String Theory

April 6, 2016 2 Contents

1 Free superstrings 7 1.1 From particle to strings ...... 7 1.2 Superstring action and symmetries ...... 11 1.3 Equations of motion and mode expansion ...... 13 1.3.1 Closed strings ...... 14 1.3.2 Open strings ...... 15 1.4 Quantization of the superstring ...... 17 1.4.1 Canonical quantization ...... 17 1.4.2 The light cone gauge ...... 20 1.5 The conformal ﬁeld theory description ...... 23 1.5.1 Closed strings ...... 26 1.5.2 Open strings ...... 27 1.5.3 The Superconformal algebra ...... 28 1.6 The bosonic string ...... 30 1.7 The critical dimension ...... 31 1.7.1 Existence of Massless states ...... 32 1.7.2 Lorentz symmetry ...... 32

2 Spectra of String States 35 2.1 String states ...... 37 2.1.1 Type IIA string ...... 38 2.1.2 Type IIB string ...... 40 2.1.3 Heterotic strings ...... 41 2.1.4 Type I string ...... 44 2.2 The vacuum amplitude and the modular group ...... 46 2.2.1 Free uncompact bosons ...... 48

3 4 CONTENTS

2.2.2 Free compact bosons ...... 48 2.2.3 Free Fermions ...... 49 2.3 String partition functions ...... 50 2.3.1 Type II Strings ...... 50 2.3.2 Heterotic Strings ...... 52 2.3.3 Type I theory ...... 52 2.4 D-branes and O-planes ...... 55 2.4.1 D-branes as boundaries of open strings ...... 56 2.4.2 D-branes as source of closed strings: tension and charge . . . 57 2.4.3 O-planes ...... 59

3 Low energy effective actions 61 3.1 Strings in curved background ...... 61 3.1.1 Type II superstrings ...... 61 3.1.2 Heterotic strings ...... 63 3.2 Low energy Supergravities ...... 64 3.2.1 Eleven dimensional supergravity ...... 65 3.2.2 Type IIA supergravity ...... 66 3.2.3 Type IIB supergravity ...... 66 3.2.4 Type I supergravity ...... 67 3.2.5 Heterotic superstrings ...... 68 3.3 Killing spinor and equations of motion ...... 68 3.4 Gravitational backgrounds ...... 69 3.4.1 Kahler and complex structures ...... 70 3.4.2 Ricci flat metrics in four dimensions ...... 71 3.4.3 Ricci flat metrics in six dimensions ...... 73 3.4.4 Waves ...... 74 3.5 Branes in 11D supergravity ...... 74 3.5.1 M2 and M5 brane ...... 74 3.6 Branes in type II supergravities ...... 75 3.6.1 The fundamental string, NS5 and Dp branes ...... 75

A Spinor and gamma matrices 77 A.1 The Cliﬀord algebras ...... 77 A.1.1 The Euclidean case ...... 77 CONTENTS 5

A.1.2 The Minkowski case ...... 78 A.2 Spinor representations ...... 79

B Anomalies 83 B.1 Anomaly polynomials ...... 83 B.2 Anomalies in ten dimensions ...... 85 B.2.1 N = 2 supergravities ...... 85 B.2.2 N = 1 supergravities ...... 86 6 CONTENTS Chapter 1

Free superstrings

1.1 From particle to strings

A free relativistic particle moving in space-time is described by an action proportional to the length of its world line Z q ˙ M ˙ N Sparticle = −m dτ −gMN X X (1.1.1) action

Here τ is a parameter describing the time evolution, XM (τ) the position of the particle at a given time τ, m the particle mass and gMN the space-time metric. The action is invariant under an arbitrary reparametrization τ → f(τ). In a similar way, the evolution of a free relativistic string is described by an action proportional to the area of the surface spanned by its evolution in spacetime Z Z p M N SNambu−Goto = −T dτ dσ − det(∂aX ∂bX gMN ) (1.1.2) actionNG where (σ, τ) are the coordinates on the string world sheet, the embedding coordinates XM (σ, τ) describe the position of the string and T is the string tension. Often the Regge slope α0 is used at the place of the string tension 1 T = . (1.1.3) 2πα0 In a quantum theory it is possible to write an object ` with the dimension of a √ s √ 0 0 length: `s = ~cα ; since we will set both ~ and c to one, we can use directly α to characterise the typical length of a microscopic string. Again reparametrization of coordinates (σ, τ) is a symmetry of the action. The string world sheet spanned

7 8 CHAPTER 1. FREE SUPERSTRINGS

Figure 1.1: Thefreestring free string world sheet.

freestring by a string moving freely in space-time is displayed in figure 1.1. We notice that this world sheet can be viewed either as the evolution of a closed string or as a openclosed loop of an open string depending on the choice of σ and τ variables, see figure 1.2. This will always be the case for string world-sheets with boundaries and will be later extensively exploited in establishing striking relations between the dynamics of open and closed strings. In the limit α0 → 0 where the typical string length actionNG is small the string2014-09-25 looks 12:18:59 like a particle and1/1 the action (1.1.2)Stringfree reduces (#4) to that of a action particle (1.1.1). String states are associated to oscillation modes of the string with √ masses growing with the oscillation number in units of 1/ α0. In the limit α0 → 0 , the infinite tower of massive string states can be integrated out and the string theory can be effectively described in terms of a standard quantum field theory with a finite number of massless degrees of freedom. We will later see that the spectrum of string states contains always a particle of spin two, so string theory is always a theory of gravity. actionNG The action (1.1.2), named as Nambu-Goto, involves fields XM in a highly non- linear form even in the case of a flat space-time metric. For many applications it is more convenient to work with an equivalent action involving fields XM in a quadratic form. To this purpose, we introduce a new field hab and the action

Z Z π √ M T ab M N S(X , hab) = − dτ dσ −hh ∂aX ∂bX gMN (1.1.4) polyakov 2 0

M The ﬁeld hab can be interpreted as the metric in the string world-sheet and X as a scalar ﬁelds in the resulting two-dimensional theory. The equivalence between polyakov actionNG the action (1.1.4), named as Polyakov action, and the Nambu-Goto action (1.1.2) 1.1. FROM PARTICLE TO STRINGS 9

can be seen by writing down the algebraic equations of motion for hab δS 1 1 = − g ∂ XM ∂ X − g h hcd ∂ XM ∂ XN = 0 (1.1.5) eom δhab 4πα0 MN a b N 2 MN ab c d eom Evaluating the determinants of the two terms in (1.1.5) one ﬁnds the square of the Nambu-Goto and Polyakov Lagrangians respectively showing the equivalence between the two actions on-shell. Let us make a digression to discuss the higher dimensional case. It is straight- actionNG forward to generalise the Nambu-Goto action (1.1.2) to an object with p-space dimensions and tension Tp. Again it is possible to introduce a classicaly equivalent action that is quadratic in the embedding coordinates Z Tp √ S(XM , h ) = − dτdpσ −h hab∂ XM ∂ XN g − (p − 1)Λ , (1.1.6) polyakovd ab 2 a b MN p where for p 6= 1 we have a non-trivial cosmological constant Λp on the worldvolume.

The equation for the hab reads 1 g ∂ XM ∂ X − h g hcd ∂ XM ∂ XN − (p − 1)Λ = 0 (1.1.7) eomd MN a b N 2 ab MN c d p M and it is easy to check that Λphab = gMN ∂aX ∂bXN is a solution. By using this polyakovd solution in (1.1.6), one recovers the standard Nambu-Goto action for p-dimensional membrane. We notice the absence of the cosmological constant in the p = 1 case. Moreover, for p = 1 the action describing a free string is unvariant under Weyl rescaling hab → λhab, where λ is an arbitrary function on the worldsheet. This symmetry plays a crucial role in the quantization and makes a theory of elementary strings more tractable than a theory of elementary membranes and so we will focus on the former case. String interactions are described by joining and splitting of strings. The basic string vertex, the pant, is represented in ﬁgure 1 and describes the splitting of a closed string into two. The strength of the interaction is measured by the string coupling constant gs. More involved string diagrams can be constructed by gluing pants and cylinder diagrams. The resulting diagram is weighted by a power of gs given by the number of pants involved in the construction of the diagram. For 0 example the cylinder representing a free string is weighted by gs while the pant is 1 generated by the insertion of a closed string state contributing gs . On the other hand the cylinder diagram can be constructed inserting two closed string states on the sphere or one on the disk, so the sphere and disk vacuum amplitudes are 10 CHAPTER 1. FREE SUPERSTRINGS

openclosed Figure 1.2: Basic string diagrams: The cylinder and the pants in the open and closed channel.

−2 −1 weighted by gs and gs respectively. In figure 2 we list all diagrams made out of 2 gluing two pants and contributing to order gs . They are obtained by gluing two pants along none, one and two boundaries respectively. String amplitudes describe scattering of strings states. They are defined by fixing a set of asymptotic string states and integrating over all possible world sheets connecting these states. In the regime where string interactions are weak the result

can be written as perturbative sum on gs that resembles the pertubative expansion on standard quantum2014-09-25 11:38:14 field theories. This is1/1 clearly not a coincidence3pt since (#2) in the limit of small string lengths α0 → 0, the string looks like a particle with string vertices reducing to a three-point interaction vertices between particles. There are however some important differences. First, unlike in standard quantum field theories, there is no a unique point in space-time where the string interaction takes place. Indeed different slicing of the string world sheet leads to different points for the splitting of the string. Second, in the limit α0 → 0 a single string diagram leads to a sum of Feynman diagram contributions associated to different kinematical channels. The various contributions arise from the expansion around different poles of the same string amplitude. Moreover the number of fields exchanged on a given channel as we will see is infinite, All these properties make of the string a better behave theory on the ultraviolet supporting the hope for a quantum consistent and finite theory of gravity. As a quantum field theory in two dimensions describing the dynamics of D bosonic fields, the string action can be naturally extended by including fermions. 1.2. SUPERSTRING ACTION AND SYMMETRIES 11

2 Figure 1.3: String diagrams at order gs .

Fermionic ﬁelds are needed in order to describe matter ﬁelds, such as electrons, quarks, etc. in nature.2014-10-11 01:41:08 The string action1/1 can be extendedG2amplitues in a(#3) symmetric fashion by including a fermionic coordinate ψM for each bosonic coordinate XM . We refer to this a the Superstring Theory, and will be the main subject of our investigations.

1.2 Superstring action and symmetries

The string action can be written as

1 Z √ S = − d2σ h ∂aX ∂ XM + i ψ¯M γa ∇ ψ 4πα0 M a a M b a M ¯ − i χ¯a γ γ ψ (2 ∂bXM − i ψM χb) (1.2.8) where XM (ψM ), M = 0, ..9, are two-dimensional bosonic(fermionic) ﬁelds transforming in the vector representation of the SO(1, 9) Lorentz group. hab and χa are the two-dimensional metric and gravitino respectively a = 1, 2, (σ0, σ1) = (τ, σ) ∈ (−∞, ∞) × [0, π] are the worldsheet coordinates and γa are the two dimensional 12 CHAPTER 1. FREE SUPERSTRINGS gamma matrices1 ! ! ! 0 1 0 1 ψ+ γ0 = i σ2 = , γ1 = σ1 = ψ = (1.2.10) paul −1 0 1 0 ψ− stringaction The two dimensional action (1.2.8) is invariant under the following symmetries:

• Reparametrization: σa → f a(σ, τ) (1.2.11) reps

• Weyl rescaling:

1 1 a a M 2 M − 2 eaˆ → λeaˆ ψα → λ ψα χα → λ χα (1.2.12)

• Supersymmetry:

M M M a M ¯M δX = i ¯ ψ δψ = γ (∂aX − i ψ χa) a a δeaˆ = 2iγ¯ χa δχaα = ∇aα (1.2.13)

• Fermionic shift:

δχa = γaη (1.2.14) ferms

a with eaˆ the two-dimensional vielbein deﬁned in terms of the metric by the relation ab a b a¯¯b b a h = ea¯e¯bδ . The fermionic shift invariance follows from the identity γaγ γ = 0 valid in two dimensions. reps ferms The 3 bosonic and 2 fermionic symmetries (1.2.11-1.2.14) can be used to ﬁx the so called conformal gauge: ! −1 0 hab = ηab = χa = 0 (1.2.15) 0 1

In this gauge the string action the string action can be written as

1 Z √ S = − d2σ h ∂aXM ∂ X + i ψ¯M γa ∂ ψ (1.2.16) Sx0 4πα0 a M a M

1ψM are Majorana spinor in 2d and we have

¯ T 0 ¯ ¯ ψ ≡ ψ γ , ψ1ψ2 = ψ2ψ1 . (1.2.9) eq:2dspinp

In 2d: there is no kinetic term for the gravitino and the covariant derivative ∇a in the kinetic term for ψ can be replaced by ∂a. 1.3. EQUATIONS OF MOTION AND MODE EXPANSION 13

with spacetime Lorentz indices are raised and lowered with ηMN = (− + + ...). Alternatively one can use the lightcone variables ! ± 1 ab 0 −2 σ = τ ± σ ∂± = (∂τ ± ∂σ) h = (1.2.17) 2 −2 0 and write 1 Z S = dσdτ 2 ∂ XM ∂ X + i ψM ∂ ψ + i ψM ∂ ψ (1.2.18) Sx 2πα0 + − M + − +M − + −M

conformalgauge ab The gauge (1.2.15) eliminates the two dimensional fields h , χa but in order to decouple completely the dynamics of these fields one should impose their equations of motion as a set of constraints, the so called “Super Virasoro constraints” on the remaining XM , ΨM fields2 2πα0 δS √ Tab = − (a eb)¯a h δea¯ M i ¯M 1 a M ¯M a = ∂aX ∂bXM + 2 ψ γ(a∂b)ψM − 2 ηab (∂ X ∂aXM + i ψ γ ∂a ψM ) = 0 0 i πα δS 1 b M Ga = − √ = γ γa ψ ∂bXM = 0 (1.2.19) h δχa 2

ab a b a a with h = ea¯ea¯, γ = ea¯γa¯ and bars denotings ﬂat indices. In components one ﬁnds +− −+ T+− = G = G = 0 and

M i M T++ = ∂+X ∂+XM + 2 ψ+ ∂+ψ+M = 0 M i M T−− = ∂−X ∂−XM + 2 ψ− ∂−ψ−M = 0 M G++ = ψ+ ∂+XM = 0 M G−− = ψ− ∂−XM = 0 (1.2.20) virasoro Sx The superstring dynamics is then described by the string action (1.2.18) with virasoro ﬁelds satisfying the super-Virasoro constaints (1.2.20).

1.3 Equations of motion and mode expansion

Sx The equations of motion following from (1.2.18) are

M M M ∂−∂+X = 0 ∂+ψ− = 0 ∂−ψ+ = 0 (1.3.21) 2Notice that the sign of the fermionic piace is diﬀerent from that of GSW but it agrees with BLT. 14 CHAPTER 1. FREE SUPERSTRINGS

The solutions can be written as

M M M + M − X = x0 + XL (σ ) + XR (σ ) M M + ψ+ = ψ+ (σ ) M M − ψ− = ψ− (σ ) (1.3.22) spmsplit

M with x0 a constant parametrising the position of the center of mass of the string. Cancellation of the boundary terms requires:

M σ=π δX ∂σXM σ=0 = 0 M M σ=π (δψ+ ψ+M − δψ− ψ−M ) σ=0 = 0 (1.3.23) bcflat The solutions to (1.3.23) depends on whether we consider open or closed strings.

1.3.1 Closed strings

Closed strings

XM (σ + π, τ) = XM (σ, τ) (1.3.24) ( M M +1 Ramond(R) ψ (σ + π, τ) = αψ (σ, τ) α = α α −1 Neveu Schwarz(NS) with α = ±. One finds four sectors according to the choices (+, −) = (±, ±). As we will see NSNS and RR sectors describe spacetime bosons while RNS and NSR correspond to spacetime fermionic excitations. The mode expansion of the fields on the cylinder worldsheet can be written as r α0 X αM XM (τ + σ) = 2α0 (τ + σ) pM + i n e−2in(τ+σ) L L 2 n n6=0 r α0 X α˜M XM (τ − σ) = 2α0 (τ − σ) pM + i n e−2in(τ−σ) R R 2 n n6=0 √ M 0 X M −2ir(τ+σ) ψ+ (τ + σ) = 2α br e r∈ −a √ Z 0 M 0 X ˜M −2ir(τ−σ) ψ− (τ − σ) = 2α br e (1.3.25) eq:closed-exp00 r∈Z−a˜0 with ( 0 Ramond a , a˜ = (1.3.26) 0 0 1 2 Neveu − Schwarz 1.3. EQUATIONS OF MOTION AND MODE EXPANSION 15 and pM pM = pM = (1.3.27) L R 2 Alternatively, the mode expansion for scalar fields can be written in the more compact form √ M 0 X M −2inσ+ ∂+X (σ+) = 2α αn e n∈ √ Z M 0 X M −2inσ− ∂−X (σ−) = 2α αñ e (1.3.28) n∈Z with √ √ M 0 M M 0 M α0 = 2α pL α˜0 = 2α pR (1.3.29)

We will later see that in the case of bosons living on a compact space pL can be diﬀerent from pR reﬂecting the fact that the string can wrap non-trivially along a one-cycle of the compact manifold.

