Lecture 24 Seiberg Witten Theory III

Transcription

1 Lecture 24 Seiberg Witten Theory III

2 Outline This is the third of three lectures on the exact Seiberg-Witten solution of N = 2 SUSY theory. The third lecture: The Seiberg-Witten Curve: the elliptic curve encoding the monodromies 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.

3 The Seiberg Witten Curve Reminder: the holomorphic coupling constant τ(u) of the effective low energy U(1) gauge theory depends on the position in modular space (parametrized by u = Trφ 2 ). The resulting τ(u) is a section (not a function) due to monodromies developing near certain singularities. This situation is managed by introducing an elliptic curve with coefficients that are genuine functions of the physical parameters. The constraint y 2 = P (x) specifying the elliptic curve has four special points: three zero s of the cubic P (x) and infinity. The singularities occur 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:

4 Electric excitations near u. This is the original description, an asymptically free gauge theory due to the non-abelian dynamics. Magnetic monopole excitations near some dynamically generated scale u = Λ 2. Dyon excitations (with both magnetic and electric charge) near the dynamically generated scale u = Λ 2. This dyon point is identified by a discrete Z 2 symmetry in the quantum theory that takes u u. The Seiberg-Witten curve is the elliptic curve with singularities in the u-plane at each of these three points: y 2 = (x Λ 2 )(x + Λ 2 )(x u). Certainly two zero s come together at u = ±Λ 2, so there are singularities at those points. Near these points, the discriminant is quadratic in u ± Λ 2 monodromy M ±Λ 2 T 2. so the

5 The singularity at is subtle: the roots of the cubic are (0, Λ 4 /(4u), u), so two sets of points approach ( 0, Λ 4 /(4u) and u, ). Change coordinates: rescale by x x (Λ 2 u), y y (Λ 2 u) 3/2, so one pair roots approaching at large u remains finite, while other is given by (±Λ/u, 1). For large u the discriminant u 2 in these variables so the monodromy is T 2. The change of variables to x y gives a factor of u to dx/y. This is odd under u e 2πi u, so the monodromy is M = T 2. Conclusion: the proposed Seiberg-Witten curve has the appropriate singularities and associated monodromies. Application: derive the complete solution for τ and exact masses of monopoles and dyons.

6 Holomorphic Coupling The ratio of the periods gives the holomorphic coupling: τ = a D a = a D/ u a/ u = ω 2 ω 1. Identify the derivatives of a and a D with the periods of the torus a D u = f(u) ω 2 = f(u) b a u = f(u) ω 1 = f(u) a where the holomorphic f(u) is arbitrary for now. Explicit computation: ω 1 = 2 Λ 2 Λ 2 ω 2 = 2 Λ 2 u dx y = 2π Λ 1+ u Λ 2 F dx y dx y, dx y. ( 1 2, 1 2, 1; 2 1+ u Λ 2 ) = πi 2Λ F ( 1 2, 1 2, 1; 1 2 (1 u Λ 2 ) ), where the hypergeometric function enters through the integral 1 0 dx (1 zx) α x β 1 (1 x) γ β 1 = Γ(β)Γ(γ β) Γ(γ) F (α, β, γ; z).,

7 Holomorphic Coupling 1/Im τ The gauge coupling g 2 over the complex u/λ2 plane. Singularities at u/λ2 ± 1 correspond to the monopole and dyon points.

8 VEV and Dual VEV The VEV a and the dual VEV a D are derivatives of the torus periods, a D u = f(u) ω 2 = f(u) b dx y, a u = f(u) ω 1 = f(u) a dx y. Define a one form λ by integration with respect to the parameter u: Comments: dλ du f(u) dx y, a D (u) = b λ, a(u) = a λ. Arbitrary integration constants in these equations are incompatible with SL(2, Z) transformation properties of a and a D. The f(u) is determined by the weak coupling limit.

9 Weak Coupling For large the u periods (already computed exactly in terms of hypergeometric functions) are approximated by ω 1 = 2π u, ω 2 = i u ln ( ) u Λ. 2 Integrating either of u a = f(u)ω 1, u a D = f(u)ω 2 and comparing with the weak coupling result we must choose f(u) = 2 2π.

