Instantonic effects in N=1 local models from magnetized branes

Transcription

1 Instantonic effects in N=1 local models from magnetized branes Marco Billò Dip. di Fisica Teorica, Università di Torino and I.N.FN., sez. di Torino L.M.U. München - A.S.C. for Theoretical Physics 6 December 007

2 Foreword This talk is mostly based on M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. Lerda and R. Marotta, Instanton effects in N=1 brane models and the Kahler metric of twisted matter, arxiv: [hep-th]. M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. Lerda and R. Marotta, Instantons in N= magnetized D-brane worlds, arxiv: [hep-th]. It, of course, builds over a vast literature The few references scattered on the slides are of course not exhaustive. I apologize for the missing ones.

3 Foreword This talk is mostly based on M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. Lerda and R. Marotta, Instanton effects in N=1 brane models and the Kahler metric of twisted matter, arxiv: [hep-th]. M. Billo, M. Frau, I. Pesando, P. Di Vecchia, A. Lerda and R. Marotta, Instantons in N= magnetized D-brane worlds, arxiv: [hep-th]. Some very recent work dealing with very similar issues R. Blumenhagen and M. Schmidt-Sommerfeld, Gauge Thresholds and Kaehler Metrics for Rigid Intersecting D-brane Models, arxiv: [hep-th].

4 Plan of the talk 1 Introduction The set-up 3 The stringy instanton calculus 4 Instanton annuli and threshold corrections 5 Holomorphicity properties

5 Introduction

6 Wrapped brane scenarios Type IIB: magnetized D9 branes Type IIA (T-dual): intersecting D6 (easier to visualize) R 1,3 D6 b D6 a CY 3 Supersymmetric gauge theories on R 1,3 with chiral matter and interesting phenomenology [review: Blumenhagen et al, Phys. Rept. 445 (007)] families from multiple intersections, tuning different coupling constants,...

7 Wrapped brane scenarios Type IIB: magnetized D9 branes Type IIA (T-dual): intersecting D6 (easier to visualize) R 1,3 D6 b D6 a CY 3 low energy described by SUGRA with vector and matter multiplets can be derived directly from string amplitudes (with different field normaliz.s) novel stringy effects (pert. and non-pert.) in the eff. action?

8 Euclidean branes and instantons Ordinary instantons E branes wrapped on the same cycle as some D6 branes are point-like in R 1,3 and correspond to instantonic config.s of the gauge theory on the D6 D6 a R 1,3 E3 a CY 3 Analogous to the D3/D(-1) system: ADHM from strings attached to the instantonic branes Witten, 1995; Douglas, ;... non-trivial instanton profile of the gauge field Billo et al, 001 N.B. In type IIB, use D9/E5 branes

9 Euclidean branes and instantons Exotic instantons E branes wrapped differently from the D6 branes are still point-like in R 1,3 but do not correspond to ordinary instantons config.s. E3 c R 1,3 D6 a CY 3 Still they can,in certain cases, give important non-pert, stringy contributions to the effective action,.e.g. Majorana masses for neutrinos, moduli stabilizing terms,... Blumenhagen et al ; Ibanez and Uranga, ; (long list)... ; Petersson Potentially crucial for string phenomenology

10 Perspective of this work Clarify some aspects of the stringy instanton calculus, i.e., of computing the contributions of Euclidean branes Focus on ordinary instantons, but should be useful for exotic instantons as well Choose a toroidal compactification where string theory is calculable. Realize (locally) N = 1 gauge SQCD in type IIB on a system of D9-branes and discuss contributions of E5 branes to the superpotential Analyze the rôle of annuli bounded by E5 and D9 branes in giving these terms suitable holomorphicity properties

11 The set-up

12 The background geometry Internal space: T (1) T () T (3) Z Z (1) T T () T (3) The Kähler param.s and complex structures determine the string frame metric and the B field: G (i) = T (i) U (i) ( ) 1 U (i) 1 U (i) 1 U (i) ( ) and B (i) 0 T (i) = 1 T (i). 1 0

13 The background geometry Complex coordinates String fields: X M (X µ, Z i ) and ψ M (ψ µ, Ψ i ), with Z i = T (i) U (i) (X i+ + U (i) X i+3 ) 10d spin fields decompose into space-time and internal parts: S A (S α S, S α S ++,..., S α S +++,...)

