Topological Amplitudes and Physical Couplings in String Theory

0.39 MB·English

by  I. Antoniadis

CERN-PH-TH/2007-024
Topological Amplitudes and Physical Couplings in String
Theory
∗
I. Antoniadisa †, S. Hoheneggera‡
aDepartment of Physics, CERN – Theory Division
7
CH-1211 Geneva 23, Switzerland
0
0
2 In these lectures, we review the main properties of the topological theory obtained by twisting the
n N = 2 two-dimensional superconformal algebra, associated to supersymmetric string compactifications.
a In particular, we describe a set of physical quantities in string theory that are computed by topological
J amplitudes. These are in general higher-dimensional F-terms in the low energy effective supergravity, or
1 fermion masses after supersymmetry breaking. We discuss N = 2 compactifications of type II strings,
3 N =1compactificationsofheteroticandtypeIstrings,aswellasN =4stringvacua. Particularemphasis
isputonalternativestringdualrepresentationsallowing calculability,andonthegeneralization ofN =2

1 holomorphicity and its anomaly.
v
0
9
Contents 4 Genus 2 Amplitudes, Supersym-
2
1 metry Breaking and Fermion
0 1 = 2 Superconformal Algebra Masses 9
N
7 and the 2d Topological Twist 2 4.1 Overview . . . . . . . . . . . . 9
0
1.1 =2 Superconformal Algebra 2 4.2 Scherk-Schwarz (SS) Mech-
/ N
h 1.2 The TopologicalTwistandthe anism for Supersymmetry
t Partition Function . . . . . . . 2 Breaking . . . . . . . . . . . . . 10
-
p 1.3 Holomorphic Anomaly Equation 3 4.3 Gravity Mediated Supersym-
e
metry Breaking . . . . . . . . . 10
h
: 2 Topological Amplitudes and = 4.4 Breaking Local Supersym-
v N
2 Higher Derivative -terms 4 metry without Breaking R-
i F
X 2.1 Physical Couplings and Topo- symmetry . . . . . . . . . . . . 12
r logical Amplitudes . . . . . . . 4 4.5 Gauge Mediated Supersymme-
a 2.2 =2HigherDerivative -terms 4 try Breaking . . . . . . . . . . 12
N F
2.3 Non-renormalization Theorem . 6
5 =4 Topological Amplitudes 13
2.4 Duality Mapping . . . . . . . . 6
N
5.1 Motivation . . . . . . . . . . . 14
5.2 Compactification on K3 . . . . 14
3 =1 Topological Amplitudes 6
N 5.2.1 = 4 Superconformal
3.1 Heterotic String Theory . . . . 6
N
Algebra . . . . . . . . . 14
3.1.1 Topological Amplitudes 6
5.2.2 Topological Amplitudes 14
3.1.2 Holomorphic Anomaly
5.2.3 Physical Couplings . . . 15
Equation . . . . . . . . 7
5.2.4 Duality Mapping . . . . 15
3.2 Type I . . . . . . . . . . . . . . 7
3.2.1 Z2 Involution . . . . . . 7 5.3 Compactification on K3×T2 . 16
5.4 Harmonicity Relation . . . . . 16
3.2.2 Scattering Amplitudes . 8
5.4.1 Harmonicity in 6 Di-
3.2.3 Holomorphic Anomaly
mensions . . . . . . . . 16
Equation . . . . . . . . 8
5.5 Harmonicity in 4 Dimensions . 17
3.2.4 Outlook to F . . . . 9
(g,h)
A Vertex Operators 18
∗Proceedings for the Cargese Summer School 2006, A.1 Type II Closed Vertex Operators 18
Lectures byI.Antoniadis
A.2 HeteroticStringVertexOpera-
†[email protected]
‡[email protected] tors . . . . . . . . . . . . . . . 18
1
2 I. Antoniadis and S. Hohenegger
1. = 2 Superconformal Algebra and 1.2. TheTopologicalTwistandthePar-
N
the 2d Topological Twist tition Function
Following [2,3,4,5] the topological field
The two-dimensional (2d) = 2 super-
N theory is obtained by twisting the energy-
conformal algebra is introduced and the topo-
momentum tensor subtracting (half of) the
logical twisting of the theory is explained.
derivative of the U(1) current:
The topological partition function is defined
and the special role of Calabi-Yau-threefold 1
T T ∂J, (7)
compactifications is shown. The holomorphic → − 2
anomaly is described, which leads to a recur-
suchthatthenewcentralchargevanishesand
sion relation for the antiholomorphic moduli
(4) of the superconformal algebra is changed
dependence of the partition function.
