LEISHMANIA CUTANEA

one obtains

Wnp ∼ gi e−2πVi. (7.3)

So hi = 2π. Instead of wrapping D3–branes on Si, we may also consider a stack of N

space-time-filling D7–branes wrapped on Si. In general orientifold compactifications,

the existence of the D7–branes is in fact forced by the tadpole cancellation conditions. Consider the open string spectrum on the D7–branes. It is in general given by an effective N = 1 supersymmetric U(N) gauge theory with some additional matter fields.

First consider pure N = 1 Yang–Mills theory with gauge group G without any matter fields. Gaugino condensation generates a nonperturbative superpotential

Wnp∼Λ3 =e

−8π2 bg2,

with eq. (7.2) we then get

Wnp ∼ gi e−

2_πVi

b , (7.4)

and hence hi = 2π/bfor pure SQCD.

Now consider N = 1 SQCD with gauge group G= SU(NC) and with NF matter

fields Q, Qe in the fundamental plus anti–fundamental representationsN_F(N_C⊕N¯_C).

For NF < NC, there is a dynamically generated superpotential (for a review see e.g.

[60]) Wnp= (Nc−Nf) Λ3NC−NF det(QQe) 1/(Nc−Nf) . (7.5)

Here, b= 3NC−NF is theN = 1 β–function coefficient of SQCD. The vacuum expec-

tation values of the meson superfieldsM ∼QQebreak the gauge groupSU(N_C) to the

non-Abelian subgroup SU(NC−NF). IfNF =NC−1, the superpotential is generated

by gauge instantons. On the other hand, the superpotential arises due to the gaug- ino condensation in the unbroken SU(NC −NF) gauge group. Therefore the gaugino

condensate is determined by the scale of the unbroken gauge group, hλλi ∼Λ3_NC−NF, where the scale ΛNC−NF of the low-energy SU(NC −NF) gauge theory can be associ-

ated to the scale Λ of the high-energy gauge theory as Λ3(NC−NF)

NC−NF = Λ

3NC−NF_/detM.

This precisely yields the effective superpotential eq. (7.5). Finally, for NF ≥NC there

is no dynamically generated superpotential of this type.

7.3

Witten’s criterion and the index on

D3–branes

Which of the divisors present in our compactification manifold give rise to a non- perturbative superpotential? The prefactor gi to the superpotential (7.1) generically

depends on the complex structure moduli and comes from a fermionic one-loop de- terminant. Unfortunately it is so far an unsolved question how to compute it for the general case1.

1_{In [61],g}

The best one can do is decide whether gi = 0 or not. In the framework of M/F

theory, Witten has shown [62] that Euclidean M5-brane instantons wrapping a divisor

e

S in a Calabi–Yau four–fold X4 give rise to a non–perturbative superpotential if the

holomorphic Euler characteristic χ(O_Se) of the divisor fulfills

χ(O_Se) =h0,0(Se)−h0,1(Se) +h0,2(Se)−h0,3(Se) = 1. (7.6)

One arrives at this criterion by studying the fermionic zero modes of the Dirac operator on the world–volume of the Euclidean M5–brane. The criterion (7.6) is fulfilled if exactly two fermionic zero modes are present.

Later on it was realized that turning on background flux can have the effect of lifting zero-modes. It can therefore happen that even if there are too many zero-modes in the original geometry to fulfill (7.6), a non-perturbative superpotential may be generated [61, 63, 64, 65, 66, 67].

Witten’s criterion can be used for models in type IIB string theory if their lift to F-theory is known. Conditions for the generation of the superpotential directly for type IIB–orientifolds without the detour of analyzing the M/F–theory case first have been worked out in [68, 69], where an index χD3 for the Dirac operator on the

D3–brane was proposed. In terms of this index, the condition for the generation of a non–perturbative superpotential for the wrapped divisor S takes the form

χD3(S) =

1

2(N+−N−) = 1, (7.7) whereN± is the number of fermionic zero modes with U(1) charge ±1₂ in the direction

normal to S. The presence of background fluxes can give rise to zero modes of mixed chirality, in which case the index changes and is not of purely geometric nature any- more. The fermionic zero modes on the world–volume of the D3–brane can be related to the Hodge numbers h(0,0)_{, h}(1,0)_{, h}(2,0) _{of S by mapping the spinors to (0, p)–forms.}

