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Home , ADHM construction, Instanton, Supersymmetry

in Gauge

Contents

1 Instantons in Gauge Theories 3 1.1 Gauge Instantons ...... 4 1.2 The k = 1 solution ...... 7

2 ADHM construction 10 2.1 The self-dual ...... 10 2.2 in the background ...... 12

3 Instantons in supersymmetric gauge theories 16 3.1 Supersymmetric gauge theories ...... 16 3.2 in the instanton space ...... 18

4 N = 1 18

4.1 SQCD with Nf = N − 1 flavors ...... 18

5 The N = 2 prepotential 21 5.1 The idea of localization ...... 21 5.2 The equivariant ...... 22 5.3 The instanton partition function and the prepotential ...... 24 5.4 Examples ...... 25

1 5.4.1 U(1) plus adjoint : k=1,2 ...... 25 5.4.2 Example: Pure SU(2) gauge : k=1,2 ...... 26

6 Seiberg-Witten curves 26

2 Outline

• Instantons in gauge theories: solutions of Euclidean . Sad- dle points of path . Instantons as self-dual connections. The k = 1 solution.

• ADHM construction. The of self-dual connections. Fermions in the instantons background.

• Supersymmetric theories. Supersymmetry in the instanton moduli space. N = 1 supersymmetric gauge theories. The Affleck-Dine-Seiberg prepotential

•N = 2 supersymmetric gauge theories. The idea of localisation. The prepo- tential. Multi-instanton calculus via localization.

• Seiberg-Witten curves from localisation.

1 Instantons in Gauge Theories

Correlators in Quantum Theories are described by path integrals over all possible field configurations

* + Z Y Y i S(φ) O(xi, ti) = Dφ O(xi, ti) e ~ (1.1) i i For a 1 Z S = − d4x F a F aµν + ... (1.2) 4g2 µν In the classical limit ~ → 0, the is dominated by the saddle point of the δS = 0. To compute the contribution of the saddle point to the integral is convenient to perform the analytic continuation

tE = i t (1.3)

3 and write 1 Z i S = − dt d3x F a F a + ... = −S (1.4) 4g2 E mn mn E

With respect to S(φ), SE(φ) has the advantage of being positive defined. The classical limit of the path integral * + Z 1 Y Y − SE (φ) O(xi, −itEi) = Dφ O(xi, −itEi) e ~ (1.5) i i is then dominated by the minima of SE(φ). A solution of the Euclidean equations of motion is called an instanton. Besides its applications in the computation of path integrals , instantons can be used also to compute tunnelling effects between different vacua of a quantum field theory. At low , the levels of a moving on a potential V (x) can be approximated by those of the harmonic oscillator for a particle moving on

ω0 2 a quadratic potential V (x) ≈ 2 (x − x0) around a minima at x = x0. In presence of a tunnelling between two vacua, each harmonic oscillator energy levels split into two with energies

1 1 −SE En = (n + 2 ± 2 ∆n) ω0 ∆n ∼ e (1.6) with SE the Euclidean action for an instanton solution describing the transition between the two minima of a particle moving in the upside-down potential VE = −V (x). The factor in front of the exponential can be computed evaluating the fluctuation of the action up to quadratic order around the instanton action. We refer to Appendix 1A for details.

1.1 Gauge Instantons

In gauge theories, instantons are solutions of the Euclidean version of the Yang-Mills action Im τ Z Re τ Z S = d4x Tr F F − i d4x Tr F F˜ E 8π mn mn 8π mn mn 1 Z = d4x Tr F F + ... (1.7) 2g2 mn mn with

Fmn = ∂mAn − ∂nAm + [Am,An] ˜ 1 Fmn = 2 mnpqFpq (1.8)

4 and θ 4π τ = + i (1.9) 2π ˜ The dual field Fmn satisfied by construction the Bianchi identity:

˜ ˜ ˜ DmFmn = ∂mFmn + [Am, Fmn] 1 = 2 mnpq (2∂m∂pAq + ∂m[Ap,Aq] + 2[Am, ∂pAq] + [Am, [Ap,Aq]]) = 0 (1.10)

The first three terms cancel between themselves using the antisymmetric property of the epsilon , while the last one cancel due to the Jacobi identity satisfied by any . The YM equation of motion can be written as

DmFmn = ∂mFmn + [Am,Fmn] = 0 (1.11)

Interestingly, this equation has precisely the same form than the Bianchi identity (1.10) with F˜ replaced by F . This implies that the YM equations can be solved by requiring that the field F is self or anti-self dual

F = ±F˜ (1.12)

We notice that this equation can be solved only in since F˜ = −F in the Minkowskian space, so eigenvalues of the Poincare dual action are ±i. On the other hand in the Euclidean space F˜ = F and ±-eigenvalues are allowed. In the language of forms

1 m n 1 ˜ m n F = 2 Fmndx dx ∗ F = 2 Fmndx dx (1.13) with F = DA = dA + A ∧ A (1.14)

The Bianchi and field equations read

DF = dF + [A, F ] = 2dA A + 2A dA + AA2 − A2A = 0 Bianchi Indentity D ∗ F = d ∗ F + [A, ∗F ] = 0 FieldEquations (1.15)

