓2(0 i/s) = (4⇡ ↵0) (2s) (⌘(i/s)) | | 8s ⌘(i/s) ⌘(i/s) ( ) 4 4 (2.48) 10 iV 8 ✓3(0 is) ✓4(0 is) =2 (⌘(is)) | | 8(8⇡2↵ )5 ⌘(is) ⌘(is) 0 ( ) 10 iV 2⇡s =2 2 5 16 + (e ) , 8(8⇡ ↵0) O the (NS, NS) part⇥ of the M¨obius⇤ strip partition function F N 1 1+( ) ⌦ ⇡ ⇡s(L0+L0) Nbc+N L0 NSNS B e bc NSNS = ± Tr NS b0c0( ) e 2s hh | |⌦ii 2s 2 Ho 2 2 e 1 iV 2 5 5 1 = N (8⇡ ↵0) (4s) n⇡ ± 8s n 2s 8 n=1 (1 ( ) e ) Y 1 1 i m ⇡ (m+ 1 ) 8 m ⇡ (m+ 1 ) 8 (1 + i( ) e 2s 2 ) (1 i( ) e 2s 2 ) ⇥ 2 "m=0 m=0 # (2.49) Y Y i 4 iV 2 5 5 1 2✓2✓4(0 2s ) = N (8⇡ ↵0) (4s) | ⌥ 16s (⌘( i )✓ (0 i ))4 (⌘( i ))2 2s 3 | 2s 2s iV 1 2✓ ✓ (0 2is) 4 = 25N 2 4 | ⌥ 8(8⇡2↵ )5 (⌘(2is)✓ (0 2is))4 (⌘(2is))2 0 3 | 5 iV 2⇡s = 2 N 2 5 16 + (e ) , ⌥ 8(8⇡ ↵0) O where the upper minus⇥ sign is for SO⇤(N)andlowerplussignforSp(N). We have used 2 identities involving theta functions to convert the sum of two infinite products into ✓2✓4/⌘ . Finally, we have the (NS, NS) part of the cylinder partition function ⇡s(L0+L0) s B e bc B = Z (1/s) NSNShh | | iiNSNS 4 cyl e 4 4 2 iV 8 ✓3(0 is) ✓2(0 is) = N (⌘(is)) | | 8(8⇡2↵ )5 ⌘(is) ⌘(is) (2.50) 0 ( ) 2 iV 2⇡s = N 2 5 16 + (e ) . 8(8⇡ ↵0) O ⇥ ⇤ 29 Altogether, we see that cancelation of massless (NS, NS) tadpole requires the choice of SO(N)Chan-PatonfactorwithN =25. 2.2 D-brane e↵ective action: the bosonic case The low energy dynamics of a Dp-brane can be captured by an e↵ective action that couples the world volume fields to the spacetime background fields. We can parameterize the world volume of the D-brane with an arbitrary non-degenerate coordinate system ⇠a, a =0, ,p, ··· and the embedding of the D-brane in the spacetime via the fields Xµ(⇠). In the absence of world volume gauge fields nor spacetime dilaton and B-field, the D-brane e↵ective action takes the Nambu-Goto form S = T dp+1⇠ det G , (2.51) p ab Z p where G (⇠) G (X(⇠))@ Xµ(⇠)@ X⌫(⇠)istheinducedmetricontheworldvolume.This ab ⌘ µ⌫ a b form of the e↵ective action is governed by reparameterization invariance on the world volume and general coordinate invariance in the target spacetime, up to higher derivative corrections. Tp is the tension of the brane. If we shift the spacetime dilaton field by a constant 0, Tp 0 which is proportional to the inverse string coupling will change by the factor e .This indicates that in a slowly varying dilaton background field (X), (2.51) will be generalized to p+1 S = T d ⇠e det G . (2.52) p ab Z p If we turn on background Bµ⌫(X) field and the world volume gauge field Aµ(⇠), the string worldsheet action changes by 1 S = B (X)dXµ dX⌫ + A (X)dXµ, (2.53) string 2⇡↵ µ⌫ ^ µ 0 Z⌃ Z@⌃ µ µ a where we have written dX = @aX (⇠)d⇠ as a di↵erential form on the worldsheet. (2.53) is invariant under the gauge transformation 1 B(X)=d⇤(X), A(⇠)= ⇤(X(⇠)), (2.54) 2⇡↵0 µ where ⇤=⇤µdX is a spacetime 1-form gauge parameter. It follows that the D-brane e↵ective action can only depend on B and A through the gauge invariant combination B +2⇡↵0F ,whereF = dA is the world volume gauge field strength 2-form. The precise form of the Fab coupling on the D-brane, for not only small but finite, slowly-varying field strength, can be argued based on a naive application of T-duality, as