Multi-Instanton and String Loop Corrections in Orbifolds

Emilian Dudas

CPhT-Ecole Polytechnique and LPT-Orsay

Multi-instanton and string loop corrections in orbifolds

P. Camara E.D, e-Print: arXiv:0806.3102 [hep-th] • (also P. Camara, E.D., T. Maillard, G. Pradisi, Nucl.Phys.B795:453- • 489,2008. e-Print: arXiv:0710.3080 [hep-th]) Outline

Motivations • The S-dual orbifold pairs • Thresholds and eﬀective action : type I side • Thresholds and eﬀective action : heterotic side • Comments on E1 instantons • Universality of = 2 instantonic corrections • N Conclusions • july 28, 2008 CERN Institute String Pheno 08

CERN-TH 1. Motivations

SO(32) heterotic -type I duality (Polchinski,Witten) was explored extensively over the last ten years. S-duality allows exact computation of E1 instanton ef- fects in type I heterotic α corrections E1 instanton corrections • 0 → NS5 eﬀects E5 instanton corrections • →

There are simple examples of dual pairs using Vafa- Witten adiabatic argument : freely-acting orbifolds (see Blumenhagen talk). Ex. heterotic-type I duality: higher-derivative in = N 4 SUSY case (Bachas, Kiritsis and coll., Lerche and Stieberger, etc) = 2 case discussed in (Antoniadis,Bachas,Fabre,Partouche N and Taylor; Bianchi, Morales). = 1 models : CDMP, Blumenhagen, Schmidt-Sommerfeld N We will mostly focus today on a standard Z2 orbifold - type I side constructed by Bianchi-Sagnotti and Gimon- Polchinski - heterotic dual pair identified by Berkooz,Leigh,Polchinski,Schwarz, Seiberg,Witten Our goal : computation of the full one-loop + E1 instanton corrections to the type I effective action . An important consistency check : comparison of the effective theory Kaplunovsky-Louis formula 2 2 4π ba MP = Refa + ln g2(µ) 4 µ2 c C (G) 4π2 T (r) + a + 2 ln a ln K 2 detZr 4 2 g (µ) − r 2 X where Ta(r) is the Dynkin index for the matter representation r with wavefunctions Zr, and

ba = Ta(r) 3C2(G) , ca = Ta(r) C2(G) r − r − X X with the one-loop string computation 4π2 4π2 = + Λ where 2 2 a , ga(µ) ga,0 2 ba Ms ∆a Λa = ln + 4 µ2 4 ∆a encodes string threshold corrections from massive string states. 2. The S-dual orbifold pairs

Simplest orbifold T 4/Z T 2, sixteen fixed points. 2 × Type I side - One twisted hypermultiplet per fixed point - Maximal gauge group U(16) U(16) . 9 × 5 Hypers in (120 + 120, 1) + (1, 120 + 120) + (16, 16) In order to have a perturbative heterotic dual, distribute 1/2 D5 brane per fixed point. D5 gauge group broken to U(1)16. Each U(1) gets mixed with twisted four forms and become massive. D9 spectrum : hypers in 120 + 120 + 16 16. × Heterotic dual

SO(32) compactified to 4d on T 4/Z T 2. Shift vector 2 × on the gauge lattice 1 V = (1, 1, 3) 4 · · · − The gauge group is U(16). Charged matter is in rep- resentations : untwisted : 120 + 120 twisted : 16 16. × 3. Effective action and quantum corrections : type I side Threshold corrections to gauge couplings depend on moduli of T 2 : √G √G + iG T = + ib , U = 12 λ G22 One finds (Bachas-Fabre, Antoniadis,Bachas,E.D.)