5 A connection satisfying (1.12) with the plus sign is called a Yang Mills instanton while the one with minus sing is called an anti-instanton. Instantons are classified by the topological 1 Z 1 Z k = − d4x tr (F F˜µν) = − d4x tr (F ∧ F ) (1.16) 16π2 µν 8π2 called, the instanton number. This number is the second Chern number k = R ch2(F ), with the Chern character defined as X iF  ch(F ) = ch (F ) = exp (1.17) n 2π n Next we show that instantons minimize the Euclidean Yang-Mills action over the space of gauge connections with a given topological number k. To see this we start from the trivial inequality Z d4x tr (F ± F˜)2 ≥ 0 (1.18) and use tr F 2 = tr F˜2 to show Z Z 2 1 4 2 1 4 ˜ 8π |k| 2 d x tr F ≥ 2 d x tr F F = (1.19) 2g 2g g2 with the inequality saturates for instantons or anti-instantons. Plugging this value in the YM action one finds that instantons are weighted by e−Sinst with 8π2k −S = 2πi k τ = − + ... (1.20) inst g2 with k > 0. On the other hands anti-instanton effects are weighted by e−2πiτ ∗ . The instanton corrections to correlators in gauge theories take then the form ∞ Z 8π2k −S X k X − 2 hOi = DA e O = ck g + dk e g (1.21) k=1 Summarizing amplitudes in gauge theories are computed by path integrals domi- nated by the minima of the Euclidean classical action and quantum fluctuations around them. Fluctuations around the trivial , i.e. where all fields are set to zero, give rise to loop corrections suppressed in powers of the gauge coupling. Instantons are non trivial solutions of the Euclidean classical action and lead to cor- rections to the correlators exponentially suppressed in the gauge coupling . These corrections are important in theories like QCD where the gauge coupling get strong at low energies. Understanding of the instanton dynamics is then crucial in ad- dressing the study of phenomena in the strong coupling regime like confinement or chiral breaking, etc.

6 1.2 The k = 1 solution

To construct self-dual connections we first observe that we can construct a self- dual two-form with values on SU(2) in terms of the 2 × 2 matrices σn = (i~τ, 1), † 1 σ¯ = σn = (−i~τ, 1)

1 1 pq σmn = 4 (σm σ¯n − σn σ¯m) = 2 mnpq σ (1.22) This implies that a self-dual SU(2) connection can be written as

2(x − x0)nσnm Am = 2 2 (1.23) (x − x0) + ρ Indeed, the field strength associated to this connection is

2 4ρ σnm Fmn = 2 2 2 (1.24) ((x − x0) + ρ ) which is self-dual due to (1.22). Moreover, the instanton number is2

4 3 1 Z ρ vol 3 Z r dr k = − d4x tr (F F˜mn) = 6 S = 1 (1.25) 16π2 mn π2 (r2 + ρ2)4

2 with volS3 = 2π the volume of the unitary sphere. The connection (1.23) represents then a k = 1 instanton. We notice that we can find a different solution by rotating A with a matrix of SU(2) A → UAU † (1.26) so the total number of parameters describing the solution is 8: 4 positions x0, 1 size ρ and 3 rotations U. For a general SU(N), we start from the SU(2) self-dual connection an embed it the 2 × 2 matrix inside the N × N matrix. The k = 1 solution is then described by N 2 − 1 rotations minus (N − 2)2 rotations in the space orthogonal to the 2 × 2 block that leave invariant the solution, i.e. 4N − 5 parameters . Together with the size and 4 positions we get 4N parameters. Finally for k instantons far from each other we expect 4kN parameters. We conclude that the dimension of the instanton moduli space is SU(N) dimMk = 4kN (1.27) In the rest of this section we construct the general solution that goes under the name of ADHM construction.

1 01 0 −i 1 0 Here τ1 = (10), τ2 = (i 0 ), τ3 = (0 −1) are the . 2 R r3 dr 1 2 1 Here we use the result (r2+ρ2)4 = 12ρ4 and trσnm = 2 .

7 Appendix 1A Instantons in

Besides its applications in the computation of path integrals , instantons can be used also to compute tunnelling effects between different vacua of a quantum field theory. To illustrate this points lets us consider a particle moving in a Double Well potential  x2 2 V (x) = V0 1 − 2 (1.28) x0

When V0 is large, the solution of the Schrodinger equation reveals that the spec- trum of energies deviates from that of the harmonic oscillator by a quantity ∆n exponentially suppressed in the height of the barrier V0, i.e.

2V0ω0 1 1 − 3 En = (n + 2 ± 2 ∆n) ω0 ∆n ∼ e (1.29)

2 8V0 and w0 = 2 . This deviation is the result of the tunnelling effect between the two x0 vacua. To see this, let us compute the probability that a particle moves from the vacuum at x = −x0 to that one at x = x0 in a time T Z  Z T  2  −HT x˙ hx0|e | − x0i = Dx(t)exp − dtE + VE(x) (1.30) 0 2 with

VE(x) = −V (x) (1.31) Notice that the Hamiltonian H is nothing but the Euclidean Lagrangian with the upside-down potential VE. The classical equation of motion are √ 0 p 2V0 2 2 −x¨ − VE(x) = 0 ⇐ x˙ = −2VE(x) = 2 (x − x0) (1.32) x0 which is solved by the kink s 2V0 x = x0 tanh tE 2 (1.33) x0 Plugging the solution into the action one finds s Z ∞  2  Z ∞ 2 x˙ 2 2V0 8x0 S = dtE − VE = dtE x˙ = (1.34) −∞ 2 −∞ 3 V0 The tunnelling amplitude  s  2 −S 2V0 8x0 A ∼ e E ∼ exp −  (1.35) 3 V0

8 gives then the correct exponential suppression factor to account for the level split- ting ∆n. The factor in front of the exponential can be computed evaluating the fluctuation of the action up to quadratic order around the instanton action.

Appendix 1B: Forms and Poincare duality

In the language of forms

1 m n 1 ˜ m n F = 2 Fmndx dx ∗ F = 2 Fmndx dx (1.36) with F = dA + A ∧ A (1.37) and m n 1 mn p q ∗dx dx = 2  pq dx dx (1.38) One finds 4 1 mn 4 1 mn ˜ d x 2 F Fmn = F ∧ ∗F d x 2 F Fmn = F ∧ F (1.39)

Appendix 1C: Chern Classes

Consider the first of a U(1) connection A on R2 Z Z i 2 i c1(F ) = d x dA = A (1.40) 2π 2π 1 S∞ At infinity A becomes a trivial gauge

¯ Am = U ∂mU (1.41)

−ikθ 1 Take U = e with θ the coordinate of the S∞ at infinity. Plugging this into (1.40) one finds i Z 2π c1(F ) = dθ (−ik) = k (1.42) 2π 0 1 So the number k measures the winding of the connection along S∞ at infinity.