The spinors living on the word–volume of the D3-brane can locally be expressed as

+ = φ|Ω>+φa¯γ¯a|Ω>+φabγ

ab_|Ω>,

− = φz¯γz¯|Ω>+φazγaz|Ω>+φ_zabγzab|Ω> . (7.8)

Here, + (−) denotes the spinor with positive (negative) chirality with respect to

the structure group SO(2) of the normal bundle of the divisor S inside the compact space. |Ω > denotes the fermionic Clifford–vacuum, while the γs are products of γ– matrices. The γ–matrices with indices a, b, z etc. act as creation operators on the Clifford vacuum. a, b label the directions in the D3–brane world–volume, z denotes the direction in the Calabi–Yau manifold normal to the wrapped divisor. Note that+

and − also carry an SO(1,3) spinor index, so the number of zero modes is doubled.

The zero–modes of negative chirality have one leg in the direction normal to the divisor

S. They can be related to the zero–modes on the world–volume using Serre–duality:

gz¯zΩ_abzφz =φe_ab, gz¯zga¯aΩ_abzφ_az =φe_a, gz¯zga¯agb

¯_b

7.3 Witten’s criterion and the index on D3–branes 103

where Ω is again the (3,0)–form of the Calabi–Yau. We see thus that we have h(1,0)

zero modes φa of positive chirality corresponding to (0,1)–forms and another h(1,0)

zero modes of negative chirality coming from φaz, also corresponding to (0,1)–forms

via Serre duality. Analogously, we haveh(0,0) _{= 1 scalar zero modes of positive chirality}

(φ) and negative chirality (φ_abz), andh(2,0) (0,2)–form zero modes of positive (φ_ab) and negative chirality (φz).

On theD3–brane, the gauge fixing of the κ–symmetry must be chosen such that it is compatible with the orientifold projection. Some of the zero modes are pure gauge and are annihilated by the κ–symmetry projector. The orientifold action projects out part of the zero modes. We can distinguish three cases regarding the position of the

O7–planes in relation to the divisor S wrapped by the EuclideanD3–brane: (a). The O7–plane wraps the divisor S.

(b). The O7–plane intersects S along one complex dimension. (c). The O7–plane is parallel to S.

The analysis of the action of the projectors of κ–symmetry and orientifold on the zero modes [9] is summarized in Table 7.1. The zero modes given in square brackets

Case (a) (b) (c) Chirality + − + − + − h(0,0) _{φ – φ – φ φ} abz h(1,0) _{– φ} az [φa] φaz [φa] φaz h(2,0) _[φ ab] – – φz [φab] φz χD3(S) 1−h (1,0) (−) + [h (2,0) (+) ] 1 + [h (1,0) (+) ] [h (1,0) (+) ]−h (1,0) (−) −h(1,0)₍₋₎ −h(2,0)₍₋₎ +[h(2,0)₍₊₎ ]−h(2,0)₍₋₎

Table 7.1: Surviving zero modes after κ–fixing and orientifold projection are the ones that are in general lifted by background flux. Furthermore, h(i,0)₍₊₎ (h(i,0)₍₋₎) denotes the positive (negative) chirality zero modes corresponding to this Hodge num- ber. Note that in case (a), where the divisor S feels the full orientifold projection,

χD3(S)≡χ(OS), i.e. the index reduces to the holomorphic Euler characteristic. This

happens because of each type of zero modes, only one chirality survives the orien- tifold projection, which matches the situation on the fourfold. In case (c) on the other hand, S does not feel the effect of the orientifold projection at all, therefore (unless zero modes are lifted by flux) the positive and negative chirality zero modes compen- sate each other, such that the index equals zero. This is analogous to the case of a Calabi–Yau manifold without orientifold for which no non–perturbative superpotential is generated.

To decide whether there are enough contributions to the non-perturbative super- potential to stabilize all moduli, it is of prime importance to know the Hodge numbers

h(1,0) _{and h}(2,0) _{for a complete set of divisors in our compactification manifold. The}

geometric methods described in part I of this thesis yield all necessary information for the case of resolved orbifolds of type T6_/

Zn and T6/Zn×Zm.