4π2 4π2 = + Λ where 2 2 a ga (µ) ga,0 1 2 4 Λa = ˜b ln[√Gµ ReU η(iU) ] −4 | | The eﬀective action is then 2 fD9 = S + ˜b ln[ReUη (iU)] + fnp K = ln(S + S¯) ln(U + U¯) ln(T + T¯) + − − − · · · 4π E(iU, 2) + Knp − 3 (T + T¯)(S + S¯) where E(U, k) is the Eisenstein series 1 (Im U)k E(U, k) ≡ ζ(2k) j + j U 2k (j ,j )=(0,0) 1 2 1 2X6 | | Last term in K needs a separate one-loop computation (ABFPT ; Berg,Haack,Kors). We expect E1 instantonic contributions

∞ 2πnT ∞ 2πnT fnp = gn(U) e− , Knp = hn(U, T +T¯) e− +c.c. nX=1 nX=1 4. Eﬀective action and quantum corrections : heterotic side

Threshold corrections to gauge couplings are given by (Kaplunovsky)

1/2 α d2τ i 1 ϑ 1 α Λ = β 2 a 4 ∂τ  h i Qa C , τ2 4π η η − 4πτ2! "β # ZF α,β=0 | | X     α Qa is the charge operator of the gauge group; C β is the internal six-dimensional partition function. h i For the S-dual of the BSGP model we ﬁnd 1 d2τ Λ = Zˆ1 ˆf −8 τ2 A ZF with

1 24 ˆ = (D10E10 48η ) Af −20η24 − and 2 Re T π(Re T ) τ Zˆ1 = exp 2πT det(A) 1 iU A , τ2 − − τ2(Re U) 1!  ni,`i − X  

n ` A = 1 1 , n2 `2! with ni and `i integers. We used methods of Dixon,Kaplunovsky, Louis and Bachas,Kiritsis et al. to evaluate Λ. We ﬁnd

π 2 4 π E(iU1, 2) Λ = Re T1 3log[(Re U1)(Re T1)µ η(iU1) ] 2 − | | − 3 T1 + T¯1 − 1 1 1 ˆ ( ) e 2πpkT1 ˆ ( ) + AK U + c.c. − 4 − Af U + ¯ k>j 0,p>0 kp " πkpT1 T1# ≥X ˆ is the almost-holomorphic modular form, AK 1 2 3 ˆ = (Eˆ2E4E6 + 2E + 3E ) AK 12η24 6 4

We have expressed the result in terms of the induced worldvolume complex structure in the E1 multi-instantons wrapping the ﬁrst 2-torus j + ipU = 1 U k Split between analytic and non-analytic terms ; the cor- rected K¨ahler potential and gauge kinetic function are 3 1V1 loop + VE1 K = log(S + S¯) log[(Ti + T¯i)(Ui + U¯i)] + − , − − 2 S + S¯ iX=1 4π E(iU1, 2) V1 loop = , − − 3 T1 + T¯1

e 2πkpT1 ˆ ( ) πkp E ( ) V = − AK U 10 U + c.c. , E1 − ( )2 + ¯ − 2 24( ) k>j 0, p>0 π kp "T1 T1 Re η # ≥X U U πT 1 e 2πkpT1 f = S + 1 12log η(iU ) − ( ) , U(16) 2 − 1 − 2 Af U k>j 0, p>0 kp ≥X where the holomorphic modular forms and , are Af AK deﬁned as before, replacing Eˆ2 by E2. 5. E1 instantons

The instantonic corrections depend on the moduli of T 2 come from E1 instantons wrapping T 2. → They are of two diﬀerent types: E1 instantons at orbifold ﬁxed points They have unitary Chan-Paton factors, U(r), with neutral sector :

- bosonic zero modes xµ, y1,2 and fermionic zero modes Θα,a, Θα,a˙ , with a = 1, 2 in the adjoint representation r¯r. α,a - bosonic zero modes y3,4,5,6 and fermionic ones λ in r(r+1) ¯r(¯r+1) the symmetric representation 2 + 2 . - fermionic zero modes ˜λα,a˙ in the antisymmetric rep- r(r 1) ¯r(¯r 1) resentation 2− + 2− . Charged zero modes between the instanton and the • 1/2 D5-brane stuck at the singularity: -bosonic zero modes µ1,2 from the R sector and fermionic zero modes ωα from the NS sector, in the representa-

