Instantons from Open Strings and D-branes

Transcription

1 Instantons from Open Strings and D-branes Alberto Lerda U.P.O. A. Avogadro & I.N.F.N. - Alessandria Munich, November 14, 2007 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

2 Foreword This talk is mainly based on: R. Argurio, M. Bertolini, G. Ferretti, A. L. and C. Petersson, JHEP 0706 (2007) 067, arxiv: [hep-th] M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. L. and R. Marotta, JHEP 0710 (2007) 091, arxiv: [hep-th] M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. L. and R. Marotta, arxiv: [hep-th] For earlier works see: M. Billo, M. Frau, F. Fucito and A. L., JHEP 0611, 012 (2006), [hep-th/ ] M. Billo, M. Frau, I. Pesando, F. Fucito, A. L. and A. Liccardo, JHEP 0302, 045 (2003), [hep-th/ ] M. B. Green and M. Gutperle, JHEP 0002 (2000) 014 [hep-th/ ] +... For recent related works, see other talks of this conference Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

3 Plan of the talk 1 Introduction and motivation Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

4 Plan of the talk 1 Introduction and motivation 2 Part I : Branes and instantons in flat space and CY singularities N = 1 SQCD Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

5 Plan of the talk 1 Introduction and motivation 2 Part I : Branes and instantons in flat space and CY singularities N = 1 SQCD 3 Part II : Wrapped magnetized D-branes and instantons N = 2 SQCD Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

6 Plan of the talk 1 Introduction and motivation 2 Part I : Branes and instantons in flat space and CY singularities N = 1 SQCD 3 Part II : Wrapped magnetized D-branes and instantons N = 2 SQCD 4 Conclusions Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

7 Introduction String theory is a very powerful tool to analyze field theories, and in particular gauge theories. Behind this, there is a rather simple and well-known fact: in the field theory limit α 0, a single string scattering amplitude reproduces a sum of different Feynman diagrams α String theory S-matrix elements = vertices and effective actions in field theory Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

8 String theory is useful not only to retrieve perturbative results! Using D-branes, also some non-perturbative properties of field theories can be analyzed in string theory, in particular INSTANTONS! Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

9 String theory is useful not only to retrieve perturbative results! Using D-branes, also some non-perturbative properties of field theories can be analyzed in string theory, in particular INSTANTONS! Instantons are responsible for a particularly tractable class of non perturbative effects in (supersymmetric) gauge theories. In string theory instantons arise as (possibly wrapped) Euclidean branes. In addition to the usual field theory effects, stringy instantons can generate additional terms in the effective superpotentials or prepotentials, thus providing a rationale for their presence that would otherwise appear rather ad hoc. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

10 Simplest example: branes in flat space Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

11 D-branes in flat space The effective action of a SYM theory can be given a simple and calculable string theory realization using D-branes in flat space: The gauge degrees of freedom are realized by open strings attached to N D3 branes. N D3-branes Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

12 D-branes in flat space The effective action of a SYM theory can be given a simple and calculable string theory realization using D-branes in flat space: The gauge degrees of freedom are realized by open strings attached to N D3 branes. N D3-branes D-instantons The instanton sector of charge k is realized by adding k D-instantons. [Witten 1995, Douglas 1995, Dorey 1999,...] Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

13 Stringy description of gauge instantons D3 D( 1) N D3-branes D3/D3 D-instantons D3/D( 1) D( 1)/D( 1) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

14 Moduli vertices and instanton parameters Open strings with at least one end on a D( 1) carry no momentum: they are moduli, rather than dynamical fields. ADHM Meaning Vertex Chan-Paton D( 1)/D( 1) (NS) a µ centers ψ µ e ϕ adj. U(k) χ a aux. ψ a e ϕ(z)... D c Lagrange mult. η c µν ψν ψ µ... D( 1)/D( 1) (R) M αa partners S αs A e 1 2 ϕ... λ αa Lagrange mult. S α S A e 1 2 ϕ... D( 1)/D3 (NS) w α sizes S α e ϕ k N w α sizes S α e ϕ N k D( 1)/D3 (R) µ A partners S A e 1 2 ϕ k N µ A... SA e 1 2 ϕ N k Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