1.3.2 Open strings bcflat We can satisfy the boundary conditions (1.3.23) in terms of open strings, in which case the terms at σ = π and σ = 0 should vanish separately. For instance at σ = 0, the boundary conditions can be solved by taking for each direction either Neumann or Dirichlet bounday conditions

µ µ µ N: ∂σX |σ=0 = 0 ψ+(0, τ) = ψ−(0, τ) (1.3.30) i i i D: ∂τ X |σ=0 = 0 ψ+(0, τ) = −ψ−(0, τ) (1.3.31)

For this choice of boundary conditions the open string end can move only along i i a (p + 1)-dimensional hyperplane located at x0 = X |σ=0 . Similarly at σ = π one can take either Neuman or Dirichlet boundary conditions with an extra sign in the fermionic boundary conditions depending on the choice of Ramond or Neveu- Schwarz sectors. bcflat In general the boundary conditions (1.3.23) can be solved by requiring

M M N M M N ∂−XR − (R0)N ∂+XL σ=0 = 0 , ψ− − (R0)N ψ+ σ=0 = 0 (1.3.32) M M N M M N ∂−XR − (Rπ)N ∂+XL σ=π = 0 , ψ− − (Rπ)N ψ+ σ=π = 0 (1.3.33), 16 CHAPTER 1. FREE SUPERSTRINGS where = 1 for the Ramond and = −1 for Neveu-Schwarz sectors. These equations can be solved by taking

M M N M M N XR (τ) = (R0)N XL (τ) ψ− (τ) = (R0)N ψ+ (τ) M M M M N XL (σ + 2π) = (R)N XL(σ) ψ+ (σ + 2π) = (R)N ψ+ (σ) (1.3.34) eq:ReflM02 with

−1 R = Rπ · R0 (1.3.35) a matrix parametrising the relative orientation of the Dp-branes at the two ends of bcflat the open string. By using these identiﬁcations in (1.3.23) we see that the boundary 3 terms vanish if the reﬂection matrices Ri=0,π satisfy at the same time :

t t RiηRi = η Riη = ηRi (1.3.36) rrone

This implies that the eigenvalues of both Ri’s must be ±1: the eigenvectors with eigenvalue 1 or −1 deﬁne directions where the string coordinates X and ψ have Neumann or Dirichlet boundary conditions respectively. We notice that it is always possible to choose a coordinate system where the Neumann (N) and Dirichlet (D) directions for one of the reﬂection matrices, for instance R0, are aligned with the Cartesian axis ! 1p+1 0 R0 = (1.3.37) eq:Rp 0 −1D−p−1

In this coordinates the matrix R takes a general form. There are two distintic case depending on whether the branes at the two ends of the open string are parallel or not, i.e. R = 1 or R 6= 1. eq:Rp In the case of parallel branes R0 = Rπ and R = 1. Taking R0 given by (1.3.37) one ﬁnds

µ µ µ µ µ µ X = x0 + XL(σ+) + XL(σ−) ψ− = ψ+ i i i i i X = x0 + XL(σ+) − XL(σ−) ψ− = −ψ+ (1.3.38)

3We will see in the next chapter that more general reﬂection matrices are allowed when considering string theory in non-trivial backgrounds. 1.4. QUANTIZATION OF THE SUPERSTRING 17 with r α0 X αµ Xµ(σ , τ) = α0pµσ + i n e−inσ+ L + + 2 n n6=0 r 0 i σ+ α X α Xi (σ , τ) = (xi − xi ) + n e−inσ+ L + 2π π 0 2 n n6=0 √ M 0 X M −2ir σ+ ψ+ (σ+) = 2α br e (1.3.39) r∈Z−a0 In the case where the branes at the two ends of the open strings are not parallel

R 6= 1, but R and R0 commutes they can be simultaneously diagonalized. We write

R = diag e2πiθ1 , e−2πiθ1 , e2πiθ2 , e−2πiθ2 ,... (1.3.40) rrreflection with πθI>1 the relative angle between the D-branes at the two ends and θ1 = −iβ characterising the relative velocity v = tanh πβ. The ﬁeld XI , I = 1,...D/2 are now periodic up to a e2πiθI -phase parametrising the angle between the two branes in the Ith complex plane. The mode expansion can be written again as

I I I I I I X = x0 + XL(σ+) + R0 XL(σ−) ψ− = R0ψ+ (1.3.41) but now r √ α0 X αI XI (σ ) = 2α0 αI σ + i n e−inσ+ L + 0 + 2 n n∈Z−θI n6=0 √ I 0 X I −irσ+ ψ+(σ+) = 2α br e (1.3.42) r∈Z−θI −a0 I Finally the zero mode α0 is present whenever the two branes are parallel in the corresponding plane  q α0 pI NN  2  I I I xπ−x α0 = √ 0 DD (1.3.43)  π 2α0  0 ND,DN

1.4 Quantization of the superstring

1.4.1 Canonical quantization

The Hamiltonian of the superstring in the conformal gauge can be written as Z π i H = dσ(q ˙ Πi − L) (1.4.44) ham0 0 18 CHAPTER 1. FREE SUPERSTRINGS

i M M with q = (X , ψ± ) the superstring coordinates, L the Lagrangian

1 0 L = (X˙ M X˙ −X M X0 +iψM ψ˙ +iψM ψ˙ +iψM ψ0 −iψM ψ0 ) (1.4.45) 4πα0 M M + M+ − M− + M+ − M− dot and prime denoting the derivative with respect to tau and sigma respectively ∂L and Πi = ∂q˙i the associated momenta 1 Π = X˙ M 2 π α0 M i Π = − ψ (1.4.46) αM 4πα0 αM cmom ham0 with α = ± labelling the two spinor components. Plugging (1.4.46) into (1.4.44) one ﬁnds the string Hamiltonian Z π 1 ˙ M ˙ 0M 0 M 0 M 0 H = 0 dσ (X XM + X XM + iψ+ ψM+ − iψ− ψM−) (1.4.47) ham 4πα 0 The non-trivial Poisson brackets are4

M 0 M 0 {X (σ, τ), ΠN (σ , τ)}P.B. = δN δ(σ − σ ) M 0 M 0 {ψα (σ, τ), ΠβN (σ , τ)}P.B. = −δαβ δN δ(σ − σ ) (1.4.49)

We notice that the Poisson brackets of i φ = Π + ψ (1.4.50) dirac0 αM αM 4π αM cmom with itself is not zero so one cannot simply set φαM to zero as required by (1.4.46).

One says that φαM = 0 is a second class constraints. In presence of second class constraints φi the Poisson brackets should be modiﬁed in such a way that the brackets of two constraints vanish. This is achieved by replacing Poisson brackets with Dirac Brackets deﬁned as

−1 {A, B}D.B. = {A, B}P.B. − {A, φi}P.B. Dij {φj,B}P.B. (1.4.51) dirac1

−1 with D the inverse of the matrix Dij = {φi, φj}P.B.. In our case i D = {φ , φ } = − g δ δ(σ − σ0) (1.4.52) αM,βN αM βN P.B. 2π MN αβ 4Poisson brackets are deﬁned as

F ∂A ∂B ∂A ∂B {A, B} = (−) i i − i (1.4.48) ∂q ∂Π ∂Πi ∂q with F denoting the spin-statistic of the ﬁeld. 1.4. QUANTIZATION OF THE SUPERSTRING 19 and5 (D−1)αM,βN = 2πi gMN δαβ δ(σ − σ0) (1.4.53) dirac2 cmom dirac2 Using (1.4.46-1.4.53) one ﬁnds the Dirac brackets

M N 0 ˙ M ˙ N 0 {X (σ, τ),X (σ , τ)}D.B. = {X (σ, τ), X (σ , τ)}D.B. = 0 M ˙ N 0 0 MN 0 {X (σ, τ), X (σ , τ)}D.B. = 2π α g δ(σ − σ) M N 0 0 MN 0 {ψα (σ, τ), ψβ (σ , τ)}D.B. = −2π i α δαβ g δ(σ − σ) (1.4.54)

M M To quantise the two-dimensional theory, we promote fields X , ψα to operators and replace Dirac brackets by commutators or anticommutators according to the spin statistics of the given field 1 { , } → { , ] (1.4.55) D.B. i One finds the non-trivial (anti)commutation relations

[XM (σ, τ), X˙ N (σ0, τ)] = 2πi α0 gMN δ(σ0 − σ) M N 0 0 MN 0 {ψα (σ, τ), ψβ (σ , τ)} = 2π α δαβ g δ(σ − σ) (1.4.56) cannonical

Using the Fourier transform of the Dirac delta function

∞ 1 X δ(σ) = e2 i n σ (1.4.57) π n=−∞ eq:closed-exp cannonical and the mode expansion (1.5.101), the (anti)commutation relations (1.4.56) translate into the following non-trivial (anti)commutation relations for the string modes

M N M N MN M N MN [αm , αn ] = [˜αm , α˜n ] = m η δm+n,0 [x , p ] = i η M N ˜M ˜N MN {br , bs } = {br , bs } = η δr+s,0 (1.4.58) comanti

The quantum theory of closed strings can then be thought as two inﬁnite sets of quantum harmonic super-oscillators, with creation operators and annihilation operators by the positive and negative modes of the strings. We deﬁne the Fock vacuum as the product |0i ⊗ |f0i of the left and right moving vacua with

M M M ˜M αn |0i = br |0i =α ˜n |f0i = br |f0i = 0 for n, r > 0 (1.4.59) annih

5 R Here we use the identity dxδ(x0 − x)δ(x − x1) = δ(x0 − x1). 20 CHAPTER 1. FREE SUPERSTRINGS

In addition the vacuum is annihilated by half of the string zero modes. For instance one can take pM as the annihilation operator and impose

pM |0i = pM |0˜i = 0 (1.4.60) annih2

In the case of the fermionic string in the Ramond sector, we have extra fermionic N ˜N zero modes b0 or b0 satisfying the Cliﬀord algebra

M N ˜M ˜N MN {b0 , b0 } = {b0 , b0 } = η (1.4.61)

Out of these operators one can built extra creation and annihilation operators 1 1 b± = √ (b2I−1 ± i b2I ) ˜b± = √ (˜b2I−1 ± i ˜b2I ) (1.4.62) I 2 0 0I I 2 0 0I with the vacuum deﬁned as − ˜− bI |0i = bI |f0i = 0 (1.4.63) annih3 The Fock space is generated by acting on the vacuum |0i ⊗ |f0i with creation operators.

1.4.2 The light cone gauge

Closed strings

The Fock space we define in the last section always contains states of negative 0 2 00 norm (ghosts) since |α−n|0i| ∼ η < 0. This is a consequence of a residual reparametrization gauge symmetry. In this section, we show how a ghost free Fock space can be defined by choosing a gauge fixing this ambiguity. The choice of conformal gauge fixes the two-dimensional metric but it leaves the residual invariance

0 M 0 M ± M σ± = f±(σ±) ψ± = ψ± + 2 (σ±) ∂±X (1.4.64)

This residual gauge invariance can be ﬁxed by choosing

+ 0 + + X = 2α p τ ψα = 0 (1.4.65) lightcone with A± denoting the light cone components 1 A± = √ (A0 ± AD−1) (1.4.66) 2 1.4. QUANTIZATION OF THE SUPERSTRING 21 of a vector AM . We write restrict here to ﬂat space and write the scalar product as

M i i + − − + A BM = A B − A B − A B (1.4.67) with indices i running over the transverse coordinates. In terms of these variables the super Virasoro constraints T±± = G± = 0 reduces to 1 ∂ X− = ∂ Xi ∂ Xi + i ψi ∂ ψi ± 2α0p+ ± ± 2 ± ± ± 1 ψ− = ψi ∂ Xi (1.4.68) ± α0p+ ± ± These equations determine the components X−, ψ− in terms of the physical degrees of freedom Xi, ψi, ψ˜i. More precisely, equation the Fourier mode coefficients on the two sides of the equations one finds ! − 1 X i i X α = √ α α + r bn−r br n 2α0p+ n−m m m∈Z r∈Z−a0 r 2 1 X b− = bi αi (1.4.69) r α0 p+ r−m m m∈Z − ˜− with similar formulas forα ñ and br replacing modes by their titled counterparts. alphablight Equations (1.4.80) determine the minus components of the bosonic and fermionic M p 0 M oscillator modes while for n = 0, using α0 = α /2p one finds ! 2 X X −p2 = αi αi + r bi bi α0 −m m −r r m6=0 r∈Z−a0 ! 4 X X = αi αi + r bi bi − a (1.4.70) α0 −m m −r r m>0 r>0 with a the normal ordering constant ! 1 X X a = − [ αi , αi ] + r { bi , bi } 2 m −m r −r m=1 r=a0 " ∞ ∞ # D − 2 X X = − m + (m − a ) = D−2 a (1 − a ) (1.4.71) 2 0 4 0 0 m=1 m=1 computed using the regularised Riemann zeta function sum

∞ X 1 1 (m − a0) = ζ(−1, −1a0) = − 12 + 2 a0(1 − a0) (1.4.72) zeropoint m=1 22 CHAPTER 1. FREE SUPERSTRINGS

p2alpha The result (1.4.70) (and its right moving counterpart) can then be written as

4 4 M 2 = −p2 = (N − a) = (N˜ − a˜) (1.4.73) masshell α0 α0 where we introduce the occupation number operators

∞ X X N = α−m · αm + r b−r · br m=1 r>0 ∞ ˜ X X ˜ ˜ N = α˜−m · α˜m + r b−r · br (1.4.74) m=1 r>0 and the zero point energies

D−2 a = 4 a0(1 − a0) D−2 a˜ = 4 a˜0(1 − a˜0) (1.4.75) lightcone Sx Plugging (1.4.78) into (1.2.18) one ﬁnds the light cone string action

1 Z S = dσdτ 2 ∂ Xi ∂ Xi + i ψi ∂ ψi + i ψi ∂ ψi (1.4.76) Sx2 2πα0 + − + − + − + − where we discard a ﬁeld independent term. Similarly for the Hamiltonian one ﬁnds ham again (1.4.77) with the indices M replaced by i running over the transverse coordinates plus the constant term −2α0p+p−. We notice that the Virasoro constraint masshell (1.4.73) implies that the Hamiltonian evaluated at the solutions (X, ψα) of the equations of motion always vanishes

H = α0 p2 + 4(N − a) + 4(N˜ − a˜) = 0 (1.4.77) ham on−shell

Open strings

For the open strings one ﬁxes again the residual gauge invariance by choosing

+ 0 + + X = 2α p τ ψα = 0 (1.4.78) lightcone and determine the minus components from the super Virasoro constraints T++ =

G++ = 0 1 ∂ X− = ∂ Xi ∂ Xi + i ψi ∂ ψi + 2α0p+ + + 2 + + + 1 ψ− = ψi ∂ Xi (1.4.79) + α0p+ + + 1.5. THE CONFORMAL FIELD THEORY DESCRIPTION 23 that now lead to ! − 1 1 X i i X α = √ α α + r bn−r br n 2 2α0p+ n−m m m∈Z r∈Z−a0 r 2 1 X b− = 1 bi αi (1.4.80) r 2 α0 p+ r−m m m∈Z and the mass-shell conditions 1 M 2 = −p2 = (N − a) (1.4.81) masshellopen α0

1.5 The conformal ﬁeld theory description

Sx The string action (1.2.18) in the conformal gauge is still invariant under chiral reparametrizations

σ+ → f+(σ+) σ− → f−(σ−) (1.5.82) chiralrepar of the string worldsheet. It is convenient to perfomed a Wick rotation on the worldsheet coordinates and introduced the complex coordinates

z = e2(τE +iσ) z¯ = e2(τE −iσ) (1.5.83) in terms of the Euclidean time τE = iτ ∈ (−∞, ∞). The map (τE, σ) → (z, z¯) maps the world sheet cylinder into the complex plane. On the other hand, reparametriza- chiralrepar tions (1.5.82) are mapped to conformal transformations

z → z0 = f(z)z ¯ → z¯0 = f¯(¯z) (1.5.84) conformal of the complex plane. Indeed, this map preserves the angles between any two vectors in the plane and rescales the distances between any two points. The set of such coordinate transformations generate the conformal group in two dimensions. In distinction from higher dimensions this group is infinite dimensional. A conformal field theory (CFT) is a field theory enjoying conformal invariance. The basic objects of a CFT are the primary fields. Primary fields transform as tensors under conformal conformal transformations (1.5.84)

0 ∂z0 −h ∂z¯0 −h φ(z, z¯) → φ0(z0, z¯0) = φ(z, z¯) (1.5.85) primary0 ∂z ∂z¯ 24 CHAPTER 1. FREE SUPERSTRINGS

primary We say that a primary field transforming as (1.5.87) is a primary field of dimension (h, h0). In addition a conformal field theory is characterised by a traceless stress energy tensor Tzz¯ = 0 with diagonal components satisfying

¯ T (z) = Tzz(z) ∂T = 0

Te(¯z) = Tz¯z¯(¯z) ∂Te = 0 (1.5.86) conformal Since conformal transformations (1.5.84) do not mix analytic and anti-analytic components we can focus on the chiral part of the conformal ﬁeld theory. We say that a chiral primary ﬁeld φ(z) has scaling dimension h if it transforms as

∂z0 −h φ(z) → φ0(z0) = φ(z) (1.5.87) ∂z

A chiral primary ﬁeld φ(z) of dimension h can be expanded as

X −n−h φ(z) = φn z (1.5.88) n∈Z Their components can be extracted from the integral I dz n+h−1 φn = φ(z) z (1.5.89) γ0 2πi around a closed contour γ0 around z = 0. In particular the field T (z) is a quasi primary operator of scaling dimension 2. In a quantum theory the generator of dilations will take the role of the Hamil- tonian and time ordering is replaced by radial ordering. The product of two fields is defined then as the radial ordering ( φ1(z) φ2(w) |z| < |w| φ1(z) φ2(w) = (1.5.90) φ1(w) φ2(z) |w| < |z|

We notice that this product jumps when two non-commuting ﬁelds φ1(z) and φ2(w) cross each other, so the product is singular when they get close. The singularity structure of the product is captured by the so called Operator Product Expansion (OPE)

φ1(z) φ2(w) ≈ [φ1(z) , φ2(w)] + ... (1.5.91) with ... denoting regular terms in the limit where z → w. The OPE codiﬁes the non-trivial commutation relations between the various ﬁelds in the CFT. The same 1.5. THE CONFORMAL FIELD THEORY DESCRIPTION 25 holds for fermionic operators with commutator replaced by anti-commutators. If

{Oi} is a complete basis of chiral operators with deﬁnite scaling dimension hi, then the OPE of two operators can be in general written in the form

k X Cij Oi(z) Oj(w) = Ok (1.5.92) (z − w)hi+hj −hk k

A distintic role between the chiral primary fields is played by chiral primary fields of dimension h = −1. A chiral primary field j(z) of dimension h = −1 defines a conserved current (∂j¯ = 0) and can be integrated into a conserved charge

I dz Q = j(z) (1.5.93) γ0 2πi with γ0 a circular contour of radius τ around the origin. Indeed, integrals over σ conformal are mapped under conformal transformations (1.5.84) into circles around the origin. The conservation of Q follows from the independence of the complex integral on the contour. The variation of a given ﬁeld under the action of a symmetry associated to the conserved current j(z) is deﬁned as

I dz δφ(w) = [Q, φ(w)] = j(z) φ(w) (1.5.94) comope γw 2πi with γw a contour around w or equivalently the diﬀerence of two contours around the origin with |z| > |w| and |z| < |w|. In particular, we can use T (z) to construct inﬁnite number of conserved currents jξ(z) = ξ(z)T (z) with ξ(z) an arbitrary holomorphic vector of scaling dimension 1. They generate the conformal transformations

I dw δz = ξ(w) T (w) z = ξ(z) T0 (1.5.95) γz 2πi primary On the other hand, the transformation properties (1.5.87) of a chiral primary ﬁeld φ of scaling dimension h requires the OPE with the stress energy tensor T (z)

h 1 T (z)φ(w) = φ(w) + ∂ φ(w) + ... (1.5.96) tphiprimary (z − w)2 (z − w) w

In presence of extra conserved currents ja(z), the conformal algebra can be sum- 26 CHAPTER 1. FREE SUPERSTRINGS marised in the OPE

c/2 2 1 T (z)T (w) = + T (w) + ∂ T (w) + ... (z − w)4 (z − w)2 (z − w) w qa 1 1 T (z)ja(w) = + ja(w) + ∂ ja(w) + ... (z − w)3 (z − w)2 (z − w) w gab f ab ja(z)jb(w) = + c jc(w) + ... (1.5.97) (z − w)2 (z − w) with dots denoting regular terms in the limit z → w. The constants c and qa are the central and the anomalous charges of the CFT. They signal a failure of the ﬁelds T (z) and ja(z) to be primary ﬁelds.