to
1.1. =2 Superconformal Algebra 2J(w) 2T(w)
N G+(z)G−(w) + +... (8)
Four-dimensional (4d) string compactifica- ∼ (z w)2 z w
− −
tions with = 1 space-time supersymmetry
N Thetwistalsoresultsinshiftingtheconformal
( =1+1fortypeIIstrings)aredescribedby
N dimension of all operators by half their U(1)
an underlying 2d =2 superconformal field
N charge:
theory (SCFT) [1]. The = 2 superconfor-
N
mal algebra contains the energy momentum
q h=conf. weight
tensor T, two (conjugate) supercharges G± h h , with
→ − 2 (cid:26) q =U(1) charge
and a U(1) current J:4
(9)
c
T(z)T(w) = + Thus, the new conformal weights are:
2(z w)4
−
2T(w) ∂ T(w) operator conf. weight U(1)
w
+ + ,(1)
(z w)2 z w T 2 0
− − G+ 1 1
3G±(w) ∂ G±(w)
T(z)G±(w) = + w ,(2) G− 2 1
2(z w)2 z w −
− − J 1 0
J(w) ∂ J(w)
w
T(z)J(w) = + , (3)
(z w)2 z w One can now identify G+ with the BRST
− − operator
2c 2J(w)
G+(z)G−(w) = + +
3(z w)3 (z w)2
− − Q = G+, (10)
BRST
2T(w)+∂wJ(w) I
+ , (4)
z w
− whichimmediatelyshowsthattheenergymo-
G±(w)
J(z)G±(w) = , (5) mentum tensor becomes BRST exact:
± z w
J(z)J(w) = −c . (6) QBRST,G− = G+G− =T. (11)
3(z w)2 { } I
−
Thus, G− can be identified with the
The constant c = 3cˆ is the central charge,
2 reparametrization anti-ghost which can be
while the conformal dimension and U(1)
sewedwith the (3g 3) Beltrami differentials
charge of the operators are displayed in the −
ofagenusg Riemannsurfacetodefine thein-
following table:
tegration measure over its moduli space .
g
M
It is then straight-forward to write down the
operator conf. weight U(1)
expression for the topological partition func-
T 2 0
G± 3/2 1 tion
±
J 1 0 3g−3
F = G−(µ )2 . (12)
g a
Z h| | i
4Wewillrestrictourconsiderationstotheleft-moving Mg aY=1
sector only, since the algebra of right-movers follows
One may think naively that this expression
triviallyinpreciselythesameway. Inourconventions,
right-moversaremarkedbyabar: (T¯,G¯±,J¯). vanishes by charge conservation. However, in
Topological Amplitudes and Physical Couplings in String Theory 3
the twisted theory, the anomaly (5) of the pression
U(1)currentprovidesa backgroundchargeof
3g−3
tcˆe(ggr−a1ti)oonnmgeenasuusrge.oIfntthheetcoapsoeloogficˆca=lp3,atrhtietiionn- ∂¯iFg =ZMgh| aY=1 G−(µa)|2I G+I G¯+φ¯ii,
function (12) alone is sufficient to cancel this (16)
deficit and the partition function can there-
one can contour-deform the surface integrals
fore be non-vanishing without additional op-
G+and G¯+,whichwhenhittingoneofthe
erator insertions. The ‘critical value’ cˆ=3 is
HG− and G¯H− give an insertion of the energy-
reached for compactifications on Calabi-Yau-
momentum tensor
threefolds (CY ). An interesting property of
3
these compactifications is the notion of spe- 3g−4
cial geometry which, although very impor- ∂¯iFg =Z h| G−(µa)T(µ3g−3)|2φ¯ii,
tantinmanyapplications,willnotbe studied Mg aY=1
(17)
in this review.
From (9), one immediately encounters that uptoanirrelevantnumericalcoefficientwhich
the fields, which fulfill (before the twist) we dropped.