10 Exact VEVs For the exact VEV and dual VEV it is best to keep the periods ω 1,2 as integrals, integrate with respect to the parameter u, and then evaluating the resulting integral of hypergeometric type: Special cases: a(u) = 2 π Λ 2 Λ 2 = 2(Λ 2 + u) F Λ 2 dx x u (x Λ2 )(x+λ 2 ) ( 1 2, 1 2, 1; 2 1+ u Λ 2 ) dx x u (x Λ2 )(x+λ 2 ) a D (u) = 2 π u ( = i 1 u 2 Λ Λ) F ( 1 2, 1 2, 2; 1 2, ( 1 u Λ 2 )). u = Λ 2 : monopole. a D (u) vanishes, as expected for a massless magnetic u = Λ 2 : a(u) = a D (u), as expected (see later) for a massless dyon with charge (n m, n e ) = (1, 1).

11 The Solution of N = 2 SYM We 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 energy effective theory, which in turn is specified by the prepotential P (a). Integration of τ = ap 2 (a) gives the instanton expansion [ P (a) = ia2 2π 2 log a2 Λ log 2 2 k=1 p ( Λ ) 4k ] k a where the first few terms k p k Impressive as this is, the most important point is the limitation: the expansion does not converge in the region of strong coupling (Λ a). In the strongly coupled region, the dual description in terms of monopoles or dyons is more effective.,

12 Central Charge in N = 2 SUSY The N = 2 SUSY algebra allows a central charge Z {Q a α, Q αb } = 2σµ αα P µδ a b, {Q a α, Q b β } = 2 2Zɛ αβ ɛ ab. Generic SUSY representations: a highest weight state that is annihilated by half of the super-charges, then the entire representation is generated by the other half of the super-charges. Short representations: a highest weight state that is annihilated by 3/4 of the super-charges, then the entire representations is generated by just the remaining 1/4 of the super-charges the representation is much smaller than a generic representation. The BPS mass: short representations are only possible when M = 2 Z.

13 The N = 2 Lagrangian realizes the N = 2 SUSY algebra with central charge related to the electric and magnetic charges n e and n m of the state as Z = a n e + a D n m. Application of BPS mass formula in Seiberg-Witten theory: the monopole and dyon states have masses M given exactly by M (1.0) = 2 a D, M (1. 1) = 2 a a D.

14 The BPS Mass M Λ u 2 Λ Solid line: a D (u), the mass (in units of Λ) of the monopole as function of real u/λ 2. Dashed line: a(u) a D (u), the analogous mass of the dyon.

15 Stability of BPS States The BPS mass formula cannot be corrected because it is determined by the algebra. but the central charge depends on position in moduli space. The number of BPS states generally remain invariant upon motion in moduli space because short representations are constrained from combination into large representations. However, there are some exceptions: those are the walls of marginal stability. Example: a dyon with charges (n m, n e ) that are not relatively prime is only marginally stable: pairs of dyons exist whose charges add up to (n m, n e ) and masses add up to 2 an e + a D n m. If n m and n e are relatively prime then the dyon is absolutely stable, if indeed it exists in the first place. The walls of marginal stability pertain to this question.

16 The BPS Spectrum The central charge formula Z = an e + a D n m is invariant under SL(2, Z) transformations M acting as: ( ) ad s = a M s, c = (n m, n e ) cm 1, and so Z = s c Z. Acting on the dyons with charge ±(1, 1) by the monodromy M identifies dyons with charges ±(1, 2n + 1) for all n Z. The complete spectrum of (generally massive) BPS states are these dyons, as well as the massive SU(2) gauge bosons (W s) with charge ±(0, 1).