1.5.1 Closed strings

Closed strings are described by a conformal ﬁeld theory of free bosons and fermions deﬁned on the complex plane with coordinates

z = e2(τE +iσ) = e2iσ+ z¯ = e2(τE −iσ) = e2iσ− (1.5.98)

We introduce the two dimensional ﬁelds

i i i i X (z, z¯) = x + XL(z) + XR(¯z) 1 i − 2 i ψ (z) = (4z) ψ+(z) 1 ˜i − 2 i ψ (¯z) = (4¯z) ψ−(¯z) (1.5.99) closez

We notice that XM are not primary ﬁelds, but its derivatives

1 1 ∂Xi(z) = ∂ Xi(σ ) ∂X¯ i(¯z) = ∂ Xi(σ ) (1.5.100) 2iz + + 2i¯z − − are primary ﬁelds of dimensions (1, 0) and (0, 1) respectively. On the other hand ψi and ψ˜i are primary ﬁelds of scaling dimensions (1/2, 0) and (0, 1/2) respectively. For simplicity we choose in this section length units such that α0 = 2 6 In these

6The dependence in α0 can be restored by rescaling dimension length quantities by pα0/2 , i.e. Xi → pα0/2Xi, ψi → pα0/2ψi and ψ˜i → pα0/2ψ˜i. 1.5. THE CONFORMAL FIELD THEORY DESCRIPTION 27

eq:closed-exp00 variables the mode expansions (1.3.25) read

X αM Xi (z) = −2i log z pi + i n z−n L L n n6=0 X α˜i Xi (¯z) = −2i logz ¯ pi + i n z¯−n R R n n6=0 1 X −r− i i 2 ψ (z) = br z r∈Z−a0 1 X −r− ˜i ˜i 2 ψ (¯z) = br z¯ (1.5.101) eq:closed-exp r∈Z−a˜0 Sx2 The string action (3.1.1) becomes7

1 Z h i S = − dz dz¯ ∂Xi ∂X¯ i + ψi∂¯ ψi + ψ˜i∂ ψ˜i (1.5.102) 4π

1.5.2 Open strings

For open strings introduce the complex coordinate

z = eτE +iσ = ei(τ+σ) z¯ = eτE −iσ = ei(τ−σ) (1.5.103)

This transformation maps the inﬁnite strip representing the open string worldsheet into the upper-half complex plane, where the σ = 0 and σ = π boundaries become the positive and the negative real axis respectively. The theory of open strings can then be viewed as a two-dimensional CFT with boundaries.

It is convenient to think about the about the open string ﬁelds X(z, z¯), ψ+(z),

ψ−(¯z) as fields defined on the whole complex plane with left and right moving modes related in such a way that boundary conditions are satisfied along the real closez line. In other words, the worldsheet fields are defined again by (1.5.99) but with left and right moving modes identified via the reflection matrix R0 and monodromies determined by the matrix R codifying the relative boundary conditions between the eq:ReflM02 two ends of the open string. More precisely we take (1.3.34)

˜ XR(z) = R0 XL(z) , ψ(z) = R0ψ(z) 2πi 2πi XL(e z) = RXL(z) ψ(e z) = −R ψ(z) (1.5.104) eq:monodX

7Here we use dσ ∧ dτ = dz ∧ dz/¯ (8zz¯). 28 CHAPTER 1. FREE SUPERSTRINGS

eq:monodX Finally we notice that using the boundary conditions (1.5.104) and the relations rrone (1.3.36) one ﬁnds that in presence of a D-brane boundary left and right moving Virasoro and supersymmetry generators are identiﬁed

T++ = T−− G+ = G− (1.5.105) i.e. D-brane preserve half of the conformal and supersymmetries of the closed string theory.

1.5.3 The Superconformal algebra cannonical From the commutation relations (1.4.56) one derive the Operator Product Expan- sion (OPE) Xi(z, z¯)Xj(w, w¯) = − δij log |z − w|2 + ... δij ψi(z) ψj(w) = + ... (1.5.106) opex z − w with dots denoting regular terms. In terms of these fields one can define the superconformal generators 1 i i i i T (z) = 2 ∂X ∂X + ψ ∂ψ i i i G(z) = 2 ψ ∂X (1.5.107) tgdef opex Using (1.5.106) one finds the OPE expansion

c 2 T (w) ∂ T (w) T (z)T (w) = 2 + + w + ... (z − w)4 (z − w)2 (z − w) 3 G(w) ∂ G(w) T (z)G(w) = 2 + w + ... (z − w)2 (z − w) c 1 T (w) G(z)G(w) = 6 + 2 + ... (1.5.108) scftope (z − w)3 (z − w) with 3 c = 2 (D − 2) (1.5.109) the central charge of the SCFT. Chiral primary fields φh of the CFT are constructed opex out of products of ∂X, ψ, eikXL and their ∂-derivatives. Indeed, using (1.5.106) and tgdef (1.5.107) one can compute the OPE of T (z) with the basic primary fields ∂X, ψ, tphiprimary eikXL finding the expected OPE (1.5.96) with scaling dimensions 1 h(∂X) = 1 h(ψ) = h(eikXL ) = k2 (1.5.110) 2 1.5. THE CONFORMAL FIELD THEORY DESCRIPTION 29

Conserved charges can be defined as contour integrals of the holomorphic currents I dz X X L − a δ = zn+1 T (z) = 1 αi αi + 1 r bi bi n n,0 2πi 2 n−m m 2 n−r r m∈Z r∈Z−a0 I dz r+ 1 1 X i i G = z 2 G(z) = α b (1.5.111) r 2πi 2 −m r+m m∈Z azero0 with a the normal ordering constant defined in (1.4.71) in the NS sector D − 2 c a = = (1.5.112) 16 24 lgr0 We notice that the ordering of operators in (1.5.111) is relevant only for L0. The charges Ln, Gr are conserved since the contour integrals do not depend on the comanti radius of the contour. Using the(anti)commutation relations (1.4.58) one finds the super-Virasoro algebra8 c [L ,L ] = (m − n) L + (m3 − m) δ m n m+n 12 m+n m [L ,G ] = − r G m r 2 m+r c {G ,G } = 1 L + r2 − 1 δ (1.5.114) supervir r s 2 r+s 12 4 r+s Alternatively, the same results can be derived from the OPE expansion of the comope currents, noticing that according to (1.5.94) the commutator of two charges can be written as the double contour integral I dz I dw I dw I dz j1(z), j2(w) = j1(z) j2(w) (1.5.115) 2πi 2πi C0 2πi Cw 2πi For fermionic operators the same identity holds with commutators replaced by anticommutators. For the super Virasoro charges one finds I I c dw dz m+1 n+1 2 2 T (w) ∂wT (w) [Lm,Ln] = z w 4 + 2 + C0 2πi Cw 2πi (z − w) (z − w) (z − w) I I 3 dw dz 1 G(w) ∂wG(w) m+1 r+ 2 2 [Lm,Gr] = z w 2 + C0 2πi Cw 2πi (z − w) (z − w) c 1 I dw I dz 1 1 T (w) r+ 2 s+ 2 6 2 {Gr,Gs} = z w 3 + (1.5.116) C0 2πi Cw 2πi (z − w) (z − w) 8To evaluate the commutators one uses the identities

[A, B C] = [A, B] C + B [A, C] = {A, B} C − B {A, C} (1.5.113)

i i i n i and the commutator relations [αm,Ln] = m αm+n,[br,Ln] = (r + 2 ) br+n. 30 CHAPTER 1. FREE SUPERSTRINGS

algebraope Evaluating the residues in (1.5.116) one reproduces the super-Virasoro algebraa supervir (1.5.114).

1.6 The bosonic string

From the CFT point of view, a string theory with excitations only of bosonic type can also be considered. Even if this theory is to interesting by itself from a spacetime point of view since it contains states of negative mass in its spectrum, a chiral bosonic string can be combined with a superstring to build hybrid and fully consistent theories knows as heterotic strings. In addition, being simpler than superstrings, bosonic strings can be used to illustrate some of hte intricate features of superstrings in a simpler context. In this section, we summarise the main features of bosonic string theories. All formulas in this section are found starting from their analog for the superstrings by simply dropping out the contributions of fermions. The action become 1 Z √ 1 Z S = − d2σ h ∂aX ∂ XM = d2σ ∂ XM ∂ X (1.6.117) 4πα0 M a πα0 + − M The Virasoro generators are

M M T++ = ∂+X ∂+XM T−− = ∂−X ∂−XM (1.6.118) or in complex variables

M ¯ ¯ M ¯ T = ∂X ∂XM T = ∂X ∂XM (1.6.119)

String coordinates split into left and right moving modes

X(z, z¯) = XL(z) + XR(z) (1.6.120) with

r 0 r 0 α α X αn X (z) = x − i log z α + i z−n L L 2 0 2 n n6=0 r 0 r 0 α α X α˜n X (z) = x − i logz ¯α˜ + i z¯−n (1.6.121) R R 2 0 2 n n6=0

M M After quantisation, the modes αn , α˜n are promoted to operators satisfying the commutation relations

M N M N MN M N MN [αm , αn ] = [˜αm , α˜n ] = m η δm+n,0 [q , p ] = i η (1.6.122) 1.7. THE CRITICAL DIMENSION 31 acting on a vacuum deﬁned by

M M ˜ αn |0i =α ˜n |0i = 0 for n > 0 (1.6.123) In the light cone gauge X+ = 2α0 p+ τ (1.6.124)

i i states are created by acting on the Fock vacuum with transverse oscillators α−n, α˜−n. The Virasoro generators

1 X i i Ln − aδn,0 = 2 αn−m αm (1.6.125) m∈Z with ∞ 1 X i i D−2 X D−2 a = − 2 [ αm, α−m ] = − 2 m = − 24 (1.6.126) m=1 m=1 The Virasoro algebra becomes c [L ,L ] = (m − n) L + m(m2 − 1) δ (1.6.127) m n m+n 12 m+n with c = D − 2.

1.7 The critical dimension

There are several ways to understand the emergence of a critical dimension in string theory at a quantum level. For instance, quantum corrections to the worldsheet action of the string reveals that conformal symmetry is generically broken at one- loop. Unlike in the case of global symmetries, a breaking of a local symmetry renders the theory inconsistent at the quantum level. The cancellation of this anomaly requires that string theory is formulated at a critical dimension D = 10 for the superstring or D = 26 for the bosonic string. The explicit computation of this anomaly in the conformal algebra will be presented in section 3, where a covariant quantisation of the string will be presented. The same conclusions will be reached in Chapter 4 where the vanishing of the beta functions governing the running of coupling constants in the two-dimensional worldsheet theory will be shown to require a formulation at the critical dimension. In this section we show that the existence of a critical dimension can be alternatively derived by requiring the existence of massless states or gauge symmetries in the string theory or by requiring Lorentz invariance of the quantised theory in the light cone gauge. 32 CHAPTER 1. FREE SUPERSTRINGS

1.7.1 Existence of Massless states masshell masshellopen According to the mass shell conditions (1.4.73) and (1.4.81), a massless states in string theory appears whenever the conditions

N − a = N˜ − a˜ = 0 (1.7.128) are met. The domains of N, N˜ and the zero point energies a, a˜ depends on the dimension of the string and the boundary conditions of the fermionic ﬁelds. From azero azerobos (1.4.75) and (1.6.126) one ﬁnds

Ramond a = 0 N = 0, 1,... D−2 1 Neveu − Schwarz a = 16 N = 0, 2 ,... (1.7.129) D−2 Bosonic string a = 24 N = 0, 1,... The existence of massless states requires then

Superstring D = 10 Bosonic string D = 26 (1.7.130)

1.7.2 Lorentz symmetry

Besides the local superconformal symmetry, the string action is also invariant under global transformations such as SO(1,D − 1) Lorentz rotations. In this section we show that in the light cone gauge this symmetry is broken by a quantum anomaly unless the string lives at a critical dimension D = 10 for the superstring and D = 26 for a bosonic string. The chiral generators of Lorentz group can be written as 4i I dz J MN = X[M ∂XN] + ψM ψN (1.7.131) α0 2πi The action in the cone gauge is manifestly invariant under the SO(D −2)-subgroup of this symmetry that rotates the transverse degrees of freedom. The invariance with respect to the remaining generators of the Lorentz group is less obvious. For lcdx instance, let us consider the Lorentz J i−. Using (1.4.79) one ﬁnds I dz 4i I dz J i− = ji(z) = X[i∂X−] + 1 ψi ψ− (1.7.132) 2πi α0 2πi 2 with 4z ji(z) = −Xi T + ψi G (z) (1.7.133) α02p+ 1.7. THE CRITICAL DIMENSION 33

The Lorentz commutation relation [J i−,J j−] = 0 requires then

I dw I dz ji(z) jj(w) = 0 (1.7.134) C0 2πi Cw 2πi One can easily see that this integral fails to be zero in an arbitrary dimension. Indeed by looking to the OPE of the integrand and focusing on terms quadratic on X’s one ﬁnds

1 D−2 Xi(z)Xj(w) ∂Xi(z)Xj(w) ∂Xi(w)∂Xj(z) XiT (z) XjT (w) = 2 − 2 + + ... α02 B B (z − w)4 (z − w)3 (z − w)2 1 (D − 2)Xi(z)Xj(w) XiT (z) XjT (w) = + ... α02 F F 2(z − w)4 1 ∂Xj(z)∂Xi(w) ψi G(z)ψj G(w) = 2 + ... α02 (z − w)2 2Xi(z)∂Xj(w) 2 ψi G(z) XiT (w) = + ... (1.7.135) B (z − w)3

1 where TB = ∂X∂X, TF = 2 ψ∂ψ are the bosonic and fermionic components of the stress energy tensor, G = ψ∂X, and dots denote higher order terms in X’s or ψ. tanom Integrating (1.7.135) around z ≈ w and evaluating the residues one ﬁnds (up to total derivatives) ∂3Xi Xj times

D − 2 D − 2 D − 10 − 1 − 1 + + (2) − (1) = (1.7.136) crit 12 12 12 tanom where the four contributions come from the four lines in (1.7.135). The cancelation is achieved for D=10. For a theory with only bosonic modes, the last three contributions should be discarded leading to the critical dimension D = 26. 34 CHAPTER 1. FREE SUPERSTRINGS Chapter 2

Spectra of String States

In this chapter we introduce the five consistent string theories in ten dimensions. We discuss the spectra of string states in flat space time and compute the string partition functions. A critical string theory can be obtained by combining two chiral critical strings. Combining two bosonic theories one finds a bosonic closed string theory living on D = 26 dimensions. Strings made of two fermionic strings are known as type II strings and live on D = 10 dimensions. Finally one can make an hybrid string by combining a bosonic string on the left with a fermionic string on the right. Theories obtained in this way are known as heterotic strings. The coordinate of the heterotic string is parametrized by the bosonic coordinate of the right moving string and ten of the 26 left moving bosons. The remaining 16 left moving bosons can be thought of as compactified on a lattice Λ16. Alternatively the 16 compact bosons can be replaced by 32 fermions contributing the same to the central charge of the theory. Consistency at the quantum level impose sever restrictions on the structure of fermionic and bosonic strings. Travelling along a loop a closed string spans a worldsheet torus. The vacuum amplitude should be invariant under global reparametrization of this torus. The group of global reparametrization is generated by the element that exchanges the two cycles of the torus σ ↔ τ and the unit shift along one of the cycles. Invariance under the exchange of the role of σ and τ requires that the vacuum amplitude of a fermionic string involving both periodic (Ramond) and anti-periodic (Neveu-Schwarz) fermions along σ should involve also periodic and anti-periodic fermions along the τ-direction. In a hamiltonian formulation of the amplitude, the partition function of the two-dimensional CFT is computed by the

35 36 CHAPTER 2. SPECTRA OF STRING STATES loop of fermion with anti-periodic boundary conditions along τ. On the other hand, a loop of a periodic fermion can be associated to the partition function with the

FL FR extra insertion of a (−) or (−) operator, with FL/FR the left/right moving fermionic numbers. The sums of the two amplitudes project the spectrum of string states onto a sector with a definite fermionic number parity. This projection goes under the name of GSO projection. As we will see in the case of type II and heterotic strings, this projection has also the effect of projecting out all states of negative mass, the tachyons, leaving a fully consistent and supersymmetric theory. We remark that such a projection is not defined for the bosonic theory, where the presence of the tachyon signal for an instability of the string vacuum making the study of a theory of solely bosonic strings less interesting. For this reason we discard the study of the bosonic string theory from now on. The choice of GSO projection leaves a freedom on the parity to take for the Ramond vacuum of the theory. In the case of heterotic strings this makes no difference but for type II strings one finds two different theories type IIA and IIB, depending on whether you assign a different or the same signs respectively to the left and right moving Ramond vacua. On the other hand, for heterotic strings there is a freedom on the choice of the 16-dimensional lattice where the extra left-moving bosons live. Consistency at the quantum level leaves only two choices for this lattice corresponding to the only groups of rank 16 with self-dual root lattices: E8 × E8 and SO(32). The same restriction follows by requiring the cancelation of anomalies sanomalies in ten dimensions (see Appendix B for details). Finally one can consider a theory of unoriented and open strings. Again the consistency of the theory at the one-loop level restricts the choices to a single theory in ten-dimensions with gauge group SO(32). This theory is known as type I. We are left with 5 inequivalent consistent string theories in ten dimensions

• Type IIA theory: Made of two chiral superstring theories with left and right moving fermion excitations of opposite spacetime chirality. The spacetime theory is N = 2 supersymmetric in ten dimensions.