Theenergy-momentumtensorinsertioncan
1 be re-written as a partial derivative with re-
h= q, with q = 1, (13)
±2 ± spect to the world-sheet moduli, leading to a
boundary contribution
play a special role and are called chiral (anti- 4∂2
chiral) primary fields φ (φ¯). They satisfy ∂¯iFg =Z ∂m ∂m¯ h| G−(µa)|2φ¯ii. (18)
[G+,φ] = 0. It follows that the chiral pri- Mg b b aY6=b
maries, which obtain conformal weight h = 0 Aboundaryinthemodulispaceofacompact
after the topological twist, are the cohomol- Riemannsurfacecorrespondstoasurfacewith
ogy states of the BRST operator: nodes,whichappeareitherbypinchingahan-
dle ora dividing geodesic, as shownin Figure
Q φ= G+φ=0, (14) 1. In both cases, the surface develops a long
BRST
I and thin tube, which eventually in the pinch-
ing limit is turned into two punctures. Com-
and span the physical Hilbert space. In the puting explicitly the contribution stemming
same way, the anti-chiral primaries acquire fromthese twosurfaces,the resultis givenby
conformal weight h = 1 after the twist and
g−1
astraeteBsRaSnTdsehxoauctld. dHeecnocuep,lethferyomartehuentpophyosloicgai-l ∂¯iFg =12C¯i¯jk¯e2KGj¯jGkk¯(cid:18) DjFhDkFg−h+
hX=1
calpartitionfunctionF . Thisishowevertrue
g
only upto ananomalythatwe discuss below. +D D F , (19)
j k g−1
(cid:19)
1.3. Holomorphic Anomaly Equation where D is the K¨ahler covariant derivative
i
Accordingtotheprevioussection,apertur- with respect to the chiral modulus field φ ,
i
bation of the action of the form and C¯i¯jk¯ are the Yukawa couplings given by
the three-point function of three chiral pri-
S S+t¯¯i G+ G¯+φ¯, (15) maries on the sphere
i
→ I I
C = φ φ φ . (20)
ijk i j k
h i
should be trivial, since the action is altered K is the K¨ahler potential and Gi¯j = ∂i∂¯jK
by a BRST-exact operator. In particular, an is the moduli metric related to C’s by spe-
anti-holomorphic derivative with respect to cial geometry. An important point is that
the parametert¯¯i onlyleadsto the insertionof although the right hand side of (19) is not
the operator G+ G¯+φ¯ in the topological zero, it only contains contributions of lower
i
partition funcHtion aHnd should simply yield a genus Fg’s. Hence, the holomorphic anomaly
vanishingresult. However,thisisnottruebe- leads to a recursion relation, which can be
cause the so-called holomorphic anomaly shown to be strong enough to yield the non-
spoils this reasoning [3,6]: Inspecting the ex- holomorphic part for all F ’s iteratively [3].
g
4 I. Antoniadis and S. Hohenegger
(a)
(b)
Figure1. BoundariesofaRiemannsurfaceinmodulispace: (a)pinchingofahandle,(b)pinching
of a dividing geodesic. In the limit of infinite length of the connecting tubes they are replaced by
punctures at their end-points. A non-vanishing result is only achieved, if the states propagating
on the tubes cancel the background charge for the sphere. In this case, they simply resemble the
Yukawa couplings C .
ijk
2. Topological Amplitudes and = 2 form [7]
N
Higher Derivative -terms
F (space-time operators)
A h i
Certain correlation functions of the gravi-
≃ (det(Imτ))d/2 ·
tational sector of type II string-theory com-
(σ-model of the internal theory) ,
pactified on Calabi-Yau threefolds are shown ·h i
(21)
tobecomputedbythepartition function ofthe
=2 topological string.
N where d is the number of non-compact space-
timedimensionsandτ is theperiodmatrixof
the (closed string) world-sheet. Although in
2.1. Physical Couplings and Topologi-
generalallthese quantities arepresent,in the
cal Amplitudes
casesweareafter,thespace-timepartcancels
After having introduced the topological
completely the (det(Imτ))-factors and the re-
partitionfunctionF ,afurthertaskistomake
g maining expression is only determined by the
contact with the = 2 compactifications of
zero-modepartoftheinternalσ-model. Thus,
N
typeII stringtheory.5 Wearehencetryingto
itcanbeidentifiedwithsomecorrelationfunc-
find string scattering amplitudes (possibly at
tion of the topological string (in the = 2
g-looplevel),whichcanbereducedtoF . Be- N
g case with the partition function).
fore plunging into the explicit calculation, let
us first give a flavor of how this can happen. 2.2. =2 Higher Derivative -terms
N F
The computation of scattering amplitudes Let us now study precisely for which type
in string-theory6 generically contains contri- ofstringamplitudestheabovementionedcan-
butionsfromthespace-timepartofthevertex cellation occurs. We consider type II the-
operators as well as from the internal (com- ory compactified on a Calabi-Yau threefold.
pact)sectorandthe(super)ghosts. Afterper- Thus,theunderlying =2SCFThascentral
N
forminga number ofcomputationalsteps (in- charge cˆ=3 entailing that in order to obtain
cluding spin-structure sum), a generic scat- a non-vanishing result our scattering ampli-
tering amplitude is roughly of the factorized tudes have to balance a backgroundchargeof
3(g 1).