follows. Suppose we compactify a spatial direction, say X1,onacircleofradiusR,namely 30 1 1 1 1 X X +2⇡R.IntheT-dualdescription,X is replaced by the worldsheet field X0 , ⇠ 1 1 1 related by X0 = X , X0 = X .SupposewehaveN Dp-branes extended along X .On L L R R the boundary of the worldsheet, while X1 is subject to Neumann boundary condition, X1 is subject to Dirichlet boundary condition, therefore in the T-dual description we have N 1 D(p 1)-branes transverse to X0 . The open strings on the N Dp-branes carry quantized momentum in the X1 direction but no conserved winding number. In the T-dual description, these open strings do not carry 1 1 momentum along X0 , but since their ends are stuck on the D-branes transverse to X0 ,they 1 can winding around the X0 circle. This shows that the statement that T-duality exchanges momentum and winding modes extends to the open strings as well. Let us inspect the e↵ect of turning on a constant background gauge field A1(⇠), which up to a U(N) gauge transformation can be put to the diagonal form 1 A = diag(✓ , ,✓ ). (2.55) 1 2⇡R 1 ··· N A gauge-invariant quantity that describes this configuration is 1 Pei A1dX =diag(ei✓1 , ,ei✓N ). (2.56) H ··· In particular, ✓i are 2⇡-periodically valued. An open string stretched between the i-th and the j-th branes acquires the coupling ✓ ✓ S = d⌧ i i @ X1, (2.57) 2⇡R ⌧ Z which leads to a shifted canonical momentum (parameterizing the open string worldsheet by astripofwidth⇡) 1 @⌧ x ✓i ✓j n p1 = + = ,nZ, (2.58) 2↵0 2⇡R R 2 where x1 1 ⇡ d X1( ,⌧ ). The spacetime momentum charge of the open string is ⌘ ⇡ 0 R 1 1 ✓ ✓ k = @ x1 = n + j i . (2.59) 1 2↵ ⌧ R 2⇡ 0 ✓ ◆ In the T-dual description, we have ⇡ 1 1 1 X0 ( = ⇡) X0 ( =0)= d @ X0 0 Z ⇡ 1 (2.60) = d @⌧ X Z0 =2⇡↵0k =2⇡R0n +(✓ ✓ )R0, 1 j i 31 1 where R0 = ↵0/R is the radius of the T-dual X0 -circle. This suggests that in the T-dual 1 description, the i-th D(p 1)-brane is shifted by X0 = ✓ R0.Inotherwords,wecanmake i the identification 1 X0 =2⇡↵0A (2.61) |@⌃ 1 between the position of the D(p 1)-brane after T-duality and the worldvolume gauge field on the Dp-brane before the T-duality. Now consider a D2-brane extended in X1 and X2 direction, with a gauge field configu- 2 ration A1 = F12X for constant field strength F12. After T-duality, we have a D1-brane 2 1 extended in X ,withX0 -coordinate given by 1 2 X0 =2⇡↵0A1 =2⇡↵0F12X . (2.62) This tilted D1-brane has action 2 1 2 S = T 0 dX 1+(@ X ) . (2.63) D1 1 2 0 Z p It should be identical to the action of the D2-brane before T-duality, which must take the form 2⇡R S = T dX1dX2 1+(2⇡↵ F )2. (2.64) D2 2 0 12 Z0 This suggests that the D-brane e↵ective Lagrangianp density in the presence of finite, slowly- varying Fab should be proportional to det(Gab +2⇡↵0Fab). Combined with B-field, we are led to the e↵ective action p p+1 S = T d ⇠e det(G + B +2⇡↵ F ). (2.65) Dp p ab ab 0 ab Z p The Dp-brane tension Tp depends on p through the relation T 0 N 1 p+1 = h | iiX,R = , (2.66) Tp 2⇡R 0 D X,R 2⇡p↵ h | ii 0 where N X,R and D X,R are the Neumann and Dirichlet boundary states for a free compact | ii | ii p 1 boson X at radius R.ThuswelearnthatT (2⇡p↵ ) g . Note that the agreement of p / 0 s (2.63) with (2.63) requires 2⇡RT R g 1= 2 = s0 . (2.67) T10 p↵0 gs That is, while T-duality between the compact boson of radius R and radius R0 = p↵0/R is an equivalence at the level of CFT, the equivalence between the dual string theories involves p↵0 arescalingofthestringcoupling,gs0 = R gs.Thiscanalsobeunderstooddirectlyatthe level of perturbative closed string amplitudes. 