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= 2⋆ Resumming instantons in theories N with arbitrary gauge groups ∗ 6 Marco Bill`o and Marialuisa Frau 1 0 2 Dipartimento di Fisica, Universit`a di Torino and I.N.F.N. - Sezione di Torino Via P. Giuria 1, I-10125 Torino, Italy b e F Francesco Fucito and Jos´e F. Morales 6 1 I.N.F.N. - Sezione di Roma 2 and Dipartimento di Fisica, Universit`a di Roma Tor Vergata ] Via della Ricerca Scientifica, I-00133 Roma, Italy h t - p Alberto Lerda e h [ Dipartimento di Scienze e Innovazione Tecnologica,Universit`a del Piemonte Orientale and I.N.F.N. - Gruppo Collegato di Alessandria - Sezione di Torino 2 Viale T. Michel 11, I-15121 Alessandria, Italy v 3 7 2 0 Abstract 0 . We discuss the modular anomaly equation satisfied by the the prepotential of 2 4-dimensional =2⋆ theoriesandshowthatitsvalidityisrelatedtoS-duality. 0 N 6 The recursion relations that follow from the modular anomaly equation allow 1 onetowritetheprepotentialintermsof(quasi)-modularforms,thusresumming v: the instanton contributions. These results can be checked against the micro- i scopic multi-instanton calculus in the case of classical algebras, but are valid X also for the exceptional E6,7,8, F4 and G2 algebras, where direct computations r are not available. a 1 Introduction These proceedings are based on the papers [1] where we studied = 2⋆ SYM N theories withagauge algebrag A˜ ,B ,C ,D ,E ,F ,G , extendingprevious r r r r 6,7,8 4 2 ∈ { } ∗Proceedings of the XIVMarcel Grossmann Meeting, Rome, Italy,July 12-18, 2015. E-mail: billo,frau,[email protected]; fucito,[email protected] 1 results obtained in [2] for the unitary groups.1 Our motivation is to shed light on the general structure of = 2⋆ SYM theories at low energy and show that the N constraints imposed by S-duality take the form of a recursion relation which allows onetodeterminetheprepotentialatanon-perturbativelevelandresumallinstanton contributions. The =2⋆ theoriesariseasdeformationsofthe = 4theorieswhentheadjoint N N hypermultiplet acquires a mass m. Their low-energy effective dynamics is entirely encoded in the prepotential, which we denote as Fg and which is a holomorphic function of the coupling constant θ 4π τ = +i , (1) 2π g2 and of the vacuum expectation value a of the scalar field in the adjoint vector multiplet. For definiteness, we take a along the Cartan directions of g, namely a = diag a ,a , ,a (2) 1 2 r ··· wherer = rank(g).2 Totreatallalgebra(cid:0)ssimultaneou(cid:1)slyitisconvenienttointroduce the parameter α α L L n = · (3) g α α S S · where α and α are, respectively, the long and the short roots of g. For the root L S system Ψ , we follow the standard conventions [1] (see also the Appendix), so that g n = 1 for g = A˜ ,D ,E , g r r 6,7,8 n = 2 for g = B ,C ,F , (4) g r r 4 n = 3 for g = G . g 2 Using this, one finds that Fg(τ,a) = n iπτa2 +fg(τ,a) (5) g where the first term is the classical contribution while fg is the quantum part. The latter has a τ-independent one-loop term 1 α a 2 α a+m 2 fg = (α a)2log · +(α a+m)2log · (6) 1 loop 4 − · Λ · Λ − αX∈Ψgh (cid:16) (cid:17) (cid:16) (cid:17) i where Λ is an arbitrary scale, and a series of non-perturbative corrections at instan- ton number k proportional to qk, where q = exp(2πiτ). 