Resumming Instantons in N =2* Theories Alberto Lerda Universit del - - PowerPoint PPT Presentation

Resumming Instantons in N=2* Theories

Alberto Lerda

Università del Piemonte Orientale and INFN – Torino, Italy

Firenze, 28 October 2016

This talk is mainly based on:

• M. Billò, M. Frau, F. Fucito, A.L. and J.F. Morales, ``S-duality and the prepoten2al in

N=2*theories (I): the ADE algebras,’’ JHEP 1511 (2015) 024, arXiv:1507.07709

• M. Billò, M. Frau, F. Fucito, A.L. and J.F. Morales, ``S-duality and the prepoten2al in

N=2*theories (II): the non-simply laced algebras,’’ JHEP 1511 (2015) 026, arXiv: 1507.08027

• M. Billò, M. Frau, F. Fucito, A.L. and J.F. Morales, ``Resumming instantons in N=2*

theories,'’ XIV Marcel Grossmann MeeWng, arXiv:1602.00273

and

• S.K. Ashok, M. Billò, E. Dell'Aquila, M. Frau, A.L. and M. Raman, ``Modular anomaly

equa2ons and S-duality in N=2 conformal SQCD,’’ JHEP 1510 (2015) 091, arXiv: 1507.07476

• S.K. Ashok, E. Dell’Aquila, A.L. and M. Raman, ``S-duality, triangle groups and

modular anomalies in N=2 SQCD,’’ JHEP 1604 (2016) 118, arXiv:1601.01827

• S.K.Ashok, M.Billò, E.Dell'Aquila, M. Frau, A.L., M.Moskovic, M.Raman, ``Chiral
• bservables and S-duality in N=2* U(N) gauge theories’’, arXiv:1607.08327 to be

published on JHEP

but it builds on a very vast literature …

• 1. IntroducWon
• 2. N=4 SYM
• 3. N=2* SYM
• 4. Conclusions

Plan of the talk

§ Non-perturbaWve effects are important:

• in gauge theories: confinement, chiral symmetry breaking, ...
• in string theories: D-branes, duality, AdS/CFT, ...

§ They are essenWal to complete the perturbaWve expansion and lead to results valid at all couplings § In supersymmetric theories, tremendous progress has been possible thanks to the developement of localizaWon techniques

(Nekrasov ‘02, Nekrasov-Okounkov ’03, Pestun ‘07, …, Nekrasov-Pestun ‘13, ….)

§ In superconformal theories these methods allowed us to compute exactly several quanWWes:

• Sphere parWWon funcWon and free energy
• Wilson loops
• CorrelaWon funcWons, amplitudes
• Cusp anomalous dimensions and bremsstrahlung funcWon

• We will focus on SYM theories in 4d with N=2

supersymmetry

• They are less constrained than the N=4 theories
• They are sufficiently constrained to be analyzed exactly
• We will be interested in studying how S-duality on the

quantum effecWve couplings constrains the prepotenWal and the observables of N=2 theories

(earlier work by Minahan et al. ’96, ‘97)

• We will make use of these constraints to obtain exact

expressions valid at all couplings

N=4 SYM

§ Consider N =4 SYM in d=4

• This theory is maximally supersymmetric (16 SUSY charges)
• The field content is
• All fields are in the adjoint repr. of the gauge group
• The β–funcWon vanishes to all orders in perturbaWon theory
• If , the theory is superconformal (i.e. invariant

under ) also at the quantum level

N=4 SYM

1 vector

(a = 1, · · · , 4)

Xi (i = 1, · · · , 6)

4 Weyl spinors 6 real scalars

G

hXii = 0 SU(2, 2|4)

λa A

§ The relevant ingredients of N =4 SYM are:

• The gauge group (or the gauge algebra )
• The (complexified) coupling constant

N=4 SYM

G

τ = θ 2π + i 4π g2 ∈ H+

g

§ The relevant ingredients of N =4 SYM are:

• The gauge group (or the gauge algebra )
• The (complexified) coupling constant

§ Many exact results have been obtained using:

• Explicit expressions of scarering amplitudes
• Integrability
• AdS/CFT correspondence
• Duality

N=4 SYM

G

τ = θ 2π + i 4π g2 ∈ H+

g

§ N =4 SYM is believed to possess an exact duality invariance which contains the electro-magneWc duality

(Montonen-Olive ‘77, Vafa-Wiren ‘94, Sen ’94, ...)

