Exotic instantons in d=8 and the heterotic/type I duality

Transcription

1 Exotic instantons in d=8 and the heterotic/type I duality Marco Billò Dip. di Fisica Teorica, Università di Torino and I.N.FN., sez. di Torino Problemi attuali di Fisica teorica Vietri sul Mare - April 8, 2009

2 Foreword Mostly based on M. Billo, M. Frau, L. Gallot, A. Lerda and I. Pesando, Classical solutions for exotic instantons?,, JHEP 03 (2009) 056, arxiv: [hep-th]. Same authors + L. Ferro, in preparation. It builds over a vast literature I apologize for missing references...

3 Plan of the talk 1 Motivations 2 Exotic instantons in type I 3 Interpretation as 8d instanton solutions 4 Rôle in type I /Heterotic duality 5 Conclusions and perspectives

4 Motivations

5 Non-perturbative sectors in brane-worlds R 1,3 D6 b D6 a CY 3 (Susy) gauge and matter sectors on the uncompactified part of (partially wrapped) D-branes chiral matter, families from multiple intersections, tuning different coupling constants...

6 Non-perturbative sectors in brane-worlds E3 a R 1,3 D6 b D6 a CY 3 (Susy) gauge and matter sectors on the uncompactified part of (partially wrapped) D-branes chiral matter, families from multiple intersections, tuning different coupling constants... Non-perturbative sectors from partially wrapped E(uclidean)-branes Pointlike in the R 1,3 space-time: instanton configurations

7 Ordinary vs. exotic E-branes identical to D-branes in the internal directions: gauge instantons ADHM from strings attached to the instantonic branes Witten, 1995; Douglas, ;... non-trivial instanton profile of the gauge field Billo et al, 2001 E-branes different from D-branes in internal directions do not represent gauge instantons They are called exotic or stringy instantons in certain cases can give important contributions to the effective action,.e.g. Majorana masses for neutrinos, moduli stabilizing terms,... Blumenhagen et al ; Ibanez and Uranga, ; (long list)... ; Petersson

8 World-sheet properties Consider the strings stretching between the gauge D-branes and the E-branes NS sector (lowest KK level) physicity condition L = N X + N ψ + 3 i=1 θ i 2 = 0, Ordinary case: internal twists θi = 0. There are bosonic moduli w α typical of ADHM construction, related to the size Exotic case: θi > 0, i.e., there are more than 4 ND directions. The moduli w α are absent. Hints at zero-size limit of some gauge field configuration.

9 World-sheet properties Consider the strings stretching between the gauge D-branes and the E-branes In the R sector, the ground states always: fermionic anti-chiral moduli λ α Ordinary case: Lagrange multipl. of fermionic ADHM constraints Exotic case: the the abelian component of the λ s is a true fermionic zero-mode since the abelian part of ADHM constraint vanishes (would cointain the w α )

10 World-sheet properties Consider the strings stretching between the gauge D-branes and the E-branes Exotic case: need to remove the fermionic zero-mode to get non-zero correlators orientifold projections Argurio et al, 2007;..., lift with closed string fluxes Blumenhagen et al, 2007; Billo et al, 2008;... other mechanisms Petersson, 2007;...

11 World-sheet properties Consider the strings stretching between the gauge D-branes and the E-branes These w.s. properies of the exotic systems must be taken into account when trying to answer the question: Do exotic instantonic branes correspond to some classical configuration of gauge/matter fields?

12 World-sheet properties Consider the strings stretching between the gauge D-branes and the E-branes We will consider this problem in a simplified setting: D(-1)/D7 in type I on T 2 It has more than 4 ND directions (eight, in fact): exotic Built-in orientifod projection that eliminates the λ α ferm. 0-modes However, the gauge theory part lives in 8 dimensions

13 Instanton counting and dualities Beside providing (new) non-perturbative couplings in the effective actions see Alberto s talk instantons (also exotic ones?) play a key rôle in duality statements, both at the field theory and the string theory level