14 The background geometry The orbifold Action of the Z Z orbifold group elements: h 1 : (Z 1, Z, Z 3 ) (Z 1, Z, Z 3 ), h : (Z 1, Z, Z 3 ) ( Z 1, Z, Z 3 ), h 3 : (Z 1, Z, Z 3 ) ( Z 1, Z, Z 3 ), The group has 4 irreducible representations:. R 0 (trivial), R 1, R, R 3

15 The geometric moduli Supergravity basis- tree level Supergravity basis: s, t (i), u (i), with Back Lüst et al, ;... Im(s) s = 1 4π e φ 10 T (1) T () T (3), Im(t (i) ) t (i) = e φ 10 T (i), u (i) = u (i) 1 + i u(i) = U(i), (real parts from suitable RR or B fields). N.B. s 1/g s. N = 1 bulk Kähler potential: K = log(s ) log(t (i) ) i=1 i=1 log(u (i) ) Antoniadis et al,

16 The geometric moduli Supergravity basis - corrections At one-loop level, there are corrections to the bulk Kähler potential (and to the Einstein term) Antoniadis et al, ;... ; Berg et al, These lead to non-holomorphic redefinitions of the supergravity fields s andt i w.r.t. the their tree-level expressions. In particular s (0) = s + δ 8π Differently from corresponding Heterotic constructions, δ in these models appears to be of order g s rather than 1. It would be interesting [see later!] to clarify if any other mechanism can induce, in the models we consider, a shift δ (0) of order 1.

17 N = 1 from magnetized branes The gauge sector Place a stack of N a fractional D9 branes ( color branes 9a). 9 a R 0 Massless spectrum of 9a/9a strings gives rise, in R 1,3, to the N = 1 vector multiplets for the gauge group U(N a ) The gauge coupling constant is given (at tree level) by 1 g a = 1 4π e φ 10 T (1) T () T (3) = s (0)

18 N = 1 from magnetized branes The gauge sector Place a stack of N a fractional D9 branes ( color branes 9a). 9 a R 0 Massless spectrum of 9a/9a strings gives rise, in R 1,3, to the N = 1 vector multiplets for the gauge group U(N a ) The Wilsonian coupling 1/ g a must correspond to (the imaginary part of) a chiral field, so it is corrected w.r.t. to the tree level: Back 1 g a = 1 g a + δ 8π

19 N = 1 from magnetized branes Adding flavors Add D9-branes ( flavor branes 9b) with quantized magnetic fluxes f (i) b = m(i) b n (i) b and in a different orbifold rep. R 1 9 b 9 a R 0 (Bulk) susy requires ν (1) b ν () b ν (3) b = 0, where f (i) (i) b /T = tan πν (i) b with 0 ν (i) b < 1, (other possibilities by sign changes) Marino et al,

20 Add D9-branes ( flavor branes 9b) with quantized magnetic fluxes f (i) b N = 1 from magnetized branes Adding flavors = m(i) b n (i) b and in a different orbifold rep. R 1 9 b q ba q 9 a R 0 9a/9b strings are twisted by the relative angles Back If ν (1) ba ν() ba ν(3) ba ν (i) ba = ν(i) b ν(i) a = 0, this sector is supersymmetric: massless modes fill up a chiral multiplet q ba in the anti-fundamental rep N a of the color group

21 N = 1 from magnetized branes Adding flavors Add D9-branes ( flavor branes 9b) with quantized magnetic fluxes f (i) b = m(i) b n (i) b and in a different orbifold rep. R 1 9 b q ba q 9 a R 0 The degeneracy of this chiral multiplet is N b I ab, where I ab is the # of Landau levels for the (a, b) intersection I ab = i=1 ( (i) m a n (i) b ) m(i) b n(i) a

22 Local vs global realization Introducing branes in a compact space requires the cancellation of the associated tadpoles. This can be achieved by a suitable orientifold projection in the string description, and severely constrains the set-up. We take a local approach, and do not discuss the global requirement of tadpole cancellation (which is however to be assumed) and the contribution of orientifolds in these models: our goal is to understand certain mechanisms of the stringy instanton calculus rather than provide phenomenological models these aspects can be taken into accout, and the picture goes through see Akerblom et al, ; Blumenhagen et al,