1Hereand in thefollowing we denoteby A˜r thealgebra of the unitary group U(r+1). 2Thespecial unitary case, corresponding to thealgebra Ar is recovered by simply imposing the tracelessness condition on a. 2 The quantum prepotential can be expanded in even powers of m as fg(τ,a) = fg(τ,a) (7) n n 1 X≥ with fg proportional to m2n. The first coefficient fg receives only a contribution at n 1 g one-loopand,thus,isindependentofτ. Forn > 1,instead,thecoefficientsf receive n contributionsalsofromtheinstantonsectors. Wheng A˜ ,B ,C ,D ,thesenon- r r r r ∈{ } perturbative terms can be computed using localization techniques [3, 4, 5, 6] as we will show in Section 4, but for the exceptional algebras they have to be derived with other methods. As a by-product, our analysis provides also an explicit derivation of all instanton contributions to the prepotential for the exceptional algebras E , F 6,7,8 4 and G , at least for the firstfew values of n. The key ingredient for this is S-duality. 2 2 S-duality In = 4 SYM theories with gauge algebra g, the duality group is generated by N 0 1/√ng 1 1 S = − and T = , (8) √ng 0 0 1 (cid:18) (cid:19) (cid:18) (cid:19) which, on the coupling constant τ, act projectively as follows 1 S(τ) = and T(τ) = τ +1 . (9) −n τ g The matrices (8) satisfy the constraints S2 = 1 and (ST)pg = 1 with n = 4cos2 π , (10) − − g pg and generate a subgroup of SL(2,R) which is known as the Heck(cid:0)e g(cid:1)roup H(p ). For g the simply laced algebras, i.e. n = 1, we have p = 3, and the duality group H(3) is g g justthe modulargroup Γ = SL(2,Z). For thenon-simply laced algebras, the duality groups H(4) and H(6), corresponding respectively to n = 2 and n = 3, are clearly g g different from the modular group but contain subgroups which are also congruence subgroups of Γ. Indeed, one can show that the following H(p ) elements g 1 0 1 1 V = STS = − and T = (11) n 1 0 1 g (cid:18) − (cid:19) (cid:18) (cid:19) generate a b Γ (n ) = Γ : c = 0 mod n Γ . (12) 0 g c d ∈ g ⊂ ( ) (cid:18) (cid:19) 3 As we will see, the modular forms of Γ (n ), which are known and classified, and 0 g have a simple behavior also under S-duality, play an important role for the = 2⋆ N SYM theories.3 Another important feature is that the duality transformations exchange electric states of the theory with gauge algebra g with magnetic states of the theory with the GNO dual algebra g , which is obtained from g by exchanging (and suitably ∨ rescaling) the long and the short roots [8]. The correspondence between g and g is ∨ given in the following table g A˜ B C D E F G r r r r 6,7,8 4 2 g A˜ C B D E F G ∨ r r r r 6,7,8 4′ ′2 whereforF andG ,the inthelasttwocolumunsmeansthatthedualrootsystems 4 2 ′ are equivalent to the original ones up to a rotation. This duality structure remains and gets actually enriched when the =4 SYM N theories are deformed into the corresponding = 2⋆ ones. Here the S transforma- N tion (8) relates the electric variable