14 Instanton counting and dualities Beside providing (new) non-perturbative couplings in the effective actions see Alberto s talk instantons (also exotic ones?) play a key rôle in duality statements, both at the field theory and the string theory level In theories with higher susy, sum over all instanton # needed to check dualities and exact results D-instanton partition function from matrix integrals Moore et al, 1998 checks against expression derived by self-duality of type IIB Green-Gutperle, 1997 Seiberg-Witten sol. by instanton counting Nekrasov, 2002

15 Instanton counting and dualities Beside providing (new) non-perturbative couplings in the effective actions see Alberto s talk instantons (also exotic ones?) play a key rôle in duality statements, both at the field theory and the string theory level In theories with higher susy, sum over all instanton # needed to check dualities and exact results D-instanton partition function from matrix integrals Moore et al, 1998 checks against expression derived by self-duality of type IIB Green-Gutperle, 1997 Seiberg-Witten sol. by instanton counting Nekrasov, 2002 Effective action at O(F 4 ) summing over D(-1) s in type I must match by string duality the Het SO(4) 4 one: 2nd part of this talk

16 Exotic instantons in type I

17 A D(-1)/D7 system in type I Type I is type IIB on T 2 modded out by Ω = ω ( 1) F L I 2 where ω = w.s. parity, F L = left-moving w.s. fermion #, I 2 = inversion on T 2 Ω has four fixed-points on T 2 where four O7-planes are placed O7

18 A D(-1)/D7 system in type I Type I is type IIB on T 2 modded out by Ω = ω ( 1) F L I 2 where ω = w.s. parity, F L = left-moving w.s. 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 O7

19 A D(-1)/D7 system in type I Type I is type IIB on T 2 modded out by Ω = ω ( 1) F L I 2 where ω = w.s. parity, F L = left-moving w.s. 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 Local RR tadpole cancellation requires 8 D7-branes at each fix point D7

20 A D(-1)/D7 system in type I Type I is type IIB on T 2 modded out by Ω = ω ( 1) F L I 2 where ω = w.s. parity, F L = left-moving w.s. 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 Local RR tadpole cancellation requires 8 D7-branes at each fix point We focus on one fix point

21 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): Aμ, Λ α, ϕ m, μ = 1,... 8, m = 8, 9 Can be assembled into a chiral superfield Back Φ(x, θ) = ϕ(x) + 2 θλ(x) θγμν θ F μν (x) +... where ϕ = (ϕ 9 + iϕ 10 )/ 2. Formally very similar to N = 2 in d = 4

22 Effective action on the D7 s Effective action in F μν an its derivatives: NABI Back S D7 = 1 Fμν F μν d 8 x Tr 8πg s (2π α ) (2π) t 2 8 F 4 + α g s d 8 x L (5) (F, DF) + = S YM + S (4) + S (5) +, The Yang-Mills action S YM has a dimensionful coupling g 2 YM 4πg s(2π α ) 4

23 Effective action on the D7 s Effective action in F μν an its derivatives: NABI Back S D7 = 1 Fμν F μν d 8 x Tr 8πg s (2π α ) (2π) t 2 8 F 4 + α g s d 8 x L (5) (F, DF) + = S YM + S (4) + S (5) +, The quartic action Detail has a dimensionless coupling λ 4 4π 3 g s : S (4) = 1 d 8 x Tr t 4!λ 4 8 F 4

24 Effective action on the D7 s Effective action in F μν an its derivatives: NABI Back S D7 = 1 Fμν F μν d 8 x Tr 8πg s (2π α ) (2π) t 2 8 F 4 + α g s d 8 x L (5) (F, DF) + = S YM + S (4) + S (5) +, Adding the WZ term, we can write Back 1 S (4) = d 8 x Tr t 4! 4π 3 8 F 4 2πi C 0 c (4) g s where c (4) is the fourth Chern number

25 Effective action on the D7 s Effective action in F μν an its derivatives: NABI Back S D7 = 1 Fμν F μν d 8 x Tr 8πg s (2π α ) (2π) t 2 8 F 4 + α g s d 8 x L (5) (F, DF) + = S YM + S (4) + S (5) +, Contributions of higher order in (α ): rôle to be discussed later

26 Adding D-instantons Add k D-instantons. D7/D(-1) form a 1/2 BPS system with 8 ND directions k D(-1) D(-1) classical action Back Back S cl = k( 2π 2πi C 0 ) 2πi kτ, g s Coincides with the quartic action Recall on the D7 for gauge fields F with c (4) = k and d 8 x Tr t 8 F 4 = 1 2 d 8 x Tr ε 8 F 4 = 4! 2 (2π)4 c (4)

27 Adding D-instantons Add k D-instantons. D7/D(-1) form a 1/2 BPS system with 8 ND directions k D(-1) D(-1) classical action Back Back S cl = k( 2π 2πi C 0 ) 2πi kτ, 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

28 The moduli spectrum Besides the classical action, we must consider the spectrum and interactions of strings ending on a D(-1).