Now consider the second Chern class. First we notice that c2(F ) is a total derivative 1 Z 1 Z k = − tr F ∧ F = − tr d(AdA + 2 A3) 8π2 8π2 3 Z Z 1 2 3 1 3 = − 2 tr (AdA + 3 A ) = 2 tr A (1.43) 8π 3 24π 3 S∞ S∞

9 where we used the fact that at infinity F → 0 and therefore dA → −A2. To evaluate the integral at infinity we write A as a total gauge 1 A = U¯ dU with U = (x + ix σi) (1.44) r 4 i with σm the pauli matrices. Consider the region where x4 ≈ R and xi are small. In this region 6 12 tr A3 → tr σ σ σ dx ∧ dx ∧ dx = dx ∧ dx ∧ dx = 12 Ω (1.45) R3 1 2 3 1 2 3 R3 1 2 3 3 3 with Ω3 the in this region. Integrating over S∞ one finds Z Z 1 3 1 k = 2 tr A = 2 Ω3 = 1 (1.46) 24π 3 2π 3 S∞ S∞ In general the instanton number k measures the winding number of the map U : 3 S∞ → g from the three sphere at infinity to the Lie algebra.

2 ADHM construction

2.1 The self-dual connection

Here we review the ADHM construction of instantons in R4. This construction exploits again the observation that any field strength 0f the form

Fmn ∼ σmn (2.47) is self-dual in virtue of (1.22). We look for a gauge connection following the ansatz ¯ ¯ Am = U∂mU UU = 1[N×N] (2.48) with U[(N+2k)×N]. If k = 0 this is simply a pure gauge. We will now show that this connection is self-dual if:

• U is a normalized kernel of a matrix ∆ of the form ! ! w 0 ∆ = a + x bn = u,iα˙ + x (2.49) n n n aiα,jα˙ σα,α˙ δi,j with i = 1, . . . k, u = 1,...N. In other words U satisfy ¯ ¯ ¯ ∆U = U∆ = 0 UU = 1[N×N] (2.50)

with ∆ of the form (2.49). By bars we will always mean hermitian conjugates.

10 • If ∆ satisfy the ADHM constraints

¯ α˙ −1 α˙ ∆i λ ∆λ,jβ˙ = fij δβ˙ (2.51) or in components3

c c wτ¯ w − iη¯mn[am, an] = 0 (2.52)

m c c with aαα˙ = amσαα˙ andη ¯mn = −i tr (¯σmnτ ).

The matrices w[N×2k], a[2k×2k] are made of pure numbers describing the instanton moduli4. Notice that the resulting connection is invariant under U(k) rotations

† am → UamU wα˙ → Uwα˙ (2.53) The moduli space of instantons is then defined by the U(k) quotient of the hyper- surface defined by (2.52) and has dimension

2 2 dimRMk = 4k(N + 2k) − 3k − k = 4kN (2.54) Notice that equation (2.50) and (2.51) imply

1 = UU¯ + ∆ f ∆¯ (2.55)

To see that the gauge connection constructed in this way is self-dual let us compute

Fmn:

Fmn = ∂mAn − ∂nAm − [AmAn] ¯ ¯ ¯ = 2∂[mU∂n]U − [U∂mU, U∂nU] (2.56) Inserting the identity (2.55) into the first term in (2.56), rewriting derivatives on U’s as derivatives on ∆’s and using (2.50) one finds ¯ ¯ ¯ ¯ Fmn = 2∂[mU∆ f ∆ ∂n]U = 2U∂[m∆ f ∂n]∆ U ! 0   ¯ † 1 = 2U f 0 σn] ⊗ [k×k] U σ[m ⊗ 1[k×k] ! ¯ 0 0 = 4U U ∼ σmn (2.57) 0 σmn ⊗ f[k×k]

3 To see this, we notice that ∆¯ ∆ = a¯ a + xn (b¯n a + a¯ bn) + xn xm b¯n bm. The b-dependent β˙ c terms are all proportional to δα˙ , while the first term leads to (2.52). Hereη ¯mn is antisymmetric, c c η¯ab = cab,η ¯4m = δmc. 4 In the mathematical literature the ADHM equations are often written as [B1,B2] + IJ = 0, † † † † [B1,B1] + [B2,B2] + II − J J = ξ 1k×k. These equations follows from the identifications B` = √1 (a + ia ) and w = (JI†). 2 2` 2`−1

11 Explicit solutions

The simplest solution: k = 1, N = 2 at aαα˙ = 0: ! ρ 1 1   [2×2] ¯ 1 ∆ = U = 1 −x[2×2] ρ [2×2] 2 2 x[2×2] (ρ + r ) 2 1 ∆¯ ∆ = (ρ2 + r2) 1 ⇒ f = r2 = x xm [2×2] ρ2 + r2 m 2x σ A = U∂¯ U = n mn m m x2 + ρ2 ! 2 ¯ 0 0 4 ρ σmn Fmn = 4U U = (2.58) σmn (ρ2 + r2)2 0 (ρ2+r2)

2.2 Fermions in the instanton background

Let us consider now the gauge theory in the background of an instanton. To quadratic order in all fields beside the gauge field Am, the Euclidean action of any supersymmetric theory can be written as

Z   1 iθ  S ≈ d4x tr F 2 + F mnF˜ E Adj 2g2 mn 16π2 mn ¯ ¯ +2 trR DnΨ¯σnΨ + trRa DnφaDnφa + ... (2.59) with Ψ = (ΛA, ψ, ψ˜) the set of all fermions in the adjoint, fundamental and anti- fundamental representations and

DmΦ = ∂mΦ + Am Φ (2.60) the covariant derivative. To this order the equations of motion of the fields are linear and can be written as

DmFmn ≈ 0 ¯ Dn(σnΨ) ≈ Dn(¯σnΨ) ≈ 0 2 D φa ≈ 0 (2.61)

12 To find solutions we propose the ansatz

¯ Am = U∂mU A ¯ A ¯ ¯ A Λα = U(M f bα − bα f M )U ¯ ψα = K f bα U ˜ ¯ ˜ ψα = U bα f K ¯ ¯ ˜¯ ΛαA˙ = ψα = ψα = φa = 0 (2.62)

