Lecture 24Seiberg–Witten Theory III

OutlineThis is the third of three lectures on the exact Seiberg-Witten solutionof N = 2 SUSY theory.

The third lecture:

• The Seiberg-Witten Curve: the elliptic curve encoding the mon-odromies of Seiberg-Witten theory.

• The exact solution to SU(2) gauge theory with N = 2 SUSY.

• Monopoles and BPS States.

• Generalization: Seiberg-Witten theory with flavor.

Reading: Terning 7.1, 13.6, 13.7.

The Seiberg–Witten CurveReminder: the holomorphic coupling constant τ(u) of the effective lowenergy U(1) gauge theory depends on the position in modular space(parametrized by u = Trφ2).

The resulting τ(u) is a section (not a function) due to monodromiesdeveloping near certain singularities. This situation is managed by in-troducing an elliptic curve with coefficients that are genuine functions ofthe physical parameters.

The constraint y2 = P (x) specifying the elliptic curve has four specialpoints: three zero’s of the cubic P (x) and infinity. The singularitiesoccur when any two of these speecial points coincide.

The physical origin of the singularities (and the associated monodromies):states become massless.

In SU(2) gauge theory with N = 2, three types of states become light,at complementary points in modular space:

• Electric excitations near u → ∞. This is the original description,an asymptically free gauge theory due to the non-Abelian dynam-ics.

• Magnetic monopole excitations near some dynamically generatedscale u = Λ2.

• Dyon excitations (with both magnetic and electric charge) nearthe dynamically generated scale u = −Λ2. This dyon point isidentified by a discrete Z2 symmetry in the quantum theory thattakes u→ −u.

The Seiberg-Witten curve is the elliptic curve with singularities in theu-plane at each of these three points:

y2 = (x− Λ2)(x+ Λ2)(x− u) .

Certainly two zero’s come together at u = ±Λ2, so there are singularitiesat those points.

Near these points, the discriminant ∆ is quadratic in u ± Λ2 so themonodromy M±Λ2 ∼ T 2.

The singularity at∞ is subtle: the roots of the cubic are' (0,Λ4/(4u), u),so two sets of points approach ( 0,Λ4/(4u) and u,∞).

Change coordinates: rescale by

x→ x′(Λ2 − u) , y → y′(Λ2 − u)3/2 ,

so one pair roots approaching at large u remains finite, while other isgiven by (±Λ/u,−1).

For large u the discriminant ∆ ∼ u−2 in these variables so the mon-odromy is ∼ T−2.

The change of variables to x′ − y′ gives a factor of√u to dx/y. This is

odd under u→ e2πiu, so the monodromy is M∞ = −T−2.

Conclusion: the proposed Seiberg-Witten curve has the appropriate sin-gularities and associated monodromies.

Application: derive the complete solution for τ and exact masses ofmonopoles and dyons.

Holomorphic CouplingThe ratio of the periods gives the holomorphic coupling:

τ = ∂aD

∂a = ∂aD/∂u∂a/∂u = ω2

ω1.

Identify the derivatives of a and aD with the periods of the torus∂aD

∂u = f(u)ω2 = f(u)∫bdxy ,

∂a∂u = f(u)ω1 = f(u)

∫adxy .

where the holomorphic f(u) is arbitrary for now.

Explicit computation:

ω1 = 2∫ Λ2

−Λ2dxy = 2π

Λ√

1+ uΛ2F(

12 ,

12 , 1; 2

1+ uΛ2

),

ω2 = 2∫ Λ2

udxy = −πi√

2ΛF(

12 ,

12 , 1; 1

2 (1− uΛ2 )),

where the hypergeometric function enters through the integral∫ 1

0dx (1− zx)−αxβ−1(1− x)γ−β−1 = Γ(β)Γ(γ−β)

Γ(γ) F (α, β, γ; z) .

Holomorphic Coupling

1

0

11

0

1

3

4

5

1

0

1

3

4

1/Im τ

-

-

The gauge coupling g2 over the complex u/Λ2 plane. Singularities atu/Λ2 ± 1 correspond to the monopole and dyon points.