• Type IIB theory: Made of two chiral superstring theories with left and right moving fermion excitations of equal spacetime chirality. The spacetime theory is N = 2 supersymmetric in ten dimensions. 2.1. STRING STATES 37

• SO(32) heterotic string : Made of a left-moving bosonic and a right moving chiral superstring. The theory lives in ten-dimensions with the 16-internal bosons living on the root lattice of SO(32) . The spacetime theory is N = 1 supersymmetric in ten dimensions and has SO(32) gauge symmetry.

• E8 ×E8 heterotic string : Made of a left-moving bosonic and a right moving chiral superstring. The theory lives in ten-dimensions with the 16-internal

bosons living on the root lattice of E8 × E8. The spacetime theory is N = 1

supersymmetric in ten dimensions and has E8 × E8 gauge symmetry.

• Type I theory: A theory of unoriented closed and open strings. It can be deﬁned as a quotient of type IIB theory by worldsheet parity. The consistency of the theory at the quantum level requires the addition of unoriented open strings. The spacetime theory is N = 1 supersymmetric in ten dimensions and has SO(32) gauge symmetry.

2.1 String states

The Fock space of string excitations is generated by acting with string oscillators on a string vacuum. States are conveniently organised according to their oscillator ˜ M + + numbers N and N. We notice that the zero modes q , bI , bI commute with N, so the vacuum states deﬁned by N = 0 are always degenerated. We can better think of the chiral vacuum as the family

M ikM x |0, kiNS = e |0iNS D 2 M Y a ikM x + I |0, k, aiR = e (bI ) |0iR I=1 D 2 M Y a˙ ikM x + I |0, k, a˙iR = e (bI ) |0iR (2.1.1) I=1 parametrised by the momentum kM and the discrete variable a ora ˙ labelling the a a˙ choices {I , I = 0, 1} with

4 4 X a X a˙ I = even I = odd (2.1.2) I=1 I=1 38 CHAPTER 2. SPECTRA OF STRING STATES

The indices a, a˙ = 1,... 8 runs over the two inequivalent spinor representations of the SO(8) Lorentz group. On the other hand, the NS vacuum transforms as a singlet. Similar arguments apply to the right moving vacua. From now on we will always omit the k-label with the understanding that any string state carry always a momentum k. The spectrum of closed string states is deﬁned as the tensor product of the two Fock spaces, the left and the right moving ones subjected to the Virasoro constraints

 1 Bosonic  ¯ 1 (L0 − a) = (L0 − a¯) = 0 a = 2 NS Fermionic (2.1.3)   0 R Fermionic alpha0nntilde l00 Summing and subtracting the two equations and using (??), (1.4.74) and (??) to write α0 p α0 p L = + N − a L = + N − a (2.1.4) 0 4 0 4 one can rewrite the two conditions in the form

Closed Strings N˜ − a˜ = N − a Level matching 4 M 2 = (N − a) On shell (2.1.5) α0 with M 2 = −k2 describing the mass of the closed string in terms of the oscillation number N. For open strings connecting parallel branes one has a single set of oscillators acting on the vacuum and (x − x )2 Lopen = α0 p2 + π 0 + N − a (2.1.6) 0 4π2α0 The physical condition reads N − a (x − x )2 Open Strings M 2 = + π 0 On shell (2.1.7) α0 4π2α02 In the following we describe the lowest string states of each of the 5 ten dimensional superstrings.

2.1.1 Type IIA string

The spectrum of Type IIA strings is built as the tensor product of a left moving and a right moving fermionic strings with the GSO projection acting with opposite 2.1. STRING STATES 39

signs on the two sides. Let us start from the left moving sector. There are two choices, the Nevau-Schwarz (NS) and the Ramond (R) sector depending on whether fermions are taken to be anti-periodic or periodic along the σ-direction. States are built by acting with negative string modes on the vacuum subjected to the GSO projection that keeps only odd number of fermionic excitations. In the following table we list the chiral string states at level ` = N − a = 0, 1. We notice that the number of bosons and fermions at each string level match. This is a signal of the supersymmetry of the eﬀective spacetime theory describing the excitations of the string.

N-a operators vacuum SO(8)

M 0 b 1 |0iNS 8v − 2 1 |0, aiR 8s tchiral

M M N [M1 M2 M3] 2 1 b 3 , α−1 b 1 , b 1 b 1 b 1 |0iNS 8v + (8v) + 56v − 2 − 2 − 2 − 2 − 2 M α−1 |0, aiR 8v × 8s M b−1 |0, a˙iR 8v × 8c

Table 2.1: Chiral fermionic string states at level ` = 0, 1

The right moving sector can be constructed in a similar way with fermionic states now projected with opposite chirality. As a result, the chiral spectrum is tchiral given again by (2.1.1) with operators replaced by tilded operators and the chirality of spinor representations ﬂipped. The spectrum of closed string states is built by tensoring left and right moving states at the same string level. At the massless level one ﬁnds The NSNS spectrum contains a singlet , the dilaton φ, a symmetric tensor, the

graviton gMN and an anti-symmetric tensor BMN . The RR spectrum contains a

one form CM and a three form CMNP . NSR and RNS sectors contains two grav- ˜ ˜ itinos ψMa˙ , ψMa and two dilatinos λa, λa˙ . We notice that the number of bosonic and fermionic ﬁelds match. This is true at any string level and it is a signal of the supersymmetry of the higher spin theory describing the eﬀective dynamics of string 40 CHAPTER 2. SPECTRA OF STRING STATES

operators vacuum SO(8) ﬁelds

M ˜N b 1 b 1 |0iNS ⊗ |f0iNS 8v × 8v = 1 + 35 + 28 φ, gMN ,BMN − 2 − 2

tchiral 1 |0, aiR ⊗ |]0, a˙iR 8s × 8c = 8v + 56v CM ,CMNP M b 1 |0iNS ⊗ |]0, a˙iR 8v × 8c = 8s + 56s λa, ψMa˙ − 2 ˜N ˜ ˜ b 1 |0, aiR ⊗ |f0iNS 8v × 8s = 8c + 56c λa˙ , ψMa − 2

Table 2.2: Massless string states of IIA theory

modes. There are two gravitinos, so we say that the theory is N = 2 supersymmetric. We notice also that fermions appear also in pairs of opposite chirality so the theory is non-chiral in ten-dimensions.

2.1.2 Type IIB string

Type IIB strings are agin given by the tensor product of two fermionic strings but now with the GSO acting on the same way on left and right moving modes. The spectrum of closed strings can then be found by tensoring two copies of the tchiral tchiralII spectra in table (2.1.1). At massless level one ﬁnds the spectrum in table 2.1.2. The

operators vacuum SO(8) ﬁelds

M ˜N b 1 b 1 |0iNS ⊗ |f0iNS 8v × 8v = 1 + 35 + 28 φ, gMN ,BMN − 2 − 2 + tchiralII 1 |0, aiR ⊗ |]0, biR 8s × 8s = 1 + 35 + 28 C,CMN ,CMNPQ M b 1 |0iNS ⊗ |]0, aiR 8v × 8s = 8c + 56c λa˙ , ψMa − 2 ˜N ˜ ˜ b 1 |0, aiR ⊗ |f0iNS 8v × 8s = 8c + 56c λa˙ , ψMa − 2

Table 2.3: Massless string states of IIB theory

NSNS spectrum coincides with that of the type IIA theory and contains a dilaton, a

graviton gMN and an anti-symmetric tensor BMN . The RR spectrum contains now forms of even rank 0, 2, 4. In particular the four form satisfy a self-dual equation in 2.1. STRING STATES 41 eight dimensions giving rise to 35 on-shell degrees of freedom. NSR and RNS sectors give rise now two a pair of gravitinos and dilatinos with the same chirality. Again the number of bosonic and fermionic fields match at any string level and effective gravity theory is N = 2 supersymmetric. Since fermions coming from the NSR and RNS sectors have the same chirality the theory is chiral in ten-dimensions. Chiral theories are potentially anomalous. Collecting the contributions to anomalies of the chiral fields: two left-moving Majorana-Weyl gravitinos and dilatinos and a self-dual four form, one finds a perfect cancelation of anomalies due to the identity h i IIB : I 3 (R) − I 1 (R) − IA(R) = 0 (2.1.8) 2 2 12 sanomalies satisfied by the anomaly polynomials (see Appendix B for details) .

2.1.3 Heterotic strings

The heterotic string is made from the tensor of a fermionic string on the right and a bosonic string on the left. The two strings share a ten-dimensional spacetime. The remaining 16 directions of the left moving bosonic string live on the root lattice of

E8 ×E8 or SO(32). Alternatively the lattices of E8 ×E8 and SO(32) can be realised in terms of 32 real fermionsχ ˜I , I = 1,... 32 with appropriate boundary conditions. Explicitly

• SO(32) lattice: States organised into two sectors withχ ˜I either anti-periodic or periodic along σ. In the case of anti-periodic fermions, the vacuum transforms as a singlet of SO(32) with N = 0 oscillation number. In the case of periodic boundary conditions the vacuum transforms in the 215 spinor representation of SO(32) group and its oscillation number is N = 2. States are created by acting on these vacua with an even number of half-integer and integer negative modes respectively.

I • E8 × E8 lattice: Fieldsχ ˜ split into two groups of sixteen ﬁelds. There are four sectors given by all possible choices of periodic-antiperiodic boundary conditions for the two groups. Sectors with AA boundary conditions appear at level N = 0 and transform as a singlet of the gauge group. Sectors with AP or PA boundary conditions appear at level N = 1 and transforms in the (27, 1) and (1, 27) spinorial representation of the SO(16) × SO(16) maximal

subgroup of E8 × E8. Finally the sector PP appears at level N = 2 and 42 CHAPTER 2. SPECTRA OF STRING STATES

transforms in the bispinorial representation (27, 27). Again excitations are created by acting on the vacua with half-integer or integer negative modes according to the boundary conditions for the fermions and only states with an even number of excitations in both groups are allowed.

SO(32) heterotic string

The spectrum of the chiral SO(32) bosonic string at level ` = 0, 1 is displayed in thetso32 table 2.1.3.

N-1 operator vacuum SO(8) × SO(32)

M [I1 I2] 2 0 α˜−1, χ˜ 1 χ˜ 1 |f0i (8v, 1) + (1, 32anti) − 2 − 2

M (M1 M2) M [I1 I2] 2 2 1 α˜−2 , α˜−1 α˜−1 , α˜−1 χ˜ 1 χ˜ 1 |f0i (8v, 1) + (8vsym, 1) + (8v, 32anti) − 2 − 2 I1 I2 [I1 I2 I3 I4] 2 4 χ˜ 1 χ˜ 3 χ˜ 1 χ˜ 1 χ˜ 1 χ˜ 1 |f0i (1, 32 ) + (1, 32anti) − 2 − 2 − 2 − 2 − 2 − 2 1 |]0, si (1, 215)

thetso32 Table 2.4: Chiral SO(32) bosonic string states at level ` = 0, 1

2 15 Here 32, 32anti and 2 label the vector, adjoint and chiral spinor representation of SO(32). The spectrum of closed strings is deﬁned by the tensor product of left tchiral moving states from the fermionic string in table 2.1.1 and right moving states from thetso32 the bosonic string in table 2.1.3 at the same string level. For example at the massless level one ﬁnds

state vacuum SO(8) × SO(32)

M N M [I1 I2] 2 b 1 α˜−1, b 1 χ˜ 1 χ˜ 1 |0iNS ⊗ |f0i (8v × 8v, 1) + (8v, 32anti) − 2 − 2 − 2 − 2 M [I1 I2] 2 α˜−1, χ˜ 1 χ˜ 1 |0, aiR ⊗ |f0i (8v × 8s, 1) + (8s, 32anti) − 2 − 2 thetso32 Table 2.5: Massles states of the SO(32) heterotic string. 2.1. STRING STATES 43

The spectrum contains again the universal NS sector made of the metric, the dilaton and an antisymmetric tensor. In addition one finds a vector field in the adjoint of SO(32). Finally one finds a single gravitino, a dilatino and a gaugino in the adjoint of the gauge group. The effective theory describes now the coupling of gravity to a gauge theory in ten dimensions. Again the number of bosons and fermions match, so the theory is N = 1 supersymmetric in ten dimensions. The sanomalies theory is chiral and anomaly free (see Appendix B for details).

E8 × E8 heterotic string

The analysis for the E8 × E8 is very similar to that of the SO(32) string but now we have two sets of oscillatorsχ ˜i,χ ˜s with i, s = 1,... 8. Interestingly one ﬁnds that the number of states at each level of mass exactly coincide with that of the SO(32)

heterotic string but now they come in representations of E8 × E8 . For example at the massless level one ﬁnds

state vacuum SO(8) × SO(16)2

M N 2 b 1 α˜−1 |0iNS ⊗ |f0i (8v , 1, 1) − 2 M α˜−1 |0, aiR ⊗ |f0i (8v × 8s, 1) M [i1 i2] M [s1 s2] 2 2 b 1 χ˜ 1 χ˜ 1 , b 1 χ˜ 1 χ˜ 1 |0iNS ⊗ |f0i (8v, 16anti, 1) + (8v, 1, 16anti) − 2 − 2 − 2 − 2 − 2 − 2 M 7 7 b 1 |0iNS ⊗ (|^0, s1i + |^0, s2i) (8v, 2 , 1) + (8v, 1, 2 ) − 2 [i1 i2] [s1 s2] 2 2 χ˜ 1 χ˜ 1 , χ˜ 1 χ˜ 1 |0, aiR ⊗ |f0i ((8s, 16anti, 1) + (8s, 1, 16anti) − 2 − 2 − 2 − 2 7 7 1 |aiR ⊗ (|^0, s1i + |^0, s2i) (8s, 2 , 1) + (8s, 1, 2 )

Table 2.6: Massles states of the E8 × E8 heterotic string. States are grouped 2 2 according to their representation under a SO(16) maximal subgroup of E8 . States 2 7 in the 16anti and 2 representation of SO(16) combine together to build the adjoint thetso32 248 representation of E8.

States in the ﬁrst line give the universal NS sector, including the metric, the dilaton and the antisymmetric form. States in the second line provides their supersymmetric partners. States in the last four lines combine together into a vector

multiplet in the adjoint representation of E8 × E8 algebra. The resulting theory is 44 CHAPTER 2. SPECTRA OF STRING STATES

N = 1 supersymmetric in ten dimensions with gauge group E8 × E8. The theory sanomalies is chiral and anomaly free (see Appendix B for details).

2.1.4 Type I string

Closed strings

The last consistent string theory in ten dimensions is the so called type I string theory. Unlike, type II and heterotic strings, type I string theory is made of unoriented closed and open strings. The theory can be viewed as a quotient of type IIB theory under the symmetry that exchange left and right moving modes of the string reversing its orientation. We introduce the worldsheet parity operator Ω acting on the oscillator string modes as

˜ Ω: αn ↔ α˜n br ↔ br (2.1.9)

Since the NS and R vacuum come with bosonic and fermionic statistics the worldsheet parity acts on the NSNS and RR vacuum with eigenvalues plus and minus tchiralI respectively. The invariant states at massless level are displayed in the table 2.1.4. The NSNS spectrum contains now only the dilaton and the metric while RR sector

states SO(8) ﬁelds

b(M |0i ⊗, ˜bN) |0i 1 + 35 φ, g − 1 NS − 1 fNS MN tchiralI 2 2 |0, [aiR ⊗ |]0, b]iR 28 CMN M ˜N b 1 |0iNS ⊗ |]0, aiR + |0, aiR ⊗ b 1 |f0iNS 8c + 56c λa˙ , ψMa − 2 − 2

Table 2.7: Massless closed string states of type I theory

provide an anti-symmetric two form CMN . From the NSR and RNS sectors only the symmetric combination survives the projection leading to a gravitino and a dilatino. Again the number of bosonic and fermionic ﬁelds match at any string level and the eﬀective theory is N = 1 supersymmetric. The resulting spectrum is chiral and anomalous. This inconsistency can be cured by the inclusion of open strings. 2.1. STRING STATES 45

Open strings

Let us consider now the inclusion of open strings. To respect the SO(10) Lorentz invariance we assign Neuman boundary conditions to each of the two ends of the open strings. Equivalently, one can think of the open strings as starting and ending on D9-branes. Let us denote by N the number of D9-branes. The open string vacua are labeled by the integer pair (i, j) with i, j = 1,...N specifying the D9-branes at the two ends of the open string. Open string excitations are generated by a single set of super-oscillators acting on this vacuum. The spectrum of open string states are then given by N 2 copies tchiral of that of the chiral superstring displayed on table 2.1.1. This spectrum should be projected onto its Ω-invariant component with Ω flipping the orientation of the string or in other words acting by exchanging the label i and j. It is conveneient to think about the N 2 copies of the open string spectrum as filling the entries of an N × N matrix. The worldsheet parity invariant states corresponds then to the symmetric or anti-symmetric components of this matrix with N(N + 1)/2 and N(N − 1)/2 independent components respectively. Since the open string vacua in the NS sector contains a vector, the number of independent components should be identified with the number of generators of the gauge group. The remaining components at the massless level fill the content of a single vector multiplet in the adjoint representation of the gauge group. Together with the gravity multiplet coming from the closed string sector one find at the massless level the field content of N = 1 supergravity with gauge group Sp(N) or SO(N). From the requirement sanomalies of cancelation of anomalies (see Appendix B for details), we conclude that SO(32) is the only consistent choice. The spectrum of massless open strings is displayed in tmasslessI table 2.1.4.