In components one writes

u ¯ u (Am)v = Uλ ∂mUλv A u ¯ u  A ¯ ¯ A (Λα )v = Uλ Mλi fij (bα)jλ − (bα)λi fij Mjλ0 Uλ0v ¯ (ψα)u = Ki fij (bα)jλ Uλu ˜ u ¯ u ¯ ˜ (ψα) = Uλ (bα)λi fij Kj (2.63)

A ˜ with λ = 1,...N + 2k, u = 1,...N, i = 1, . . . k. The matrices Mλi, Ki, Ki are made of Grassmanian numbers and5

¯αα˙ α˙ α ¯αα˙ ¯ β˙ α˙ β˙ ¯α ∂ ∆β˙ = δβ˙ b ∂ ∆ =  b (2.64)

¯αα˙ 1 αα˙ with ∂ = 2 σ¯n ∂n. To evaluate the Dirac equations on the we should evaluate the action of the covariant derivative on the fermions. To this aim we use the writing of the identity 1 = UU¯ + ∆ f ∆¯ in order to translate derivatives on U into derivatives on ∆’s. First we notice that

¯ ¯ ¯ DnU = (UU + ∆ f ∆)∂nU + UAn = −∆ f ∂n∆ U (2.65) and similarly ¯ ¯ ¯ Dn U = −U ∂n∆ f ∆ (2.66) Using these equations one finds

αα˙ A ¯  A ¯αα˙ ¯ A ¯ ¯αα˙ ¯ ¯αα˙ ¯ A ¯ D Λα = U M ∂ f bα − M f bα∆ f ∂ ∆ − ∂ ∆ f ∆ M f bα ¯αα˙ ¯ A ¯αα˙ ¯ ¯ A ¯ A ¯αα˙ ¯ − bα ∂ f M + ∂ ∆ f ∆ bα f M + bα f M ∆ f ∂ ∆ U αα˙ ¯αα˙ ¯ ¯ ¯αα˙ ¯ D ψα = K (∂ f bα − f bα ∆ f ∂ ∆) U αα˙ ˜ ¯ ¯αα˙ ¯αα˙ ¯ ˜ D ψα = U (bα ∂ f − ∂ ∆ f ∆ bα f) K (2.67)

5Here we use σ σ = 2   ,σ ¯αα˙ σn = 2δα δα˙ . n,αα˙ n,ββ˙ αβ α˙ β˙ n ββ˙ β β˙

13 The first two terms in each line cancel between each other in virtue of the identities (2.64) and

∂¯αα˙ f = − f ¯bα∆α˙ f = − f ∆¯ α˙ bα f ¯bα ∆α˙ = ∆¯ α˙ bα (2.68) that follows from (2.64) and the bosonic ADHM constraint in (2.69). The cancela- tion of the last terms in the first two lines requires the fermionic ADHM constraints

∆¯ M + M¯ ∆ = 0 (2.69)

In components, writing ! ! w µA ∆ = u,iα˙ MA = u,j (2.70) λ,αj˙ λ,j A aiα,jα˙ + xαα˙ δij Mαi,j the fermionc constraints (2.69) reduce to

¯ A A A A αA Mα = Mα µ¯ wα˙ − w¯α˙ µ + [M , aαα˙ ] = 0 (2.71) Summarizing, the moduli space of instantons in a supersymmetric gauge theory is characterised by the following zero modes:

c c c Vector : (aαα,ij˙ , wα,ui˙ , w¯α,iu˙ ) D =wτ ¯ w − iη¯mn[am, an] = 0 gauge 2 2 2 dim Mk,N = 4k + 4kN − 3k − k = 4kN A A A A A A αA Adj. fermions : (µui, µ¯iu,Mα,ij) λα˙ =µ ¯ wα˙ − w¯α˙ µ + [M , aαα˙ ] = 0 Adj.matter 2 2  dim Mk,N = (# of Adj ferm.) × 2kN + 2k − 2k = 2kN Fund matter Fund matter : Ki dim Mk,N = (# of Fund. ferm.) × (k) (2.72) with i = 1, . . . k, u = 1,...N. We notice that the components of matrices aαα˙ and A Mα proportional to the identity, 1 A A 1 am = x0,m k×k + ...Mα = θ0,α k×k + ... (2.73) do not enter on the ADHM constraints. They are exact zero modes of the instanton action parametrising the position of the instanton in the -time. They can α be reabsorbed in a shift of the space-time supercoordinates (xm, θα) Finally the asymptotic behaviour or various fields at infinity can be found from the asymptotics6 !   1 δ ¯ α˙ αα˙ ij ∆ ≈ w¯ x¯ U ≈ α˙ fij ≈ (2.74) xαα˙ w¯ x2 − x2 6We use thatσ ¯αα˙ σ = δ δα˙ . (m n)αβ˙ mn β˙

14 leading to 1 2x 1 A ≈ w x¯ σ w¯ − n w w¯ F ≈ w x¯ σ x w¯ n x2 n x2 mn x4 mn 1 ΛA ≈ (w x¯α˙ µ¯A − µA x w¯α˙ ) α x4 α˙ α αα˙ 1 1 ψ ≈ − x K w¯α˙ ψ˜ ≈ − x¯α˙ w K˜ (2.75) α x4 αα˙ α x4 α α˙

Appendix 2A: Dirac matrices

Here we collect our conventions for Dirac matrices. We define

† σn = (i~τ, 1)σ ¯ = σn = (−i~τ, 1) (2.76) with 01 0 −i 1 0 τ1 = (10) τ2 = (i 0 ) τ3 = (0 −1) (2.77) We introduce the

1 1 σmn = 4 (σm σ¯n − σn σ¯m)σ ¯mn = 4 (¯σm σn − σ¯n σm) (2.78) satisfying the (anti)self duality conditions

1 pq 1 pq σmn = 2 mnpq σ σ¯mn = − 2 mnpq σ¯ (2.79)

The t’Hooft symbols are defined as

c c c c ηmn = −i tr (¯σmnτ ) ηab = abc ηm4 = δmc c c c c η¯mn = −i tr (¯σmnτ )η ¯ab = abc η¯4m = δmc (2.80)

We write n ∂ 1 αα˙ xαα˙ = xnσαα˙ ∂αα˙ = = 2 (σn) ∂n (2.81) ∂xαα˙ The following identities will be used along the text

αα˙ α˙ σ¯(m σn)αβ˙ = δmn δβ˙ (2.82) αα˙ n α α˙ σ¯n σββ˙ = 2δβ δβ˙ (2.83) αα˙ ββ˙ αβ α˙ β˙ σn,αα˙ σn,ββ˙ = 2 αβ α˙ β˙ σ¯n σ¯n = 2   (2.84)

12 where 12 = − = 1 and by by T(mn) we denoted the symmetric part of a tensor.