VEV and Dual VEVThe VEV a and the dual VEV aD are derivatives of the torus periods,

∂aD

∂u = f(u)ω2 = f(u)∫bdxy , ∂a

∂u = f(u)ω1 = f(u)∫adxy .

Define a one form λ by integration with respect to the parameter u:

dλdu ≡ f(u)dxy ,

⇒ aD(u) =∫bλ , a(u) =

∫aλ .

Comments:

• Arbitrary integration constants in these equations are incompatiblewith SL(2,Z) transformation properties of a and aD.

• The f(u) is determined by the weak coupling limit.

Weak CouplingFor large the |u| periods (already computed exactly in terms of hyperge-ometric functions) are approximated by

ω1 = 2π√u, ω2 = i√

uln(u

Λ2

).

Integrating either of ∂ua = f(u)ω1, ∂uaD = f(u)ω2 and comparing withthe weak coupling result we must choose

f(u) =√

22π .

Exact VEVsFor the exact VEV and dual VEV it is best to keep the periods ω1,2 asintegrals, integrate with respect to the parameter u, and then evaluatingthe resulting integral of hypergeometric type:

a(u) = −√

2π

∫ Λ2

−Λ2dx√x−u√

(x−Λ2)(x+Λ2)

= −√

2(Λ2 + u)F(− 1

2 ,12 , 1; 2

1+ uΛ2

),

aD(u) = −√

2π

∫ Λ2

udx√x−u√

(x−Λ2)(x+Λ2)

= −i 12

(uΛ − Λ

)F(

12 ,

12 , 2; 1

2

(1− u

Λ2

)).

Special cases:

• u = Λ2: aD(u) vanishes, as expected for a massless magneticmonopole.

• u = −Λ2: a(u) = aD(u), as expected (see later) for a masslessdyon with charge (nm, ne) = (1,−1).

The Solution of N = 2 SYMWe have computed the holomorphic coupling τ(u) and the VEV a(u),as functions (sections, actually) of u, the coordinate on moduli space.

The “exact solution” in the present case means finding the low energyeffective theory, which in turn is specified by the prepotential P (a).

Integration of τ = ∂2aP (a) gives the instanton expansion

P (a) = ia2

2π

[2 log a2

Λ2 − 6 + 8 log 2−∑∞k=1 pk

(Λa

)4k],

where the first few terms

k 1 2 3 4 5pk

125

5214

3218

1469231

4471234·5

Impressive as this is, the most important point is the limitation: theexpansion does not converge in the region of strong coupling (Λ� a).

In the strongly coupled region, the dual description in terms of monopolesor dyons is more effective.

Central Charge in N = 2 SUSYThe N = 2 SUSY algebra allows a central charge Z

{Qaα, Q†αb} = 2σµααPµδ

ab ,

{Qaα, Qbβ} = 2√

2Zεαβεab .

Generic SUSY representations: a “highest weight” state that is anni-hilated by half of the super-charges, then the entire representation isgenerated by the other half of the super-charges.

Short representations: a “highest weight” state that is annihilated by3/4 of the super-charges, then the entire representations is generated byjust the remaining 1/4 of the super-charges⇒ the representation is muchsmaller than a generic representation.

The BPS mass: short representations are only possible when

M =√

2|Z| .

The N = 2 Lagrangian realizes the N = 2 SUSY algebra with centralcharge related to the electric and magnetic charges ne and nm of thestate as

Z = ane + aD nm .

Application of BPS mass formula in Seiberg-Witten theory: the monopoleand dyon states have masses M given exactly by

M(1.0) =√

2|aD| ,M(1.−1) =

√2|a− aD| .

The BPS Mass

Λ

MΛ

2

u

-4 -2 2 4

0.5

1

1.5

2

2.5

3

Solid line: |aD(u)|, the mass (in units of Λ) of the monopole as functionof real u/Λ2.Dashed line: |a(u)− aD(u)|, the analogous mass of the dyon.

Stability of BPS StatesThe BPS mass formula cannot be corrected because it is determined bythe algebra. but the central charge depends on position in moduli space.

The number of BPS states generally remain invariant upon motion inmoduli space because short representations are constrained from combi-nation into large representations.