N-a states ﬁelds SO(8) × SO(32)

M tmasslessI 0 b 1 |0, i, jiNS AM (8v, 496) − 2 |0, a, i, jiR λa (8s, 496)

Table 2.8: Massless open string states in type I theory

We notice that the total spectrum of type I theory at massless level coincides 46 CHAPTER 2. SPECTRA OF STRING STATES with that of the SO(32) string theory. We will later see that this is not a coincidence and that the two theories are indeed equivalent at the quantum level.

2.2 The vacuum amplitude and the modular group

A closed string during its evolution along the loop span a worldsheet two-torus. The torus is deﬁned by the identiﬁcations

σ ∼ σ + 1 σ ∼ σ + τ (2.2.10) with σ = σ0 +i σ1 a complex coordinate on the string world sheet and τ = τ1 +i τ2 a complex number (with positive imaginary part), the complex structure, parametrising the torus shape. The vacuum amplitude is deﬁned as an integral over all posibles shapes τ of the torus. It is important to remark that not all possible choices of τ represent inequivalent tori. For example a torus with τ → −1/τ can be viewed as the same torus with σ0 and σ1 coordinates exchanged. Similarly two torus with complex structures τ and τ + 1 can be related to each other by cutting one of the two along a circle and gluing back the two ends after a single twist. These two transformations, referred as T and S 1 T : τ → τ + 1 ,S : τ → − (2.2.11) τ generate a group Γ = SL(2, Z) known as the modular group. The moduli space of inequivalent tori is then deﬁned by the quotient

F = C/SL(2, Z) (2.2.12) The partition function is given by an integral over the fundamental domain Z 2 Z 2 Z d τ d τ ˜ S(τ|X,ψ,ψ˜) Z = 2 Z(τ, τ¯) = 2 DX Dψ Dψ e (2.2.13) F τ2 F τ2 with S(τ|X, ψ, ψ˜) the action for a string in the light cone gauge evaluated on a torus world sheet with complex structure τ. We introduce the SL(2, Z) invariant 2 2 measure d τ/τ2 . Consequently, the function Z(τ, τ¯), often referred as “the partition function” should be also invariant under the modular group. The partition function Z(τ, τ¯) can be alternatively written, in the Hamiltonian formalism, as a trace over the Fock space of physical states

L0−a L¯0−a˜ Z(τ, τ¯) = TrH q q¯ (2.2.14) 2.2. THE VACUUM AMPLITUDE AND THE MODULAR GROUP 47 with q = e2πiτ , α0p2 α0p2 L = + N L¯ = + N˜ (2.2.15) 0 4 0 4 and the trace running over the Hilbert space H of oﬀ-shell physical states. In the case of open string , the vacuum amplitude is related to the open string partition function Z ∞ dt πit (Lopen−a) A = TrH e 0 (2.2.16) 0 t with (x − x )2 Lopen = α0p2 + π 0 + N (2.2.17) 0 4π2α0 The modular invariant function Z(τ, τ¯) will be written in terms of the so called “theta functions” deﬁned as

∞ 1 Y n n 2 ϑ1(q) = (1 − 1) q 8 (1 − q )(1 − q ) n=1 ∞ 1 Y n n 2 ϑ2(q) = (1 + 1)q 8 (1 − q )(1 + q ) n=1 ∞ Y n n− 1 2 ϑ3(q) = (1 − q )(1 + q 2 ) n=1 ∞ Y n n− 1 2 ϑ4(q) = (1 − q )(1 − q 2 ) (2.2.18) n=1 and the Dedekin eta function

∞ 1 Y n η(q) = q 24 (1 − q ) (2.2.19) n=1

These functions have deﬁnite modular transformation properties

πi ϑ2(τ + 1) = e 4 ϑ2(τ) ϑ3(τ + 1) = ϑ4(τ) ϑ4(τ + 1) = ϑ3(τ) 1 √ 1 √ 1 √ ϑ (− ) = iτ ϑ (τ) ϑ (− ) = iτ ϑ (τ) ϑ (− ) = iτ ϑ (τ) 2 τ 4 3 τ 3 4 τ 2 πi √ 12 1 η(τ + 1) = e η(τ) η − τ = iτ η(τ) (2.2.20) and provide the basic building block to construct modular invariant forms in elliptic geometry. 48 CHAPTER 2. SPECTRA OF STRING STATES

2.2.1 Free uncompact bosons

The contributions of free uncompact bosons to the partition functions are of two types. First, one has an integral over the d = D − 2 light cone momenta kI

d Z d k 0 2 − d Γ = V e−π τ2 α k = v τ 2 (2.2.21) lzero d (2π)d d 2

Vd with Vd the volume of the space and vd = (2π)dα0 . Second, one has a discrete sum over the states created by acting on the vacuum with the creation operators α−n andα ˜−n. This later contribution factors into a holomorphic and anti-holomorphic function describing the contribution of left and right moving excitations respectively. The left moving Hilbert space is deﬁned as

∞ d M M 2 HL = ⊗n=1 ⊗M=1 {1, α−n, (α−n) ,,...}|0i (2.2.22)

The contribution to the trace can then be written as

∞ Y 1 Tr qN = (1 + qn + q2n + ...)d = (2.2.23) HL (1 − qn)d n=1

− d Including the zero point energy q 24 and putting together contribution from left lzero and right moving oscillators and zero modes (2.2.21) one ﬁnds the total contribution

1 d Zboson(τ, τ¯) = vd √ (2.2.24) zbos τ2 η(q) η(¯q) zbos The result (2.2.24) is modular invariant, as can be seen using the modular trans- modular 2 formation property (2.2.20) and the fact that τ2 → τ2/|τ| under τ → −1/τ.

2.2.2 Free compact bosons

I I Let us consider now d compact bosons with bosonic coordinates (XL,XR) leaving on an O(d, d) lattice Λd,d. The zero modes of the closed string in this case are ∗ parametrised by a vector (pL, pR) leaving on the dual lattice Λd,d and parametrising the Kaluza-Klein momenta and the wrapping numbers of the string. The zero mode partition function is now given by the sum

X α0p2 α0p2 Γ = vd q L q R (2.2.25) ∗ (pL,pR)∈Λd,d 2.2. THE VACUUM AMPLITUDE AND THE MODULAR GROUP 49 that replaces the integral over momenta in the uncompact case. The spectrum of non-zero modes exactly coincides with that of d uncompact bosons so one ﬁnds for the total partition function

d 1 X α0p2 α0p2 Zboson(τ, τ¯) = vd q L q R (2.2.26) zbos2 η(q) η(¯q) ∗ (pL,pR)∈Λd,d zbos2 The result (2.2.28) is again modular invariant since

1 1 d/2 Γ − τ , − τ¯ = (ττ¯) Γ(τ, τ¯) (2.2.27) as can be seen after Poisson resummations on left and right moving modes. Similarly for a chiral, let us say left moving, boson one ﬁnds

1 X 0 2 α pL Zchiral boson(τ, τ¯) = d q (2.2.28) zbos2 η(q) ∗ pL∈Λd

∗ In this case modular invariance requires that the lattice Λd be self-dual.

2.2.3 Free Fermions

Fermionic contributions to the string partition function always factor into a left and right moving parts. We can then focus on the chiral partition function. The Fock space has two sectors: the Neveu-Schwarz and Ramond sectors corresponding to taking anti-periodic and periodic boundary conditions respectively for the fermions along the σ direction. Since fermionic excitations are Grassmanian there are only two possibilities for a given excitation: it is either present or not. The left moving Hilbert states are then deﬁned as

NS ∞ d M HL = ⊗n=1 ⊗M=1 {1, b 1 }|0iNS −n+ 2 2d/2 R X ∞ d M HL = ⊗n=1 ⊗M=1 {1, b−n ...}|0, αiR (2.2.29) α=1

The contribution to the partition function depends on whether we consider anti- periodic or periodic boundary conditions along the τ-direction. In the later case,

FL fermionic excitations are weighted by (−) with FL counting the number of left moving fermionic excitations. Including also the contribution of the zero point 50 CHAPTER 2. SPECTRA OF STRING STATES energy one ﬁnds1

∞ d 2 L −a − d Y n− 1 d ϑ3 Tr q 0 = q 48 (1 + q 2 ) = NS η n=1 ∞ d 2 F L −a − d Y n− 1 2 ϑ4 Tr (−) L q 0 = q 48 (1 − q 2 ) = NS η n=1 ∞ d 2 L −a d Y n 2 ϑ2 Tr q 0 = q 24 (1 + 1) (1 + q ) = R η n=1 ∞ d 2 F L −a d Y n 2 ϑ1 Tr (−) L q 0 = q 24 (1 − 1) (1 + q ) = = 0 (2.2.30) R η n=1 amplitudes It is often convenient to combine amplitudes (2.2.30) into characters collecting states with a deﬁnite fermionic parity. These are characters of the Lorentz SO(d) algebra and are given by ϑn + ϑn ϑn − ϑn ϑn + i−nϑn ϑn − i−nϑn O = 3 4 V = 3 4 S = 2 1 C = 2 1 (2.2.31) characterson 2n 2ηn 2n 2ηn 2n 2ηn 2n 2ηn with O2n, V2n, S2n and C2n starting with a scalar, a vector, a spinor left and a spinor right of the SO(2n) algebra. We notice that S2n and C2n numerically coincide since

ϑ1 = 0.

2.3 String partition functions

In this section we display the partition functions for the ﬁve ten-dimensional string theories. For simplicity, in this section we will always omit the volume factors v10 common to any string partition function in ten dimensions.

2.3.1 Type II Strings

The partition function of type II strings is deﬁned as

FL FR 1 − (−) 1 − (−) ˜ Z(τ, τ¯) = Tr qL0−a qL0−a˜ (2.3.32) ztrace NS+R,NS+R 2 2 We counts spacetime bosons with a plus and spacetime fermions with a minus sign, so states from the NSR and RNS sectors come with opposite sign with respect to

1The contribution to the zero point energy −a of a fermion is minus half the right hand side zeropoint 1 1 of (1.4.72), i.e. − 48 in the NS and 24 in the Ramond sector. 2.3. STRING PARTITION FUNCTIONS 51 those in the NSNS and RR sectors. In addition, we choose the GSO projection such that only states with odd number of fermionic excitations are kept. We will see that only for this choice one finds a modular invariant answer for the partition function. There is an ambiguity in the assignment of the fermionic numbers FL,R of the Ramond ground states. There are two inequivalent choices depending on whether we assign different signs (IIA) and equal signs (IIB) to the left and right moving ground states. Still, the two choices lead to the same number of states, so the partition function of IIA and IIB coincide. The various contributions to the ztrace zbos amplitudes trace in (2.3.32) can be read directly from (2.2.24) and (2.2.30) with d = 8. Putting altogether one finds 4 4 4 2 1 ϑ3 − ϑ4 − ϑ2 Z(τ, τ¯) = 4 12 (2.3.33) zii τ2 2η zii We notice that the result (2.3.33) is invariant under modular transformations as can modular be seen using the transformations properties (2.2.20) of the theta functions. Notice zii that signs in (2.3.33 ) are fixed by modular invariance, so only this choice of GSO projection is consistent with modular invariance. Moreover Z(τ, τ¯) is exactly zero since theta functions satisfy the so called “abstruse identity”

4 4 4 ϑ3 − ϑ4 − ϑ2 = 0 (2.3.34)

This is a manifestation of supersymmetry that requires that the number of bosons zii and fermions match at any string level. The result (2.3.33) can also be written in characterson terms of the character of the SO(8) Lorentz algebra defined in (2.2.31). One finds 1 ZIIA(τ, τ¯) = 4 8 8 (V8 − S8)(q)(V8 − C8)(¯q) τ2 η η¯ 1 2 ZIIB(τ, τ¯) = 4 8 8 |V8 − S8| (2.3.35) τ2 η η¯ where we used the fact that the projections in type IIA and IIB keep fermions of different and equal charities respectively. Clearly, the number of states in S8 and zii C8 coincide, so the two formulas match with (2.3.33). Using the small q expansion of the SO(8) characters

1 V8 = q 3 [8v + (8v + 56v) q + ...] 1 S8 = q 3 [8s + (8v × 8c) q + ...] 1 C8 = q 3 [8c + (8v × 8s) q + ...] −8 − 1 η = q 3 [1 + 8v q + ...] (2.3.36) 52 CHAPTER 2. SPECTRA OF STRING STATES one ﬁnds

2 2 ZIIA(τ, τ¯) = (8v − 8s)(8v − 8c) + q q¯(8v − 8s) (8v − 8c) + ... 2 2 ZIIB(τ, τ¯) = (8v − 8s)(8v − 8s) + q q¯(8v − 8s) (8v − 8c) + ... (2.3.37) with dots referring to non-level matched and higher massive states. We remark that only at the massless level the spectra of type IIA and type IIB theories are diﬀerent.

2.3.2 Heterotic Strings

The partition function of heterotic strings is deﬁned as

FR 1 − (−) ˜ Z(τ, τ¯) = Tr qL0−a qL0−a˜ (2.3.38) ztracehet NS+R 2 The contributions to the trace of spacetime bosons and fermions are given again by zbos amplitudes (2.2.24) and (2.2.30) respectively with d = 8. On the other hand the contribution of the 16 internal bosons on the SO(32) or E8 × E8 lattice can be easily computed using the description of the lattice in terms of 32 fermions and is given by the modular invariant formulas ϑ16 + ϑ16 + ϑ16 Λ = 2 3 4 SO(32) 2η16 ϑ16 + ϑ8 + ϑ8 2 Λ = 2 3 4 (2.3.39) E8×E8 2η8 Putting altogether one ﬁnds

4 4 4 1 ϑ3 − ϑ4 − ϑ2 ZG(τ, τ¯) = 4 12 ΛG (2.3.40) τ2 2η lso32e8e8 with ΛSO(32) and ΛE8×E8 given by (2.3.39).

2.3.3 Type I theory

The type I partition function is made of four diﬀerent string amplitudes corresponding to the four diﬀerent (in general unoriented) world sheets of genus one: the torus T , the Klein K, the annulus A and the Moebius M amplitudes. The one-loop partition function is then given by the sum

1 1 Z = 2 (T + K) + 2 (A + M) (2.3.41) 2.3. STRING PARTITION FUNCTIONS 53 with the torus and Klein amplitudes involving traces over closed string states Z d2τ L0−a L¯0−a˜ T = 2 Trclosed q q¯ τ2 Z dτ 2 L0−a L¯0−a˜ K = 2 Trclosed Ω q q¯ (2.3.42) τ2 while the annulus and the Moebius amplitudes given in terms of trace over the open string states Z dt A = Tr qL0−a t2 open Z dt M = Tr Ω qL0−a (2.3.43) t2 open The torus amplitude coincide with that of type IIB theory and therefore

Z 2 4 4 4 2 d τ 1 ϑ3 − ϑ4 − ϑ2 T = 2 4 12 (2.3.44) torusamp τ2 τ2 2η In computing the Klein amplitude we notice that states made of two diﬀerent excitations on the left and right moving sides of the string can be combined into a symmetric and an anti-symmetric combination with eigenvalues +1 and −1 for Ω, so the net contribution to the partition function is zero. Only diagonal states con- torusamp tribute to the Klein amplitude so the total contribution follows from (2.3.44) by identifying left and right moving modes Z dt ϑ4 − ϑ4 − ϑ4 K = 3 4 2 (2it) (2.3.45) kleindir t6 2η12 The open string amplitudes are given by similar formulas since the open string spectrum is isomorphic to N 2 copies that of the chiral superstring. The energy of the open string excitation is half that of the chiral one so the argument of the it modular function is τ = 2 . Finally, the trace of the worldsheet parity operator Ω over open string states receives contributions only from diagonal states with the two ends on the same brane, so the Moebius amplitude is proportional to N with minus sign for the SO(N) projection. The resulting amplitudes read Z dt ϑ4 − ϑ4 − ϑ4 A = N 2 3 4 2 ( it ) t6 2η12 2 Z dt ϑ4 − ϑ4 − ϑ4 M = −N 3 4 2 it + 1 (2.3.46) t6 2η12 2 2 (2.3.47) 54 CHAPTER 2. SPECTRA OF STRING STATES

Tadpole cancelations

It is important to notice that changing the role of σ and τ, the one-loop amplitudes K, A and M can be alternatively thought as tree level diagrams representing exchanges of closed string states between boundaries and/or crosscaps. We can think of the crosssscap as a ten dimensional mirror reflecting the orientation of the open string, and we will refer to this surface as an O9 orientifold plane). In the limit where the cylinder is very long these diagrams factorize into two disk or projective plane diagrams with the insertion of a massless closed string state. The one-point diagrams describe the rate of emission of massless closed string states from D9 branes and O9 planes and are known as tadpoles. A consistent vacuum has no tadpole, so better these contributions cancel between boundaries and crosscaps. In this section we show that this cancelation is achieved precisely for N = 32. Notice that this is precisely the number of branes needed in order to get an anomaly free theory in ten dimensions. We will later see that this is not a coincidence. and indeed theories without tadpoles are always free of anomalies. To compute the tadpoles of the theory, one can start from the open string amplitudes, and rewrite the integral in terms of the new variable ` ∼ 1/t . This is nothing but a change of variables inside the integral so the resulting amplitudes in the open and closed string channels coincide. We will still denote the closed string amplitudes with a tilde to emphasise their different meaning. Using the modular modular transformation properties (2.2.20) the amplitudes in the closed string channel can be written as Z ϑ4 − ϑ4 − ϑ4 Ke = 25 d` 3 4 2 (i`) 2η12 Z ϑ4 − ϑ4 − ϑ4 Ae = 2−5 N 2 d` 3 4 2 (i`) 2η12 Z 4 4 4 ϑ3 − ϑ4 − ϑ2 1 Mf = −2N d` i` + (2.3.48) 2η12 2 with `K = 1/(2t), À = 2/t and `M = 1/(2t). Klein and Annulus amplitudes in the open and closed string channels are related by S-duality, while Moebius amplitudes in the open and closed string channel are connected by the chain of dualities it 1 2 2 2it i 1 i 1 1 + −→S − −→T −→S − −→T + = i` + (2.3.49) 2 2 it + 1 it + 1 2t 2 2t 2 2 or equivalently 1 it 1 i` + = P + (2.3.50) 2 2 2 2.4. D-BRANES AND O-PLANES 55

2 1 with P = TST S. We notice that the extra 2 -shift in the argument of the Moebius amplitude produce alternating signs in the string oscillator levels. The sum of the three amplitude becomes then proportional to (N ± 32)2 with minus and plus signs for even and odd oscillator levels. The fact that the total amplitude reconstructs a sum of squares reﬂects the fact that cylinder amplitudes always factorize into identical contributions describing the emission of the exchanged closed string state form N D9 and one O9 plane. In the limit where the cylinder get very long ` → ∞, the total closed string amplitude reduce to

Z ∞ −5 2 Ke + Ae + Mf = 2 (N − 32) d` (8v − 8s) + ... (2.3.51) 0

The two terms 8v and 8s describe tadpoles for diagonal components of the NSNS and RR fields. The cancelation of massless tadpoles is achieved for N = 32 ! On the other hand for generic N there is cancelation between the contribution associated to an exchange of the NSNS field (attractive gravitational force) and the exchange of the RR field (repulsive ”electric” force).