32 The overall constant coe cient can be determined by comparison of the D-brane e↵ective action expanded to linear order in the closed string background fields with the disc 1-point amplitudes. Equivalently, we can compare the amplitude of graviton and dilaton exchange between a pair of D-branes against the cylinder amplitude computed in the previous section. The latter is more convenient for keeping track of the normalization. Let us consider the brane-bulk e↵ective action of massless fields in critical bosonic string theory, 1 26 2 µ⌫ p+1 S = d xp Ge [R +4G @ @ + ] T d ⇠e det(G + ), 22 µ ⌫ ··· p ab ··· Z Z p (2.68) where the omitted terms involve B-field and the world volume gauge fields that won’t play any role in the tree level one-closed string exchange amplitude. It is convenient to work in Einstein frame, in which the kinetic terms for the graviton and dilaton are canonically normalized, by defining G e 6 G , (2.69) µ⌫ ⌘ µ⌫ so that the e↵ective action can be written as e 1 26 1 µ⌫ p+1 p+1 1 S = d x G R G @ @ + T d ⇠e( 12 ) det(G + ). 22 6 µ ⌫ ··· p ab ··· Z q Z q (2.70) e e e e We can expand the metric around Minkowskian background Gµ⌫ = ⌘µ⌫ + hµ⌫.Toleading order in perturbation theory, we will keep the quadratic terms in the bulk action for the fluctuation fields hµ⌫, , e 1 2 S = d26x (@ h )2 2(@⌫h )2 +2@µh@⌫h (@ h)2 + (@ )2 + , bulk 82 ⇢ µ⌫ µ⌫ µ⌫ µ 3 µ ··· Z (2.71) where h hµ , and keep the linear terms in the brane action (expanded around the flat ⌘ µ brane extended in X0, ,Xp directions) ··· p p 11 1 S = T dp+1x ha + . (2.72) brane p 12 2 a ··· a=0 ! Z X Note that the linearized action is compatible with linearized gauge transformation hµ⌫ = @µ⇠⌫ + @⌫⇠µ.Wecanproceedbyaddingagaugefixingterm 1 1 2 S = d26x 2 @⌫h @ h (2.73) GF 82 µ⌫ 2 µ Z ✓ ◆ to the bulk action, so that the kinetic term takes the simple and non-degenerate form 1 1 2 S = d26x (@ h )2 (@ h)2 + (@ )2 . (2.74) bulk,GF 82 ⇢ µ⌫ 2 µ 3 µ Z 33 This leads to the graviton propagator 2i2 1 h (p)h (q) =(2⇡)26 26(p + q) ⌘ ⌘ + ⌘ ⌘ ⌘ ⌘ (2.75) h µ⌫ ⇢ i p2 µ ⌫⇢ µ⇢ ⌫ 12 µ⌫ ⇢ ✓ ◆✓ ◆ and the dilaton propagator 6i2 (p) (q) =(2⇡)26 26(p + q) . (2.76) h i p2 ✓ ◆ The 1-point vertex coming from (2.72) does not constrain the transverse momentum p to ? the brane. The massless exchange amplitude is 2 2 2 2 p i Tp p 11 1 ab cd 1 (p )= 2 6 + 2 ⌘ ⌘ 2⌘ad⌘bc ⌘ab⌘cd A ? p " 12 2 12 # ? ✓ ◆ ✓ ◆ a,b,c,dX=0 ✓ ◆ (2.77) 6i2T 2 = p . p2 ? By matching this result with the Fourier transform of the part of /V (2.31) due to Acyl p+1 massless closed string exchange, one finds 11 p p⇡ 2 T = (4⇡ ↵0) 2 . (2.78) p 16 In our convention for gs in bosonic string theory, matching the gravitational 3-point ampli- tude with that of the bulk e↵ective action gives the identification =2⇡gs. On the