15 Disk amplitudes and effective actions D3 disks Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

16 Disk amplitudes and effective actions D3 disks D( 1) disks Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

17 Disk amplitudes and effective actions D3 disks D( 1) disks Mixed disks Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

18 Disk amplitudes and effective actions D3 disks Disk amplitudes D( 1) disks field theory limit α 0 Effective actions D3 disks D( 1) and mixed disks Mixed disks SYM action ADHM measure Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

19 Collecting all diagrams D( 1) and mixed disk diagrams with insertion of all moduli vertices, we can extract the instanton moduli action { S 1 = tr where χ AB = χ a (Σ a ) AB. [a µ, χ a ] 2 i 4 MαA [χ AB, Mα] B + χ a w α w α χ a + i 2 µa µ B χ AB ( ) id c w α (τ c ) β α w β + i ηc µν[a µ, a ν ] ( )} + iλa α µ A w α + w α µ A + σ µ β α [MβA, a µ ] S 1 is just a gauge theory action dimensionally reduced to d = 0 in the ADHM limit. The last two lines in S 1 correspond to the bosonic and fermionic ADHM constraints. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

20 The interactions of the charged moduli and the scalar fields X a of the gauge theory are described by { S 2 = tr w α X a X a w α + i 2 (Σa ) AB µ A X a µ B} { 1 = tr 8 ɛabcd w α X AB X CD w α + i 2 µa X AB µ B} = tr{ 1 2 w α{ Φ i, Φ i } w α + i 2 µi Φ i µ4 i 2 µ4 Φ i µi i 2 ɛ ijk µ i Φ j µ k} where X AB = X a (Σ a ) AB in the 6 of SU(4), or in SU(3) notation (which will be the most convenient) Φ i 1 2 ɛijk X jk, Φ i X i4 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

21 An instanton contribution is given by the integral over ALL MODULI of the total action S mod S 1 + S 2 : Z d{a, χ, M, λ, D, w, w, µ, µ} e S mod Some zero modes are special: x µ = tra µ and θ αa = trm αa are the Goldstone modes of the broken (super)translations and S mod does not depend on them. The other neutral (anti-chiral) fermionic zero-modes λ αa are the Lagrangian multipliers for the fermionic ADHM constraints. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

22 In the case of reduced SUSY, some of the θ αa will not be present. N = 2 Z = dx 4 dθ 4 F where F d{remaining moduli} e S mod N = 1 Z = dx 4 dθ 2 W where W d{remaining moduli} e S mod Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

23 In the case of reduced SUSY, some of the θ αa will not be present. N = 2 Z = dx 4 dθ 4 F where F d{remaining moduli} e S mod N = 1 Z = dx 4 dθ 2 W where W d{remaining moduli} e S mod The integral over the anti-chiral zero modes λ αa enforces the fermionic ADHM constraints. In general, one must investigate under what conditions these instanton contributions to the prepotential F or to the superpotential W are non vanishing. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

24 N = 1 example Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

25 N = 1 SYM theories from fractional branes They can be realized by the massless d.o.f. of open strings attached to fractional D3-branes in the orbifold background where the orbifold group acts as R 1,3 C 3 / (Z 2 Z 2 ) g 1 : {z 1, z 2, z 3 } {z 1, z 2, z 3 }, g 2 : {z 1, z 2, z 3 } { z 1, z 2, z 3 } In this orbifold there 4 types of fractional D3 branes, giving rise to the following quiver gauge theory N 1 N 2 N 3 N 4 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

26 This quiver theory has the advantage of being non-chiral It is automatically free of gauge anomalies. It has a large freedom in fractional D3 branes content: (N 1, N 2, N 3, N 4 ). If we take N 1 = N c, N 2 = N f, N 3 = N 4 = 0, the theory living on the N c D3 branes is N = 1 SQCD with gauge group SU(N c ) and N f flavors and is represented by the following (simplified) quiver N c Q Q N f Now we add instantons! Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

27 Exotic Instantons Let us choose the fractional D3 brane content to be D3 : (N 1, N 2, N 3, N 4 ) (N c, N f, 0, 0) and the fractional D-instanton content to be D( 1) : (k 1, k 2, k 3, k 4 ) (0, 0, 1, 0) The instanton is associated to a node that is not occupied in the gauge theory! Q Q e N c N f µ, µ; no bosons µ, µ ; no bosons 1 x µ, θ α and λ α Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