2.4 D-branes and O-planes

The five consistent string theories in ten-dimensions exhaust the list of string vacua enjoying SO(10) invariance and supersymmetry. Less symmetric vacua can be constructed by considering string theories on non-trivial backgrounds. Alternatively, symmetries can be broken by introducing boundaries or D-branes in the string world sheets or by quotienting a string theory by a global symmetry. In this section, we study type II string theories in the presence of infinite parallel Dp-branes and/or Op-planes. Infinite D-branes and O-planes can be thought in a first approximation as rigid objects, and a vacuum containing D-branes and/or O-planes as a theory of open and closed strings in flat space time. Dp-branes are (p + 1)-dimensional surfaces where open strings can end. Alter- natively one can think of a D-brane as a source of closed strings. The vacuum amplitude of an open string connecting two Dp-branes counts the number of excitations of the open string. The same diagram, viewed as a cylinder describes the brane potential generated by the exchange of closed string states between the two Dp-branes. Op-planes are (p + 1)-dimensional surfaces where the orientation of a string get 56 CHAPTER 2. SPECTRA OF STRING STATES reversed. In presence of O-planes the string theory is unoriented and the vacuum amplitude include the contributions of string world sheets with the topology of a Klein-bottle and a Moebius strip. The resulting theory can be viewed as a quotient of a type II string, by a symmetry combining the worldsheet parity operator and a reflection of the string transverse to the Op-plane. The presence of D-branes and O-planes breaks Lorentz invariance down to SO(p + 1) × SO(9 − p) and preserves half of the supersymmetries of the parent type II string theory.

2.4.1 D-branes as boundaries of open strings

An open string ending on a Dp-brane satisfies Neumann boundary conditions along the D-brane and Dirichlet boundary conditions along the transverse directions. The boundary conditions identify left and right moving up to the action of the reflection eq:Rp matrix R0 (1.3.37). This matrix acts on the chiral string modes as a reflection I9−p along the (9-p)-dimensional plane transverse to the brane. On the other hand, I9−p acts as the chirality operator on spinor fields. Since the chirality operator preserves the chirality on a space of even dimensions and flip the chirality otherwise, this action is a symmetry only if p is even in type IIA and p odd in type IIB. Indeed for this choice a spinor on the left moving sector of the string is mapped to a spinor of the same chirality on the right. The spectrum of excitations of an open string connecting two parallel Dp-branes tchiral coincide with that of the chiral string (see table 2.1.1) with the only difference that the momentum of the string is now restricted to the (p + 1)-dimensional plane. The spectrum of string states can be found from that of the open string connecting two D9-branes in ten-dimensions dimensionally reduced to (p + 1)-dimensions. For example at the massless level one finds the bosonic fields

j j NS (Aµ)i , (ϕI )i (2.4.52) massa with µ = 0, . . . p and I = p+1,... 9 the Lorentz vectors indices along and transverse to the brane respectively and i, j = 1,...N labelling the Dp-branes connected by the string. This is the bosonic content of the maximally supersymmetric Yang- Mill supermultiplet in (p+1)-dimensions with the Chan-Paton labels i, j ﬁlling the adjoint of a U(N) gauge group. The fermionic components of this multiplet arise 2.4. D-BRANES AND O-PLANES 57 from the Ramond sector and can be obtained again by dimensional reduction from a single Majorana-Weyl fermion in ten-dimensions down to (p+1) dimensions. Now let us consider the vacuum amplitude for an open string stretched between two Dp-branes separated by a distance Y = xπ − x0. Open string states carry a momentum k along the Dp-brane and ﬁll a space of regularised volume Vp+1. The zero mode contribution to the string partition function is given by

Z p+1 2 d k −πt α0p2+ Y Vp+1 4π2α0 Γ = Vp+1 p+1 e = p+1 (2.4.53) (2π) (4π2α0) 2

Together with the contribution of massive oscillators one ﬁnds

Z 2 4 4 4 Vp+1 dt − t Y ϑ3 − ϑ4 − ϑ2 4πα0 it Ap = p+1 p+3 e 12 ( 2 ) (2.4.54) (4π2α0) 2 t 2 2η

States come again in multiplets of maximally supersymmetry in (p + 1)-dimensions but unlike in the case of coinciding Dp-branes they carry an extra contribution 2 Y 2 to the mass δM = 4π2α02 coming from the ground states energy of the stretched string. This provides a nice picture of the Higgs mechanism in the effective Yang- Mills theory describing the low energy dynamics of the open string. Indeed, starting from a stack of N coinciding Dp-branes, the U(N) gauge symmetry can be broken down to U(1)N (or to any unitary subgroup of U(N) ) by separating the Dp-branes along the transverse directions. The mass of the off-diagonal components of the U(N) vector field is given by the distance between the two Dp-branes.

2.4.2 D-branes as source of closed strings: tension and charge annulusp The annulus diagram (2.4.54) can be alternatively viewed as a cylinder diagram describing the exchange of closed string states between two parallel Dp-branes. As we have seen for the case of D9-branes, D-branes exchange both NSNS and RR ﬁelds. The cylinder diagram describes the brane potential mediated by this exchange. In this section, we show that this potential is generated by gravitational and electromagnetic interactions associated to the fact that branes have a mass and a charge. More precisely, since Dp-branes are inﬁnitely extended object, we say that branes have a density of mass (tension) Tp and a density of charge µp. We will compute the tension and density of charges by factorisation of the closed string diagram. 58 CHAPTER 2. SPECTRA OF STRING STATES

annulusp The cylinder diagram follows from (2.4.54) after deﬁning ` = 2/t and using the modular modular transformation properties (2.2.20) to write2

Z 2 4 4 4 Vp+1 d` − Y ϑ3 − ϑ4 − ϑ2 2πα0` Aep = p+1 9−p e 12 (i`) (8π2α0) 2 ` 2 2η Z 2 Vp+1 d` − Y 2πα0` = p+1 (8 − 8) 9−p e + ... (8π2α0) 2 ` 2 2 0 3−p = 2π(4π α ) (1 − 1)Vp+1 G9−p(|Y |) + ... (2.4.55) with dots standing for massive string state contributions and Z 1 dt −Y 2t Gd(|Y |) = d 4−d e (2.4.56) 4π 2 t 2 representing the massless propagator in d-dimensions. In the limit where Y is large (or equivalently ` large) only massless states contribute and the diagram factorizes into two disks with a closed string insertion on each and a massless propagator connecting them. This shows that Dp-brane boundaries are charge under both NSNS and RR ﬁelds and that both interactions compensate each other leading to a net zero potential. A(p + 1)-dimensional object couples minimally to a (p + 1)-form, e.g. a particle couples to a vector, a string to a two-form, etc. There is indeed a p+1-form Cp+1 in the RR spectrum of the closed strings, so we say that Dp-branes carry RR charge. On the other hand, the mass of the Dp-brane should be proportional to its volume, so the Dp-brane world volume action can be written as Z √ Z p+1 −Φ SDp = −Tp d ξe detG − i µp Cp+1 (2.4.57) actiondp

The factor e−Φ follows from the fact that the coupling of the Dp-brane to the metric appears at the disk level. The constants Tp and µp describe the tension and density charge of the Dp-brane. To compute these quantities from the cylinder diagram, we need also the propagator of the massless closed string fields. In the next section we will describe the low energy effective action describing the dynamics of these fields. As we will see, the low energy action includes the standard Einstein-Hilbert term, kinetic terms for the dilaton and RR fields and Wess-Zumino like interactions. Restricted to the fields entering in the closed string diagram the action reads Z 1 10 −2Φ 2 1 2 SII = 2 d x e R + 4(∂Φ) − Fp+2 (2.4.58) actionii 2κ10 2(p + 2)!

2 We recall that Aep = Ap. 2.4. D-BRANES AND O-PLANES 59

actiondp actionii with κ10 a constant. The exchange amplitude following from (2.4.57) and (2.4.58) is

2 2 ANSNS = 2 κ10 Tp Vp+1 G9−p(|Y |) 2 2 ARR = −2 κ10 µp Vp+1 G9−p(|Y |) (2.4.59) cylinderp Comparing this with the string result (2.4.55) we conclude that

2 0 3−p −2 Tp = µp = π(4π α ) κ10 (2.4.60)

2.4.3 O-planes

Let us consider now the inclusion of an Op-plane at the origin. The Op-plane can be thought as a mirror reflecting the orientation of the strings. The resulting theory can be viewed as a quotient of type II string by ΩI9−p, with Ω the worldsheet parity operator and I9−p a reflection along the plane orthogonal to the O-plane. The spectrum of closed string states coincide with that of the type I theory (type IIB theory with an O9-plane) but now fields with odd number of legs transverse to the brane come in antisymmetric (rather than symmetric) combinations. For example, the bosonic massless sectors is given by

NSNS φ, gµν, gij, bµi

RRCµν,Cij,Cµijk (2.4.61)

States in the RNS and NSR sectors come always in even/odd pairs, so the projection keeps exactly the same number of states as in type I theory, i.e. a ten- dimensional Majorana-Weyl gravitino and a dilatino dimensionally reduced down to (p + 1)-dimensions. On the other hand, acting on open strings connecting Dp-branes on top of the O-plane, the orientifold action projects open strings onto symmetric or antisymmetric representations filling the adjoint representations of Sp(N) and SO(N) gauge massa groups respectively. The spectrum of massless string states is again given by (2.4.52) but now the fields transform in the adjoint representations of the Sp(N) or SO(N) depending on the sign of the orientifold projection. On the other hand for a stack of N Dp-branes away from the Op-plane (and their images at the other side of the mirror) one finds a the field content of maximally supersymmetric U(N) gauge theory. Notice that this theory can be viewed as a Higgs broken phase of an O(2N) 60 CHAPTER 2. SPECTRA OF STRING STATES or Sp(2N) corresponding to the collision of the N Dp-branes and their images at the Op-plane. The vacuum amplitude of the unoriented theory include now besides the oriented torus and annulus amplitudes, the Klein bottle and the Moebius strip unoriented amplitudes. They describe the exchange of closed string states between two crosscaps (Op-Op) and a boundary/crosscap (Dp-Op) respectively. As in the case of the Dp-Dp annulus, these diagrams are singular due to the presence of tadpoles of massless closed string states. The computation is identical to that of type I theory with the only modification that now open and closed string states move only along a (p + 1)-dimensional plane and therefore the integral over the momenta leads to − p+3 kleindir amdir −6 a power t 2 in the direct amplitudes (2.3.45) and (2.3.46) instead of t . As a result one finds for the tadpole

2 Z ∞ − p+1 p+1 Ke + Ae + Mf = 2 2 N − 2 2 d` (8v − 8s) + ... (2.4.62) 0

p+1 We notice that there is an exact cancelation of the tadpole for N = 2 2 Dp-branes. This cancelation shows that Op-planes behave eﬀectively as objects of negative tension and charge opposite to that of Dp-branes. More precisely, one ﬁnds the relation p+1 TOp = µOp = −2 2 Tp (2.4.63) Chapter 3

Low energy eﬀective actions

In this chapter we consider the evolution of strings on curved spacetime backgrounds. We show that conformal invariance of the two-dimensional worldsheet theory requires that string backgrounds satisfy the equations of motion of supergravity theories with N = 1, 2 super symmetries in ten dimensions. We describe the relevant supergravity actions, equations of motion, supersymmetry variations and their basic solitonic solutions.

3.1 Strings in curved background

3.1.1 Type II superstrings

In the background geometry with metric gM N, a NSN two form BMN and a dilaton, the type II superstring action can be written as 1 Z h√ i S = − d2σ hhabG (X) + abB (X) ∂aXM ∂ XN + i ψ¯M γa ∂ ψ II 4πα0 MN MN a a M 1 Z √ + d2σ hR2 Φ(X) (3.1.1) 4π with 12 = 1 the epsilon tensor and R2 the two-dimensiona curvature scalar. The ﬂat spacetime corresponds to the choice where all ﬁelds are constant

GM N(X) = ηMN BMN (X) = 0 Φ(X) = Φ0 (3.1.2) We remark that for Φ(X) constant the last term in the action is topological. Indeed, for any two dimensional surface the integral 1 Z √ χ = d2σ hR2 = 2 − 2g (3.1.3) 4π

61 62 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS is known to be a topological number, the Euler number, counting the genus g (number of holes) of the surface. String amplitudes deﬁned on a worldsheet of genus g 2−2g are weighed by gs with the string coupling gs deﬁned as

hΦi gs = e (3.1.4) with hΦi the vacuum expectation value of the dilaton field, i.e. its value at infinity. The dynamical part of the string action describes a two-dimensional theory of free bosons and fermions. The worldsheet theory becomes interacting if we consider a background of varying spacetime fields. Indeed, for a slow varying fields, starting from a given point in spacetime we can always choose coordinates such that

1 P Q GMN (X) = ηMN − 3 RMNPQ X X + ... 1 P 1 P Q BMN (X) = 3 HMNP X + 6 ∇QHMNP X X + ... P 1 P Q Φ(X) = Φ0 + ∂P Φ X + 2 ∇P ∇QΦ X X + ... (3.1.5) leading to cubic, quartic vertices and so on. These interactions can break some of the symmetries of the classical theory at a quantum level. Indeed, it is easy to see that for a general background, conformal invariance is broken at one-loop. Indeed a worldsheet computation at one loop reveals that the trace of the energy momentum tensor is given by √ √ a 1 Φ (2) 1 G ab B ab a M N ¯M a Ta = 2 β hR + 2α0 (βMN hh + βMN ) ∂ X ∂aX + i ψ γ ∂a ψM (3.1.6) with

G 0 1 P1P2 βMN = α RMN − 4 HMP1P2 HN + 2∇M ∇N Φ + ... B 0 1 P P βMN = α − 2 ∇ HPMN + HMNP ∇ Φ + ... Φ 1 0 2 1 2 1 2 β = 4 (D − 10) + α (∇Φ) − 2 ∇ Φ − 4 H + ... (3.1.7) and dots referring to higher α0 (loop) corrections. Conformal invariance is then broken unless all beta functions vanish

G B Φ βMN = βMN = βMN = 0 (3.1.8) beta0

This is the main requirement on a string background. We conclude that string beta0 theory is consistent only on backgrounds satisfying the equations (3.1.8). It is 3.1. STRINGS IN CURVED BACKGROUND 63

beta0 important to observe that equations (3.1.8) are the equations of motion of a ten- dimensional gravity theory with action Z √ 10 2 1 2 SNSNS = d x −G R + 4(∇Φ) − 12 H (3.1.9)

The couplings of the string to the NSNS background can then be effectively described by an effective ten dimensional gravity theory. We refer to the theory as the spacetime theory. Similarly higher loop corrections to the two-dimensional beta functions will translate into α0-corrections to the spacetime field equations and will be described by higher derivative couplings in the gravity theory. On the other hand one can consider the effect of turning on a non-trivial RR background. Unfortunately, the coupling of RR fields to fundamental strings is non- minimal and can be written in a closed form. For slow varying fields, the effects of a non-trivial RR background can be extracted from string amplitudes describing small excitations of the RR field in the flat geometry. Alternatively, the low energy effective action can be derived exploiting the spacetime supersymmetry. Indeed, the two-derivative action of an N = 2 supersymmetric gravity theory in ten dimensions is completely determined by supersymmetry and go under the name of type IIA and IIB supergravities depending on whether the two super symmetries share or not their chirality. The two supergravities describe the low energy limit of type IIA and type IIB superstrings respectively.

3.1.2 Heterotic strings

For heterotic strings, the couplings of the NSNS fields g, B, Φ to the worldsheet Sx2 bosons XN is given again by (3.1.1), while for gauge fields one finds the extra coupling (in the fermionic formulation) 1 Z √ S = − d2σ hhabA (X)∂aXM (λT I λ) (3.1.10) A 4πα0 MI with T I the generators of the gauge group. To order α0, this extra coupling does betas not modify the equations of motion (3.1.7), while the beta function associated to this coupling results into

A 0 1 N 1 LP N βM = α 2 D FNM + 4 HM FLP + FM ∂N Φ + ... (3.1.11) The requirement of vanishing of this beta function gives the equation of motion of the vector ﬁeld in the low energy spacetime theory. Again higher loop corrections 64 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS provide α0 corrections corresponding to higher derivative couplings in the spacetime theory. Once again supersymmetry can be used to determine the complete low energy action. Indeed, the form of the two-derivative eﬀective action of an N = 1 supergravity theory in ten dimensions is completely determined by supersymmetry and will be described in the next section

3.2 Low energy Supergravities

In this section we discuss the supereravities in ten and eleven dimensions. Eleven dimension is the maximal dimension where a supergravity theory can be deﬁned, and the theory is unique. Under dimensional reduction to ten dimensions it reduces to type IIA supergravity. Ten-dimensional supergravities describe the low energy dynamics of ten dimensional string theories. These theories provide an eﬀective description of the low energy dynamics of massless string modes when the coupling

2 7 04 2k10 = (2π) α (3.2.12) describing the strength of gravitational interactions is small. There are two different looking for the supergravity action depending on whether we use the so called String frame or the Einstein frame. In the string frame, kinetic terms for NSNS spacetime fields are weighted by a factor e−2Φ consistent with the fact that the minimal coupling of a graviton to NSNS fields arise from three point functions on the sphere. Similarly, kinetic terms for open string fields are weighted by e−Φ since they couple at the level of the disk. Finally RR fkinetic terms carry no dilaton dependence. On the other hand the Einstein frame the Einstein Hilbert term R comes with no-dilaton dependence. The two frames are related by the metric rescaling Φ string Einstein − 2 gMN = e gMN (3.2.13) and consequently1

√ −2Φ √ 9 2 9 2 gst e Rst = gE RE − 2 (∂Φ) − 2 ∇ Φ (3.2.15) 1Here we use the identities

γ 2 2 RMN = RMN − (d − 2) [∇M ∂N γ − ∂M γ∂N γ] + gMN ∇ γ − (d − 2)(∂γ) Rγ = e−γ R − (d − 1)(d − 2)(∂γ)2 + (d − 1)∇2γ (3.2.14) relating the metrics g and gγ = eγ g in d dimensions. 3.2. LOW ENERGY SUPERGRAVITIES 65

The last term is a total derivative and can be discarded from the action. We will write the ten dimensional supergravity actions and the supersymmetry variations in the string frame and the equations of motion in the Einstein frame. For simplicity we will display only the bosonic part of the actions.