a of the g theory with the magnetic variable a D of the dual g theory ∨ 1 ∂Fg∨ 1 ∂fg∨ a = τ a+ , (13) D ≡ 2πin ∂a 2πin τ ∂a g g (cid:16) (cid:17) according to S aD = 0 −1/√ng aD = −a/√ng . (14) a √ng 0 a √ngaD (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) In other words, the S transformation exchanges the description based on a with its Legendre-transformed one, based on a : D S[Fg]= [Fg∨] , (15) L where the Legendre transform is defined as Fg∨ Fg∨ a ∂Fg∨ = n πiτa2 a ∂fg∨ +fg∨ . (16) g L ≡ − · ∂a − − · ∂a (cid:2) (cid:3) Thus, as is clear from (15), S-duality is not a symmetry of the effective theory since it changes the gauge algebra; nevertheless, as we shall see, it is powerful enough to constrain the form of the prepotential at the non-perturbative level. 3It is interesting to observe that for ng = 3, the matrices T and V2 generate the subgroup Γ (3),whosemodular formsplayarole intheN =2SYMtheorywith gauge groupSU(3)andsix 1 fundamental hypermultiplets[7]. 4 3 The modular anomaly equation If one uses eq.s (9), (13) and (14) to evaluate S[Fg], the requirement (15) can be recast in the following form: fg −n1gτ,√ngaD = 4πi1ngτ ∂∂fag∨ 2+fg∨ (17) (cid:16) (cid:17) (cid:16) (cid:17) where the r.h.s. is evaluated in τ and a. Eq. (17) can be solved assuming that the coefficients f in the mass expansion n (7) of the quantum prepotential depend on τ only through quasi-modular forms of Γ (n ). The ring of these quasi-modular forms is generated by 0 g E ,E ,E for n = 1 , 2 4 6 g { } (18) E ,H ,E ,E for n = 2,3 , 2 2 4 6 g { } where E (τ) are the Eisenstein series while n 1 1 H (τ) = ηng(τ) λg +λng ηng(ngτ) λg −ng (19) 2 g η(n τ) η(τ) " g # (cid:16) (cid:17) (cid:16) (cid:17) 6 where η is the Dedekind η-function and λ = nng(ng−1). Thus, λ = 8,3 for n = 2,3 g g g g respectively. All these forms admit a Fourier expansion in terms of the instanton weight q, which starts as 1+O(q). This means that their perturbative part is just 1. Being able to express the prepotential in terms of quasi-modular forms entails resumming its istanton expansion. The modular forms (18) transform in a simple way also under S; in fact H2 − n1gτ = − √ngτ 2H2 , (20a) E2(cid:0)− n1gτ (cid:1)= √(cid:0)ngτ 2(cid:1)E2+(ng−1)H2+δ , (20b) E4(cid:0)− n1gτ(cid:1) = (cid:0)√ngτ(cid:1)4hE4+5(ng −1)H22+i(ng−1)(ng −4)E4 , (20c) E6(cid:0)− n1gτ(cid:1) = (cid:0)√ngτ(cid:1)6hE6+ 27(ng−1)(3ng −4)H23 i h (cid:0) (cid:1) (cid:0) (cid:1) 1(n 1)(n 2)(7E H +2E ) , (20d) − 2 g− g− 4 2 6 i where δ = 6 . Thus a quasi-modular form of Γ (n ) with weight w is mapped πiτ 0 g under S to a form of the same weight with a prefactor (√ngτ)w, up to the δ-shift introduced by E . 