tion r 1 + ¯r1. − r -bosonic zero modes µ10 ,2 in the representation +1 + ¯r 1. − E1-D9 strings : bosonic zero mode ν in the represen- • tation rn¯ + ¯rn. E1 instantons oﬀ the orbifold ﬁxed points It can be argued that these instantons do not contribute to threshold corrections. In order the instantons to contribute to the gauge kinetic function, only four fermionic neutral zero modes should be massless (the “goldstinos” ; Akerblom,Blumenhagen, Lust and Schmidt-Sommerfeld) most of the above → zero modes should be lifted by interactions. Possible qualitative picture : a U(1) instanton on top of a singularity correspond is • a “gauge” instanton for the U(1) gauge theory inside the corresponding half D5-brane. These instantons are analogous to the ones discussed by Petersson, with the extra fermionic zero modes being lifted by couplings in- volving the D5-branes. They should be responsible for the 1-instanton (N = 1) contribution. For the N-instanton contribution, when all the instan- • tons are on top of the same singularity, the instanton CP is enhanced to U(N) and only four zero modes sur- vive, with the extra zero modes presumably lifted by interactions with the D5-branes. (Dedicated works on lifting of zero modes ; Blumen- hagen,Cvetic,Richter,Weigand; Garcia-Etxebarria and Uranga) 5. Universality of = 2 instantonic corrections N

We shall try to generalize the instantonic corrections • to the effective action in case where the heterotic S- dual description is unknown. Framework: orbifold compactifications, with orbifold action G containing subgroups Gi leaving unrotated a given complex plane. The contribution of these sectors to threshold corrections : 1 2 d τ ˆ â Λa = Zi f,i . −8 τ2 A Xi ZF The gauge group is given by a product G = a Ga. Q T , U are moduli of the corresponding unrotated plane • i i â M a/η24, with M a an almost holomorphic mod- • Af,i ∼ i i ular form of degree 24. By imposing the absence of tachyons in the spectrum, we obtain

a a γi 24 ˆ = 2b + D10E10 528η , Af,i i 20η24 − h i where ba is the β-function coeﬃcient of the = 2 i N gauge theory associated to a would-be T 6/Gi orbifold

and γi is a model dependent (but gauge group indepen- dent) coefficient. We get a π(bi + 6γi) πγi E(iUi, 2) Λa = Re Ti + ( 12 3 Ti + T¯i − Xi 1 1 γ ˆ ( ) 2πpkTi â i K i e− f,i( i) A U + c.c. −4  kp "A U − πkp Ti + T¯i #  − k>j X0,p>0  ≥ a  bi 4 2 log η(iUi) + log[(Re Ui)(Re Ti)µ ] , − 4 | | ) The Kähler potential and gauge kinetic functs are i i 1V1 loop + VE1 K = log(S + S¯) log[(Ti + T¯i)(Ui + U¯i)] + − , − − ( 2 S + S¯ ) Xi i 4πγi E(iUi, 2) V1 loop = , − 3 Ti + T¯i γ e 2πkpTi ˆ ( ) 2ikp E ( ) V i = i − AK Ui 10 Ui + c.c. , E1 ( )2 + ¯ − ¯ 24( ) π k>j 0, p>0 kp " Ti Ti i i η i # ≥X U − U U a 2πkpTi π(bi + 6γi)Ti a e− a fa = S + bi log η(iUi) f,i( i) . ( 12 − − 2kp A U ) Xi kX,j,p γi are model-dependent. Conclusions

We computed E1 instanton corrections in the T 4/Z • 2× T 2 type I orbifold model : holomorphic corrections f. → non-holomorphic corrections K. → We expect similar results for the superpotential. Sup- pose S, U moduli are stabilized and there are ﬁeld-theory (E5) nonperturbative eﬀects on D9 branes (gaugino condensation, racetrack). Then

B(f+fnp) ∞ 2πnT Wnp = Ae− = A0 dne− nX=1 When combined with suitable ﬂuxes and/or E5 ef- • fects, they generate moduli stabilization and SUSY break- ing, changes in the soft spectra. Similar to the perturbative threshold corrections , the • E1 instantonic corrections to the gauge kinetic func- tions from N = 2 sectors have some universal proper- ties. Similar results therefore for N = 1 models with

N = 2 sectors : Z6, Z60 , etc. S-dual of stringy instantonic eﬀects in type I could • teach us more about the heterotic string dynamics. The instantonic corrections to f and K deserve more • phenomenological studies.