−
Approaching the problem in a systematic
way [7], we consider which operators are nec-
essary to balance this charge. In the absence
5WeconsiderthecaseofN =(1,1)supersymmetry.
of external vertices, there are (2g 2) in-
6Throughout these lectures we will use the RNS −
sertions of picture changing operators (PCO)
(Ramond-Neveu-Schwarz) formalismtocompute am-
plitudes. which stem from the integration over super-
Topological Amplitudes and Physical Couplings in String Theory 5
moduli of the world-sheet. Since they will in the effective action:
only contribute with their internal part (in
thiscaseG−G¯−),theymakeupalready(2g− Z d4θ(Wa1b1,ijWa2b2,klǫa1a2ǫb1b2ǫijǫkl)g. (25)
2) units of the background charge. The re-
Upon performing the superspace integration,
maining (g 1) have to come from addi-
− this action term yields (among other con-
tionalPCOsbalancingthesuper-ghostcharge
tributions) precisely one term which corre-
of some additional vertex operators7. Since
sponds to the amplitude (23), involving two
we want to compute amplitudes of the grav-
self-dual Riemann tensors and (2g 2) self-
itational sector, one possibility to provide a dual graviphoton field strengths, R2−T2g−2.
total super-ghost charge of (g 1) is to in- + +
− − We can now return to the explicit evalua-
sert(2g 2)graviphotonsinthe( 1/2)-ghost
− − tionofthe amplitude (23). Thisisdoneusing
picture. In order to cancel alsothe chargeas-
the RNS formalism [10] and performing the
sociated to space-time fermionic coordinates,
spin structure sum. The result is [7]:
wechoose(g 1)ofthemtocomewithapos-
−
itive 4d helicity (+ sign in the 1st and 2nd g d2x g d2y
plane) andthe other half to come with a neg- Fg =Qi=1R(det(iImQτj=))12R j|detωi(xj)|2·
ativehelicity. Thecorrelationfunctionisthen
3g−3
schematically of the form
detω (y )2 G−(µ )2 ,
·| i j | Z h| a1 | i
g−1 g−1 Mg aY=1
V(−1/2)(T ) V(−1/2)(T ) (26)
−− ++
h(cid:16) (cid:17) (cid:16) (cid:17) ·
·VP3Cg−O3i. (22) where µa are the (3g −3) Beltrami differen-
tials parameterizing changes in the moduli of
However, it turns out that this amplitude a genus g Riemann surface. Comparing to
gives a vanishing result because of supersym- (21), the space-time correlationfunction is:
metry8. Thus, we inserttwo additionalgravi-
g g
tons in the 0-ghost picture: d2x d2y detω (x )2 detω (y )2,
i j i j i j
Z Z | | | |
Yi=1 jY=1
2 g−1
V(0)(G) V(−1/2)(T ) (27)
−−
h(cid:16) (cid:17) (cid:16) (cid:17) ·
g−1 whereω arethe g abeliandifferentialsdefined
V(−1/2)(T ) V3g−3 . (23)
·(cid:16) ++ (cid:17) PCO i on the world-sheet. The points xi and yi are
the insertion points of the self-dual gravipho-
Before we explicitly evaluate this expres-
ton and graviton vertex-operators of the two
sion, let us examine more closely what term
respectivehelicities,andtheintegrationisun-
of the effective action this amplitude actu-
derstood over the homology cycles. It can be
ally represents. The gravitons as well as the
shown, that these integrals give (up to a con-
graviphotons are part of the 4 dimensional
stant factor which we neglect)
=2 supergravity multiplet. Supplemented
N g g
by two spin 3/2 gravitini, the component ex-
d2x d2y detω (x )detω (y )2
pansion of this Weyl chiral superfield reads Z i Z j| i j i j | ≃
Yi=1 jY=1
[8,9]
(det(Imτ))2, (28)
≃
Wab,ij =Tab,ij +θc,[iψcja]b+ canceling as advertised the corresponding
+(σµν) bθi,c(σρτ) dθjR , (24) space-time zero-mode contribution in the de-
a c d µνρτ nominator of (26). The final expression is
hence given by the topological expression
where a,b,c,d = 1,2 are (chiral) spinor in-
dices, i,j = 1,2 are SU(2) indices labeling 3g−3
the supersymmetries and µ,ν,ρ,τ = 0,...,3 F = G−(µ )2 . (29)
g a
Z h| | i
are 4 dimensional space-time indices. On can Mg aY=1
then constructthe following 1/2-BPS -term As already mentioned, the remarkable fact
F
aboutthisresultisthatitispreciselythepar-
7FordefinitionofvertexoperatorsseeAppendixA.