2 g Suppose moreover that the coefficients f enjoy the following property: n fng −n1gτ,a = √ngτ 2n−2 fng∨(τ,a) E2 E2+δ . (21) (cid:16) (cid:17) (cid:12) → (cid:0) (cid:1) (cid:12) (cid:12) 5 If we use this relation in the l.h.s. of eq. (17) and take into account eq. (13), upon formally expanding in δ we obtain ∂fg∨ 1 ∂fg∨ ∂fg∨ + = 0 ; (22) ∂E 24n ∂a · ∂a 2 g of course, since we considered a generic case, we could have equivalently written it in terms of fg. This equation governs the appearance in the quantum prepotential of terms containing the second Eisenstein series E , which is the only source of a 2 quasi-modular behaviour. Using the mass expansion (7), this “modular anomaly” equation becomes a recursion relation ∂fng = 1 n−1∂fℓg ∂fng−ℓ . (23) ∂E −24n ∂a · ∂a 2 g ℓ=1 X 3.1 Exploiting the modular anomaly g g Startingfromf ,wecanusetherelation(23)todeterminethepartsofthef ’swhich 1 n g explicitly contain E . The remaining terms of f are strictly modular; we fix them 2 n g by comparison with the result of the explicit computation of f via localization n techniques, when available, up to instanton order (d 1) where d is the 2n 2 2n 2 − − − number of independent modular forms of weight (2n 2). Once this is done, the − resulting expression is valid at all istanton orders. We stress that the modular anomaly implements a symmetry requirement and does not eliminate the need of a dynamical input; yet it is extremely powerful as it greatly reduces it. The mass expansion of the one-loop prepotential (6) reads g m2 α a 2 ∞ m2n g g f = log · L +S (24) 1 loop 4 Λ − 4n(n 1)(2n 1) 2n 2 2n 2 − − − αX∈Ψg (cid:16) (cid:17) nX=2 − − (cid:16) (cid:17) where we introduced the sums 1 Lg = , n;m1···mℓ (α a)n(β1 a)m1 (βℓ a)mℓ αX∈ΨLg β16=···Xβℓ∈Ψg(α) · · ··· · (25) 1 Sg = , n;m1···mℓ αX∈ΨSg β16=···Xβℓ∈Ψ∨g(α) (α·a)n(β1∨·a)m1···(βℓ∨·a)mℓ which are crucial in expressing the results of the recursion procedure. Here ΨL and g ΨS denote, respectively, the sets of long and short roots of g, and for any root α we g have defined Ψ (α) = β Ψ : α β = 1 , g g ∨ ∈ · (26) Ψ (α) = β Ψ : α β = 1 ∨g (cid:8) g ∨ (cid:9) ∈ · (cid:8) (cid:9) 6 with α being the coroot of α. For the ADE algebras (n = 1) all roots are long and ∨ g g only the sums of type L exist. Thus, in all subsequent formulæ the sums n;m1 mℓ g ··· S are to be set to zero in these cases. n;m1 mℓ ··· g The initial condition for the recursion relation (23) is f . Since this receives 1 contribution only at one-loop, it can be read from the term of order m2 in eq. (24). Then, the first step of the recursion reads ∂fg 1 ∂fg ∂fg m4 α β m4 1 2 = 1 1 = · = Lg + Sg (27) ∂E −24n ∂a · ∂a −96n (α a)(β a) −24 2 n 2 2 g g α,Xβ∈Ψg · · (cid:18) g (cid:19) where the last equality follows from the properties of the root system Ψ . g g For n = 1 there are no forms of weight 2 other than E (see (18)), and thus f g 2 2 g only depends on E . For n = 2,3, instead, f may contain also the other modular 2 g 2 g form of degree 2 that exists in these cases, namely H . The coefficient of H in f 2 2 2 is fixed by matching the perturbative term with the m4 term in eq. (24), namely m4(Lg + Sg). In this way we completely determine the expression of fg. The −24 2 2 2 process can be continued straightforwardly to higher orders in the mass expansion, though of course the structure gets rapidly more involved. In [1] we gave the results uptoorderm10 forthesimply-laced algebras, anduptom8 forthenonsimply-laced ones. Here, for the sake of brevity we only reportthe results upto order m6, namely g g f and f : 2 3 m4 m4 g g g f = E