3.2.1 Eleven dimensional supergravity

The ﬁeld content of eleven dimensional supergravity is made of a metric gMN , a three form AMNP and a gravitino ΨMΛ. with M = 0,... 10 and Lambda = 1,... 32 running in the vector and spinorial representations of the SO(1, 10)) Lorentz group. The bosonic part of the supergravity action can be written as Z Z 1 10 √ 1 2 1 SIIA = 2 d x g R − F4 − 2 A3 ∧ F4 ∧ F4 (3.2.16) 2κ11 2 4! 12κ11 with

F4 = dA3 (3.2.17) The theory is invariant under the supersymmetry variations

a a δeM = i ¯Γ ΨM (3.2.18) δA = 3i ¯Γ Ψ MNP [MN P ] √ 1 2 δΨ = D (w ˆ) + Γ NP QR − 8 δN ΓP QR Fˆ M κ M 288 M M NP QR with i ¯ PQ wˆMab = wMab − 4 ΨP ΓNab ΨQ ˆ ¯ FMNPQ = 4∂[M ANPQ] − 3Ψ[M ΓNP ΨQ] (3.2.19) The bosonic equations of motion can be written as

(4) RMN − gMN R = TMN √ 1 ∂ ( gF MNPQ) = − NP QM1...M8 F F (3.2.20) M 2 4!2 M1...M4 M5...M8 (4) with TMN the stress energy tensor of the four form field. In general, we define the (n) stress energy tensor TMN of an n-form field strength as 1 1 T (n) = F F P2...Pn − g F 2 (3.2.21) MN 2 (n − 1)! MP2...Pn N 2n MN On the other hand supersymmetry solutions are given as solutions of the Killing spinor equations

δΨM = 0 (3.2.22) 66 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

3.2.2 Type IIA supergravity

Type IIA supergravity can be found dimensional reduction of eleven dimensional supergravity down to ten dimensions. More precisely, the eleven dimensional metric and three-form ﬁeld are related to the ﬁelds of ten-dimensional type IIA supergravity via the dctionary

(11) 2Φ (10) 4Φ 2 M N − 3 M N 3 10 M 2 ds11 = gMN dx dx = e gMN dx dx + e (dx + AM dx ) (11) 2 (11) CMN10 = 3 BMN CMNP = CMNP (3.2.23)

Taking all eleven-dimensional ﬁelds independent of the x10 coordinate and reducing along this direction one ﬁnds the type IIA supergravity bosonic action " # 1 Z √ X 1 S = d10x g e−2Φ R + 4(∂Φ)2 − 1 H2 − F 2 IIA 2k2 12 2n! n 10 n=2,4 1 Z − 2 B ∧ dC3 ∧ dC3 (3.2.24) 4k10

2 2 with k10 = k11/(2πR),

n H = dB Fn = dCn−1 − H ∧ Cn−3 ∗ F10−n = (−) 2 ∗ Fn (3.2.25) and Cp+1, B2 the RR and NSNS forms respectively. Supersymmetry variations can susy11 be found from (3.2.18) after dimensional reduction down to ten dimensions.

3.2.3 Type IIB supergravity

The bosonic action of type IIB supergravity can be written as " # 1 Z √ X 1 1 S = d10x g e−2Φ R + 4(∂Φ)2 − 1 H2 − F 2 − F 2 IIB 2k2 12 2n! n 45! 5 10 n=1,3 1 Z − 2 C4 ∧ H ∧ F3 (3.2.26) 4k10 with

n−1 H = dB Fn = dCn−1 − H ∧ Cn−3 ∗ F10−n = (−) 2 ∗ Fn (3.2.27) and Cn, B the RR and NSNS forms respectively. We notice that the ﬁve-form ﬁeld

F5 deﬁned in this way is self-dual. This condition should be imposed in addition to the equations of motion of type IIB theory if we look for solutions. 3.2. LOW ENERGY SUPERGRAVITIES 67

SL(2, Z)-covariant action

The type IIB supergravity action has a hidden SL(2, Z) invariance that becomes manifest in the Einstein frame where the action can be written as Z M 1 10 √ ∂M τ∂ τ¯ 1 2 1 2 SIIB = 2 d x g R − 2 − |G3| − F5 2κ0 2τ2 2 3! 4 5! Z 1 1 ¯ + 2 C4 ∧ G3 ∧ G3 (3.2.28) 8iκ0 τ2 with −Φ −Φ τ = C0 + ie G3 = F3 − ie H3 = dC2 − τ dB2 (3.2.29)

The action is invariant under the SL(2, Z) symmetry ! ! ! a τ + b C a b C τ → 2 → 2 (3.2.30) c τ + d B2 c d B2

This symmetry is a symmetry of the classical action and as we will later see an exact symmetry of the full quantum theory.

3.2.4 Type I supergravity

The bosonic action of type I supergravity couple to a N = 1 vector multiplet in the adjoint of a group G can be written as Z 1 10 √ −2Φ 2 1 2 SI = 2 d x g e R + 4(∂Φ) − F3 2κ10 2 3! Z 1 10 √ −Φ 2 − 2 d x g e Tr F (3.2.31) 2g10 with 1 α0 2 = 2 (3.2.32) g10 4κ0 and α0 F = dC − [ω (A) − ω (Ω)] 3 2 4 3 3 2i ω3(A) = Tr A ∧ dA − 3 A ∧ A ∧ A 2 ω3(Ω) = Ω ∧ dΩ + 3 Ω ∧ Ω ∧ Ω (3.2.33) where A, Ω are the gauge and spin connections. 68 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

3.2.5 Heterotic superstrings

The low energy eﬀective action describing ten-dimensional heterotic strings with gauge group G = E8 × E8,SO(32) is described by the bosonic action Z 2 1 10 √ −2Φ 2 1 2 k10 2 SI = 2 d x g e R + 4(∂Φ) − H − 2 Tr F 2κ10 2 3! 2g10 (3.2.34) with α0 H = dB − ω (A) (3.2.35) 4 3 where A the gauge connection.

3.3 Killing spinor and equations of motion

In this section we collect some solutions of ten and eleven dimensional supergravities preserving half of the super symmetries of the supergravity theory. The solutions will typically involve a non-trivial metric and a non-trivial n-form ﬂux and in the ten dimensional case also a non-trivial dilaton. Supersymmetric solutions are found by solving the Killing spinor equations deﬁned by setting to zero all fermionic variations in a bosonic background. For a type II string the Killing spinor equations can be written in the form 1 / eΦ P / δΨM = ∇M + 4 HM P + 16 F n ΓM Pn = 0 (3.3.36) n / 1 / eΦ P n / δλ = ∂Φ + 2 H P + 8 (−1) (5 − n)F n Pn = 0 (3.3.37) n with ΨM , and λ some doublets containing two Majorana–Weyl spinors of opposite(same) chirality for type IIA (IIB) ! ! L L Γ11 = (3.3.38) R ∓R with upper and lower signs referring to type IIA and type IIB respectively. ∇M is the standard covariant derivative, and ΓM are the ten-dimensional Dirac matrices. The sum runs over n ∈ {0,... 10} with n even for type IIA and odd for type IIB,

n−1 Fn = dCn−1 − H ∧ Cn−3 ∗ F10−n = (−1) 2 Fn (3.3.39) 3.4. GRAVITATIONAL BACKGROUNDS 69 and 1 H/ = 1 H ΓMNO H/ = 1 H ΓNO F/ = F ΓM1...Mn 3! MNO M 2 MNO n n! M1...Mn (3.3.40)

The projection matrices P, Pn are given by

3 n/2 1 IIA P = σ Pn = σ3 σ 3 1 2 IIB P = −σ P1,5,9 = σ P3,7 = i σ . (3.3.41)

Killing spinor equations imply the equations of motion.The converse is not true and indeed non-supersymmetric solutions of the equations of motion can in general be found. For a supergravity background involving a non-trivial metric , dilaton Φ, and a ﬂux Fn the bosonic action in this frame read

1 Z √ eanΦ S = ddx g R − 1 (∂Φ)2 − F 2 (3.3.42) 2κ 2 2n! n with a = −1 for a NS-NS ﬂux F = dB and an = (5 − n)/2 for RR ﬂuxes. The equations of motion in the Einstein frame

1 1 anΦ P2..Pn n−1 2 RMN = 2 ∂M Φ ∂N Φ + 2n! e n FMP2...Pn FN − d−2 gMN Fn 1 √ a 2Φ = √ ∂ ( g∂M Φ) = n ean Φ F 2 g M 2 n! n √ anΦ MN2..Nn ∂M ( ge F ) = 0 (3.3.43)

3.4 Gravitational backgrounds

The simplest solutions to the supergravity equations of motions are those with no ﬂuxes. For this choice any supergravity action reduces to the Einstein-Hilbert term Z 1 10 √ S = 2 d x g R (3.4.44) 2k0 and the equations motion require that the metric be Ricci-Flat.

RMN = 0 (3.4.45) ricciflat

Equivalently, in absence of ﬂuxes the Killing spinor equations require the existence of a covariantly constant spinor

∇M = 0 (3.4.46) constspinor 70 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

The existence of a covariantly constant spinor imply that the metric is Ricci ﬂat. constspinor To see this we can act on (3.4.46) with ∇N to ﬁnd [∇M , ∇N ] = 0 or

PQ RMNPQΓ = 0 (3.4.47)

Contracting with ΓM and using the identities

ΓM ΓPQ = ΓMPQ + gMP ΓQ − gMQΓP

RMNPQ + RMQNP + RMP QN = 0 (3.4.48) one finds Q Γ RNQ = 0 (3.4.49) that implies Ricci flat RNQ = 0. In the following we describe the main properties of Ricci flat manifolds and display some simple examples. We will first consider 1,d−1 spacetime of the form R × M2n with M2n a 2n dimensional Ricci flat manifold. After discussing the general properties of Euclidean Ricci flat manifolds we list some simple examples in four and six dimensions. Finally we display some elementary examples of Ricci flat Minkowskian metrics.

3.4.1 Kahler and complex structures

In a manifold of complex dimension n, the existence of a covariantly constant spinor implies the existence of a 2-form J and an n form Ωn deﬁned as

† Jmn = −2i γmn Ωm1...mn = −2i γm1...mn (3.4.50) constspinor The Killing spinor equations (3.4.46) imply that these forms are closed while Fierz identities can be used to show that

J ∧ Ω = 0 (3.4.51) jomega0

The form Ωn can be used to deﬁne a complex structure, i.e. a map I : TM → TM squaring to minus one. Indeed, given Ωn one can deﬁne

( MM2...M2n M n odd c (Re Ωn)NM2...Mn (Re Ωn)Mn+1...M2n I N = (3.4.52) MM2...M2n n even c (Re Ωn)NM2...Mn (Im Ωn)Mn+1...M2n with M1...M2n the completely antisymmetric tensor with 12...2n = 1 and c a normal- isation function chosen such that

P N N IM IP = −δM (3.4.53) 3.4. GRAVITATIONAL BACKGROUNDS 71

A manifold admitting a complex structure is called a complex manifold. A complex manifold can be described locally in terms of complex coordinates

(zi, z¯i) with i = 1, . . . n obtained by diagonalising the complex structure matrix I. In these coordinates, that map I can be written in the simple form

I : zi → i zi z¯i → −iz ¯i (3.4.54) and the form Ωn can be written as

Ωn = dz1 ∧ dz2 ... ∧ dzn (3.4.55)

On the other hand a manifold admitting a closed two form J is said to be a Kahler manifold. A manifold that is both complex and Kahler, admits a Ricci ﬂat metric given as

P gMN = −JMP I N . (3.4.56) where I is a complex structure induced by Ωn. This metric is called the Kahler metric. In complex coordinates, the Kahler metric and the two form can be written in the form

i j J = Ji¯j dz ∧ dz¯

gi¯j = i Ji¯j gij = g¯i¯j = 0 (3.4.57)

Moreover the conditions dJ = dΩ = 0 implies that the Kahler metric can be locally written as derivatives of a function K(zi barzi), known as the Kahler potential, according to ∂ ∂ gi¯j = K(z, z¯) (3.4.58) ∂zi ∂z¯j

3.4.2 Ricci ﬂat metrics in four dimensions

Flat space

Ricci flat metrics have been mostly studied in four dimensions where they can be viewed as the instanton solutions of gravity. The simplest Ricci-flat metric is the flat metric 4 2 X 2 ds = dxi (3.4.59) i=1 72 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

It is often convenient to use polar coordinates

θ i (ψ+φ) x + ix = r cos e 2 1 2 2 θ i (ψ−φ) x + ix = r sin e 2 (3.4.60) 3 4 2 with 0 < θ ≤ π 0 < φ ≤ 2π 0 < ψ < 4π (3.4.61) In this coordinates the ﬂat metric can be written in the form 2 2 2 r 2 2 2 Flat space : ds 4 = dr + (σ + σ + σ ). (3.4.62) R 4 3 1 2 in terms of the SU(2) Maurer Cartan forms

σ1 = sin ψdθ − cos ψ sin θdφ,

σ2 = cos ψdθ + sin ψ sin θdφ,

σ3 = dψ + cos θdφ. (3.4.63)

These forms satisfy

dσ1 = −σ2 ∧ σ3, dσ2 = −σ3 ∧ σ1, dσ3 = −σ1 ∧ σ2. (3.4.64)

Egushi-Hanson and Taub-Nut metrics maurersu2 The forms (3.4.63) can be used to write some simple examples of Ricci-ﬂat but not ﬂat metric. The Egushi-Hanson (EH) metric reads

a −1 r2 a r2 ds2 = 1 − dr2 + 1 − σ2 + (σ2 + σ2) (3.4.65) EH r4 4 r4 3 4 1 2 with 0 < θ ≤ π 0 < φ ≤ 2π 0 < ψ < 2π (3.4.66) We notice that for r large the metric reduces to the ﬂat metric but the period of ψ 4 is half, so the space looks like R /Z2. The Taub-Nut (TN) metric reads 1 r + n r − n 1 ds2 = dr2 + n2σ2 + (r2 − n2)(σ2 + σ2) (3.4.67) TN 4 r − n r + n 3 4 1 2 with n an integer. The two metrics are symmetric under SO(3) rotations of the 2 2 two sphere spanned by σ1 + σ2. 3.4. GRAVITATIONAL BACKGROUNDS 73

ALE and ALF-spaces

The Egushi-Hanson and Taub-Nut metrics admit a multi-center generalisation, the so called A-series of the ALE metric: (asymptotically locally Euclidean)

2 −1 i 2 2 dsALE = V (dτ − Aidy ) + V dyi N X Qi V = ∂ A = ∂ V (3.4.68) |y − y | [i j] ijk k i=1 i with τ ∈ [0, 2π]. The case N = 2 is equivalent to the Egushi-Hanson metric. On the other hand the multi-center Taub-Nut, known also as ALF space is deﬁned by the metric

2 −1 i 2 2 dsALF = V (dτ − Aidy ) + V dyi N X Qi V = 1 + ∂ A = ∂ V (3.4.69) |y − y | [i j] ijk k i=1 i

2 2 2 We remark that the ﬁve-dimensional ds5 = −dt + dsALF is known as the ﬁve- dimensional Kaluza-Klein monopole.

3.4.3 Ricci ﬂat metrics in six dimensions

Flat space

The ﬂat metric in six-dimensions can be written in polar coordinates as

2 2 2 r 2 2 2 2 2 2 2 ds 6 = dr + cos θ(σ + σ + σ ) + r (σ + σ ). (3.4.70) R 4 1 2 3 b1 b2 with

2 2 2 2 2 σ1 + σ2 = δθ + sin θ dφ (3.4.71) 2 2 2 2 2 σb1 + σb2 = δθb + sin θb dφb (3.4.72)

σ3 = dψ + cos θdφ, (3.4.73) with 0 < θ, θb ≤ π 0 < φ, φb ≤ 2π 0 < ψ < 4π (3.4.74) 74 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

Conifold and Cones over Yp,q

Here we list some classical examples of Ricci ﬂat metrics in six dimensions: Conifold: r2 r2 ds2 = dr2 + (σ + cos θdφ)2 + σ2 + σ2 + σ2 + σ2 (3.4.75) Conifold 9 3 6 1 2 b1 b2 Cone over Y p,q: 2 2 2 2 dy 1 − y 2 2 ds p,q = dr + r + σ + σ C(Y ) w(y)q(y) 6 1 2 q(y) + σ2 + w(y)(dτ + f(y)σ )2 9 3 3 with 2(a − y2) a − 3y2 + 2y3 a − 2y + y2 w(y) = q(y) = f(y) = (3.4.76) 1 − y a − y2 6(a − y2) and 1 p2 − 3q2 p a = a = − 4p2 − 3q2 (3.4.77) p,q 2 4p3 given in terms of two coprime integers p, q.