15 3 Instantons in supersymmetric gauge theories

In this section we describe the moduli space of instantons in supersymmetric gauge theories.

3.1 Supersymmetric gauge theories

Supersymmetric theories are theories symmetric under the action of fermonic gen- erators exchanging bosonic and fermionic degrees of freedom. In a supersymmet- ric theory, states organise in with equal number of bosonic and fermionic degrees of freedom related to each other by the action of supersymme- try. In the case of a gauge theory with N = 1 supersymmetry there are two basic multiplets:

¯ Vector multiplet V = (Aµ, Λα, Λα˙ ,D)Adj

Chiralmultiplet C = (φ, ψα,F )rep (3.85)

It is convenient to pack them in the so called Vector V and Chiral Φ superfields

µ ¯ ¯¯ ¯¯ 1 ¯¯ V = θσ θ Aµ(x) − i θθθ Λ(x) + iθθθΛ(x) + θθθθD(x) √ 2 Φ = φ(y) + 2θψ(y) + θθF (y) yµ = xµ − iθσµθ¯ (3.86) √ m ¯ i m ¯ 1 ¯¯ = φ(x) + 2θψ(x) + θθF (x) + i θσ θ∂mφ − √ θθ ∂mψ(x)σ θ + θθθθ  φ 2 4 The variation of a superfield under supersymmetry is defined by

α ¯α˙ δ =  Qα + ¯α˙ Q (3.87) with ∂ ∂ Q = − i σm θ¯α˙ ∂ Q¯α˙ = − i θ σm β˙α˙ ∂ (3.88) α α αα˙ m ¯ α αβ˙ m ∂θ ∂θα˙ The resulting supersymmetry transformations are

¯ δAn = −iΛ¯σn + i¯σ¯nΛ mn δΛ = −σ  Fmn + i  D ¯ mn δΛ = ¯σ¯ Fmn − i¯ D m ¯ m δD = −σ ∂mΛ − ∂mΛσ ¯ (3.89)

16 for the vector and √ δφ = 2  ψ √ √ m δψ = i 2 σ ¯ ∂mφ + 2  F √ m δF = i 2 ¯σ¯ ∂mψ (3.90) for the chiral multiplets. The general N = 1 supersymmetric gauge theory can be written as Z Z  2 2 ¯ ¯ V 2  τ α  L = tr d θd θ K(Φ, e Φ) + tr d θ 16π W Wα + W (Φ) + h.c. (3.91) with K(Φ¯, eV Φ) a real function, the Kahler potential, W (Φ) a , the , θ 4π τ = + i 2 (3.92) 2π gYM is the complexified gauge coupling and

1 ¯ ¯ β mnβ 2 m ¯ α˙ Wα = − 4 DDDαV = Λα(y) + (iDδα − Fmn σ α) θβ + i θ σαα˙ ∂mΛ α 1 µν i µν σρ 2 ¯ m W Wα|θ2 = 2 FµνF + 4 µνσρF F − D + 2iΛσ ∂mΛ (3.93) When not say explicitly we will restrict to the simplest choice of Kahler potential ¯ V ¯ V K(Φ, e Φ) = trR Φ e Φ (3.94) corresponding to the case where the scalar manifold is flat. Theories with extended supersymmetry can be seen as special case of N = 1 supersymmetric theories where vector and chiral multiplets combined into bigger multiplets entering in the action in a symmetric fashion

N = 2 : VN =2 = (V + C)Adj H = (C + C¯)rep

N = 4 : VN =4 = (V + 3C)Adj (3.95) The form of the Kahler potential and the superpotential is restricted by the extra supersymmetry. For example in the case of pure N = 2 supersymmetry, the action can be written in terms of a single holomorphic function F(Φ) and is given by (3.91) with ∂2F 1 ∂F τ(Φ) = K(Φ¯, Φ) = Im tr Φ¯ W (Φ) = 0 (3.96) ∂Φ2 4π Adj ∂Φ In the case of N = 4 the complete action is fixed and is given again by (3.91) with 3 ¯ V X ¯ V K(Φa, e Φa) = trAdj Φa e Φa W (Φa) = trAdj Φ1[Φ2, Φ3] (3.97) a=1

17 3.2 Supersymmetry in the instanton moduli space

The space-time supersymmetry induces a supersymmetry on the instanton moduli space. In particular a symmetry under

αα˙ QAn = ¯αA˙ σ¯n Λα + ... (3.98) implies that the moduli matrices ∆ and M describing the zero modes of vector and fields A δmod∆λ,αj˙ = Mλ,j = ¯αA˙ Mλ,j (3.99) We will write

δmod∆ = M (3.100) Acting on U one finds ¯ ¯ ¯ δmodU = (U U + ∆ f ∆) δmodU = Uα − ∆ f M U (3.101) ¯ with α = U δmodU. We notice that the α-dependent term in the right hand side is a gauge transformation δαU = −Uα. So if we define

Q = δmod + δα (3.102) with α a gauge transformation with parameter α, we find the supersymmetry trans- formation rule QU = −∆ f M¯ U (3.103) Similarly Q U¯ = −U¯ M f ∆¯ (3.104) Acting on the gaugino one finds

QA = QUdU¯ + U¯ d(QU) = −U¯ M f ∆¯ dU − U¯ d(∆ f M¯ U) = U¯( M f d∆¯ − d∆ f M¯ ) U = Λ (3.105) reproducing the right space-time supersymmetry transformations.