23 N = 1 from magnetized branes Engineering N = 1 SQCD Introduce a third stack of 9c branes such that we get a chiral mult. q ac in the fundamental rep N a and that R 1 9 b q ba q q ac q 9 c R 1 N b I ab = N c I ac N F 9 a R 0 This gives a (local) realization of N = 1 SQCD: same number N F of fundamental and anti-fundamental chiral multiplets, resp. denoted by q f and q f

24 N = 1 from magnetized branes Engineering N = 1 SQCD Introduce a third stack of 9c branes such that we get a chiral mult. q ac in the fundamental rep N a and that R 1 9 b q ba q q ac q 9 c R 1 N b I ab = N c I ac N F 9 a R 0 Kinetic terms of chiral mult. scalars from disks N X F f =1 n D µq f D µ q f + D µ q f D µ q o f Sugra Lagrangian: different field normaliz. s N X F f =1 n K Q D µq f D µ Q f + K Q D µ Qf D µ Q f Related via the Kähler metrics: q = K Q Q, q = K Q Q o Back Back

25 Non-perturbative sectors from E5 Adding ordinary instantons Add a stack of k E5 branes whose internal part coincides with the D9a: ordinary instantons for the D9a gauge theory would be exotic for the D9b, c gauge theories R 1 9 b 9 a 9 c R 1 5 a R0 New types of open strings: E5 a /E5 a (neutral sector), D9 a /E5 a (charged sector), D9 b /E5 a or E5 a /D9 c (flavored sectors, twisted) These states carry no momentum in space-time: moduli, not fields. [Collective name: M k ] charged or neutral moduli can have KK momentum

26 Non-perturbative sectors from E5 The spectrum of moduli Sector ADHM Meaning Chan-Paton Dimension 5 a/5 a NS a µ centers adj. U(k) (length) D c Lagrange mult.. (length) R M α partners. (length) 1 Lagrange mult.. (length) 3 λ α 9 a/5 a NS w α sizes N a k (length) 5 a/9 a w α... k Na... 9 a/5 a R µ partners N a k (length) 1 5 a/9 a µ. k N a... 9 b /5 a R µ flavored N F k (length) 1 5 a/9 c µ... k NF...

27 Non-perturbative sectors from E5 Some observations Among the neutral moduli we have the center of mass position x µ 0 and its fermionic partner θα (related to susy broken by the E5a): Back a µ = x µ 0 1 k k + y µ c T c, M α = θ α 1 k k + ζ α c T c, R 1 9 b µ µ 9 c R 1 In the flavored sectors only fermionic zero-modes: µ f (D9 b /E5 a sector) µ f (E5 a /D9 c sector) 5 a R 0

28 The stringy instanton calculus

29 Instantonic correlators The stringy way In presence of Euclidean branes, dominant contribution to correlators of gauge/matter fields from one-point functions. Polchinski, 1994; Green and Gutperle, ; Billo et al, 00; Blumenhagen et al, 006 E.g., a correlator of chiral fields q q... is given by q q... ( ) Disks: = 8π g a k + S mod (M k ) (with moduli insertions) Annuli: A 5a (no moduli insert.s, otherwise suppressed)

30 The effective action in an instantonic sector The various instantonic correlators can be obtained shifting the moduli action by terms dependent on the gauge/matter fields. In the case at hand, q q S mod (q, q; M k ) = ( = tr k id c w α (τ c ) α β w β + i η µν c ˆaµ ν, a iλ α µw α + w α µ + ˆa µ, M α σ µ α α X n + tr k w αˆq f q f + q f q f w α i µ q f µ f + i µ f q f µo. f o

31 The effective action in an instantonic sector There are other relevant diagrams involving the superpartners of q and q, related to the above by susy Ward identities. Complete result: in S mod (q, q; M k ). q(x 0 ), q(x 0 ) q(x 0, θ), q(x 0, θ) The moduli have to be integrated over

