Table Of Content= 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,lerda@to.infn.it; fucito,morales@roma2.infn.it
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