4 N = 1 Superpotentials

4.1 SQCD with Nf = N − 1 flavors

In this section we consider a N = 1 U(N) gauge theory with Nf chiral superfields ˜ Qf and Nf superfields Q in the fundamental and antifundamental representations

18 respectively. In the background of the instanton the effective action is given by the moduli space integral Z Z X k kβ −Sinst (M) 4 2 ˜ Seff = q Λ dM e k = d x0 d θ0 W (Q, Q) (4.106) k with

Q = ϕ(x0) + θ0 ψ(x0) + ... (4.107) the and antiquark superfields in the moduli space,

Z kβ kβ −S (M) Λ W (ϕ) = Λ dMc e instk = (4.108) ϕkβ−3 the superpotential and 4 2 dM = d x0 d θ0 dMc (4.109) The factor Λkβ is included in order to make the action dimensionless. Since have length dimension one and fermions half, kβ is nothing but the number of bosons nB minus half the number of fermions nF ,

1 1 k β = nB − 2 nF = 4kN − 2 (2 k N + 2 k Nf ) = k (3N − Nf ) (4.110)

We notice also that

X 1 β = 3cAdj − cR cAdj = N cfund = 2 (4.111) R is also the one-loop coefficient of the gauge theory. Indeed one can write β qk = qk Λk β with τ = τ + ln Λ (4.112) eff eff 2πi so Λ can be interpreted as the dynamically generated scale of the theory and τeft as the running gauge coupling at scale Λ.

At the instanton background with zero vevs one finds Sinstk (M) = 0. In order to soak the fermionic zero mode integrals we have to turn on a vev for ϕ in Q and brings down Yukawa term interactions from the action. Since Yukawa terms combine an Adjoint and a fundamental zero mode, a non-trivial result is found only if the number of adjoint and fundamental fermionic zero modes match besides the two extra θ’s, i.e.

nAdj − nfund = 2 k (N − Nf ) = 2 ⇒ Nf = N − 1 , k = 1 (4.113)

19 The instanton measure becomes

c 4N 2N N−1 N−1 ˜ dMc = δ(D )δ(λα˙ ) d w d µ d K d K (4.114)

Finally let us evaluate Sinstk (M). We first notice that ,once evaluated at the solutions of the equations of motion, the YM action is a total derivative. Schemat- ically Z Z M 4 V ¯ 4 2 Sinstk ( ) = d x Q e Q θ2θ¯2 + ... = d x (|Dϕ| + ψ Λϕ ¯ + ...) Z 4  m 2  = d x ∂ (ϕ ¯ Dmϕ) + (−D ϕ + ψ Λ)ϕ ¯ + ... (4.115)

We can compute then Sinstk (M) by using the asymptotic expansion of the instanton solution. From (2.75) one finds 1 1 ϕA¯ A ϕ ∼ ϕ¯ w w¯α˙ ϕ ψα Λ ϕ¯ ∼ K µ¯ ϕ¯ (4.116) m m x4 α˙ α x4 with similar expressions for tilted fields. Notice that both contributions are quadratic in the instanton moduli, so the integrals become Gaussian. To simplify further the computation we take the vev matrix ϕ,ϕ ˜ in the diagonal form:

s s u u ϕu = ϕs δu ϕ˜s =ϕ ˜s δs (4.117) with u = 1,...N, s = 1,...N − 1. For this choice, the zero modes µN , µ¯N ap- pear only on δ(λα˙ ) and can be solved in favour of the others. The integral over ˜ Ks, Ks, µs, wsα˙ , lead to the determinant

N−1 Y ϕ¯s ϕ¯˜s 1 W (ϕ) ∼ = (4.118) |ϕ |2|ϕ¯˜ |2 det m s=1 s s with u mss0 = ϕs ϕus0 (4.119) the matrix. The remaining integrals are ϕ-independent, so they give a numerical factor. Collecting all pieces one finds

Z Λ2N+1 S = c d4xd2θ (4.120) eff detM(x, θ) with c a constant and M the super field version of the meson matrix m.

20 5 The N = 2 prepotential

In this section we will compute the instanton corrections to the prepotential of N = 2 theories. The instanton corrections to the prepotential are given by the moduli space integral Z Z X k Smod(Φ) 4 4 Seff = q dMk e = d xd θFnon−pert(Φ) (5.121) with Z X k Smod(au) Fnon−pert(au) = q dMcke (5.122)

Here we denote by q = Λβ e2πiτ , with Λβ compensating for the length dimension of instanton moduli space measure. The matrix of gauge couplings is defined as ∂2F τuv = (5.123) ∂au∂av

5.1 The idea of localization

Theorem of Localization:

• Let g = U(1) a group action on a manifold M of complex dimension ` specified i ∂ by the vector field ξ = ξ (x) ∂xi ,

i i δξx = ξ (x) (5.124)

s i s with isolated fixed points x0, i.e. points where ξ (x0) = 0 ∀i.

• Let Qξ an equivariant derivative,

Qξ ≡ d + iξ (5.125)

with d the and

i i i j i j i j iξdx ≡ δξx iξ(dx ∧ dx ) = δξx dx − dx δξx (5.126)

a contraction with ξ.

• Let α an equivariantly closed form, i.e. a form satisfying

Qξ α = 0 (5.127)

21 Then, the integral of alpha is given by the localisation formula Z s ` X α0(x0) α = (−2π) 1 (5.128) 2 2 s M s det Qξ(x0) with α0 the zero-form part of α. We notice that

2 Qξ = diξ + iξd = δξ (5.129)

2 j j 2 and therefore Q i = ∂iξ can be viewed as the map Q : Tx0 M → Tx0 M induced by the action of the vector field ξ. Example: Gaussian integral via U(1) localization on R2. Let us consider the integral Z I = e−a(x2+y2) dxdy (5.130) R2 We first notice that R2 admit the action of the rotation group induced by the vector field !  ∂ ∂  0 − ξ =  x − y ⇒ Q2 = (5.131) ∂y ∂x  0 This action has only the origin x = y = 0 as a fixed point. An equivariantly closed form is given by 2 2  2 2 α = e−a(x +y ) dxdy − e−a(x +y ) (5.132) 2a Using the Localization formula one finds

−a(x2+y2) Z  e 0 0 π I = α = 2π = (5.133) R2 2a a with x0 = y0 = 0 the critical point. Notice that the right hand side does not depend on  as expected.