29 The moduli spectrum Spectrum: Back Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length)

30 The moduli spectrum Spectrum: Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length) The SO(k) rep. is determined by the orientifold projection Ω

31 The moduli spectrum Spectrum: Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length) Abelian part of a μ, M α Goldstone modes of the (super)translations on the D7 broken by D(-1) s Identified with coordinates x μ,θ α

32 The moduli spectrum Spectrum: Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length) Abelian part of λ α would be a dangerous 0-mode Removed by the orientifold projection

33 The moduli spectrum Spectrum: Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length) One needs auxiliary fields D m, m = 1,... 7 to disentangle the quartic interaction a μ, a ν [a μ, a ν ]. Octonionic analogue of the fact that for D(-1)/D3 systems one needs D c, c = 1, 2, 3

34 The moduli spectrum Spectrum: Sector Name Meaning Chan-Paton Dimension 1/ 1 NS a μ centers symm SO(k) (length) χ, χ adj SO(k) (length) 1 D m Lagr. mult.. (length) 2 R M α partners symm SO(k) (length) 1 2 λ α Lagr. mult. adj SO(k) (length) 3 2 1/ 7 R μ 8 k (length) For mixed strings, no bosonic moduli from the NS sector This is a characteristic of exotic instantons

35 The moduli action Beside the classical action Recall we have Back S = S quartic + S cubic + S mixed

36 The moduli action Beside the classical action Recall we have Back S = S quartic + S cubic + S mixed The quartic part can be (partly) disentangled with auxiliary fields D m : S quartic = 1 g tr 2 D md m D m(τ m ) μν [a μ, a ν ] a μ, χ [a μ, χ] [ χ, χ] 2

37 The moduli action Beside the classical action Recall we have Back S = S quartic + S cubic + S mixed The cubic part reads: S cubic = 1 g 2 0 tr iλ α (γ μ ) αβ a μ, M β iλ α [χ, λ α ] im α [ χ, M α ].

38 The moduli action Beside the classical action Recall we have Back S = S quartic + S cubic + S mixed From diagrams with mixed b.c. s μ t S mixed = 1 g 2 tr i μ T χμ ; 0 χ μ

39 The moduli action Beside the classical action Recall we have Back S = S quartic + S cubic + S mixed In the case k = 1 all of these contributes vanish (no adjoint moduli!). We are left with the classical part only S cl = 2πiτ Coincides with the quartic action on the D7 for gauge fields F s.t. Back c(4) = 1 Tr(t8 F 4 ) = 1/2Tr(ε 8 F 4 ) Suggests interpretation as instantons but...

40 No gauge field emission Ordinary instantonic brane systems (such as D(-1)/D3): classical instanton profile due to the emission of gauge field from mixed disks Billo et al, 2001 x ν A i μ = 2ρ2 η i μν x (SU(2), sing. gauge, large- x, 2ρ 2 = tr w αw α ) p A μ (p) w w

41 No gauge field emission Ordinary instantonic brane systems (such as D(-1)/D3): classical instanton profile due to the emission of gauge field from mixed disks Billo et al, 2001 x ν A i μ = 2ρ2 η i μν x p A μ (p) w w (SU(2), sing. gauge, large- x, 2ρ 2 = tr w αw α ) In the exotic D(-1)/D7 system there is no emission diagram for A μ because there are no bosonic mixed moduli w