32 The instanton partition function as an integral over moduli space Summarizing, the effective action has the form (Higgs branch) S k = C k e 8π ga k e A 5a dm k e S mod (q, q;m k )

33 The instanton partition function as an integral over moduli space Summarizing, the effective action has the form (Higgs branch) S k = C k e 8π ga k e A 5a dm k e S mod (q, q;m k ) In A 5 a the contribution of zero-modes running in the loop is suppressed because they re already explicitly integrated over Blumenhagen et al, 006

34 The instanton partition function as an integral over moduli space Summarizing, the effective action has the form (Higgs branch) S k = C k e 8π ga k e A 5a dm k e S mod (q, q;m k ) C k is a normalization factor, determined (up to numerical constants) counting the dimensions of the moduli M k : Back C k = ( α ) (3N a N F )k (ga ) Nak. Notice the appearing of the β-function coeff. b 1

35 Instanton induced superpotential In S mod (q, q; M k ), the superspace coordinates x µ 0 and θα appear only through superfields q(x 0, θ), q(x 0, θ),... Recall We can separate x, θ from the other moduli Mk writing S k = d 4 x 0 d θ W k (q, q), with the effective superpotential W k (q, q) = C k e 8π ga k e A 5a d M k e S mod (q, q; c M k )

36 Issues of holomorphicity A superpotential is expected to be holomorphic. We found W k (q, q) = C k e 8π ga k e A 5a d M k e S mod (q, q; c M k )

37 Issues of holomorphicity A superpotential is expected to be holomorphic. We found W k (q, q) = C k e 8π ga k e A 5a d M k e S mod (q, q; c M k ) S mod (q, q; M k ) explicitly depends on q and q. This dependence disappears upon integrating over M k as a consequence of the cohomology properties of the integration measure. However, we have to re-express the result in terms of the SUGRA fields Q and Q Recall

38 Issues of holomorphicity A superpotential is expected to be holomorphic. We found W k (q, q) = C k e 8π ga k e A 5a d M k e S mod (q, q; c M k ) The prefactors should combine into a dynamically generated holomorphic scale Λ hol, obtained by integrating the Wilsonian β-function of the N = 1 SQCD Novikov et al, 1983; Dorey et al, 00;... To this effect, it is crucial that A 5 a can introduce a non-holomorphic dependence on the complex and Kähler structure moduli of the compactification space. Back Our aim is to consider the interplay of all these observations. For this we need the explicit expression of the mixed annuli term A 5 a

39 The ADS/TVY superpotential To be concrete, let s focus on the single instanton case, k = 1. In this case, the integral over the moduli can be carried out explicitly. Balancing the fermionic zero-modes requires N F = N a 1 The end result is Dorey et al, 00; Akerblom et al, 006; Argurio et al, 007 W k=1 (q, q) = C k e 8π ga k e A 1 5a det ( qq )

40 The ADS/TVY superpotential To be concrete, let s focus on the single instanton case, k = 1. In this case, the integral over the moduli can be carried out explicitly. Balancing the fermionic zero-modes requires N F = N a 1 The end result is Dorey et al, 00; Akerblom et al, 006; Argurio et al, 007 W k=1 (q, q) = C k e 8π ga k e A 1 5a det ( qq ) Same form as the ADS/TVY superpotential Affleck et al, 1984; Taylor et al, 1983;

41 The ADS/TVY superpotential To be concrete, let s focus on the single instanton case, k = 1. In this case, the integral over the moduli can be carried out explicitly. Balancing the fermionic zero-modes requires N F = N a 1 The end result is Dorey et al, 00; Akerblom et al, 006; Argurio et al, 007 W k=1 (q, q) = C k e 8π ga k e A 5a 1 det ( qq ) We ll see how these factors conspire to give an holomorphic expression in the sugra variables Q and Q

42 Instanton annuli and threshold corrections

43 The mixed annuli The amplitude A 5a is a sum of cylinder amplitudes with a boundary on the E5a (both orientations) = + + A 5a A 5a;9 a A 5a;9 b A 5a;9 c

