Exotic instantons and duality Marco Billò Dip. di Fisica T eorica, Università di T orino and I.N.F .N., sez. di T orino 15-th European Workshop on String Theory ETH, Zurich - September 8, 2009
Foreword Mostly based on M. Billo, L. Ferro, M. Frau, L. Gallot, A. Lerda and I. Pesando, “Exotic instanton counting and heterotic/type I’ duality,” JHEP 0907 (2009) 092, arXiv:0905.4586 [hep-th]. M. Billo, M. Frau, L. Gallot, A. Lerda and I. Pesando, “Classical solutions for exotic instantons?,”, JHEP 03 (2009) 056, arXiv:0901.1666 [hep-th]. It builds over a vast literature ◮ I apologize for missing references...
Plan of the talk Introduction and motivations 1 “Exotic” instantons in type I’ 2 3 Interpretation as 8d instanton solutions The effective action 4 Conclusions and perspectives 5
Introduction and motivations
Non-perturbative sectors in brane-worlds ◮ (Susy) gauge and matter sectors on the uncompactified part of (partially wrapped) D-branes ◮ gauge couplings involve 1 /g s × different volumes → string scale not tied to 4d Planck scale ◮ chiral matter, families from multiple intersections,... D7 b D7 a R 1 , 3 CY 3
Non-perturbative sectors in brane-worlds ◮ (Susy) gauge and matter sectors on the uncompactified part of (partially wrapped) D-branes ◮ gauge couplings involve 1 /g s × different volumes → string scale not tied to 4d Planck scale ◮ chiral matter, families from multiple intersections,... E3 a D7 b D7 a R 1 , 3 CY 3 ◮ Non-perturbative sectors from partially wrapped E(uclidean)-branes ◮ Pointlike in the R 1 , 3 space-time: “instanton configurations” ◮ Tractable in String Theory, with techniques in rapid development
Ordinary instantons W.r.t. the gauge theory on a given D-brane stack, D7 a E3 a R 1 , 3 CY 3 ◮ E-branes identical to D-branes in the internal directions: gauge instantons ◮ ADHM from strings attached to the instantonic branes Witten, 1995; Douglas, 1995-1996; ... ◮ non-trivial instanton profile of the gauge field Billo et al, 2001 ◮ Rules and techniques to embed the instanton calculus in string theory have been constructed Polchinksi, 1994; Green-Gutperle, 2000, ...; T urin/Rome/Münich/UPenn/Madrid,...
Exotic instantons W.r.t. the gauge theory on a given D-brane stack, E3 c D7 a R 1 , 3 CY 3 ◮ E-branes different from D-branes in internal directions do not represent gauge instantons; they are called exotic or stringy instantons ◮ May explain important terms in the effective action: neutrino Majorana masses, moduli stabilizing terms, . . . Blumenhagen et al 0609191; Ibanez and Uranga, 0609213; ... ; ◮ Exponentially suppressed but not just exp ( − 1 /g 2 ) , can involve volumes of different internal cycles ◮ Need to understand their status in the gauge theory and to construct precise rules for the “exotic” instanton calculus
Our strategy ◮ Select a simple example: D(-1)/D7 in type I’ theory, sharing many features of stringy instantons ◮ Investigate the field-theory interpretation of D(-1)’s in this 8d gauge theory Billo et al, 2009a; ◮ Compute the non-perturbative effective action on the D7’s extending the rules of stringy instanton calculus to this “exotic” case. ◮ Check against the results in the dual Heterotic SO ( 8 ) 4 theory. Impressive quantitative check of this string duality. Billo et al, 2009b ◮ Apply the technology to tractable example leading to 4d models Work in progress, T urin + T or Vergata
“Exotic” instantons in type I’
A D(-1)/D7 system in type I’ ◮ T ype I’ is type IIB on a two-torus T 2 modded out by Ω = ω ( − 1 ) F L I 2 where ω = w.s. parity, F L = left-moving fermion #, I 2 = inversion on T 2 ◮ Ω has four fixed-points on T 2 where four O7-planes are placed O 7
