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 in- stanton 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 repre- sentation 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 : pG pG + 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[pGµ ReU η(iU) ] −4 j j 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 j j Last term in K needs a separate one-loop computation (ABFPT ; Berg,Haack,Kors). We expect E1 instan- tonic contributions 1 2πnT 1 2πnT fnp = gn(U) e− ; Knp = hn(U; T +T¯) e− +c:c: nX=1 nX=1 4. Effective action and quantum corrections : het- erotic side Threshold corrections to gauge couplings are given by (Kaplunovsky) 1=2 α d2τ i 1 # 1 α Λ = β 2 a 4 @τ 0 h i1 Qa C ; τ2 4π η η − 4πτ2! "β # ZF α,β=0 j j X B C @ A 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 = Z^1 ^f −8 τ2 A ZF with 1 24 ^ = (D10E10 48η ) Af −20η24 − and 2 Re T π(Re T ) τ Z^1 = exp 2πT det(A) 1 iU A ; τ2 2− − τ2(Re U) 1! 3 ni;`i − X 4 5 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 − j j − 3 T1 + T¯1 − 1 1 1 ^ ( ) e 2πpkT1 ^ ( ) + AK U + c.c. − 4 − Af U + ¯ k>j 0;p>0 kp " πkpT1 T1# ≥X ^ is the almost-holomorphic modular form, AK 1 2 3 ^ = (E^2E4E6 + 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 ^ ( ) 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 E^2 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 neu- tral 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 !α 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 ki- netic 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 correc- tions : 1 2 d τ ^ ^a 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 ^a 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 ^ = 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 ^ ( ) 2πpkTi ^a i K i e− f;i( i) A U + c.c. −4 0 kp "A U − πkp Ti + T¯i # 1 − k>j X0;p>0 @ ≥ a A bi 4 2 log η(iUi) + log[(Re Ui)(Re Ti)µ ] ; − 4 j j ) The K¨ahler 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 ^ ( ) 2ikp E ( ) V i = i − AK Ui 10 Ui + c.c.