42 No gauge field emission Ordinary instantonic brane systems (such as D(-1)/D3): classical instanton profile due to the emission of gauge field from mixed disks Billo et al, 2001 x ν A i μ = 2ρ2 η i μν x p A μ (p) w w (SU(2), sing. gauge, large- x, 2ρ 2 = tr w αw α ) In the exotic D(-1)/D7 system there is no emission diagram for A μ because there are no bosonic mixed moduli w The classical profile vanishes outside the location of the D(-1)

43 Interpretation as 8d instanton solutions

44 Expected features A D(-1) inside the D7 s should correspond to the zero-size limit of some instantonic configuration of the SO(8) gauge field such that has 4-th Chern number c(4) = 1 the quartic action reduces to the D(-1) action Recall preserves SO(8) Lorentz invariance corresponds to a 1/2 BPS config. in susy case

45 Expected features A D(-1) inside the D7 s should correspond to the zero-size limit of some instantonic configuration of the SO(8) gauge field such that has 4-th Chern number c(4) = 1 the quartic action reduces to the D(-1) action Recall preserves SO(8) Lorentz invariance corresponds to a 1/2 BPS config. in susy case Many generalizations of 4d instantons to 8d In particular linear instantons [Corrigan, 1982; Fubini-Nicolai, 1985;...] F μν T [μνρσ]f ρσ = 0, Bianchis imply YM e.o.m D μ F μν = 0 do not fully preserve the SO(8) Lorentz group, and are less than 1/2 BPS

46 The SO(8) instanton All our requirements met by the SO(8) instanton [Grossmann e al, 1985] Aμ (x) αβ = (γ μν ) αβ x ν r 2 + ρ 2 with ρ = instanton size and r 2 = x μ x μ, while αβ adjoint of the SO(8) gauge group. is self-dual in the sense that F F = (F F) satisfies t8 F 4 = 1/2ε 8 F 4 from Clifford Algebra has c(4) = 1 and S (4) = 2πiτ

47 The SO(8) instanton All our requirements met by the SO(8) instanton [Grossmann e al, 1985] Aμ (x) αβ = (γ μν ) αβ x ν r 2 + ρ 2 with ρ = instanton size and r 2 = x μ x μ, while αβ adjoint of the SO(8) gauge group. is self-dual in the sense that F F = (F F) satisfies t8 F 4 = 1/2ε 8 F 4 from Clifford Algebra has c(4) = 1 and S (4) = 2πiτ However, it is not a solution of Y.M. e.o.m. in d = 8: D μ F μν (x) = 4(d 4)ρ2 (r 2 + ρ 2 ) 3 γ μνx ν.

48 Consistency conditions Eff. action on the D7 is the NABI action Recall To keep the quartic action and the instanton effects the field-theory limit must be α 0, g s fixed

49 Consistency conditions Eff. action on the D7 is the NABI action Recall To keep the quartic action and the instanton effects the field-theory limit must be α 0, g s fixed This limit is dangerous on the YM action S YM since g 2 YM g sα 2. On the SO(8) instanton, however, we have (R regulates the volume): ρ4 S YM α 2 log ρ/r, g s which vanishes in the zero-size limit ρ 0 if ρ 2 /α 2 0 (done before removing R)

50 Consistency conditions Eff. action on the D7 is the NABI action Recall To keep the quartic action and the instanton effects the field-theory limit must be α 0, g s fixed Consider the higher order α corrections to the NABI action. On the SO(8) instanton, by dimensional reasons, must be ρ d 8 α n a n, n=1 The coefficients a n should vanish for consistency! ρ 2

51 O(F 5 ) terms in the NABI The first coefficient a 1 arises from the integral of L (5) (F, DF), i.e.the term of order α 3 w.r.t to the YM action. We would like to check that it vanishes. Crucial point: which is the form of L (5) (F, DF)?

52 O(F 5 ) terms in the NABI The first coefficient a 1 arises from the integral of L (5) (F, DF), i.e.the term of order α 3 w.r.t to the YM action. We would like to check that it vanishes. Crucial point: which is the form of L (5) (F, DF)? Various proposals in the literature Refolli et al, Koerber-Sevrin, Grasso, Barreiro-Medina,..., obtained by different methods differing by terms which vanish on-shell, i.e. upon use of the YM e.o.m.