44 The mixed annuli The amplitude A 5a is a sum of cylinder amplitudes with a boundary on the E5a (both orientations) = + + A 5a A 5a;9 a A 5a;9 b A 5a;9 c Both UV and IR divergent. The UV divergences (IR in the closed string channel) cancel if tadpole cancellation assumed. Regulate the IR with a scale µ

45 The mixed annuli The amplitude A 5a is a sum of cylinder amplitudes with a boundary on the E5a (both orientations) = + + A 5a A 5a;9 a A 5a;9 b A 5a;9 c There is a relation between these instantonic annuli and the running gauge coupling constant Back A 5a = 8π k ga(µ). 1 loop Abel and Goodsell, 006; Akerblom et al, 006 Indeed, in susy theories, mixed annuli compute the running coupling by expanding around the instanton bkg Billo et al, 007

46 Computing the YM effective action using different backgrounds There are two gauge backgrounds on which string theory is computable and yields the effective action for the gauge fields Constant gauge field f (turned on a color D9-brane) At tree level, the YM action 1 g d 4 x Tr 1 a F µν evaluates to S(f ) = Vol 4 f f f ga At loop level, we have ( are threshold corrections) S(f ) 1 loop = ( ) b1 16π log α µ + Vol 4 f Vol 4 f = ga(µ) 1 loop f f

47 Computing the YM effective action using different backgrounds There are two gauge backgrounds on which string theory is computable and yields the effective action for the gauge fields Instanton background (realized by k E5 branes) At tree level, the YM action evaluates to Back S inst = 8π k g a At loop level, we have the analogous relation: 8π k S inst 1 loop = ga(µ) = A 5a 1 loop With susy, the 1-loop determinants of the non-zero-modes cancel out: the only effect is the renormalization of the gauge coupling constant. Dadda et al, 1977;...

48 Expression of the annuli Outline of the computation The explicit computation of the annuli confirms the relation of these annuli to the running coupling. Imposing the appropriate b.c. s and GSO one starts from 0 dτ τ ( ) ( )] [Tr NS P GSO P orb. q L 0 Tr R P GSO P orb. q L 0 For A 5a;9 a, KK copies of zero-modes on internal tori T (i) give a (non-holomorphic) dependence on the Kähler and complex moduli Lüst and Stieberger, 003. For A 5a;9b and A 5 a;9 c, the modes are twisted and the result depends from the angles ν (i) ba and ν(i) Recall ac.

49 Expression of the annuli Explicit result Back [ A 5a;9a = 8π 3Na k 16π log(α µ ) + N ( a 16π log U (i) T (i) (η(u(i) ) 4)], i ( A 5a;9b + A 5a;9c = 8π NF k 16π log(α µ ) + N F 3π log (Γ ba Γ ac ) ),

50 Expression of the annuli Explicit result Back [ A 5a;9a = 8π 3Na k 16π log(α µ ) + N ( a 16π log U (i) T (i) (η(u(i) ) 4)], i ( A 5a;9b + A 5a;9c = 8π NF k 16π log(α µ ) + N F 3π log (Γ ba Γ ac ) β-function coefficient of SQCD: 3N a N F ),

51 Expression of the annuli Explicit result Back [ A 5a;9a = 8π 3Na k 16π log(α µ ) + N ( a 16π log U (i) T (i) (η(u(i) ) 4)], i ( A 5a;9b + A 5a;9c = 8π NF k 16π log(α µ ) + N F 3π log (Γ ba Γ ac ) Non-holomorphic threshold corrections ),

52 Expression of the annuli Explicit result Back [ A 5a;9a = 8π 3Na k 16π log(α µ ) + N ( a 16π log U (i) T (i) (η(u(i) ) 4)], i ( A 5a;9b + A 5a;9c = 8π NF k 16π log(α µ ) + N F 3π log (Γ ba Γ ac ) ), Γ ba = Γ(1 ν(1) ba ) Γ(ν (1) ba ) Γ(ν () ba ) Γ(1 ν () ba ) Γ(ν (3) ba ) Γ(1 ν (3) ba ) Lüst and Stieberger, 003 Akerblom et al, 007