other hand, comparison of the disc amplitude of massless open string fields with tree amplitude computed from the brane e↵ective action gives 1 2 0 2 =2⇡ ↵ Tp. (2.79) gp This then determines the open string coupling gp on the Dp-brane in terms of the closed string coupling gs. 2.3 D-brane e↵ective action: the supersymmetric case Dp-branes in type II string theory admit an e↵ective action of the massless open string degrees of freedom consisting of the collective coordinates, world volume gauge field, and gauginos, coupled to massless closed string background fields. In Minkowskian background, ageneralizationofthebosonicD-branee↵ectactionthatnonlinearrealizesthe10Dsuper- Poincar´einvariance can be constructed as follows. 34 Let ⇠a (a =0, ,p)betheworldvolumecoordinatesasbefore.Theworldvolumefields ··· include a gauge field A ,thescalarsXµ (µ =0, , 9), and fermions ✓ that are understood a ··· to be a Majorana but not Weyl spinor of so(1, 9) in the type IIA case (i.e. containing chiral and anti-chiral spinor components) and a pair of Majorana-Weyl spinors in the type IIB case. In the latter case, we will typically suppress the label for the two of M-W spinors, and in various formulae an explicit sum over the two copies is understood. The full 32 spacetime supersymmetries act on these D-brane world volume fields as µ µ ✏✓ = ✏, ✏X = ✏ ✓, (2.80) where ✏ is a constant 32-component Grassmannian spinor parameter. Clearly, the expressions @a✓ and ⇧µ = @ Xµ ✓ µ@ ✓, ⇧ ⇧ d⇠a, (2.81) a a a ⌘ a are invariant under (2.80). An invariant quantity that generalizes the notion of the induced metric in the bosonic case is = ⌘ ⇧µ⇧⌫. (2.82) Gab µ⌫ a b The generalization of the expression B +2⇡↵0F turns out to be =2⇡↵0F + b , Fab ab ab 1 (2.83) b b d⇠ad⇠b = ✓ µd✓ dXµ + ✓ µd✓ . ⌘ 2 ab 11 ^ Here Fab is the ordinary field strength of Aa.Onecanverifythatthesupersymmetryvariation b is a total derivative, and A is chosen to cancel it so that is supersymmetry invariant. ✏ ✏ Fab The fully supersymmetric Dp-brane e↵ective action is S = T dp+1⇠ det( + )+ ⌦ , (2.84) p Gab Fab p+1 Z p Z where ⌦p+1, known as the Wess-Zumino term, can be constructed as follows. Formally, viewing ⌦p+1 as a (p +1)-formextendedtop +2dimensions,itsexteriorderivativecanbe written as I d⌦ =( )p+1d✓ W d✓, (2.85) p+2 ⌘ p+1 ^ p ^ where the spinor-matrix valued p-form Wp is given by the generating formula 1 n+1 2n µ W = eF , ⇧ , p (2n)! 11 ⌘ µ (2.86) even p n 0 X X in the type IIA case, and 1 n 2n+1 W = eF p (2n +1)! 3 1 (2.87) odd p n 0 X X 35 in the type IIB case, where 1 and 3 are Pauli matrices acting on the two M-W spinor components of ✓. While ⌦p+1 is clearly supersymmetry invariant, its precise form as stated above is deter- mine by demanding that S is invariant under the so-called -symmetry, which is a fermionic gauge symmetry that reduces the fermionic degree of freedom from 32 to 16, corresponding µ to Goldstinos on the D-brane. The kappa symmetry acts on X and Aa by µ µ X = ✓ ✓, µ 1 µ 1 µ (2.88) 2⇡↵0 A = ✓ ✓ ⇧ + ✓ d✓ ✓ ✓ ✓ d✓ . 