§ If the gauge algebra is simply laced (ADE)

• maps the theory to itself but with electric and magneWc

states exchanged

• It is a weak/strong duality, acWng on the coupling by
• Together with ( ), it generates

the modular group :

N=4 SYM

g

T(τ) = τ + 1

S S

θ → θ + 2π

S = ✓0 −1 1 ◆ , T = ✓1 1 1 ◆ ; S2 = −1 , (ST)3 = −1

S(τ) = −1/τ Γ = SL(2, Z)

§ N =4 SYM is believed to possess an exact duality invariance which contains the electro-magneWc duality

(Montonen-Olive ‘77, Vafa-Wiren ‘94, Sen ’94, ...)

§ If the algebra of the gauge group is non-simply laced (BCFG) duality relaWon sWll exist, but they are more involved… (see Billò et al. ‘15 and Ashok et al.‘16) § For simplicity I will only describe the case of simply laced algebras , but all the arguments can be generalized to include also the non-simply laced cases

N=4 SYM

S

g

G

g

Let us decompose the N=4 mulWplet into

• one N=2 vector mulWplet
• one N=2 hypermulWplet

N=4 SYM as a N=2 theory

2 Weyl fermions 1 complex scalar

A , ψI , φ

1 vector

q , ˜ q , χ , ˜ χ

2 Weyl fermions 2 complex scalars

Let us decompose the N=4 mulWplet into

• one N=2 vector mulWplet
• one N=2 hypermulWplet

By introducing the v.e.v.

• we break the gauge group
• we spontaneously break conformal invariance
• we can describe the dynamics in terms of a holomorphic

prepotenWal , as in N=2 theories

N=4 SYM as a N=2 theory

2 Weyl fermions 1 complex scalar

A , ψI , φ

1 vector

q , ˜ q , χ , ˜ χ

2 Weyl fermions 2 complex scalars

hφi = a = diag(a1, ..., an)

G → U(1)n

F(a)

• The prepotenWal of the N=4 theory is simply
• The dual variables are defined as
• S-duality relates the electric variable to the magneWc

variable :

N=4 SYM as a N=2 theory

a

aD

S ✓aD a ◆ = ✓0 −1 1 ◆ ✓aD a ◆ = ✓−a aD ◆

F = i π τ a2

aD ≡ 1 2πi ∂F ∂a = τ a

• Let’s find the S-dual prepotenWal:
• S-duality exchanges the descripWon based on with its

Legendre-transform, based on :

N=4 SYM as a N=2 theory

a

aD

S(F) = i π ⇣ − 1 τ ⌘ (aD)2 = −i π 1 τ a2

D

L(F) = F − a ∂F ∂a = i π τ a2 − 2π i a aD = −i π 1 τ a2

D

• Let’s find the S-dual prepotenWal:
• S-duality exchanges the descripWon based on with its

Legendre-transform, based on :

• Thus

N=4 SYM as a N=2 theory

a

S(F) = i π ⇣ − 1 τ ⌘ (aD)2 = −i π 1 τ a2

D

L(F) = F − a ∂F ∂a = i π τ a2 − 2π i a aD = −i π 1 τ a2

D

S(F) = L(F) aD

• Let’s find the S-dual prepotenWal:
• S-duality exchanges the descripWon based on with its

Legendre-transform, based on :

• Thus
• This structure is present also in N=2 theories and has

important consequences on their strong coupling dynamics!