8Technically, this manifests itselfinthe factthat the titionfunctionofthe =2topologicalstring
N
sumoverspinstructuresgiveszeroresult. defined in (12).
6 I. Antoniadis and S. Hohenegger
2.3. Non-renormalization Theorem 6 dimensions. There, it was observed that
In order to study possible perturbative or the moduli space of the heterotic string com-
non-perturbative corrections to the above- pactified on T4 is identical to that of type
mentioned topological amplitudes, it is nec- IIAcompactifiedonK3,andthe twotheories
essarytofigureoutthe precisedilatondepen- can be identified upon inversion of the six-
dence of the action term dimensional (6d) string coupling. From this
resultonecanderivefurtherdualitiesbycom-
√GR2T2g−2. (30)
pactifying down to 4 dimensions. In particu-
lar, one can establish a duality between type
In the string frame, this expressionis propor-
IIAonaK3-fiberedCalabi-Yauthreefoldand
tional to
theheteroticstringonK3 T2. Keepingtrack
×
e2ϕ(g−1) e2ϕ(g−1) =e2ϕ(2g−2). (31) of the 4d dilaton, which resides in an = 2
· N
hypermultiplet on the type II side, one finds
loop-factor RR-insertions
| {z } | {z } that it belongs to an = 2 vector multiplet
The second factor stems from the fact that on the heterotic side.NThe reason is that the
the graviphotons come from the Ramond- heterotic dilaton is mapped to the volume of
Ramondsectorandarethereforeaccompanied thebaseoftheK3-fibrationinthetypeIIside.
byfactorsofthestringcouplinggs =eϕ,with For the Fg’s, this means that they are exact
ϕ the 4d dilaton field. Switching to the Ein- onthe IIA side (as mentionedabove)and ap-
steinframe,themetricbecomesdependenton pear already at genus g = 1 on the heterotic
thestringcoupling. Thepreciserelationisde- side, where however they receive corrections.
rived by studying the Einstein-Hilbert term Despite its non exactness, their one-loop het-
eroticrepresentationis veryusefulandallows
string frame Einstein frame
→ many properties to be uncoveredand studied
√GRe−2ϕ √GR
→ in a simple way [11].
Therefore, we read off that the metric is
rescaled as 3. =1 Topological Amplitudes
N
G Ge2ϕ. (32) A reduction of the = 2 topological am-
→ N
plitudes to = 1 is achieved in two ways:
To establish the full dilaton dependence, we (i)theheterNoticdualisstudied, beingcaptured
now have to count the metrics contained in by a torus integral; (ii) Z world-sheet involu-
2
(30). The measure factor √G behaves like tions oftheclosed stringamplitudes arefound
G2 just like the Riemann tensor. To con- to result in type I open string amplitudes. For
tracttwoRiemanntensors,oneneeds fourin- both theories the analog of the holomorphic
verse metrics, which yields another factor of anomaly equation is derived.
(G−4). Finally, to contract the graviphotons,
another (2g 2) inverse metrics are deployed 3.1. Heterotic String Theory
−
giving G−(2g−2). All together, they behave 3.1.1. Topological Amplitudes
as G−(2g−2), yielding an additional contribu- We choose the left moving side of the het-
tionofe2ϕ(−2g+2)whichpreciselycancels(31). erotic string to be the supersymmetric one
Hence, we see that there is no dilaton depen- and thus, the topological twist will only be
dence in the Einstein frame at all. performed on this side. To match the ampli-
This result is consistent with the well tudes of the type IIA side, we have to insert
known fact that in = 2 type II compact- (2g 2) left-moving spin fields. Possible can-
N −
ifications in 4 dimensions, the dilaton resides didates are gauginos, gravitinos or dilatinos.