L E +(n 1)H S , (28) 2 −24 2 2− 24n 2 g− 2 2 g h i m6 m4 fg = 5E2+E Lg E2 E Lg 3 −720 2 4 4− 576 2 − 4 2;11 m6h i h i 5E2+E +10(n 1)E H −720n2 2 4 g− 2 2 g h +5n (n 1)H2 +(n 1)(n 4)E Sg (29) g g− 2 g− g − 4 4 m6 i E2 E +2(n 1)E H −576n2 2 − 4 g− 2 2 g h +(n 1)(n 6)H2 (n 1)(n 4)E Sg . g− g− 2 − g− g − 4 2;11 i g Consistency requires that the f ’s obtained from the recursion procedure satisfy n eq. (21). For the ADE algebras (n = 1), using the modular properties of the g Eisenstein series, it is not difficult to show that they do. On the other hand, for the non-simply laced algebras (n = 2,3), using the properties of the root systems, one g can prove that Lg = 1 n+m1+···+mℓSg∨ , n;m1···mℓ √ng n;m1···mℓ (30) (cid:16) (cid:17) Sng;m1 mℓ = √ng n+m1+···+mℓLgn∨;m1 mℓ . ··· ··· (cid:0) (cid:1) 7 Thesedualityrelations, together withthemodulartransformations(20),ensurethat the expressions in eq.s (28) and (29), as well as those arising at higher mass orders, indeed obey eq. (21). 3.2 One-instanton contributions By considering the instanton expansion of the modular forms appearing in the ex- pression of the fg’s, one can see that at the one-instanton order, i.e. at order q, n g the only remaining terms involve the sums of type L . In fact it can be argued 2;1 1 from the recursion relation that this is the case at any o··r·der in the mass expansion. Thus, the one-instanton prepotential reads m2ℓ Fg = m4 Lg k=1 ℓ! 2;1...1 ℓ 0 X≥ ℓ m4 m2ℓ 1 = |{z} (31) (α a)2 ℓ! (β a) (β a) 1 ℓ αX∈ΨLg · Xℓ≥0 β16=···=6Xβℓ∈Ψg(α) · ··· · m4 m = 1+ (α a)2 β a αX∈ΨLg · β∈YΨg(α)(cid:18) · (cid:19) g wherethe intermediate step follows from thedefinition (25) of the sums L . The 2;1 1 number of factors in the product above is given by the order of Ψ (α). W···hen α g is a long root, this is (2h 4) where h is the dual Coxeter number of g (see the ∨g ∨g − Appendix). Thus, in (31) the highest power of the mass is m2h∨g. This is precisely the only term which survives in the decoupling limit q 0 and m with qm2h∨g Λ2h∨g fixed , (32) → → ∞ ≡ in which the = 2⋆ theory reduces to the pure = 2 SYM theory. Indeed, 2h N N b ∨g is the one-loop β-function coefficient for the latter. In this case the one-instanton prepotential is qFk=1 =2 = Λ2h∨g (α1a)2 β1a . (33) (cid:12)(cid:12)N αX∈ΨLg · β∈YΨg(α) · b (cid:12) This expression perfectly coincides with the known results present in the literature (see for example [9] and in particular [10]), while (31) represents the generalization thereof to the = 2⋆ theories with any gauge algebra g. N 4 Multi-instanton results from localization For a classical algebra g A˜ ,B ,C ,D one can efficiently apply the equivariant r r r r ∈ { } localization methods [3, 4, 5, 6] to compute the instanton prepotential, order by 8 orderintheinstanton numberk. Evenifstraightforward inprinciple,thesemethods become computationally quite involved as k increases, and thus they are practical only for the first few values of k. Nonetheless the information obtained in this way is extremely useful since it provides a benchmark against which one can test the results predicted using the recursion relation