53 Holomorphicity properties

54 The holomorphic gauge coupling Computing the pure instantonic disks and annuli yields the gauge coupling up to 1 loop in the form Recall A 1 loop = 8π k g a(µ) = 8π k g a + A 5a The very general Kaplunovsky-Louis formula expresses the one-loop gauge coupling in terms of the wilsonian coupling 1/ g a = s and of other tree-level quantities in the effective action

55 Kaplunovsky-Louis relation at one loop Dixon et al, 1991; Kaplunovsky and Louis, ;... 1 g (µ) = 1 g + 1 [ b 8π r log µ M P n r T (r) log K r ] f (1) c K + T (G) log 1 g Here T A = generators of the gauge group, n r = # chiral mult. in rep. r and ( ) T (r) δ AB = Tr r TA T B, T (G) = T (adj) b = 3 T (G) r n r T (r), c = T (G) r n r T (r),

56 Kaplunovsky-Louis relation at one loop Dixon et al, 1991; Kaplunovsky and Louis, ;... 1 g (µ) = 1 g + 1 [ b 8π r log µ M P n r T (r) log K r ] f (1) c K + T (G) log 1 g Holomorphic function

57 Kaplunovsky-Louis relation at one loop Dixon et al, 1991; Kaplunovsky and Louis, ;... 1 g (µ) = 1 g + 1 [ b 8π r log µ M P n r T (r) log K r ] f (1) c K + T (G) log 1 g Non-holomorphic corrections

58 Kaplunovsky-Louis relation at one loop Dixon et al, 1991; Kaplunovsky and Louis, ;... 1 g (µ) = 1 g + 1 [ b 8π r log µ M P n r T (r) log K r ] f (1) c K + T (G) log 1 g Inside the square bracket the bulk Kähler potential K and the Kähler metrics for the matter multiplets K r are at tree level

59 Kaplunovsky-Louis relation at one loop Dixon et al, 1991; Kaplunovsky and Louis, ;... 1 g (µ) = 1 g + 1 [ b 8π r log µ M P n r T (r) log K r ] f (1) c K + T (G) log 1 g The only place where the shift δ Recall in the holomorphic coupling matters is the tree-level term. Moreover only a shift δ (0) of order 1 in g s is relevant at this level!

60 Instantonic annuli in Kaplunovsky-Louis form The result for the instantonic annuli Recall can be recast in the following form: A 5a = 8π k g a + N a N F with (similarly for Z ac ) [ + k 3N a N F log µ M P N a 3 i=1 K + N a log g a δ (0) + N F log(z baz ac ) log (η(u (i) ) ) Z ba = ( ) 1 4π s ( 4 t (1) ) 1 ( t() t(3) 4 u (1) ) 1 ( ) 1 u() u(3) Γ ba ] If δ (0) = 0, Z ba coincides with the Kähler metric K ab of the twisted matter

61 Instantonic annuli in Kaplunovsky-Louis form The result for the instantonic annuli Recall can be recast in the following form: A 5a = 8π k g a + N a N F [ + k 3N a N F log µ M P N a 3 i=1 K + N a log g a δ (0) + N F log(z baz ac ) log (η(u (i) ) ) If there is some one-loop shift of s of order 1, i.e., δ (0) 0, then we have K ab = χ ab Z ba ] with δ (0) + N F log χ abχ bc = 0

62 The Kähler metric for twisted matter Thus, up to possible factors χ due to one-loop shifts δ (0), the Kähler metric of chiral multiplets Q arising from twisted D9 a /D9 b strings is given by Back with K Q = ( ) 1 4π s ( 4 t (1) ) 1 ( t() t(3) 4 u (1) ) 1 ( ) 1 u() u(3) Γ ba Γ ba = Γ(1 ν(1) ba ) Γ(ν (1) ba ) This is very interesting because: Γ(ν () ba ) Γ(1 ν () ba ) Γ(ν (3) ba ) Γ(1 ν (3) ba ) for twisted fields, the Kähler metric cannot be derived from compactification of DBI

63 The Kähler metric for twisted matter Thus, up to possible factors χ due to one-loop shifts δ (0), the Kähler metric of chiral multiplets Q arising from twisted D9 a /D9 b strings is given by Back with K Q = ( ) 1 4π s ( 4 t (1) ) 1 ( t() t(3) 4 u (1) ) 1 ( ) 1 u() u(3) Γ ba Γ ba = Γ(1 ν(1) ba ) Γ(ν (1) ba ) This is very interesting because: Γ(ν () ba ) Γ(1 ν () ba ) Γ(ν (3) ba ) Γ(1 ν (3) ba ) the part dependent on the twists, namely Γ ba, is reproduced by a direct string computation Lüst et al, 004; Bertolini et al, 005 the prefactors, depending on the geometric moduli, are more difficult to get directly: the present suggestion is welcome!