11 µ 2 2 11 µ ✓ ◆ The kappa symmetry transformation of ✓ itself takes the form ✓ = (1 ⌥(p+1)), (2.89) where is an arbitrary 32-component local (i.e. ⇠-dependent) spinor parameter. The matrix ⌥(p+1) is defined by p+1 ⌥(p+1) = ⇤P (2.90) det( + ) G F (p+1) 2 and obeys (⌥ ) =1,wherethematrixvalued(p p +1)-form p+1 is formally constructed P from the generating formula 1 n+1 2n+1 = eF (IIA), Pp+1 (2n +1)! 11 even p n 0 X X (2.91) 1 n+1 2n = eF (IIB). Pp+1 (2n)! 3 1 odd p n 0 X X While the expression (2.84) for the e↵ective action is manifestly supersymmetric, its physical content is rather obscure due to the kappa symmetry redundancy. A much more useful form of the action is obtained by working in the static gauge, i.e. identifying the world volume coordinates ⇠a = Xa, a =0, ,p,andfixingthekappagaugesymmetrybysetting ··· one M-W component of ✓ to zero. The other nontrivial component of ✓ will be renamed and becomes the Goldstino. The resulting static gauge e↵ective action turns out to be S = T dp+1⇠ det ⌘ + @ Xi@ Xi +2⇡↵ F 2 ¯( + @ Xi )@ +( ¯ µ@ )( ¯ @ ) . p ab a b 0 ab a a i b a µ b Z q ⇥ (2.92) ⇤ Note that in the gauge fixed form (2.92), the supersymmetry transformations of the fields are rather complicated and it is nontrivial to directly verify the supersymmetric invariance i of the action. When expanded to quadratic order in the fields X ,Aa, ,(2.92)reducesto the action of the maximally supersymmetric U(1) gauge theory. 36 Now let us consider a Dp-brane in a massless closed string background, i.e. with nontrivial RR spacetime metric Gµ⌫, B-field Bµ⌫, dilaton , as well as RR fields Cq+1.Itispossibleto describe such a supergravity background in terms of the super-vielbein and super-B-field etc. and write a manifestly supersymmetric e↵ective action for the Dp-brane analogous to (2.84), but we will not discuss this general construction here. Instead, we will simply consider the bosonic part of the action, p+1 B+2⇡↵ F RR S = T d ⇠e det(G + B +2⇡↵ F )+µ e 0 C . bos p ab ab 0 ab p ^ q (2.93) q Z p Z X Here the RR fields are normalized such that the bulk supergravity action contains the RR kinetic term 1 S d10xp G F RR 2, (2.94) bulk 42 | p+2| Z RR RR where Fp+2 = dCp+1.ApairofparallelDp-branes in type II string theory preserve the same set of 16 supersymmetries, and the force between them due to the exchange of (NS, NS) and (R, R) closed string fields cancel. This leads to the relation µp = Tp,whichamountsto the statement that the flat Dp-brane saturates the BPS bound. Furthermore, comparison of the (NS, NS) exchange contribution to the cylinder amplitude against the corresponding exchange amplitude computed from the brane+bulk e↵ective action determines 3 p p⇡ 2 T = (4⇡ ↵0) 2 . (2.95) p ⇡ 2 Recall that is further related to the closed string coupling gs by = 2 gs. Comparison with the 4-gauge boson disc amplitude, on the other hand, determines the relation between the brane tension and the open string coupling, 1 1 2 0 2 = ⇡ ↵ Tp. (2.96) gp 2 Let us note that the definitions for gs and gp replied on our specific conventions for superstring perturbation theory, whereas brane tension Tp and the gravitational coupling are physical and unambiguously defined. In type IIB string theory, a more standard notion of the string coupling is the ratio between the fundamental string and D1-brane tension, 1 g g = = s . (2.97) 7 2 5 2 ⌘ 2⇡↵0T1 8⇡ 2 ↵0 16⇡ 2 ↵0 RR The couplings in (2.93) that involve RR fields Cq and world volume gauge field strength F can be understood through T-duality, similarly to the discussion of section 2.2, now that 2Recall that this relation di↵ers from that of Polchinski Vol 2 by a factor of 4, due to a di↵erent normal- ization convention for the picture changing operator. 