53 O(F 5 ) terms in the NABI The first coefficient a 1 arises from the integral of L (5) (F, DF), i.e.the term of order α 3 w.r.t to the YM action. We would like to check that it vanishes. Crucial point: which is the form of L (5) (F, DF)? Various proposals in the literature Refolli et al, Koerber-Sevrin, Grasso, Barreiro-Medina,..., obtained by different methods differing by terms which vanish on-shell, i.e. upon use of the YM e.o.m. One proposal is singled out by admitting a susy extension Collinucci et al, 2002

54 Check at O(F 5 ) in the NABI The bosonic part of the supersimmetrizable O(α 3 ) lagrangian is L (5) = ζ(3) 2 Tr 4 F μ1 μ 2, F μ3 μ 4 Fμ1 μ 3, F μ2 μ 5, Fμ4 μ F μ1 μ 2, F μ3 μ 4 Fμ1 μ 2, F μ3 μ 5, Fμ4 μ F μ1 μ 2, D μ5 F μ1 μ 4 Dμ5 F μ2 μ 3, F μ3 μ 4 2 F μ1 μ 2, D μ4 F μ3 μ 5 Dμ4 F μ2 μ 5, F μ1 μ 3 + F μ1 μ 2, D μ5 F μ3 μ 4 Dμ5 F μ1 μ 2, F μ3 μ 4

55 Check at O(F 5 ) in the NABI Plugging the instanton profile into α g s d 8 x L (5) we get [Using the CADABRA program by Kasper Peeters] α ζ(3) 2 d/2+9 πd/2 Γ(9 d/2) g s 9! ρ 10 d (d 1)(d 2)(d 4) d 9 d + d + 2 d 2 2 namely a result proportional to d(d 1)(d 2)(d 4)(d 8) The quintic action vanishes on the SO(8) instanton! The check is successful

56 D(-1) s and type I /Heterotic duality

57 Type I /Heterotic duality In the web of string dualities, the S-duality between Het. SO(32) and Type I plays a fundamental rôle Upon compactification (e.g. on T 2 ), other dualities follow from it 11-d sugra Type IIA Het E 8 E 8 Type I Het SO(32) Type IIB The Type I theory is S-dual to Het SO(8) 4 (obtained with Wilson lines on T 2 ) The mapping of parameters involve the relation τ T τ: (complexified) string coupling in type I T: the Kähler param. of the torus in the Het.

58 The heterotic F 4 effective action Het SO(8) 4 is a theory of closed strings. Focus on the effective action for one of the SO(8) gauge factors The BPS-saturated t 8 F 4 terms arise just at 1-loop. Threshold corrections organize as a series in q e 2πiT

59 The heterotic F 4 effective action Het SO(8) 4 is a theory of closed strings. Focus on the effective action for one of the SO(8) gauge factors The BPS-saturated t 8 F 4 terms arise just at 1-loop. Threshold corrections organize as a series in q e 2πiT At even powers in q one gets contributions to the following structures: Lerche-Stieberger, Gutperle,... Back t 8 Tr F q 2k 1 1 q 4k 2 l 2 l k l k l k t 8 (Tr F)) q 2k 1 1 q 4k 4 l 8 l k l k l k

60 The heterotic F 4 effective action Het SO(8) 4 is a theory of closed strings. Focus on the effective action for one of the SO(8) gauge factors The BPS-saturated t 8 F 4 terms arise just at 1-loop. Threshold corrections organize as a series in q e 2πiT For SO(8) there is another quartic gauge invariant, Pf (F) = ε a1...a 8 F a 1a 2... F a 7a 8 (Lorentz indices omitted). Gets contributions at odd powers in q Gava et al 1 8 t 8 Pf (F) q k l kodd l k

61 The type I F 4 effective action Under the Het/type I duality, q = e 2πiT q = e 2πiτ The series of threshold corrections maps to a series of non-perturbative corrections These corrections can be provided by D-instantons: indeed q k is the weight e S cl for k D(-1) Recall The explicit check that the coefficients of the expansion agree would represent a direct, highly non-trivial, test of this string duality

62 The type I F 4 effective action Under the Het/type I duality, q = e 2πiT q = e 2πiτ The series of threshold corrections maps to a series of non-perturbative corrections These corrections can be provided by D-instantons: indeed q k is the weight e S cl for k D(-1) Recall The explicit check that the coefficients of the expansion agree would represent a direct, highly non-trivial, test of this string duality Much discussed in this setting Gutperle, 1999 or in the T-dual one (D1/D9 systems in Type I) Bachas et al, 1997; Kiritsis-Obers, 1997;...