A D(-1)/D7 system in type I’ ◮ T ype I’ is type IIB on a two-torus T 2 modded out by Ω = ω ( − 1 ) F L I 2 where ω = w.s. parity, F L = left-moving fermion #, I 2 = inversion on T 2 ◮ Ω has four fixed-points on T 2 where four O7-planes are placed O 7 ◮ Admits D(-1), D3 and D7’s transverse to T 2
A D(-1)/D7 system in type I’ ◮ T ype I’ is type IIB on a two-torus T 2 modded out by Ω = ω ( − 1 ) F L I 2 where ω = w.s. parity, F L = left-moving fermion #, I 2 = inversion on T 2 ◮ Ω has four fixed-points on T 2 where D 7 four O7-planes are placed ◮ Admits D(-1), D3 and D7’s transverse to T 2 ◮ Local RR tadpole cancellation requires 8 D7-branes at each fix point
A D(-1)/D7 system in type I’ ◮ T ype I’ is type IIB on a two-torus T 2 modded out by Ω = ω ( − 1 ) F L I 2 where ω = w.s. parity, F L = left-moving fermion #, I 2 = inversion on T 2 ◮ Ω has four fixed-points on T 2 where four O7-planes are placed ◮ Admits D(-1), D3 and D7’s transverse to T 2 We focus on one ◮ Local RR tadpole cancellation requires fix point 8 D7-branes at each fix point
The gauge theory on the D7’s ◮ From the D7/D7 strings we get N = 1 vector multiplet in d = 8 in the adjoint of SO ( 8 ) : � � A μ , Λ α , ϕ m , μ = 1 , . . . 8 , m = 8 , 9 ◮ Can be assembled into a “chiral” superfield � 1 θγ μν θ F μν ( x ) + . . . Φ( x , θ ) = ϕ ( x ) + 2 θ Λ( x ) + 2 � where ϕ = ( ϕ 9 + i ϕ 10 ) / 2. ◮ Formally very similar to N = 2 in d = 4
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s ◮ The quadratic Yang-Mills term S ( 2 ) has a dimensionful � coupling g 2 α ′ ) 4 YM ≡ 4 π g s ( 2 π
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s ◮ Contributions of higher order in α ′ , whose rôle will be discussed later
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s ◮ The quartic term has a dimensionless coupling: � 1 � F 4 � d 8 x t 8 Tr S ( 4 ) = − 96 π 3 g s
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s ◮ Adding the WZ term, we can write � 1 � F 4 � d 8 x t 8 Tr S ( 4 ) = − − 2 π iC 0 c ( 4 ) 4 ! 4 π 3 g s where c ( 4 ) is the fourth Chern number � 1 � � c ( 4 ) = Tr F ∧ F ∧ F ∧ F 4 !( 2 π ) 4
Effective action on the D7 (tree level) ◮ Effective action in F μν and its derivatives: NABI Back S = S ( 2 ) + S ( 4 ) + S ( 5 ) + · · · � F 2 � � F 4 � � � Tr � 1 t 8 Tr � � d 8 x 3 ( 2 π ) 2 + α ′ L ( 5 ) = ( 2 π ) 4 α ′ 2 − F , DF + · · · 8 π g s ◮ Adding the fermionic terms, can be written using the superfield Φ( x , θ ) as � � i π τ Φ 4 � 1 d 8 x d 8 θ Tr + c . c . S ( 4 ) = ( 2 π ) 4 12 i where τ = C 0 + g s is the axion-dilaton.
Adding D-instantons ◮ Add k D-instantons. ◮ D7/D(-1) form a 1/2 BPS system with 8 k D(-1) ND directions ◮ D(-1) classical action 2 π S cl = k ( − 2 π iC 0 ) ≡ − 2 π ik τ , g s ◮ Coincides with the quartic action on the D7 for gauge fields F with c ( 4 ) = k and � � � t 8 F 4 � 1 � ε 8 F 4 � 4 ! ( 2 π ) 4 c ( 4 ) d 8 x Tr d 8 x Tr = − = − 2 2
Adding D-instantons ◮ Add k D-instantons. ◮ D7/D(-1) form a 1/2 BPS system with 8 k D(-1) ND directions ◮ D(-1) classical action 2 π S cl = k ( − 2 π iC 0 ) ≡ − 2 π ik τ , g s ◮ Analogous to relation with self-dual YM config.s in D3/D(-1) ◮ Suggests relation to some 8d instanton of the quartic action
The size of the instanton solution (“gauge” instantons) D3 ◮ For ordinary instantons, e.g. D3/D(-1), there are moduli w ˙ α related to the size D(-1) ρ of the instanton profile ◮ They com from the NS sector of mixed D3/D(-1) strings
The size of the instanton solution (“gauge” instantons) D3 ◮ For ordinary instantons, e.g. D3/D(-1), there are moduli w ˙ α related to the size D(-1) ρ of the instanton profile ◮ They com from the NS sector of mixed D3/D(-1) strings
The size of the instanton solution (“gauge” instantons) D3 ◮ For ordinary instantons, e.g. D3/D(-1), there are moduli w ˙ α related to the size ρ of the instanton profile ◮ They com from the NS sector of mixed ρ D3/D(-1) strings
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