However, there are some exceptions: those are the walls of marginalstability.

Example: a dyon with charges (nm, ne) that are not relatively prime isonly marginally stable: pairs of dyons exist whose charges add up to(nm, ne) and masses add up to

√2|ane + aDnm|.

If nm and ne are relatively prime then the dyon is absolutely stable, ifindeed it exists in the first place. The walls of marginal stability pertainto this question.

The BPS SpectrumThe central charge formula Z = ane +aDnm is invariant under SL(2,Z)transformations M acting as:

~s =(aDa

)→ M~s ,

~c = (nm, ne) → ~cM−1 ,

and so

Z = ~s · ~c→ Z .

Acting on the dyons with charge ±(1,−1) by the monodromyM∞ iden-tifies dyons with charges ±(1, 2n+ 1) for all n ∈ Z.

The complete spectrum of (generally massive) BPS states are thesedyons, as well as the massive SU(2) gauge bosons (W’s) with charge±(0, 1).

The Wall of Marginal StabilityImportant refinement: the BPS spectrum just established is valid atweak coupling.

Near the origin, the VEVs a, aD are small enough that decays involvingmonopoles and dyons are possible, eg.

(0, 1)→ (1, 0) + (−1, 1) .

Generally such decays are prevented by the triangle inequality

|Zne1+ne2 ,nm1+nm2| ≤ |Zne1 ,nm1

|+ |Zne2 ,nm2| ,

but for u such that aD(u)/a(u) is real all the Z have the same phase,independently of the charges.

The wall of marginal stability:

C : = {u : aD(u)a(u) ∈ R} .

The monopole and the dyon are the only stable states inside the wall ofmarginal stability.

Physics of ConfinementConsider the point u1, where aD vanishes.

The low-energy effective theory: monopole and the dual photon.

Add a mass term mTrφ2 ⇒ the effective N = 1 superpotential for thedual adjoint and monopoles:

Weff =√

2ADMM +mf(AD) ,

⇒ e.o.m.:√

2MM +mf ′(AD) = 0 , aDM = 0 , aDM = 0 .

For m = 0: aD arbitrary, M = 0, M = 0 ⇒ N = 2 moduli space.

For m 6= 0: aD = 0, M2 = M2

= −mf ′(0)/√

2. VEV of charged field M⇒ mass to the magnetic photon ⇒ electric charge confinement throughthe dual Meissner effect.

This is a concrete realization of confinement with a mass gap.

BPS States as Classical SolutionsIn the weak coupling regime, monopoles can be realized as classical so-lutions.

Example: SO(3) gauge theory with scalar field in the vector repre-sentation and a Higgs potential such that gauge symmetry is brokenSO(3)→ U(1):

L =∫d4x− 1

4FaµνF

aµν + (Dµφ)aDµφa − λ4 (φa∗φa − v2)2 .

Classical solutions with magnetic charge∫S2∞Bai φ

adSi = vg ,

have energy that can be rewritten as a lower bound that identifies thetopological class

E = 12

∫d3x

[Bai B

ia + (Diφa)∗Diφ

a]

≥ vg + 12

∫d3x|Bai −Diφ

a|2 .

The PS solutions are those that saturate the bound Bai = Diφa.

Extended SUSY adds further structure: the topological charge is identi-fied with the central charge of the extended SUSY algebra.

In this context the solutions are BPS.

The exact (or approximate) classical solutions are instructive becausethey make properties of BPS states concrete.

The logic of such an effort is typically that once the symmetry structureis clarified then results can be extended away from the classical region.

Donaldson TheoryThe Poincare conjecture: compact 2-manifolds are classified by the num-ber of handles ⇒ conjecture that the same situation holds in 3D.

Generalization to n-manifolds, proven for n 6= 3.

For n = 3: Thurston conjectured a classification of all 3-manifolds.Perelman proved Thurston’s conjecture (using an RG analog) ⇒ thePoincare conjecture.

For n = 4: Poincare conjecture proven, but no classification proposed.

The best one can do is to study topological invariants: different invariants⇒ different manifolds.

Donaldson constructed invariants of four-manifolds by studying instan-tons.