N=4 SYM as a N=2 theory

a

S(F) = i π ⇣ − 1 τ ⌘ (aD)2 = −i π 1 τ a2

D

L(F) = F − a ∂F ∂a = i π τ a2 − 2π i a aD = −i π 1 τ a2

D

S(F) = L(F) aD

N=2* SYM

§ The N=2* theory is a mass deformaWon of the N=4 SYM § Field content:

• one N=2 vector mulWplet for the algebra
• one N=2 hypermulWplet in the adjoint rep. of with

mass m

§ Half of the supercharges are broken, and we have N=2 SUSY § The β-funcWon sWll vanishes, but the superconformal invariance is explicitly broken by the mass m

The N=2* set-up

g g m → 0 N = 2∗ pure N = 2 SYM N = 4 SYM

decoupling m → ∞

§ The N=2* prepotenWal contains classical, 1-loop and non- perturbaWve terms § The 1-loop term reads

• is the set of the roots α of the algebra
• is the mass of the W-boson associated to the root α

§ The non-perturbaWve contribuWons come from all instanton sectors and are proporWonal to and can be explicitly computed using localizaWon for all classical algebras

(Nekrasov ‘02, Nekrasov-Okounkov ‘03, …, Billò et al 15, ...)

Structure of the N=2* prepoten@al

1 4 X

α∈Ψg

h −(α · a)2 log ⇣α · a Λ ⌘2 + (α · a + m)2 log ✓α · a + m Λ ◆2i Ψg g

α · a

qk

F = i π τ a2 + f with f = f1−loop + fnon−pert

§ The dual variables are defined as § Applying S-duality we get § CompuWng the Legendre transform we get

S-duality and the prepoten@al

aD ≡ 1 2πi ∂F ∂a = τ ✓ a + 1 2πiτ ∂f ∂a ◆ S(F) = i π ⇣ − 1 τ ⌘ a2

D + f

⇣ − 1 τ , aD ⌘ L(F) = F − 2iπ a · aD = i π ⇣ − 1 τ ⌘ a2

D + f(τ, a) +

1 4iπτ ✓∂f ∂a ◆

2

§ The dual variables are defined as § Applying S-duality we get § CompuWng the Legendre transform we get

S-duality and the prepoten@al

aD ≡ 1 2πi ∂F ∂a = τ ✓ a + 1 2πiτ ∂f ∂a ◆ S(F) = i π ⇣ − 1 τ ⌘ a2

D + f

⇣ − 1 τ , aD ⌘ L(F) = F − 2iπ a · aD = i π ⇣ − 1 τ ⌘ a2

D + f(τ, a) +

1 4iπτ ✓∂f ∂a ◆

2

§ Requiring implies Modular anomaly equaWon! § This constraint has very deep implicaWons!

S-duality and the prepoten@al

S(F) = L(F) f ✓ −1 τ , aD ◆ = f(τ, a) + 1 4iπτ ✓∂f ∂a ◆

2

§ Requiring implies Modular anomaly equaWon! § This constraint has very deep implicaWons! § The modular anomaly equaWon is related to the holomorphic anomaly equaWon of the local CY topological string descripWon of the low-energy effecWve theory (BCOV ‘93, Wiren ‘93, … Aganagic et al ’06, Gunaydin et al ‘06, Huang et al 09, Huang ‘13, … )

S-duality and the prepoten@al

S(F) = L(F) f ✓ −1 τ , aD ◆ = f(τ, a) + 1 4iπτ ✓∂f ∂a ◆

2

§ We organize the quantum prepotenWal in a mass expansion § From explicit calculaWons, one sees that:

• is only 1-loop and thus τ-independent
• are both 1-loop and non-perturbaWve. They are

homogeneous funcWons (This is because the prepotenWal has mass dimension 2)

Modular anomaly equa@on

f(τ, a) = X

n=1

fn(τ, a) with fn ∝ m2n f1 f1(a) = m2 4 X

α∈Ψg

log ⇣α · a Λ ⌘2 fn (n ≥ 2) fn(τ, λ a) = λ2−2n fn(τ, a) f

§ The modular anomaly equaWon implies

Modular anomaly equa@on

f ⇣ − 1 τ , aD ⌘ = f(τ, a) + δ 24 ✓∂f ∂a ◆

2

, δ = 6 iπτ fn ⇣ − 1 τ , aD ⌘ = fn(τ, a) + · · ·

§ The modular anomaly equaWon implies § n = 1

• Using and

requiring that under S-duality , we have

Λ → τ Λ

X

Modular anomaly equa@on

f ⇣ − 1 τ , aD ⌘ = f(τ, a) + δ 24 ✓∂f ∂a ◆

2

, δ = 6 iπτ fn ⇣ − 1 τ , aD ⌘ = fn(τ, a) + · · · f1(a) = m2 4 X

α∈Ψg

log ⇣α · a Λ ⌘2 f1 (aD) = f1 (τa + · · · ) = f1 (a) + · · ·

§ n = 2

• Using the definiWon of the dual variable and the homogeneity

property, we have

• In order to solve the equaWon, we must require that
• i.e. should have modular weight 2 under S-duality !