and S-duality. The essential ingredient is the instanton partition function Kg dχ Zg = i zgaugezmatter (34) k 2πi k k I i=1 Y where K is the number of integration variables given by g k for g = A˜ ,B ,D , r r r K = (35) g ( k2 for g = Cr , whilezgauge andzmatter are,respec(cid:2)tiv(cid:3)ely, thecontributionsofthegaugevector multi- k k plet and thematter hypermultipletin theadjoint representation of g. Thesefactors, which are different for the different algebras, depend on the vacuum expectation valueaandonthedeformationparametersǫ , ,ǫ ,andaretypicallymeromorphic 1 4 ··· functions of theintegration variables χ . Theintegrals in (34) are computed by clos- i ing the contours in the upper-half complex χ -planes after giving the ǫ-parameters i an imaginary part with the following prescription Im(ǫ ) Im(ǫ ) Im(ǫ ) Im(ǫ )> 0 . (36) 4 3 2 1 ≫ ≫ ≫ In this way all ambiguities are removed and we obtain the instanton partition func- tion Zg = 1+ qkZg . (37) inst k k 1 X≥ At the end of the calculations we have to set ǫ +ǫ ǫ +ǫ 1 2 1 2 ǫ = m , ǫ = m (38) 3 4 − 2 − − 2 in order to express the result in terms of the hypermultiplet mass m in the normal- ization of the previous sections. Finally, the non-perturbative prepotential of the = 2⋆ SYM theory is given by N Fg = lim ǫ ǫ logZg = qkFg . (39) inst ǫ1,ǫ2 0 − 1 2 inst k → (cid:16) (cid:17) Xk≥1 Wenowprovidetheexplicitexpressionsofzgauge andzmatter forallclassicalalgebras. k k The details on the derivation of these expressions can be found in [1, 2] (see also, for example, [9] and [5]). 9 The unitary algebras A˜ In this case the localization techniques yield r • ( 1)k (ǫ +ǫ )k ∆(0)∆(ǫ +ǫ ) k 1 gauge 1 2 1 2 z = − , (40a) k k! (ǫ1ǫ2)k ∆(ǫ1)∆(ǫ2) i=1 P χi+ǫ1+2ǫ2 P χi−ǫ1+2ǫ2 Y (cid:0) (cid:1) (cid:0) (cid:1) (ǫ +ǫ )k(ǫ +ǫ )k ∆(ǫ +ǫ )∆(ǫ +ǫ ) k zmatter = 1 3 1 4 1 3 1 4 P χ +ǫ3 ǫ4 P χ ǫ3 ǫ4 k (ǫ ǫ )k ∆(ǫ )∆(ǫ ) i −2 i− −2 3 4 3 4 i=1 Y (cid:0) (cid:1) (cid:0) (cid:1) (40b) where r+1 k P(x) = x a ) , ∆(x) = x2 (χ χ )2 . (41) u i j − − − u=1 i<j Y(cid:0) Y(cid:0) (cid:1) The orthogonal algebras B and D In these cases we find r r • ( 1)k (ǫ +ǫ )k ∆(0)∆(ǫ +ǫ ) k 4χ2 4χ2 (ǫ +ǫ )2 zgauge = − 1 2 1 2 i i − 1 2 , (42a) k 2kk! (ǫ1ǫ2)k ∆(ǫ1)∆(ǫ2) i=1 P χi +(cid:0) ǫ1+2ǫ2 χi− ǫ1+2(cid:1)ǫ2 Y (cid:0) (cid:1)(cid:0) (cid:1) (ǫ +ǫ )k(ǫ +ǫ )k ∆ ǫ +ǫ ∆ ǫ +ǫ zmatter = 1 3 1 4 1 3 1 4 k (ǫ ǫ )k ∆ ǫ ∆ ǫ 3 4 (cid:0) 3(cid:1) (cid:0) 4 (cid:1) k P χi+(cid:0) ǫ(cid:1)3−2ǫ(cid:0)4 P(cid:1) χi − ǫ3−2ǫ4 , (42b) × 4χ2 ǫ2 4χ2 ǫ2 i=1 (cid:0) i − 3(cid:1) (cid:0) i − 4 (cid:1) Y (cid:0) (cid:1)(cid:0) (cid:1) where k ∆(x) = x2 (χ χ )2) x2 (χ +χ )2 , i j i j − − − i<j Y(cid:0) (cid:1)(cid:0) (cid:1) (43) r r P(x) = x x2 2a2) for B , P(x) = x2 a2) for D . u r u r − − u=1 u=1 Y(cid:0) Y(cid:0) The symplectic algebras C Finally, for the symplectic algebras we have r • ( 1)k (ǫ +ǫ )k ∆(0)∆(ǫ +ǫ ) 1 gauge 1 2 1 2 z = − (44a) k 2k+νk! (ǫ1ǫ2)k+ν ∆(ǫ1)∆(ǫ2) P ǫ1+2ǫ2 ν k [2] 1 (cid:0) (cid:1) , × P χ + ǫ1+ǫ2 P χ ǫ1+ǫ2 (4χ2 ǫ2) 4χ2 ǫ2 i=1 i 2 i− 2 i − 1 i − 2 Y (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 10

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