in a hypermultiplet which does not couple to For reasons which will become clear below,
vector multiplets through local effective ac- we will opt for the first ones and we will sup-
tionterms. Fromthis, weconcludethatthere plement them by two gauge fields. Thus, the
shouldbenocorrections(perturbativeornot) transition from the type IIA side is given by
to the F ’s given by the expression (29).
g
graviphoton gaugino,
→
2.4. Duality Mapping
graviton gauge field, (33)
The starting point for a duality mapping →
=2 Weyl W b,ij =1 gauge Wa,
between heterotic string and type IIA is in N a → N α
Topological Amplitudes and Physical Couplings in String Theory 7
which is given by the chiral multiplet a derivative with respect to an anti-
holomorphic modulus simply leads to an in-
i
Waα =−iλαa − 2(σµσ¯ν)abθbFµαν +.... (34) sertionofthisoperatorintheamplitude. The
naive vanishing of the resulting expression is
As before a,b = 1,2 are spinor indices, while
again spoiled by the appearance of (anoma-
α labels the gauge group factor.
lous) boundary terms, which take the form
The corresponding expression to the type
I1I/A2-eBffPeSctFiv-etearcmtio[n12t]e:rm (25) is the following ∂¯iFg = F¯ig,¯1jG¯jjDjFg2. (40)
g1+Xg2=g
d2θFHET(TrW2)g =FHETF2(λλ)g−1. The new feature of this equation, compared
Z g g
to the corresponding one in type IIA theory,
(35)
isthe appearanceofadditionalobjectsonthe
An additional complication arising here is right hand side which are of the form [12]
that besides the det(Imτ) factors, there are
extra contributions stemming from double 3g−3+n
poles of the operator product expansion of F¯ig,¯j =Z h (µaG−)(µ¯a¯b)(detQbij)·
the right-moving gauge charge operators in- Mg,n aY=1
vofoltvheodseincotnhteacatmteprlmitus,dae.suIintaobrldeedrifftoeregnetceriodf ·(detQcij)Z φ¯¯iZ φ˜¯¯ji. (41)
twogaugegroupswithnochargemattermust
Thesenew objects indicate thatthe recursion
be taken:
relations do not close within the topological
d2θFHET[(Tr Tr )W2]g, (36) partition function, but one has to allow for
Z g 1− 2 operatorsinthetwistedtheory,whicharenot
where Tr denotes tracing with respect to the in the kernel of the (twisted) BRST operator
i
i-th gauge group factor. Thus, finally, only QBRST.
the Kac-Moody currents of the vertex oper- Theadditionaltermscanbecomputedfrom
ators are left to deal with. These contribute
FgΠnW2g, (42)
only with their zero-mode part, given by n
g wherethe Fg are arbitraryfunctions ofchiral
Qaω¯ , (37) n
i i superfields and the Π’s denote the chiral pro-
Xi=1 jectionof non-holomorphicfunctions ofchiral
where Qai is an operator measuring the superfields.10 The component expansionof Π
a-th charge of the state that propagates is given as
on the i-th loop of the world-sheet, and
the abelian differentials ω¯ provide the miss- Π=λ¯λ¯+θ2((∂Z¯)2+∂2Z¯)+..., (43)
ing anti-holomorphic contributions to cancel
(upon integration) the det(Imτ)-factors. where Z and λ denote the bosonic and
fermionic components of chiral superfields.
With these subtleties, the topological am-
plitudes take the form [12] However, the introduction of these new ob-
jectsalonedoesnotseemtosolvetheproblem
3g−3
of integrability, which remains open.
FHET = (µ G−)(µ¯ ¯b)(detQbj)
g Z h a a i ·
Mg aY=1 3.2. Type I
(detQcj) . (38) 3.2.1. Z Involution
· i i 2
The simplest way to compute open string
3.1.2. Holomorphic Anomaly Equation
scattering amplitudes is to take Z involu-
An interesting puzzle appears when trying 2
tions of closed (type IIB) string amplitudes
torepeatthecomputationleadingtotheholo-
[13,14,15]. This is done by identifying ho-
morphic anomaly equation. After perturbing
mology cycles of the world-sheet and taking
the action by the BRST exact operator9
an appropriate “square root”. In this man-
{QBRST,φ¯¯i}, (39) ner, the previously closed world-sheet gets a
9Note,thatonlyoneBRSToperatorispresentincon- 10Inthissenseitcanbeviewedasageneralizationof
trasttotwoofthemin(15). theD¯2 operatorofrigidsupersymmetry.