§ The only quanWty with this property is the second Eisenstein series E2 (quasi-modular)

Modular anomaly equa@on

f2 ⇣ − 1 τ , aD ⌘ = f2 ⇣ − 1 τ , τ(a + · · · ) ⌘ = τ −2 f2 ⇣ − 1 τ , a + · · · ⌘ f2 ⇣ − 1 τ , a + · · · ⌘ = τ 2 f2 ⇣ τ, a + · · · ⌘ = τ 2 f2 ⇣ τ, a ⌘ + · · · f2(τ, a)

§ Generic n

• The previous analysis can be easily generalized to arbitrary n.
• In order to be able to solve the equaWon, we must have
• Thus we must require that depends on τ through “modular”

funcWons with weight , i.e. where are the Eisenstein series

f g

n

2n − 2 E2(τ), E4(τ), E6(τ)

Modular anomaly equa@on

fn ⇣ − 1 τ , a + · · · ⌘ = τ 2n−2 fn ⇣ τ, a ⌘ + · · · fn(τ, a) = fn ⇣ E2(τ), E4(τ), E6(τ), a ⌘

§ The Eisenstein series are “modular” forms with a well-known Fourier expansion in : § E4 and E6 are truly modular forms of weight 4 and 6 § E2 is quasi-modular of weight 2 § Thus a modular form of weight is mapped under S into a form of weight Wmes , up to shivs induced by E2

Eisenstein series

E2(τ) = 1 − 24q − 72q2 − 96q3 − 168q4 + · · · E4(τ) = 1 + 240q + 2160q2 + 6720q3 + 17520q4 + · · · E6(τ) = 1 − 504q − 16632q2 − 122976q3 − 532728q4 + · · · E4 ⇣ − 1 τ ⌘ = τ 4 E4(τ) , E6 ⇣ − 1 τ ⌘ = τ 6 E6(τ)

E2 ⇣ − 1 τ ⌘ = τ 2 ⇥ E2(τ) + δ ⇤ , δ = 6 iπτ

q = e2iπτ τ w

w w

§ S-duality § Modular anomaly equaWon

Recursion rela@on

f ⇣ − 1 τ , aD ⌘ = f(τ, a) + δ " 1 24 ✓∂f ∂a ◆

2#

f ⇣ − 1 τ , aD ⌘ = f ✓ E2(−1 τ ), E4(−1 τ ), E6(−1 τ ), τ

• a + δ

12 ∂f ∂a ◆ = f ✓ E2 + δ, E4, E6,

• a + δ

12 ∂f ∂a ◆ = f (τ, a) + δ " ∂f ∂E2 + 1 12 ✓∂f ∂a ◆2# + O(δ2)

§ S-duality § Modular anomaly equaWon

Recursion rela@on

f ⇣ − 1 τ , aD ⌘ = f(τ, a) + δ " 1 24 ✓∂f ∂a ◆

2#

f ⇣ − 1 τ , aD ⌘ = f ✓ E2(−1 τ ), E4(−1 τ ), E6(−1 τ ), τ

• a + δ

12 ∂f ∂a ◆ = f ✓ E2 + δ, E4, E6,

• a + δ

12 ∂f ∂a ◆ = f (τ, a) + δ " ∂f ∂E2 + 1 12 ✓∂f ∂a ◆2# + O(δ2)

§ We thus obtain which implies the following recursion relaWon

(Minahan et al ’97)

• This allows us to determine from the lower coefficients up to E2-

independent terms. These are fixed by comparison with the perturbaWve expressions (or the first instanton correcWons).

• The modular anomaly equaWon is a symmetry requirement; it does

not eliminate the need of a dynamical input

§ Once this is done, the result is valid to all instanton orders.