28 The distinctive features of these exotic D-instantons are: there are no bosonic moduli in the charged sectors ( no w s nor w s) there are no ADHM-like constraints Thus S 1 = 0, S 2 = i 2 µ Q f µ f i 2 µ f Q f µ (notice that the Q dependence is holomorphic). Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

29 The distinctive features of these exotic D-instantons are: there are no bosonic moduli in the charged sectors ( no w s nor w s) there are no ADHM-like constraints Thus S 1 = 0, S 2 = i 2 µ Q f µ f i 2 µ f Q f µ (notice that the Q dependence is holomorphic). But W d{λ, D, µ, µ, µ, µ } e S 2 0 because of the λ integration!! These exotic configurations do NOT contribute to the superpotential, unless the λ s are removed! see however C. Petersson s talk Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

30 Orientifold Projection Ω In addition to the Z 2 Z 2 orbifold, let us introduce an O3-plane generated by Ω = ω I 6 where ω is the world-sheet parity, and I 6 is the parity in the transverse space. For the D3 branes, we choose an anti-symmetric representation for Ω on the open string Chan-Paton factors γ (Ω) = diag (ɛ 1... ɛ 4 ) with ɛ T i = ɛ i, ɛ 2 i = 1 The gauge theory fields must satisfy A µ = γ (Ω)A T µγ (Ω) 1 implying that the gauge groups are now USp(N). For example, USp(N c) Q Q USp(N f ) with Q = ɛ 2 Q T ɛ 1 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

31 Now let us add D-instantons For consistency, on the D-instantons we must choose a symmetric (orthogonal) representation for Ω This implies γ + (Ω) = 1 a µ = (a µ ) T, M α = (M α ) T, λ α = (λ α ) T, µ µ T Thus, for any single fractional instanton, the λ zero-modes are projected out! For a single exotic instanton in this orientifold, the quiver diagram is USp(N c) Q USp(N f ) µ µ 1 x µ, θ α and no λ α s! Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

32 Exotic contributions to superpotential Now we have S 1 = 0, S 2 = i µ T ɛ 1 Q f µ f and W dµdµ e S 1 S 2 = dµdµ e iµt ɛ 1 Φµ δ Nf,N c det(q) Introducing the USp(N c ) meson matrix M = Qɛ 1 Q in the antisymmetric of USp(N f ), we can write W pf(m) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

33 Exotic contributions to superpotential Now we have S 1 = 0, S 2 = i µ T ɛ 1 Q f µ f and W dµdµ e S 1 S 2 = dµdµ e iµt ɛ 1 Φµ δ Nf,N c det(q) Introducing the USp(N c ) meson matrix M = Qɛ 1 Q in the antisymmetric of USp(N f ), we can write W pf(m) The condition N f = N c is the same as for the existence of contributions from ordinary instantons [Intriligator and Pouliot, 1995] In this case, the ADS-like term can be computed from the (1, 0, 0, 0) fractional instanton (just as in the SU(N) case) and one finds W Λ2Nc+3 pf(m) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

34 Wrapped branes scenarios Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

35 Wrapped D-branes One can also use systems of wrapped D-branes in R 1,3 CY 3 to engineer 4-d SYM theories with chiral matter and interesting phenomenology: Type IIB : magnetized D9 branes Type IIA : intersecting D6 (easier to visualize) R 1,3 D6 b D6 a CY 3 families from multiple intersections, tuning different coupling constants low energy described by SUGRA with vector and matter multiplets can be derived directly from string amplitudes Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

36 Instantons and Euclidean branes (I): gauge instantons In type IIA Euclidean D2 branes that wrap the same 3-cycle as the D6 branes are point-like in R 1,3 and correspond to ordinary gauge instantons of the SYM theory on the D6 branes This is analogous to the D3/D(-1) system in flat space: the ADHM construction is obtained from strings attached to the instantonic branes [Witten 1995, Douglas 1995, Dorey 1999,...] D6 a R 1,3 E2 a CY 3 In type IIB we should consider instead D9 a /E5 a branes wrapped on the entire CY space. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