3.4.4 Waves

In any dimension d, a Ricci ﬂat metric is given by the so called wave metric

2 2 2 2 ds = −dt + H(y)(dx − dt) + dyi (3.4.78) with i = 1, ..(d − 2) and d−2 Q X H = r2 = y2 (3.4.79) rd−4 i i=1 3.5 Branes in 11D supergravity

3.5.1 M2 and M5 brane

There are two basic brane solutions in eleven dimensional supergravity: M2-brane:

2 1 2 − 3 2 3 2 ds = H dxk + H dx⊥ −1 Fm012 = −∂mH (3.5.80) 3.6. BRANES IN TYPE II SUPERGRAVITIES 75 with c N H = 1 + M2 r2 = x2 c = 25 π2`6 (3.5.81) r6 ⊥ M2 11 M5-brane:

1 2 2 − 3 2 3 2 ds = H dxk + H dx⊥

Fm1...m5 = m1...m5m6 ∂m6 H (3.5.82) with c N H = 1 + M5 r2 = x2 c = π `3 (3.5.83) r3 ⊥ M2 11

3.6 Branes in type II supergravities

3.6.1 The fundamental string, NS5 and Dp branes

There are three basic brane solutions in type II supergravities: The fundamental string:

2 −1 2 ds = H dxk + dx⊥ 2ϕ 2 −1 e = gs H −1 Hm01 = ∂mH (3.6.84) with c N H = 1 + F 1 r2 = x2 c = 25 π2 (α0)3 g (3.6.85) r6 ⊥ F 1 s

The solution preserves half the super symmetries with the Killing spinor = (L, R) satisfying 01 01 L = Γ L R = −Γ R (3.6.86) The NS ﬁvebrane:

2 2 2 ds = dxk + H dx⊥ 2ϕ 2 e = gs H

Hmnp = mnpq∂qH (3.6.87) with mnpq the ﬂat epsilon tensor and c N H = 1 + NS5 r2 = x2 c = α0 (3.6.88) r2 ⊥ NS5 76 CHAPTER 3. LOW ENERGY EFFECTIVE ACTIONS

The solution preserves half the super symmetries with the Killing spinor = (L, R) satisfying 0...5 0...5 L = Γ L R = ±Γ R (3.6.89) with upper and lower signs referring to type IIA and type IIB respectively. Dp-branes :

1 1 2 − 2 2 2 2 ds = H dxk + H dx⊥ 3−p 2ϕ 2 2 e = gs H −1 Fm0,..p = ∂mH (3.6.90) with

cpN 2 2 07−p 5−p 7−p H = 1 + r = x c = g α 2 (4π) 2 Γ (3.6.91) r7−p ⊥ p s 2 In the case of the D3 brane, the formulas for the metric and the dilaton are still valid but the ﬁve form ﬂux is replaced by the self-dual form

−1 F5 = (1 + ∗10) dH (3.6.92)

Dp-brane solutions preserve half the super symmetries with the Killing spinor =

(L, R) satisfying 0...p L = Γ R (3.6.93) Appendix A

Spinor and gamma matrices

A.1 The Cliﬀord algebras

A.1.1 The Euclidean case

The Cliﬀord algebra in d Euclidean directions is deﬁned by

{γM , γN } = 2δMN 1 , (M,N = 1, . . . d) . (A.1.1) cliffd

The γ-matrices can be chosen to satisfy

† γM = γM . (A.1.2) propgammaeuc

Similarity transformations by means of unitary matrices,

γM → γ0M = U −1γM U, (A.1.3) uneq cliffd propgammaeuc preserve the algebra (A.1.1) and the hermiticity properties (A.1.2). In even dimensions d = 2n we can introduce the chirality matrix

γ = (−i)nγ1γ2 . . . γ2n , (A.1.4) chir2n with the following properties:

γ, γM = 0, γ2 = 1 , γ† = γ . (A.1.5) chirprop

Given a representation of the 2n-dimensional Cliﬀord algebra by means of γm, with m = 1,... 2n, one immediately obtains a representation (acting on a car- rier space of the same dimension) of the 2n + 1-dimensional algebra by taking, in

77 78 APPENDIX A. SPINOR AND GAMMA MATRICES additions to the above matrices, γ2n+1 = ±γ, namely setting

γM = (γm, γ) (A.1.6) g2np1 orγ ˆM = (γm, −γ). The two choices lead to two inequivalent representations. It is possible to show that when d = 2n all Clifford algebra representations of the same dimension are unitarily equivalent; while for d = 2n + 1, instead, there are two classes of inequivalent representations, of which two representatives are given by the γM andγ ˆM defined above. In even dimension we can define the “charge conjugation matrix” C satisfying1

T CγM C−1 = (−)n γM . (A.1.7) Cfirst

The same charge conjugation matrix can be used In d = 2n + 1.

A.1.2 The Minkowski case n the Minkowskian space, with a metric ηMN of signature (1, d − 1), we have

γM , γN = 2ηMN 1 . (A.1.8) cliffdm

The Minkowskian Cliﬀord algebra can be obtained from the Euclidean one by sending γd → γ0 = −iγd . (A.1.9) wickgamma

After the Wick rotation to the Minkowskian case, we have the same for M 6= 0, and 2 † γ0 = −1 , γ0 = −γ0 ; (A.1.10) propgamma0 altogether we can then write the hermiticity properties of the Minkowsian matrices as † −1 γM = γ0γM γ0 = −γ0γM γ0 . (A.1.11) hermmink

The hermitean conjugation involves thus, on the representation space on which these matrices act (i.e., on the spinors) a multiplication by γ0.

1 More generally one can introduce two charge conjugation matrices C± satisfying M −1 M T C±γ (C±) = ∓ γ . Here for concreteness we will always take C+ for n odd and C− for n even. For this choice the charge conjugation matrices in d = 2n and d = 2n + 1 can be taken to be the same. A.2. SPINOR REPRESENTATIONS 79

For the chirality matrix we take

γ = (−i)n−1γ0γ1γ2 . . . γ2n−1 , (A.1.12) chir2nm

It enjoys the same properties as in the Euclidean case; in fact, it is the same matrix, rewritten in terms of the Wick-rotated matrix γ0.

A.2 Spinor representations

The construction of Cliﬀord algebra representations is instrumental to the study of so-called spinorial representations (brieﬂy, spinors) of rotation groups. Spinors of SO(d) are representations of its covering group Spin(d) which, as representations of SO(d), are multi-valued. d=2 dimensions

A representation of the Cliﬀord algebra In two dimensions is given by

γ1 = σ1 , γ2 = σ2 , γ = σ3 ,C = σ2 (A.2.13) gs with ! ! ! 0 1 0 i 1 0 σ1 = σ2 = σ3 = . (A.2.14) 1 0 −i 0 0 −1 the Pauli matrices. It is convenient to introduce the combinations γ1 ± iγ2 γ± = (A.2.15) gpm 2 or in matrix notation ! ! 0 1 0 0 γ+ = , γ− = . (A.2.16) gsisf 0 0 1 0

The operators γ± acts as the creation and annihilation operators on a two-dimensional fermionic Fock space with states ! ! 0 1 |−i = , |+i = (A.2.17) foisv 1 0 80 APPENDIX A. SPINOR AND GAMMA MATRICES

SO(2n) spinors: Spin basis

Explicit representations of the Cliﬀord algebra can be realised in terms of γ-matrices in 2n-dimensions constructed as follows:

γ1 = σ1 ⊗ 1 ⊗ 1 ⊗ ... ⊗ 1 γ2 = σ2 ⊗ 1 ⊗ 1 ⊗ ... ⊗ 1 γ3 = σ3 ⊗ σ1 ⊗ 1 ⊗ ... ⊗ 1 γ4 = σ3 ⊗ σ2 ⊗ 1 ⊗ ... ⊗ 1 (A.2.18) gm2bis ... γ2n−1 = σ3 ⊗ σ3 ⊗ σ3 ⊗ ... ⊗ σ1 γ2n = σ3 ⊗ σ3 ⊗ σ3 ⊗ ... ⊗ σ2 .

The chirality matrix is given by

γ = σ3 ⊗ σ3 ⊗ σ3 ⊗ .... (A.2.19) chir2ne

This is a diagonal matrix, so we have constructed a chiral representation of the

Cliﬀord algebra. In particular a state identiﬁed by the signs {si} has chirality +1 if it has an even number of minuses. The charge conjugation matrices can be written as

C = σ2 ⊗ (σ1 ⊗ σ2)m n = 2m + 1 (A.2.20) C = σ2 ⊗ (σ1 ⊗ σ2)m n = 2m

In the Minkowskian case, instead of γ2n we have

γ0 = −iσ3 ⊗ σ3 ⊗ σ3 ⊗ ... ⊗ σ2 . (A.2.21) gm0sf

SO(2n) spinors: Chiral basis

Except in the d = 2 case, the spin-ﬁeld basis described above is not chiral, because chir2ne the chirality matrix (A.2.19) is diagonal, but not in a block form (1, −1). A representation where the chirality matrix takes this block form is given by the so called A.2. SPINOR REPRESENTATIONS 81 chiral basis. γ1 = σ1 ⊗ σ1 ⊗ σ1 ⊗ ... ⊗ σ1 ⊗ σ1 , γ2 = σ2 ⊗ σ1 ⊗ σ1 ⊗ ... ⊗ σ1 ⊗ σ1 , γ3 = σ3 ⊗ σ1 ⊗ σ1 ⊗ ... ⊗ σ1 ⊗ σ1 , γ4 = 1 ⊗ σ2 ⊗ σ1 ⊗ ... ⊗ σ1 ⊗ σ1 , γ5 = 1 ⊗ σ3 ⊗ σ1 ⊗ ... ⊗ σ1 ⊗ σ1 , (A.2.22) gm2bisc γ6 = 1 ⊗ 1 ⊗ σ2 ⊗ ... ⊗ σ1 ⊗ σ1 , ... γ2n−2 = 1 ⊗ 1 ⊗ 1 ⊗ ... ⊗ σ2 ⊗ σ1 , γ2n−1 = 1 ⊗ 1 ⊗ 1 ⊗ ... ⊗ σ3 ⊗ σ1 , γ2n = 1 ⊗ 1 ⊗ 1 ⊗ ... ⊗ 1 ⊗ σ2 . The chirality matrix takes the simple form

γ = 1 ⊗ 1 ⊗ ... ⊗ 1 ⊗ σ3 . (A.2.23) chirs2n

The charge conjugation matrix can be written as C = (σ2 ⊗ σ3)m ⊗ σ2 n = 2m + 1 (A.2.24) C = (σ2 ⊗ σ3)m n = 2m We remark that the γ matrices in this basis can be deﬁned iteratively following the recursion rule µ µ 1 γ(d+2) = γ(d) ⊗ σ µ = 1, . . . , d , d+1 1 γ(d+2) = γ(d) ⊗ σ , (A.2.25) 2nto2nm2 d+2 2 γ(d+2) = 1d ⊗ σ , 1 1 2 2 with d = 2n and the “initial condition”: γ(2) = σ and γ(2) = σ . 2nto2nm2 Eq. A.2.25 can also be written as follows: ! 0 Σ¯ M γM = ,M = 1,..., (d + 2) (A.2.26) dec2nblocks (d+2) ΣM 0 with M µ Σ = (γ(d), γ(d), i1) , (A.2.27) SigmaM ¯ M µ Σ = (γ(d), γ(d), −i1) . For example in d = 4 one ﬁnds ! 0 σ¯µ γµ = , (A.2.28) odg4 σµ 0 82 APPENDIX A. SPINOR AND GAMMA MATRICES with Euclideanσ ¯µ = (σ1, σ2, σ3, −i1) , σµ = (σ1, σ2, σ3, i1) . (A.2.29) sigmamu Minkowskianσ ¯µ = (σ1, σ2, σ3, −1) , σµ = (σ1, σ2, σ3, 1) . Appendix B

Anomalies sanomalies

B.1 Anomaly polynomials

Symmetries of classical field theories can be broken by quantum effects known as anomalies. The origin of these effects can be traced on ill-behaved Feynman diagrams involving conserved currents that do not admit a regulator compatible with the simultaneous conservation of all currents. Anomalies in local conservation laws such as gauge invariance or general covariance cause a theory to be inconsistent. Anomalous diagrams arise only at tree level and one-loop. The tree level anomalous diagrams are always reducible since a presence in the action would break explicitly the symmetry. Reducible contributions should cancel against similar contributions at one-loop. On the other hand irreducible anomalous diagrams arises only at one- loop and should cancel by itself. The anomalous diagrams at one-loop origins from chiral particles running along the loop. Since chiral particles exist only in even dimensions, theories in odd dimensions are always anomaly free.

Gauge and gravitational anomalies in D-dimensions are characterised by a ho-

mogenous polynomia lD+2 of order D/2 + 1 in the curvature two form R and the gauge ﬁeld strength F . The polynomial computes the anomalous one-loop diagrams P involving D/2+1 currents. The form of the generating function I(R,F ) = D ID+2 for the anomaly polynomials in any dimension is particularly simple. For left mov- 1 3 ing Weyl fermions of spin 2 , 2 and antisymmetric two forms the contributions to

83 84 APPENDIX B. ANOMALIES the generating function of the gravitational anomaly polynomials read

D 2 xi Y 2 I 1 (R) = x 2 sinh i i=1 2 D X2 I 3 (R) = (−1 + 2 cosh xi)I 1 (R) 2 2 i=1 D 2 Y xi I (R) = 1 A 8 tanh x i=1 i (B.1.1)

with xi the oﬀ-diagonal eigenvalues of R. Right moving particles contribute the same with opposite sign while Majorana Weyl spinors contribute half the result for a Weyl spinor of the same chirality. Similarly, gauge and mixed anomalies for a 1 Weyl spinor of spin 2 are codiﬁed in the polynomial

iF I 1 (F,R) = tr e I 1 (R) (B.1.2) 2 2

The anomaly polynomials can be conveniently written in terms of the SO(D) and gauge invariant traces

1 f = tr F n n n! D 1 2(−)n X2 r = tr R2n = x2n (B.1.3) 2n 42n 4n i i=1 with the subscript indicating the degree on F and R. One ﬁnds

2 3 r2 r2 2 r4 r2 2 r2 r4 32 r6 I 1 (R) = 1 + + + + + + + ... (B.1.4) 2 3 18 45 162 135 2835 (d − 25) (D − 49) 2 2 (D + 239) I 3 (R) = (D − 1) + r2 + r2 + r4 2 3 18 45 (D − 73) 2 (D + 215) 32 (D − 505) + r3 + r r + r 162 2 135 2 4 2835 6 1 r 56 r 4 r2 32 r3 448 r r 15872 r I (R) = − + 2 + 4 − 2 + 2 − 2 4 + 6 + ... A 8 3 45 9 81 135 2835 B.2. ANOMALIES IN TEN DIMENSIONS 85 and n r2 f1 r2 I 1 (R,F ) = n + i f1 + −f2 + + i −f3 + (B.1.5) 2 3 3 f r n r2 2 n r f r f r2 2 f r + f − 2 2 + 2 + 4 + i f − 3 2 + 1 2 + 1 4 4 3 18 45 5 3 18 45 f r f r2 n r3 2 f r 2 n r r 32 n r + −f + 4 2 − 2 2 + 2 − 2 4 + 2 4 + 6 + ... 6 3 18 162 45 135 2835 with n denoting the dimension of the representation under which the spinor ﬁeld transforms. We collect inside parenthesis polynomials of homogenous degree p in F and R contributing to the anomaly polynomial on D = 2p − 2 dimensions. Can- celation of anomalies requires that the irreducible part of the anomaly polynomials exactly cancel. This requires that either fD/2+1 and rD/2+1 are reducible or their coeﬃcients exactly cancel. In addition, the remaining part of the polynomial should exactly factories into the product of two polynomials (or a sum of factorized terms) and the theory should contain tree level anomalous couplings involving the two factors in such a way that the one-loop contributions exactly cancel against tree level anomalous diagrams. We will see that these conditions impose severe constraints on the chiral spectrum of the theory.

B.2 Anomalies in ten dimensions

The gravitational, gauge and mixed anomaly polynomials in D = 10 are given by h i I12(R) = ∆n 3 I 3 (R) + ∆n 1 I 1 (R) + ∆nA IA(R) 2 2 2 2 12 h i I12(R,F ) = ∆n 1 I 1 (R,F ) (B.2.6) 2 2 12 with ∆n the net number of chiral Weyl or self-dual forms in the theory, and the subscript denoting the restriction on the rank of the polyform. The explicit form of anomalyI anomalyI2 the polyform can be extract from (B.1.4) and (B.1.5) by restricting to polynomials of degree 6, i.e. polynomials inside the parenthesis involving r6 and f6.

B.2.1 N = 2 supergravities

There are two quantum field theories with N = 2 supersymmetry in ten dimensions: IIA and IIB supergravity. The two theories are unique, once the supersymmetry is 86 APPENDIX B. ANOMALIES chosen. The field content in each case fill a single multiplet containing the graviton: the supergravity multiplet. IIA supergravity is non-chiral and therefore anomaly free. The chiral content of IIB supergravity contains two left-moving Majorana- Weyl gravitinos, two right moving Majorana Well dilatinos and a self-dual four form. The total anomaly polynomial vanishes due to the identity h i IIB : I 3 (R) − I 1 (R) − IA(R) = 0 (B.2.7) 2 2 12 anomalyI satisfied by the anomaly polynomials (B.1.4).

B.2.2 N = 1 supergravities

The chiral content of the N = 1 supergravity multiplet in ten dimensions includes h i a left moving gravitino and a right moving dilatino. Since I 3 (R) − I 1 (R) is not 2 2 12 vanishing the pure N = 1 supergravity theory is anomalous. This can be cured by adding N = 1 gauge multiplets. The anomaly polynomial becomes

1 h i f6 16 (n − 496) r6 I12 = 2 I 3 (R) − I 1 (R) − I 1 (R,F ) = − + (B.2.8) 2 2 2 12 2 2835 (n + 224) (n − 64) f r f r2 f r + r r + r3 + 4 2 − 2 2 − 2 4 135 2 4 324 2 6 36 45

In SO(10) the trace r6 is irreducible, so cancellation of gravitational anomalies requires that the total number of vector multiplets is n = 496. Cancelation of 3 gauge anomalies requires f6 to be reducible: f6 = c1f4f2 + c2f2 . The coeﬃcients c1,2 can be ﬁxed by requiring that the whole polynomial factorizes into two pieces

I12 = X4 X8. One ﬁnds, that this happen if

n = 496 1 1 3 f6 = 720 f2 f4 − 1296000 f2 (B.2.9)

There are two groups satisfying these two requirements: G = SO(32) and G =

E8 × E8. The anomaly polynomial takes the factored form

1 2 2 I12(F,R) = 2592000 (f2 − 240 r2) f2 − 1800 f4 + 240 f2 r2 − 14400 r2 − 57600 r4 (B.2.10)