37 T-duality exchanges IIA with IIB string theories and also changes the degree of the RR forms. An important feature of the D-brane charges is that they obey ⇡ µpµ6 p = . (2.98) 2 While the Dp-brane is electrically charged with respect to CRR ,theD(6 p)-brane is magnet- p+1 ically charged. It follows from the linearized brane+bulk e↵ective action that the magnetic flux out of a D(6 p)-brane is RR 2 Fp+2 =2 µ6 p. (2.99) p+2 ZS 2 Dirac quantization condition demands that µp 2 µ6 p to be an integer multiple of 2⇡.This · condition is saturated by (2.98), which means that the Dp-branes in fact carry the minimal possible RR charges. In this sense, a D-brane is a fundamental object of string theory that cannot be divided further. The low energy excitations of N (nearly) coincident Dp-branes, where the the massless open string states carry U(N)Chan-Patonfactor,canbecapturedbyane↵ectiveaction that involves a U(N)gaugefieldA and the massless scalar fields Xi, i = p +1, , 9, and a ··· fermionic spinor fields , all transforming in the adjoint representation of the gauge group, of the form (in Minkowskian background) p+1 1 i 2 1 2 ab 1 2 i j S = T d ⇠ tr (D X ) + (2⇡↵0) F F (2⇡↵0) [X ,X ][X ,X ] p 2 a 4 ab 4 i j Z ⇢ (2.100) i ¯ aD ¯ i[X , ]+ a 2⇡↵ i ··· 0 i i where Fab = @aAb @bAa + i[Aa,Ab] is the non-Abelian field strength, DaX = @aX + i 1 2 a b i[A ,X ], etc. The scalar potential V (X)= T (2⇡↵0) tr[X ,X ][X ,X ]andthescalar- a 4 p a b fermion coupling can be understood as follows. The configuration in which the N D-branes are parallel and separate in the transverse directions is described by a set of mutually com- muting matrices Xi,whichuptoaU(N)gaugerotationcanbeputinthediagonalform Xi =diag xi , ,xi .Expandingaroundthisfieldconfiguration,theo↵-diagonalentries { 1 ··· N } of the world volume fields which correspond to the lowest open string modes stretched be- 1 tween the m-th and the n-th D-brane acquires mass ~x m ~x n . Note that 8 p of such 2⇡↵0 | | i bosonic physical degrees of freedom come from the scalar fields Xmn (modulo gauge trans- formation), whereas p bosonic degrees of freedom come from the now-massive vector boson (Aa)mn. 1 i 3 1 If we assign mass dimension 1 to Aa and X , to ,thenalldimension4operators 2⇡↵0 2 2⇡↵0 made out of these fields and their derivatives are captured by the action (2.100), which 38 admits 16 linearly realized supersymmetries. While the higher dimensional operators can ··· in principle be determined by comparison with open string tree amplitudes, an extension of this non-Abelian action that nonlinearly realizes the full spacetime super-Poincar´esymmetry is not explicitly known (in contrast to the Abelian case). 2.4 D-branes at angles and scattering 2.5 D-instanton e↵ects 39