63 The type I F 4 effective action Under the Het/type I duality, q = e 2πiT q = e 2πiτ The series of threshold corrections maps to a series of non-perturbative corrections These corrections can be provided by D-instantons: indeed q k is the weight e S cl for k D(-1) Recall The explicit check that the coefficients of the expansion agree would represent a direct, highly non-trivial, test of this string duality The explicit derivation of the coefficients in the type I side has never been performed (to our knowledge)

64 Interaction with the multiplet Φ How can D(-1) contribute to the F 4 effective action? There s no emission diagram leading to a classical profile, but there are mixed disks (related by SUSY) involving D7/D7 fields μ t μ t μ t ϕ Λ F θ θ θ μ Net effect: moduli action Recall dependence on the superfield Φ(x, θ) Recall μ S (k) [Φ] = 2πiτk + S quartic + S cubic + S mixed + Tr μ t Φμ μ

65 The moduli integration The effective action for the gauge fields is obtained integrating, for each k, over the D-instanton moduli M (k) = (x, θ, M (k) ): Recall Z[ϕ, Λ, F] = d 8 xd 8 θ d M (k) e S (k)[φ(x,θ)] = k d 8 xd 8 θ quartic inv. (Φ(x, θ)) Second step from dimensionality: [d M (k) ] = l 4 The integration over θ yields then a t 8 F 4 term: d 8 θ (θγ μ 1ν 1 θ)f μ1 ν 1... (θγ μ 4ν 4 θ)f μ4 ν 4 = t 8 F 4

66 Integration at k = 1 At k = 1 spectrum of moduli is extremely reduced Z 1 = e 2πiτ d 8 x d 8 θ d 8 μe Tr μtφμ = e 2πiτ d 8 x d 8 θ Pf (Φ) To go to higher k, exploit the susy of the moduli action leading to an (equivariant) cohomological BRS structure localization of the integrals (upon suitable deformations from closed string backgrounds) Similar kind of techniques to those used for D(-1) partition function in type IIB Moore et al,...;... resummation of instantons in N = 2 SYM leading to the SW solution Nekrasov, 2002 re-obtained by D3/D(-1) on orbifold Billo et al, 2006

67 BRS reformulation Single out one of the supercharges Q α, say Q = Q 8. After relabeling some of the moduli: M α M μ (M m, M 8 ), λ α (λ m, η) (λ m, λ 8 ) one has Qa μ = M μ, Qλ m = D m, Q χ = i 2η, Qχ = 0, Qμ = W

68 BRS reformulation Single out one of the supercharges Q α, say Q = Q 8. After relabeling some of the moduli: M α M μ (M m, M 8 ), λ α (λ m, η) (λ m, λ 8 ) one has Qa μ = M μ, Qλ m = D m, Q χ = i 2η, Qχ = 0, Qμ = W On any modulus, Q 2 = T(χ) +R(ϕ) T(χ) = SO(k) rotation in the appropriate rep R(ϕ) a gauge SO(8) rot.

69 BRS reformulation Single out one of the supercharges Q α, say Q = Q 8. After relabeling some of the moduli: M α M μ (M m, M 8 ), λ α (λ m, η) (λ m, λ 8 ) one has Qa μ = M μ, Qλ m = D m, Q χ = i 2η, Qχ = 0, Qμ = W On any modulus, Q 2 = T(χ) +R(ϕ) T(χ) = SO(k) rotation in the appropriate rep R(ϕ) a gauge SO(8) rot. The complete moduli action is Q-exact S = QΞ

70 Deformations from RR background To perform the computation, it is convenient to simply cosider the part. function with Φ(x, θ) ϕ = Φ. Integration over the moduli x, θ would then diverge Introduce suitable deformations that regulate the divergence help to fully localize the integral Arise from RR field-strengths 3-form with one index on T 2 F μν F μνz, F μν F μνz Disk diagrams RR insertions modify the moduli action F λ λ