64 The Kähler metric for twisted matter Thus, up to possible factors χ due to one-loop shifts δ (0), the Kähler metric of chiral multiplets Q arising from twisted D9 a /D9 b strings is given by Back with Γ ba = Γ(1 ν(1) ba ) Γ(ν (1) ba ) This is very interesting because: Γ(ν () ba ) Γ(1 ν () ba ) Γ(ν (3) ba ) Γ(1 ν (3) ba ) We have checked this expression against the known results for Yukawa couplings of magnetized branes: perfect consistency! Cremades et al, 004 N.B. This check also severely constrains the possible extra pre-factors χ ba,...

65 Back to the instanton calculus Getting holomorphicity Beside being related to the gauge thresholds, the instantonic annuli A 5a are relevant because they enter the stringy instanton calculus In particular, the form of the A 5a annuli is crucial for the holomorphicity properties of E5 non-perturbative contributions We consider for definiteness the ADS/TVY case.

66 Back to the ADS/TVY superpotential Making it holomorphic We found ( recall that N F = N a 1 in this case) W k=1 (q, q) = C k e 8π ga k e A 1 5a det ( qq ) Insert the expression of the annuli, from which we must subtract the contrib. of the zero-modes running in the loop, which are responsible for the IR divergences. Use the natural UV cut-off of the low-energy theory, the Planck mass M P = 1 α e φ 10 s and write A 5a = k b 1 log µ M P + A 5a

67 Back to the ADS/TVY superpotential Making it holomorphic W k=1 (q, q) = C k e 8π ga k e A 1 5a det ( qq ) Make explicit the prefactor C k Recall Allow for a possible shift in the gauge coupling: 1 g a = 1 g a + δ 8π

68 Back to the ADS/TVY superpotential Making it holomorphic In this way we obtain W k=1 = e K / 3 ) (η(u (i) ) Na i=1 ( K QK Q ) Na 1 1 det ( qq ) ( α ) b 1 e 8π eg a Rescale the chiral multiplet to their sugra counterparts assuming K Q, K Q are the matter Kähler metrics Recall Introduce the invariant scale in the Wilsonian scheme hol = ( α ) b 1 e 8π eg a Λ b 1

69 Back to the ADS/TVY superpotential Making it holomorphic We get thus W k=1 = e K / 3 ) (η(u (i) ) Na i=1 e K / Λ N a+1 hol 1 det( QQ) Λ Na+1 hol 1 det( Q Q) In the second step the moduli dependent factors of η(u (i) ) are readsorbed by a holomorphic redefinition of the scale A part from the prefactor e K /, the final expression is holomorphic in the variables of the Wilsonian scheme

70 Back to the ADS/TVY superpotential Making it holomorphic We get thus W k=1 = e K / 3 ) (η(u (i) ) Na i=1 e K / Λ N a+1 hol 1 det( QQ) Λ Na+1 hol 1 det( Q Q) The rôle of the annuli in these non-perturbative considerations leads to equivalent information on the Kähler metric of the twisted matter as the comparison with the perturbative KL formula

71 Remarks and conclusions Also in N = toroidal models the instanton-induced superpotential is in fact holomorphic in the appropriate sugra variables if one includes the mixed annuli in the stringy instanton calculus Akerblom et al, 007; Billo et al, 007 W.r.t. to the color D9 a branes, the E5 a branes are ordinary instantons. For the gauge theories on the D9 b or the D9 c, they would be exotic (less clear from the field theory viewpoint) The study of the mixed annuli and their relation to holomorphicity can be relevant for exotic, new stringy effects as well. R 1 9 b µ 5 a µ 9 c R 0 R 1