D-Brane Instantons

Instead of filling out four-dimensional space–time and wrapping a cycle of the compacti- fication manifold, it is also possible that a D-brane wraps an internal cycle only such that it is point-like from the four-dimensional point of view. One then speaks of a (Euclidean) D-brane instanton or E-instanton as it affects the calculation of correlation functions in a similar fashion as the well-known instantons in field theory. In fact, special configurations of D-branes and D-brane instantons precisely reproduce in the low-energy limit a four- dimensional gauge theory including the effects of gauge instantons [].

As in the field theory case, the effects of D-brane instantons come with the typical expo- nential factor of∼exp(−S_inst)_{and are thus highly suppressed. Only if certain couplings are}

absent in perturbation theory, for instance due to non-renormalization theorems, instanton effects may furnish the leading order contribution. The exponential suppression can poten- tially be used to explain some of the small and large hierarchies present in the standard model

 . T IIB S T  M B

and its supersymmetric extensions. Examples include Majorana masses for the right-handed neutrinos [, , ], theMSSMµ-term [, ] and top quark Yukawa couplings inSU(5) GUTtheories [, ].

In this work, we will be mainly interested in the generation of mass terms for the so far massless Kähler moduli. Intuitively it is clear that this should be possible as the instanton action depends on the volume of the internal cycle it wraps.

As there does not exist a complete second quantized version of string theory yet, the instanton calculus cannot be derived from first principles. Instead one relies on analogies to the field theory case. Doing so, we expect a contribution to the four-dimensional effective action of the form:

S_n.p.=

∫

dMe−SE(0)−SEint.(M), _(2.55)

where Mdenotes the collection of all instanton zero modes, S_E(0) _{is the classical instanton}

effective action and Sint.

E its interaction part. The integral over the fermionic zero modes vanishes if more than one zero mode is pulled down from the exponent as they are described by Graßmann variables. Hence only instantons with a very specific fermionic zero mode structure can contribute to the effective action. We encounter a similar distinction of cases as with D-branes since various E-instantons can intersect with themselves, D-branes and orientifold planes, giving rise to different sets of zero modes. In detail, there exist:

Universal zero modes. They arise from strings starting and ending on the same instanton. There are four bosonic ones xµ parameterizing the (point-like) position of the instanton in four-dimensional space–time. They can be understood as being the Goldstone bosons corresponding to the breakdown of the four-dimensional translational invariance. Univer- sal fermionic zero modes arise from the breakdown of supersymmetries. Here one has to distinguish several cases: firstly, the instanton can wrap a cycle which is not invariant under the orientifold projection and thus an image instanton has to be included. In such a config- uration, there are two chiral Goldstinos θα as well as anti-chiral ones τ¯α˙. Consequently, a contribution to an F-term can only arise if the extraτ¯α˙ are saturated. If the instanton wraps a cycle already populated by a D-brane, the extra anti-chiral modes are soaked up in a way reproducing the celebratedADHM constraints for gauge instantons. Such a configuration is in fact equivalent to an ordinary gauge instanton in the field theory living on the D-brane in question. For anSU(Nc)N = 1supersymmetricQCDwithNf =Nc−1flavors engi- neered on a stack of D-branes it was shown in [–] that indeed the Affleck–Dine–Seiberg superpotential is generated by an E-instanton. Finally, the D-brane instanton may be on top of the orientifold plane. A stack of ordinary D-branes on such a cycle can have gauge group

SO(N)_or SP(N)_{. For a D-brane instanton on this cycle the gauge group is swapped to} SP(N)_orSO(N)_{. In the first case, there are}N(N−1)/2_{zero modes}θα

andN(N+ 1)/2 ¯

.. D-B I  the case of most interest areO(1) _{instantons on top of an orientifold plane because we end}

up with the right number of two universal zero modes θα_{. For the case of E3-instantons} wrapping a four-cycleΓ4 in the internal space, which we will mainly consider in the later

chapters, the contribution to the superpotential of the low-energy effective action was al- ready pioneered in [] and turns out to be of the form:

W_inst ∼e−2πT, (2.56)

whereT is the Kähler modulus corresponding to the four-cycleΓ4.

Deformation zero modes. They stem from the transversal modes of open strings starting and ending on a particular D-brane instanton, describing infinitesimal deformations of the instan- tonic brane. Each complex valued deformation leads to one complex bosonic zero mode. One chiral and one anti-chiral Weyl spinor comprise the fermionic zero modes. Whether a cycle can be deformed or not is a question of topology. The number of deformations is counted by the Betti numbersb1 _{and b}2 _{of the cycle in question. They count the Wilson-}

line moduli and the transversal deformations respectively. Since these zero modes come in addition to the universal ones, instantons on cycles with deformations contribute only if the extra zero modes are soaked up, or lifted by flux [–, –]. The extra modes are absent if the cycle simply has no deformations, i. e. is rigid.

Charged zero modes appear at the intersection of a D-brane instanton with an ordinary, space-filling D-brane. They are called “charged” as one end of the open strings at the intersection locus is attached to the (stack of) D-brane(s) and hence is charged under the four-dimensional gauge group of it. We specialize now to the case of an O(1) instanton. The number of charged zero modes is then counted by the intersection number IE,Da,

defined in (.). They carry the total U(1)_{a charge} Qa(E) = Na(IE,Da − IE,Da0). In

order to contribute to the superpotential, the additional charged zero modesλ _{have to be}

soaked up. This works with the help of couplings of the formλE aiΦaibiλbiE appearing in Sint.

E . Hence the saturation of theλzero modes pulls down charged matter fieldsΦaibi in the

fermionic integral in such a way that theU(1)_a charges are preserved. Consequently there appear terms in the superpotential of the form

W = M ∏ i=1 Φaibie −SE_. (2.57) Here, the charge of the product of the matter fields is canceled by the sum of the charges of the zero modes:

M

∑

i=1

Qa(Φaibi) = −Na(IE,Da−IE,D0a), (2.58)

 . T IIB S T  M B a charged superpotential (.) will be discussed in chapter  in detail.

Multi-instanton zero modes. In the same way a D-brane instanton can intersect a D-brane, it can also intersect another E-instanton. Then zero modes arise at the intersection locus and are therefore called multi-instanton zero modes. For the case we are interested in, namely an E3-instanton with gauge groupO(1) _{(hence lying on the orientifold plane), the number}

of zero modes is given by the same expression as for D-brane case, given in table ..