5.2 The equivariant charge

For N = 2 gauge theories if we identify the R-symmetry index A withα ˙ , the supersymmetry parameter ξαA˙ become a scalar αA˙ = ξαA˙ . The supersymmetry variation relate then fields with the same quantum numbers. This identification is known as a topological twist since the resulting supersymmetry variation satisfies δ2 = 0, defining a BRS charge. By construction the moduli space action is a δ- variation of something, so the resulting theory in the moduli space is topological. The action of the BRS charge can be written in general as

QΦ = Ψ QΨ = Q2Φ = 0 (5.134)

22 The list of multiplets (Φ, Ψ), transformation properties, and the eigenvalues λΦ for gauge and matter moduli are displayed in table 5.2. The second and third columns display the multiplets and the of their lowest component. The fourth and fifth columns display the transformation properties under the symmetry groups.

F 2 (Φ, Ψ) (−) φ RG SU(2) λΦ

¯ Gauge (aαα˙ ; Mαα˙ ) + kk (2, 2) χij + 1,χij + 2 1 ¯ 2 (λc; Dc) − kk (1, 3) χij +  , χij 1 ¯ 2 (¯χ, λ) + kk (1, 1) χij 1 (wα˙ ; µα˙ ) + k N (1, 2) χi − au + 2  ¯ 1 (w ¯α˙ ;µ ¯α˙ ) + k N (1, 2) au − χi + 2 

a a ¯ Adjoint (Mα; hα) − kk (2, 1) χij + 1 − m,χij + 2 − m ¯ (χαa˙ ; λαa˙ ) + kk (1, 2) χij − m , χij +  − m ¯ 1 (µa; ha) − k Nc (1, 1) χi − au + 2  − m ¯ ¯ 1 (¯µa; ha) − k Nc (1, 1) au − χi + 2  − m

Fund (K; h) − k Nf (1, 1) χi − mf

˜ ˜ Anti-F (K; h) − N˜f k (1, 1) m˜ f − χi

Table 1: Instanton moduli space for N = 2 gauge theories.

˜ a a We introduce the auxiliary fieldsχ ¯, h, h, hα, h to account for the extra degrees ˜ a a of freedom in λα˙ β˙ , K, K, Mα andµ ¯ respectively. Here a = 3, 4. The sign indicates the spin statistics of the given field. In particular, multiplets with negative signs are associated to constraints subtracting degrees of freedom. We notice that for Nf = 2N or N = 4 we have the same number of bosonic and fermionic supermultiplets as expected for a conformal theory. To evaluate the integral over the instanton moduli space with the help of the

23 localisation formula we should first find a BRS equivariant charge Qξ = δ + iξ. For N = 2 gauge theories the instanton moduli space is invariant under the symmetry

G = U(k) × U(N) × SO(4) × U(Nf ) × U(Nf˜) (5.135)

We parametrize the Cartan of this group by the parameters

(χi, au, `, mf , m˜ f ) i = 1, . . . k , u = 1, . . . N , , ` = 1, 2 , f = 1,...Nf (5.136) and the (mf , m˜ f ) and M for the case of (anti)fundamental and adjoint matter respectively. The action of the equivariant BRS charge on the moduli space can then be written generically as

2 QξΦ = Ψ QξΦ = δξΦ = λΦ Φ (5.137) with Φ a complex field that can be either a in the case of a multiplet containing a physical field or a fermion for multiplets involving auxiliary fields. The list of multiplets (Φ, Ψ) and their eigenvalues ΛΦ are displayed in table 5.2. We use the shorthand notation  = 1 + 2.

5.3 The instanton partition function and the prepotential

We will regularize the volume factor by introducing some 1,2-deformations of the four-dimensional geometry and recover the flat space result from the limit 1,2 → 0. More precisely we will find

Fnon−pert(au, q) = − lim 12 ln Z(`, au, q) (5.138) `→0 with Z X k Smod(au,`) Z = q Zk Zk = dMke (5.139)

R 4 4 1 The factor 12 in (5.138) takes care of the volume factor d xd θ ∼ . 12 The k-instanton partition function Zk is given by the moduli space integral Z Z Z k dχ Y −(−1)F 1 Y dχi Z = dM e−Smod = λ = zgaugezmatter k k vol U(k) Φ k! 2πi k k Φ i=1 (5.140) with Z k dχ 1 Y dχi Y = χ (5.141) volU(k) k! 2πi ij i=1 i6=j

24 Using the eigenvalues in Tab. 5.2, one finds

1−δij Y χij (χij + ) Y 1 zgauge = (−1)k k (χ +  )(χ +  ) (χ − a +  )(−χ + a +  ) i,j ij 1 ij 2 i,u i u 2 i u 2

Y (χij + 1 − m)(χij + 2 − m) Y zAdjoint = (χ − a +  − m)(−χ + a +  − m) k (χ − m)(χ +  − m) i u 2 i u 2 i,j ij ij i,u fund Y zk = (χi − mf ) i,f anti−fund Y zk = (−χi +m ˜ f ) (5.142) i,f with  = 1 + 2. The integral over χi has to be thought of as a multiple contour integral around M-independent poles with

Im 1 >> Im 2 >> Imau > 0 (5.143)

Poles are in one-to-one correspondence of N-sets of two-dimensional Young Tableaux

Y = {Yu} with total number of k boxes. Given Y , the eigenvalues χi can be written as χY = χYu = a + (I − 1 ) + (J − 1 ) (5.144) i Iu,Ju u u 2 1 u 2 2

th with Iu,Ju running over the rows and columns of the u tableaux Yu. The partition function can then be written as X Zk = ZY (5.145) Y ;k=|Y | with gauge matter Y (−1)F +1 ZY = Resχ=χY zk zk = λΦ,Y (5.146) Φ