Recursion rela@on

∂f ∂E2 + 1 24 ✓∂f ∂a ◆2 = 0 ∂fn ∂E2 = − 1 24

n−1

X

`=1

∂f` ∂a ∂fn−` ∂a fn

§ Using this recursive procedure we find where and are root laxce sums of defined as

with

Exploi@ng the recursion

Cg

2 =

X

α2Ψg

1 (α · a)2 Cg

2;1,1 =

X

α2Ψg

X

β16=β22Ψg(α)

1 (α · a)2(β1 · a)(β2 · a)

Cg

2;1,1

Cg

2

g

Ψg(α) = {β ∈ Ψg : α · β = 1}

f2 = −m4 24 E2 Cg

2

f3 = − m6 720 (5E2

2 + E4)Cg 2 − m6

576 (E2

2 − E4)Cg 2;1,1

§ For example and thus § From the Fourier expansion of E2 we get the perturbaWve and all non-perturbaWve contribuWons to the prepotenWal at

• rder m4 !

§ There are no free parameters !

Exploi@ng the recursion

CU(2)

2

= 1 (a1 − a2)2 CU(3)

2

= 1 (a1 − a2)2 + 1 (a1 − a2)2 + 1 (a2 − a3)2 f U(2)

2

= −m4 24 E2(τ) CU(2)

2

f U(3)

2

= −m4 24 E2(τ) CU(3)

2

§ For the classical algebras A, B, C and D

• the ADHM construcWon of the k instanton moduli spaces is avaliable
• the integraWon of the moduli acWon over the instanton moduli

spaces can be performed à la Nekrasov using localizaWon techniques

§ In principle straighyorward; in pracWce computaWonally rather intense. Not many explicit results for the N=2* theories in the literature. § We worked it out:

• for An and Dn with n<6, up to 5 instantons;
• for Cn with n<6, up to 4 instantons;
• for Bn with n<6, up to 2 instantons.

§ The results match the q-expansion of those obtained above § For the excepWonal algebras our results are predicWons!

Checks on the results

§ These results can be extended to non-flat space-Wmes by turning-on the so-called background which actually was already present in the localizaWon calculaWons § For one finds that the generalized prepotenWal

• beys a generalized modular anomaly equaWon

Generaliza@ons

Ω

    ✏1 −✏1 ✏2 −✏2    

✏1, ✏2 6= 0

(Nekrasov ‘02)

F = i ⇡ ⌧ a2 + f(a, ✏)

@f @E2 + 1 24 ✓@f @a ◆2 + ✏1✏2 24 @2f @a2 = 0

§ These results can be extended to non-flat space-Wmes by turning-on the so-called background which actually was already present in the localizaWon calculaWons § For one finds that the generalized prepotenWal

• beys a generalized modular anomaly equaWon

Generaliza@ons

Ω

    ✏1 −✏1 ✏2 −✏2    

✏1, ✏2 6= 0

(Nekrasov ‘02)

F = i ⇡ ⌧ a2 + f(a, ✏)

@f @E2 + 1 24 ✓@f @a ◆2 + ✏1✏2 24 @2f @a2 = 0

§ In the ADE case, this equaWon can be used to prove that S- duality acts on the prepotenWal as a Fourier transform § This is consistent with viewing

• and as canonically conjugate variables
• S-duality as a canonical transformaWon and

as a wave-funcWon in this space with as Planck’s constant, in agreement with the topological string

Generaliza@ons

a

aD

Z(a, ✏) = exp ⇣ − F(a, ✏) ✏1✏2 ⌘

✏1✏2

(BCOV ‘93, Wiren ‘93,Aganagic et al ‘06, Gunaydin et al ‘06 …)

exp ✓ −S[F](aD) ✏1✏2 ◆ = ⇣ i ⌧ ✏1✏2 ⌘n/2 Z dnx exp ✓2⇡ i aD · x − F(x) ✏1✏2 ◆

(Billo et al ‘13)

§ Using Pestun’s localizaWon formula and our modular anomaly equaWon, one can easily prove that the parWWon funcWon on the sphere is modular invariant (a result that was expected on general grounds) § From one can compute (by simply doing gaussian integraWons) several interesWng observables

• Wilson loops
• Zamolodchikov metric
• CorrelaWon funcWons
• …

Applica@ons

ZS4 ZS4

(Pestun ’07, ... , Baggio, Papadodimas et al ‘14, ‘16 Fiol et al ’15, Gerchkovitz, Gomis, Komargodski et al ‘16)