71 BRS deformation Let us parametrize the RR backround as follows: F μν = 1 2 f mn(τ mn ) μν + h m (τ m ) μν, F μν = 1 2 fmn (τ mn ) μν, The moduli action is modified to S = QΞ fmn only appear in the gauge fermion Ξ : the final result does not depende on them f mn, h m parametrize SO(7) SO(8) (Lorentz) with spinorial embedding and modify the action of Q Q 2 = T(χ) + R(ϕ) + G(F) where G is the appropriate SO(7) action

72 Symmetries of the moduli The Action of the BRS charge Q is determined by the symmetry properties of the moduli SO(k) SO(7) SO(8) a μ symm 8 s 1 M μ symm 8 s 1 D m adj 7 1 λ m adj 7 1 χ adj 1 1 η adj 1 1 χ adj 1 1 μ k 1 8 v

73 Scaling to localization The BRS structure allows to suitably rescale the bosonic and fermionic moduli in such a way that the (super)jacobian of the rescaling is one (the measure is unaffected) one can take a limit in which the exponent reduces to a quadratic expression The integration over some moduli is trivially done, and one is left with (at inst. # k) Z k = N k e 2πiτk {dχda μ dm μ dd ˆm dλ ˆm dμ} e tr g 2 D ˆm D ˆm g 2 λ ˆm Q 2 λ ˆm + t 4 a μ F μν Q 2 a ν + t 4 M μ F μν M ν trμ t Q 2 μ = N k e 2πiτk {dχ} Pf (adj,6 7,1)(Q 2 )Pf(k,1,8v )(Q 2 ) det 1/2 (symm,8 s,1) (Q 2 )

74 General expressions Considering ϕ in the Cartan of SO(8), f in the Cartan of SO(7) and bringing χ to the Cartan of SO(k) one gets (here for k = 2n + 1) Back Z 2n+1 = Ñ 2n+1 e 2πiτ(2n+1) (f 1f 2 f 3 ) n n dχ l E (n+1) 2πi l=1 k χ2 k R ϕ(χ k )R f (χ k ) i<j (χ ij )2 (χ + ij )2 R f (χ ij )R f (χ + ij ) m R E(2χ m )R E (χ m ) p<q R E(χ pq )R E(χ pq + )

75 General expressions Considering ϕ in the Cartan of SO(8), f in the Cartan of SO(7) and bringing χ to the Cartan of SO(k) one gets (here for k = 2n + 1) Back Z 2n+1 = Ñ 2n+1 e 2πiτ(2n+1) (f 1f 2 f 3 ) n n dχ l E (n+1) 2πi l=1 k χ2 k R ϕ(χ k )R f (χ k ) i<j (χ ij )2 (χ + ij )2 R f (χ ij )R f (χ + ij ) m R E(2χ m )R E (χ m ) p<q R E(χ pq )R E(χ pq + ) Here χ ± ij = χ i ±χ j, E 1 = 1/2( f 1 + f 2 + f 3 ),..., E = E 1 E 2 E 3 E 4 and 4 R ϕ (x) x 2 ϕ 2 u, u=1 3 R f (x) x 2 f 2 a, a=1 RE (x) 4 x 2 E 2 A A=1

76 General expressions Considering ϕ in the Cartan of SO(8), f in the Cartan of SO(7) and bringing χ to the Cartan of SO(k) one gets (here for k = 2n + 1) Back Z 2n+1 = Ñ 2n+1 e 2πiτ(2n+1) (f 1f 2 f 3 ) n n dχ l E (n+1) 2πi l=1 k χ2 k R ϕ(χ k )R f (χ k ) i<j (χ ij )2 (χ + ij )2 R f (χ ij )R f (χ + ij ) m R E(2χ m )R E (χ m ) p<q R E(χ pq )R E(χ pq + ) Analogous expression for k = 2n