37 Instantons and Euclidean branes (II): exotic instantons Type IIA E2 branes wrapped differently from the color D6 branes are still point-like in R 1,3 but they do not correspond to ordinary gauge instanton configurations. Their field theory interpretation is not yet clear but still they may give important non-perturbative contributions to the effective action, e.g. Majorana masses for neutrinos, moduli stabilizing terms,... [Blumenhagen et al, 2006; Ibanez and Uranga, 2006;...] E2 b R 1,3 D6 a CY 3 In type IIB we consider instead systems with D9 a and E5 b branes wrapped on the entire CY space but with different magnetizations (i.e. a b). Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

38 N = 2 example Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

39 The background geometry Internal space: T (1) 2 T (2) 2 T (3) 2 Z 2 T (1) = T (1) 1 + it (1) 2 U (1) = U (1) 1 + iu (1) 2 T (2) = T (2) 1 + it (2) 2 U (2) = U (2) 1 + iu (2) 2 T (3) = T (3) 1 + it (3) 2 U (3) = U (3) 1 + iu (3) 2 Orbifold action: (Z (1), Z (2), Z (3) ) ( Z (1), Z (2), Z (3) ) Compactification moduli in the supergravity basis: [Lüst et al, 2004;...] Bulk Kähler potential: Im(s) s 2 = 1 4π e φ 10 T (1) 2 T (2) 2 T (3) 2 Im(t (i) ) t (i) 2 = e φ 10 T (i) 2, u (i) = u (i) 1 + i u(i) 2 = U(i) K = log(s 2 ) i log(t (i) 2 ) i log(u (i) 2 ) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

40 N = 2 SQCD from magnetized branes The gauge sector Place a stack of N a fractional D9 branes in the representation R 0 of the orbifold group ( color branes 9 a ) The massless spectrum of the 9 a /9 a strings gives rise, in R 1,3, to N = 2 vector multiplets for the gauge group SU(N a ) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

41 N = 2 SQCD from magnetized branes The gauge sector Place a stack of N a fractional D9 branes in the representation R 0 of the orbifold group ( color branes 9 a ) The massless spectrum of the 9 a /9 a strings gives rise, in R 1,3, to N = 2 vector multiplets for the gauge group SU(N a ) The gauge action is S gauge = { 1 } d 4 x Tr 2ga 2 Fµν K Φ D µ ΦD µ Φ + where 1 g 2 a = 1 4π e φ 10 T (1) 2 T (2) 2 T (3) 2 = s 2, K Φ = 1 t (3) 2 u(3) 2 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

42 N = 2 SQCD from magnetized branes The matter sector Add D9-branes in the representation R 1 of the orbifold group ( flavor branes 9 b ) with quantized magnetic fluxes along the first two torii f (i) b = m(i) b n (i) b m (i) b, n(i) b Z for i = 1, 2 N = 2 susy requires ν (1) b = ν (2) b, where f (i) (i) b /T 2 = tan πν (i) b with 0 ν (i) b < 1 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

43 N = 2 SQCD from magnetized branes The matter sector Add D9-branes in the representation R 1 of the orbifold group ( flavor branes 9 b ) with quantized magnetic fluxes along the first two torii f (i) b = m(i) b n (i) b m (i) b, n(i) b Z for i = 1, 2 N = 2 susy requires ν (1) b = ν (2) b, where f (i) (i) b /T 2 = tan πν (i) b with 0 ν (i) b < 1 The 9 b /9 a strings are twisted by the relative angles θ (i) ba ν(i) b ν(i) a and their massless modes fill up N = 2 hypermultiplets in the fundamental representation of SU(N a ). Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

44 N = 2 SQCD from magnetized branes The matter sector N a Qba e Qab N b The degeneracy of these hypermultiplet is N b I ab where I ab is the # of Landau levels for the (a, b) intersection Thus in N = 1 language I ab = 2 i=1 ( (i) m a n (i) b ) m(i) b n(i) a {Q f, Q f, χ f, χ f } with f = 1,..., N f = N b I ab Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

45 N = 2 SQCD from magnetized branes The matter sector N a Qba e Qab N b The matter action is S matter = d 4 x N F f =1 { } K Q D µ Q f D µ Q f + D Qf µ D µ Q f +... together with a superpotential W = f term Q f Φ Q f and possibly a mass The Kähler metric for Q and Q is e K /2 K 1 Q = g a KΦ K Q = 1 ( (1) t 2 t (2) 2 u(1) 2 u(2) 2 ) 1/2 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