ZS4 = Z dnx

• exp

⇣ − F(a, ✏) ✏1✏2 ⌘

• 2
• a=i x;✏1=✏2= 1

R

§ Using Pestun’s localizaWon formula and our modular anomaly equaWon, one can easily prove that the parWWon funcWon on the sphere is modular invariant (a result that was expected on general grounds) § From one can compute (by simply doing gaussian integraWons) several interesWng observables § Our S-duality results could be used to promote these calculaWons to the fully non-perturbaWve regime

Applica@ons

ZS4 ZS4 ZS4 = Z dnx

• exp

⇣ − F(a, ✏) ✏1✏2 ⌘

• 2
• a=i x;✏1=✏2= 1

R

§ Other observables of the theory are the chiral correlators § They can be computed using equivariant localizaWon § The results can be expressed in terms of modular funcWons and laxce sums

Chiral correlators

(Bruzzo et al. 03, Losev et al. 03, Flume et al. 04, Billò et al. ’12)

< Trφn >=

N

X

i=1

an

i + · · ·

(Ashok et al. ‘16)

§ Using the explicit results for , it is possible to change basis and find the quantum symmetric polynomials in the a’s that transform as modular form of weight n

Chiral correlators

S(An) = τ n An

A1 = X

ii

ai1 A2 = X

ii<i2

ai1ai2 + ✓N 2 ◆m2 12 E2 + m4 288 (E2

2 − E4)C2 + · · ·

< Tr φn > An(τ, a) = X

i1<i2<···<in

ai1ai2 · · · ain + · · ·

§ These expressions coincide with the soluWon of the modular anomaly equaWon saWsfied by the ’s that can be obtained directly from their S-duality properWes!

Chiral correlators

A1 = X

ii

ai1 A2 = X

ii<i2

ai1ai2 + ✓N 2 ◆m2 12 E2 + m4 288 (E2

2 − E4)C2 + · · ·

∂An ∂E2 + 1 24 ∂An ∂a ∂f ∂a = 0

An

Conclusions

§ The requirement that the duality group acts simply as in the N=4 theories also in the mass-deformed cases leads to a modular anomaly equaWon § This allows one to efficiently reconstruct the mass-expansion

• f the prepotenWal and the chiral correlators resumming all

instanton correcWons into (quasi-)modular forms of the duality group § The existence of such modular anomaly equaWons seems to be a general feature for many observables!

Conclusions

§ The requirement that the duality group acts simply as in the N=4 theories also in the mass-deformed cases leads to a modular anomaly equaWon § This allows one to efficiently reconstruct the mass-expansion

• f the prepotenWal and the chiral correlators resumming all

instanton correcWons into (quasi-)modular forms of the duality group § A similar parern (although a bit more intricate) arises in N=2 SQCD theories with Nf=2Nc fundamental flavours, where it has been possible to describe the structure of the low energy effecWve theory at the special vacuum

Conclusions

(Ashok et al. ’15 and ‘16)

§ This approach can be profitably used in other contexts to study the consequences of S-duality on:

• theories formulated in curved spaces (e.g. S4)
• correlaWon funcWons of chiral and anW-chiral operators
• other observables (e.g. Wilson loops, cusp anomaly, … )
• more general extended observables (surface operators, ...)
• …

with the goal of studying the strong-coupling regime

Conclusions

§ This approach can be profitably used in other contexts to study the consequences of S-duality on:

• theories formulated in curved spaces (e.g. S4)
• correlaWon funcWons of chiral and anW-chiral operators
• other observables (e.g. Wilson loops, cusp anomaly, … )
• more general extended observables (surface operators, ...
• …

with the goal of studying the strong-coupling regime

Thank you for your arenWon

Conclusions

In May 2016 we celebrated the 10th anniversary of the GGI:

30 extended workshops with more than 3500 par@cipants

30 Workshops Strings, AdS/CFT, ... ParWcle Phenomenology AstroparWcle, Cosmology StaWsWcal Mech. Other 100 200 300 400 500 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 # Visitors

Marchesini MarWnelli Lerda

Happy birthday to Supergravity !! Happy birthday to GGI !!