Seiberg–Witten theory allows for much simpler invariants since BPSmonopoles, unlike instantons, cannot shrink to arbitrarily small size.

Adding to Flavor Seiberg–WittenMatter: F N = 2 hypermultiplets in the spinor representation of SO(3),a.k.a. the fundamental of SU(2).

The β-function: b = 2T (Ad)− 2F · T ( ) = 2N − F .

N = 2 SUSY require interactions encoded in the N = 1 superpotential:

W =√

2QiAQi .

The Qi, Qi combine symmetrically (they have even “parity”), so theyform a vector of SO(3).

U(1)R charges: recall that A has R-charge 2 ⇒ squark R-charge mustvanish for superpotential to have R-charge 2.

U(1)R symmetry is anomalous: there are 2 · 2T (Ad)− 2F · 2T ( ) = 2bzero-modes ⇒ R-transformation equivalent to Λb1 → eiα·2bΛb1 ⇒ assignscale Λ1 a spurious R-charge of 2.

The Seiberg-Witten CurveThe weak coupling (Λ1 → 0) limit has elliptic curve y2 = x2(x− u) andu ∼ a2 has R-charge 4 ⇒ x has R-charge 4 and y has R-charge 6.

Consider F = 1 flavor, and add a mass term mQiQi ⇒ m has spuriousR-charge of 2, and odd “parity” odd.

Instanton corrections ∝ Λb1 = Λ31 involve one zero-mode from each of

QiQi, combined antisymmetrically ⇒ it has odd “parity”.

“Parity” symmetry ⇒ allowed corrections involve even instanton num-ber, or else odd instanton number and an odd power of m.

The most general form of the elliptic curve is

y2 = x3 − ux2 + tΛ61 +mΛ3

1(ax+ bu) + cm3Λ31 .

The coefficients a, b, c, and t must be determined.

The Seiberg-Witten CurveThe theory with doublets has particles with half-integral electric charge⇒ change convention: rescale ne by 2, a by 1

2 , τ by 2.

In these conventions, the Seiberg-Witten curve with no flavors is

y2 = x3 − ux2 + 14Λ4x+ 1

4uΛ4 .

Taking the mass m of the one flavor m large the low energy scale becomesΛ4 = mΛ3

1. Taking m→∞ with Λ held fixed, the curve must reduce tothe no flavor case ⇒ a = b = 1

4 , c = 0.

A large u corresponds to a VEV for a that is simply a mass term. Whenu = m2 the masses cancel and there is indeed a singularity for largeu ≈ −m2/(64t) (the b term was absorbed in t) ⇒ t = −1/64.

In summary, the correct curve is

y2 = x3 − ux2 + m4 Λ3

1x− 164Λ6

1 .

Massless FlavorThe curve

y2 = x3 − ux2 + m4 Λ3

1x− 164Λ6

1 .

The coupling constant diverges when two roots coincide, which is whenthe discriminant vanishes. As m→ 0

∆ = (4A3 − 27C)C ,

vanishes when

u = e2πik/3 3 Λ21

4·22/3 .

Interpretation: for F = 1 flavor there is a Z3 symmetry on the modulispace.

Monodromies at these points are conjugate to T .

More FlavorsCurves for more flavors F are obtained similarly.

For F flavors the monodromy at ∞ is determined by the β function as

M∞ = −TF−4 .

The central charge Z of the N = 2 algebra is more complicated withF 6= 0: it depends on the masses of the flavors as well as on global U(1)charges.

Massless FlavorsThe final result for singularities and the associated monodromies in thecase of F massless flavors:

F monodromies BPS charges (nm, ne)0 STS−1, D2TD

−12 (1, 0), (1, 2)

1 STS−1, D1TD−11 , D2TD

−12 (1, 0), (1, 1), (1, 2)

2 ST 2S−1, D1T2D−1

1 (1, 0), (1, 1)3 ST 4S−1, (ST 2S)T (ST 2S)−1 (1, 0), (2, 1)

The conjugation matrices are Dn = TnS.

The physical origin of monodromy DnTkD−1

n : k massless dyons withcharge (1, n).

In each case the product of the monodromies multiply to M∞.