46 Now let us add instantons! Add k E5 branes whose internal structure is the same as that of the color branes (5 a branes ordinary gauge instantons) The quiver diagram becomes N a Q ba e Qab N b k 5 a where the dotted lines represent the ADHM instanton moduli. Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

47 Moduli and instanton parameters ADHM Meaning Chan-Paton 5 a/5 a (NS) a µ centers adj. U(k) χ aux.. D Lagrange mult.. 5 a/5 a (R) M αa partners. λ αa Lagrange mult.. 5 a/9 a (NS) w α sizes k N a w α sizes N a k 5 a/9 a (R) µ A partners k N a µ A... Na k 5 a/9 b (R) µ A partners k N f µ A... Nf k Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

48 Instanton contributions to the effective action We first compute the moduli action from all (mixed) disk diagrams φ φ S mod (φ, φ; M k ) = Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

49 Instanton contributions to the effective action We first compute the moduli action from all (mixed) disk diagrams φ φ S mod (φ, φ; M k ) = We then integrate over the moduli and get the instanton induced effective action Seff k = C k d 4 x d 4 θ d M k e S mod (φ, φ; M c k ) where with C k = Λ k(2na N F ) e A 5a = Λ kb 1 e A 5a [Blumenhagen et al, 2006;...] A 5a (no moduli insertions) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

50 Instanton contributions to the effective action The integral over the instanton moduli can be evaluated using localization methods and the explicit form of the effective action S k eff can be obtained [Nekrasov, 2002;...; Fucito et al., 2004;...] For our purposes it is convenient to write the result in a N = 1 superspace notation { [ Seff k = Λ kb 1 1 ] e A 5a d 4 x 0 d 2 θ 2ga 2 τuv(φ) k Wu α W αv [ 1 ] } + d 4 x 0 d 2 θ d 2 θ φ ga 2 u σu(φ) k where the instanton induced couplings τ k uv(φ) and σ k u(φ) are holomorphic functions of the canonically normalized scalar φ of the gauge multiplet Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

51 Instanton contributions to the effective action Changing to the SUGRA effective fields Φ, and using the homogeneity properties of the instanton induced couplings, we obtain { [ Seff k = Λ kb 1 e A 5a (g a KΦ ) kb 1 1 ] d 4 x 0 d 2 θ 2ga 2 τuv(φ) k Wu α W αv + [ ] } d 4 x 0 d 2 θ d 2 θ K Φ Φu σu(φ) k where K Φ = (t (3) 2 u(3) 2 ) 1 is the Kähler metric of the adjoint scalars. All non-holomorphic terms in the prefactor coming from should cancel!! the annulus amplitude A 5 a and the Kähler metric K Φ Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

52 The annulus amplitude A 5a is a sum of annulus amplitudes with a boundary on the 5 a branes = + A 5a A 5a;9 a A 5a;9 b Imposing the appropriate b.c. s and GSO, these annulus amplitudes are given by 0 dτ [ ( TrNS PGSO P orb. q L ) ( 0 TrR PGSO P orb. q L 0)] 2τ Both in the colored amplitude A 5a;9 a and the flavored amplitude A 5a;9 b, the contributions from all excited string states cancel Only the KK copies of the instanton zero-modes on the untwisted torus T (3) 2 give a contribution leading to a (non-holomorphic) dependence on the Kähler and complex structure moduli t (3) 2, u(3) 2 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

53 The annulus amplitude A 5a is a sum of annulus amplitudes with a boundary on the 5 a branes = + A 5a A 5a;9 a A 5a;9 b These annulus amplitudes are UV and IR divergent. The UV divergences (IR in the closed string channel) cancel if tadpole cancellation assumed. The IR divergence is regulated with a scale µ The explicit result of the string calculation is ( 1 A 5a = kb 1 2 log(α µ 2 ) + log ( η(u (3) ) ) 2 1 ( ) ) + 2 log U (3) 2 T (3) 2 [Billo et al. 2007; see also Lüst and Stieberger, 2003] Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