77 General expressions Considering ϕ in the Cartan of SO(8), f in the Cartan of SO(7) and bringing χ to the Cartan of SO(k) one gets (here for k = 2n + 1) Back Z 2n+1 = Ñ 2n+1 e 2πiτ(2n+1) (f 1f 2 f 3 ) n n dχ l E (n+1) 2πi l=1 k χ2 k R ϕ(χ k )R f (χ k ) i<j (χ ij )2 (χ + ij )2 R f (χ ij )R f (χ + ij ) m R E(2χ m )R E (χ m ) p<q R E(χ pq )R E(χ pq + ) The χ integration are actually contour integrals to be done with certain prescriptions on the Im parts of the poles Moore et al,... (follows from BRS structure)

78 The prepotential Cosider the complete part. function Z(ϕ, f ) = dm (k) e S (k)(ϕ,f) = Ẑ k e 2πiτ k = Ẑ k q k k To a given order in q, contribute also disconnected configurations (instantons of lower numbers k i, with k i = k). To isolate the connected components, take the logarithm k k

79 The prepotential Cosider the complete part. function Z(ϕ, f ) = dm (k) e S (k)(ϕ,f) = Ẑ k e 2πiτ k = Ẑ k q k k In dm (k) we have included the center of mass dx μ and dθ integrals. Without deformations these would diverge (with ϕ constant), now they give 1 E k k

80 The prepotential Cosider the complete part. function Z(ϕ, f ) = dm (k) e S (k)(ϕ,f) = Ẑ k e 2πiτ k = Ẑ k q k k k k The effective action for Φ(x, θ) is written as d 8 x d 8 θ F(Φ(x, θ), f = 0) where F(ϕ, f ) = E log (1 + Z(ϕ, f )) F(ϕ, f ) = F k (ϕ, f )q k k

81 Explicit results at low k By direct integration of the expression of Z k Recall and taking the log we get 1 F = Tr Φ 4 2 q q Tr Φ q q Pf Φ q q q in perfect agreement with the Heterotic results! Recall The fact that for F is finite in the f 0 limit is highly non-trivial, requires very delicate cancellations

82 Explicit results at low k By direct integration of the expression of Z k Recall and taking the log we get 1 F = Tr Φ 4 2 q q Tr Φ q q Pf Φ q q q in perfect agreement with the Heterotic results! Recall The fact that for F is finite in the f 0 limit is highly non-trivial, requires very delicate cancellations If we keep the RR background turned on to the prepotential, we compute also gravitational corrections of the form t 8 tr R 4 and t 8 (tr R 2 ) 2

83 Conclusions and perspectives

84 Conclusions and perspectives D(-1) s on D7, exotic from the w.s. point of view, seen as zero-size limit of a 8d instanton solution Does such an interpretation carry on, and is it useful, to compactified cases relevant to 4d eff. theories? Is there some similar interpretation for other exotic instantons (wrapped E-branes at angles,...)?

85 Conclusions and perspectives D(-1) s on D7, exotic from the w.s. point of view, seen as zero-size limit of a 8d instanton solution Does such an interpretation carry on, and is it useful, to compactified cases relevant to 4d eff. theories? Is there some similar interpretation for other exotic instantons (wrapped E-branes at angles,...)? Consistency condition: the SO(8) instanton must be a solution of the full NABI action check at O(F 5 ) singles out the supersimmetrizable action of Collinucci et al, 2002

86 Conclusions and perspectives D(-1) s on D7, exotic from the w.s. point of view, seen as zero-size limit of a 8d instanton solution Does such an interpretation carry on, and is it useful, to compactified cases relevant to 4d eff. theories? Is there some similar interpretation for other exotic instantons (wrapped E-branes at angles,...)? Consistency condition: the SO(8) instanton must be a solution of the full NABI action check at O(F 5 ) singles out the supersimmetrizable action of Collinucci et al, 2002 Integrating over the D(-1) moduli reproduces the F 4 effective action of the dual Het SO(8) 4 theory Checked up to k = 5 (next step: all k proof) Gravitational corrections to be checked against Heterotic

87 The tensor t 8 We have Back Tr t 8 F tμ 1μ 2 μ 7 μ 8 8 Tr F μ1 μ 2 F μ7 μ 8 =Tr F μν F νρ F λμ F ρλ F μνf ρν F ρλ F μλ 1 4 F μνf μν F ρλ F ρλ 1 8 F μνf ρλ F μν F ρλ