5.4 Examples

5.4.1 U(1) plus adjoint matter: k=1,2

From (5.140) and (5.142) one finds

(m +  )(m +  ) Z = 1 2 ,   1 2 (5.147) (m + 1)(m + 2)(m + 2 − 1)(m + 21) Z = 2 , 212(2 − 1)

25 which lead to the non-perturbative prepotential

h 2 1 2 i 2 3 2 2 Fn.p. = − lim 12 q Z + q Z + Z − Z = −q m − q m + ··· . 1,2→0 2 2 (5.148)

5.4.2 Example: Pure SU(2) gauge theory: k=1,2

−1 For pure SU(2) gauge theory we take Ta1 = Ta2 = Ta. For the first few tableaux one finds

T( , •) = T1 + T2 + T−2a + T1T2T2a

2 T2 2 T−2a T( , •) = T1 + T1 + T2 + + T−2a + T1T2T2a + T1 T2T2a + T1 T1 T( , ) = 2T1 + 2T2 + T1T2a + T2T2a + T1T−2a + T2T−2a (5.149) with similar expressions for T(•, ),T(•, ) obtained from the first two lines re- placing a → −a , for T( , •) obtained from the second line exchanging 1 ↔ 2  and T •, obtained from the second line after replacing a → −a and 1 ↔ 2 simultaneously. For the partition functions one finds 1 Z( ,•) = − 2a 12(2a + 1 + 2) 1 Z( ,•) = 2 4a12(2 − 1)(2a + 1 + 2)(2a + 21 + 2)(2a + 1) 1 Z( , ) = 2 2 2 2 2 2 12(4a − 1)(4a − 2) (5.150) leading to the prepotential   2 1 2 Fnp = − lim 12 q Z( ) + q (Z( ) + Z( ) − 2 Z( )) `→0 q 5 q2 = + + ... (5.151) 2a2 64 a6

6 Seiberg-Witten curves

For simplicity we take pure N = 2 SYM with gauge group SU(N). We write

k k k Z gauge Z 1 X k X q Y dχi ln z X Y dχi Hk(χi) Z(q) = q Z = e k = e 12 (6.152) k k! 2πi 2πi k k i=1 k i=1

26 with

1−δij Y χij (χij + ) Y 1 zgauge = (6.153) k (χ +  )(χ +  ) P (χ +  )P (χ −  ) i,j ij 1 ij 2 i i 2 i 2 and N Y  P (x) = P (x − au + 2 ) (6.154) u=1

In the limit ` → 0 one finds "   # X χij(χij + ) X H (χ ) =   ln − ln P (χ +  )P (χ −  ) + k ln q k i 1 2 (χ +  )(χ +  ) i 2 i 2 ij ij 1 ij 2 i " # X 1 X ≈ −22 − 2  ln P (χ ) + k  ln q (6.155) 1 2 χ2 1 2 i 1 2 ij ij i

Introducing the density function X ρ(x) = 12 δ(x − χi) (6.156) i one can rewrite (6.155) as

Z ρ(x)ρ(z) Z Z H (ρ) = − dxdz − 2 dzρ(z) ln P (z) + ln q dzρ(z) (6.157) k (x − z)2

The Saddle point equation becomes

Z N dHk(ρ) ρ(z) [ = −2 dz 2 − 2 ln P (x) + ln q = 0 x ∈ Σu (6.158) dρ(x) R (x − z) u=1 Defining  Z  ρ(z) − y(x) = exp − dz − 2 − ln P (x + i0 ) (6.159) R (x − z + i0 ) The saddle point equation can be written as

N 2 [ |y(x)| = q x ∈ Σu (6.160) u=1

−N with boundary condition limx→∞ y(x) = x . A solution can be written as   1 p 2 y(x) = 2 PN (x) − PN (x) − 4 q (6.161)

27 that satisfy the saddle point equation in the intervals

N − + 2 Y − + Σu = [αu , αu ] PN − 4q = (x − αu )(x − αu ) (6.162) u=1

Here PN (x) is a polynomial of order N that we will be write as

N Y PN (x) = (x − eu) (6.163) u=1 We notice that y(x) is solution of the quadratic equation

2 y(x) − PN (x)y(x) + q = 0 (6.164) that is called Seiberg-Witten equaltion. We notice that using the definition (6.159), one can identify au with the period integrals 1 Z au = x λ(x) (6.165) 2πi Σu with

0 0 0 PN (x) PN (x) 2qPN (x) λ(x) = −dx ∂x ln y(x) = dx = + + ... (6.166) p 2 3 PN (x) − 4q PN (x) P (x) leading to 0 xPN (x) au = eu + 2q Resx=eu 3 + ... (6.167) PN (x)

These relations can be inverted to find eu in terms of au. To leading order xP 0(x) e = a − 2q Res + ... (6.168) u u x=au P (x)3

Similarly one can compute chiral correlators

N X Z X  xJ P 0 (x)  htr ΦJ i = xJ λ(x) = eJ − 2q Res N + ... (6.169) u x=eu P (x)3 u Σu u=1 N In particular the prepotential F can be found from the relation ∂F htr Φ2i = 2q (6.170) ∂q

28 Example: pure SU(2) gauge theory

We take 2 2 PN (x) = x − e (6.171) leading to

 2  2 0 1 2 q 6q q 15 q a = Resx=ex PN + 3 + 7 + ... = e − 3 − 7 + ... (6.172) PN PN PN 4e 64 e Inverting this relation one finds

q 3 q2 e = a + + + ... (6.173) 4a3 64 a7 One the other hand for the chiral correlator one finds q 5 q2 htr Φ2i = 2e2 = 2a2 + + + ... = 2a2 + 2 F q + 4 F q2 + ... (6.174) a2 16 a6 1 2 leading to q 5q2 F = + (6.175) inst 2a2 64a6 in agreement with (5.151).

29