54 The annulus amplitude A 5a is a sum of annulus amplitudes with a boundary on the 5 a branes = + A 5a A 5a;9 a A 5a;9 b There is a relation between instantonic annulus amplitudes and threshold corrections to the gauge coupling constant [Abel and Goodsell, 2006; Akerblom et al, 2006] Indeed in SUSY theories the mixed annulus amplitudes compute the running coupling constant by expanding around the instanton background [Billo et al, 2007] A 5a = 8π2 k g 2 a(µ) 1 loop Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

55 The annulus amplitude A 5a is a sum of annulus amplitudes with a boundary on the 5 a branes = + A 5a A 5a;9 a A 5a;9 b Introducing the Planck mass M 2 P = 1 α e φ 10 s 2 and using the effective field theory variables, we get A 5a = kb 1 2 log M2 P µ 2 + A 5 a = kb 1 2 log M2 P µ 2 + kb 1 ( log ( η(u (3) ) ) log(g2 a) log K Φ ) Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

56 The holomorphic life of wrapped D-brane instantons Thus and e A 5a = ( η(u (3) )) 2kb1 (ga KΦ ) kb 1 S k eff = Λ kb 1 e A 5a (g a KΦ ) kb 1 + { [ 1 ] d 4 x 0 d 2 θ 2ga 2 τuv(φ) k Wu α W αv [ ] } d 4 x 0 d 2 θ d 2 θ K Φ Φu σu(φ) k Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

57 The holomorphic life of wrapped D-brane instantons Thus and e A 5a = ( η(u (3) )) 2kb1 (ga KΦ ) kb 1 S k eff = ( Λ η(u (3) ) 2) kb 1 + { [ 1 ] d 4 x 0 d 2 θ 2ga 2 τuv(φ) k Wu α W αv [ ] } d 4 x 0 d 2 θ d 2 θ K Φ Φu σu(φ) k The instanton induced effective action has the correct holomorphic structure as required by SUSY! There is a holomorphic rescaling of the Wilsonian scale Λ into Λ = Λ η(u (3) ) 2 Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

58 Further considerations Using the N = 2 bulk Kähler potential K log(s 2 ) log(t (3) 2 ) log(u(3) 2 ) = K 2 log K Q the instantonic annulus amplitude can be written as [ A = k f + b ( 1 log M2 P 2 µ 2 + K ) ] [see Bachas et al, 1997;...; Berg et al, 2005] with f a holomorphic function log[η(u (3) )] This is a particular case of the general N = 2 Kaplunovsky-Louis formula " A = k f + b «1 2 log M2 P 1 T (G) log µ 2 g 2 + T (G) log (K Φ ) X # N r T (r) K e r where T (r) δ AB = Tr r `TA T B, T (G) = T (adj), N r = # of hypers in rep. r [see de Wit et al. 1995] Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

59 Conclusions D3/D( 1) systems at orbifold singularities provide very efficient string set-ups to perform explicit instanton calculations in gauge theories The (ordinary or exotic) instanton corrections to the superpotential or the prepotential can be computed from mixed disk diagrams Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

60 Conclusions D3/D( 1) systems at orbifold singularities provide very efficient string set-ups to perform explicit instanton calculations in gauge theories The (ordinary or exotic) instanton corrections to the superpotential or the prepotential can be computed from mixed disk diagrams Also systems of wrapped magnetized D9/E5 branes are very useful to study instanton effects in toroidal orbifold/orientifold compactifications of type IIB string theory In this case the mixed annulus diagrams are crucial to provide the correct holomorphic structure of the effective action required by SUSY Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40

61 Conclusions D3/D( 1) systems at orbifold singularities provide very efficient string set-ups to perform explicit instanton calculations in gauge theories The (ordinary or exotic) instanton corrections to the superpotential or the prepotential can be computed from mixed disk diagrams Also systems of wrapped magnetized D9/E5 branes are very useful to study instanton effects in toroidal orbifold/orientifold compactifications of type IIB string theory In this case the mixed annulus diagrams are crucial to provide the correct holomorphic structure of the effective action required by SUSY The analysis of N = 2 SQCD can be generalized to N = 1 SQCD or other N = 1 theories. In this case the annulus amplitudes contain interesting information on the Kähler metric for twisted chiral matter [Akerblom et al. 2007, Billo et al, 2007, Blumenhagen et al 2007] Alberto Lerda (U.P.O.) Instantons Munich, November 14, / 40