8 I. Antoniadis and S. Hohenegger
numberofboundaries,whicharepreciselythe g = h 1. For simplicity, we will focus on
−
fixed points of the involution, and the F ’s the case where all vertex operator insertions
g
are generalized to F (in the simplest case are on the boundaries [16]. In this case, in
(g,h)
of oriented surfaces). We will preliminary fo- order to make contact with the type II cal-
cus on world-sheets which have no remaining culation, we have to use two gauge fields and
handles. Letusstudythiscasemorecarefully 2h 4 gauginos. The reasonis that upon the
with the following example: Z −involution the graviton becomes a gauge
2
field and the graviphoton a gaugino, as de-
Example We consider a genus 3 Riemann scribed in Section 3.1.1. One then finds the
surface as the double cover (that is as ‘the physical coupling
square’)ofaRiemannsurfacewithnohandles
but 4 boundaries (see Figure 2). The canon- d2θF (TrW2)h−2, (46)
(0,h)
Z
where W is the gauge supermultiplet defined
α α
in (34). Since the vertex insertions are path
2 3
a a a ordered,theinsertionofmorethantwoonthe
1 2 3
same boundary would lead to rather compli-
α α cated expressions. Hence, we distribute the
1 4 fields as follows
oneachoftheh 2boundariesweinsert
ZZ2 • −
b b b two gauginos
1 2 3
on one boundary we insert the two
•
Figure 2. Example of a Z involution of a gauge fields
2
genus 3 Riemann surface. The (a ,b) are
i i one boundary remains ‘empty’ and
the canonical homology basis of the Riemann •
serves as a spectator
surface and the α with j = 1,...,4 form
j
the boundaries of the open Riemann surface. UsingfurthermoreNeumannboundarycondi-
Note,thatupondifferentidentificationsofthe tions11 the amplitudes are given by
homology cycles different open surfaces ap-
3h−6
pear. F = (µ G−) (lattice sum).
(0,h) a
Z h Z i·
Mh aY=1
ical homology basis for this surface is given
(47)
by the cycles (a ,b ), with i = 1,2,3, while
i i
the boundaries of the open string surface ob- Itcanbeshownthatthisresultisdualto(38)
tainedupon aspecial Z involution arethecy- by heterotic/type I duality.
2
cles α ,...,α . From Figure 2 we read off
1 4
3.2.3. Holomorphic Anomaly Equation
that they are given by
Performing the same computation as be-
α1 = a1, fore and taking anti-holomorphic derivatives
α = a a−1, of the amplitude, one has to consider again
2 2 1
α = a a−1, (44) boundary contributions in the moduli space
3 3 2 of open string world-sheets [16]. In contrast
α = a−1.
4 3 tothetypeII computationhowever,this time
The special Z involution which is displayed onehastoconsiderthreecases,whicharedis-
2
in Figure 2 is defined to act on the cycles as playedinFigure3. Whileinthefirsttwocases
follows the surface develops a long and thin strip, in
the last case it develops a cylinder which cor-
a a, b b. (45)
i i i i responds to a closed string exchange between
→ →−
the two new surfaces. Just as in the heterotic
case,thereappearnewobjectswhentakingan
3.2.2. Scattering Amplitudes anti-holomorphicderivative of F , spoiling
(0,h)
We now consider a surface with h bound- the closure of the recursion relation.
aries, that is obtained from the Z2 involu- 11Thischoiceisdictatedbythenecessityofcanceling
tion of a closed Riemann surface of genus thedet(Imτ)factors.
Topological Amplitudes and Physical Couplings in String Theory 9
(a)
(b)
(c)
Figure 3. Boundaries of an open string world-sheet.
3.2.4. Outlook to F trix. Furthermore, the field insertions on the
(g,h)
A question which immediately arises is closed world-sheet are initially two gravitons
whether it is possible to relax the constraint (R)andtwo graviphotons(T). Depending on
of g = 0. In other words, are there physi- whether they are inserted in the bulk or on
calstringcouplingswhichcorrespondtoF the boundary, they either stay gravitons and
(g,h)
with g = 0 = h? The main problem in this graviphotons or get converted to gauge fields
6 6
respect is that due to the lower number of (F) and gauginos (λ), respectively, according
boundaries, a distribution of the vertex in- to(33). Bothdiagramscommunicateabreak-
sertions as in F is not possible. One has ing of supersymmetry in some hidden sector
(0,h)
to either put more than two vertices on some of the theory (e.g. in the bulk or in some
boundaries, or consider field insertions in the boundary) to the “observable” sector (e.g. a
bulk. Inbothoptionsthepositionintegralsdo gauge theory living on some supersymmetric
not quite cancel the (det(Im)τ) factors which –to lowest order– boundary).
arise from the space-time part. Thus, these Indeed, the diagram to the left of Figure 4
amplitudes are not precisely topological. has three boundaries with two gauginos on
one of them and two gauge fields on another,
givingrisetoF . Replacingthegaugefield
4. Genus 2 Amplitudes, Supersymme- (0,3)
strengths by their D-term auxiliary compo-
try Breaking and Fermion Masses
nents, one obtains a gaugino mass when su-
The Z involutions of genus 2 topological persymmetry is broken by a D-term expec-
2
amplitudes describe the communication of su- tation value. This provides an example of
persymmetry breaking in some “hidden” sec- gauge mediation [16] that will be studied be-
tor to a gauge theory living on the bound- low, in subsection 4.5. It is also worth notic-
ary. More precisely, they compute the result- ing that the holomorphic anomaly of F(0,3)
ing gaugino or fermion masses. brings a term ΠW2, generated at one loop
from the annulus diagram. Using the compo-
4.1. Overview nent expansion (43), and replacing the gauge
Studying in detail the genus 2 case, there field strengths by D-term auxiliary compo-
are two possible inequivalent involutions, nents, one obtains fermionmassesof the type
which together with their involution matrices µ-term, needed in the supersymmetric stan-
areschematicallydisplayedinFigure4. These dard model.
involutions lead to a different identification Ontheotherhand,thediagramtotheright
of cycles of the homology basis, correspond- of Figure 4 has one handle and one bound-
ing to a different form of the involution ma-
10 I. Antoniadis and S. Hohenegger
1 0 0 0 0 1 0 0
 0 1 0 0   1 0 0 0 
Z = Z =
1 0 0 1 0 2 0 0 0 1
 −   − 
 0 0 0 1   0 0 1 0 
 −   − 
Figure 4. Possible involutions of a genus 2 compact Riemann surface and their involution ma-
trices acting on the homology cycles C = (a ,a ,b ,b ). For instance Z CT reproduces the
1 2 1 2 1
transformation (45).
ary with two gauginos. Moreover, there is a periodicity (48), one finds that the Kaluza-
graviton insertion in the bulk that can be re- Klein (KK) momentum takes the form
placedbyitsauxiliarycomponent,generating
againa gravitino mass related to F , upon n+Q
(1,1) p= , (49)
supersymmetry breaking in the gravity sec- R
tor [17]. Thus, this provides an example of
gravity mediation that we study next, in the forintegern,whichleadstoamass-shiftofthe
case where supersymmetrybreaking arisesby KK modes by Q/R. In particular, the grav-
compactification via the Scherk-Schwarz(SS) itino zero mode gets a mass Q/R and (local)
mechanism [18]. supersymmetry is broken.
This method is generalized in closed string
4.2. Scherk-Schwarz (SS) Mechanism theory using world-sheet modular invari-
for Supersymmetry Breaking ance [19,20]. For open strings, the effect is
Apossibilitytobreaksupersymmetryspon- identical to field theory, for Neumann bound-
taneously with the help of compact dimen- ary conditions. For Dirichlet conditions on
sions in field theory was first introduced in the other hand, describing a D-brane perpen-
[18]. Considering (in the simplest case) com- dicular to the SS coordinate, supersymme-
pactification on a circle of radius R, one can try remains unbroken, since there are no KK
allow the fields to be periodic only up to an modes [21].
R-symmetry transformation
4.3. Gravity Mediated Supersymmetry
Φ(y+2πR)=UΦ(y), with U =e2πiQ,
Breaking
(48)
Consider a generic 1-loop two-point func-
whereQisthecorrespondingR-charge. Since, tion with some gravitational exchange in the
upon compactification, R-symmetry is re- effectivefieldtheory(seeFigure5). Sinceeach
strictedtodiscretesubgroups(thatisUN =1 ofthe twogravitationalvertices comes witha
for Z ), Q is quantized. Moreover, from the factor of the inverse Planck mass, M−1, the
N p