17 The Wall of Marginal Stability Important refinement: the BPS spectrum just established is valid at weak coupling. Near the origin, the VEVs a, a D are small enough that decays involving monopoles and dyons are possible, eg. (0, 1) (1, 0) + ( 1, 1). Generally such decays are prevented by the triangle inequality Z ne1 +n e2,n m1 +n m2 Z ne1,n m1 + Z ne2,n m2, but for u such that a D (u)/a(u) is real all the Z have the same phase, independently of the charges. The wall of marginal stability: C : = {u : a D(u) a(u) R}. The monopole and the dyon are the only stable states inside the wall of marginal stability.

18 Physics of Confinement Consider the point u 1, where a D vanishes. The low-energy effective theory: monopole and the dual photon. Add a mass term mtrφ 2 the effective N = 1 superpotential for the dual adjoint and monopoles: e.o.m.: W eff = 2A D MM + mf(a D ), 2MM + mf (A D ) = 0, a D M = 0, a D M = 0. For m = 0: a D arbitrary, M = 0, M = 0 N = 2 moduli space. For m 0: a D = 0, M 2 = M 2 = mf (0)/ 2. VEV of charged field M mass to the magnetic photon electric charge confinement through the dual Meissner effect. This is a concrete realization of confinement with a mass gap.

19 BPS States as Classical Solutions In the weak coupling regime, monopoles can be realized as classical solutions. Example: SO(3) gauge theory with scalar field in the vector representation and a Higgs potential such that gauge symmetry is broken SO(3) U(1): L = d 4 x 1 4 F a µνf aµν + (D µ φ) a D µ φ a λ 4 (φa φ a v 2 ) 2. Classical solutions with magnetic charge S 2 B a i φa ds i = vg, have energy that can be rewritten as a lower bound that identifies the topological class E = 1 2 d 3 x [ Bi abia + (D i φ a ) D i φ a] vg d 3 x Bi a D iφ a 2.

20 The PS solutions are those that saturate the bound B a i = D iφ a. Extended SUSY adds further structure: the topological charge is identified with the central charge of the extended SUSY algebra. In this context the solutions are BPS. The exact (or approximate) classical solutions are instructive because they make properties of BPS states concrete. The logic of such an effort is typically that once the symmetry structure is clarified then results can be extended away from the classical region.

21 Donaldson Theory The Poincaré conjecture: compact 2-manifolds are classified by the number of handles conjecture that the same situation holds in 3D. Generalization to n-manifolds, proven for n 3. For n = 3: Thurston conjectured a classification of all 3-manifolds. Perelman proved Thurston s conjecture (using an RG analog) the Poincaré conjecture. For n = 4: Poincaré 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 instantons. Seiberg Witten theory allows for much simpler invariants since BPS monopoles, unlike instantons, cannot shrink to arbitrarily small size.

22 Adding to Flavor Seiberg Witten Matter: 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 = 2 Q i AQ i. The Q i, Q i combine symmetrically (they have even parity ), so they form a vector of SO(3). U(1) R charges: recall that A has R-charge 2 squark R-charge must vanish for superpotential to have R-charge 2. U(1) R symmetry is anomalous: there are 2 2T (Ad) 2F 2T ( ) = 2b zero-modes R-transformation equivalent to Λ b 1 e iα 2b Λ b 1 assign scale Λ 1 a spurious R-charge of 2.

23 The Seiberg-Witten Curve The weak coupling (Λ 1 0) limit has elliptic curve y 2 = x 2 (x u) and u a 2 has R-charge 4 x has R-charge 4 and y has R-charge 6. Consider F = 1 flavor, and add a mass term m Q i Q i m has spurious R-charge of 2, and odd parity odd. Instanton corrections Λ b 1 = Λ 3 1 involve one zero-mode from each of Q i Q i, combined antisymmetrically it has odd parity. Parity symmetry allowed corrections involve even instanton number, or else odd instanton number and an odd power of m. The most general form of the elliptic curve is y 2 = x 3 ux 2 + tλ mλ 3 1(ax + bu) + cm 3 Λ 3 1. The coefficients a, b, c, and t must be determined.

24 The Seiberg-Witten Curve The theory with doublets has particles with half-integral electric charge change convention: rescale n e by 2, a by 1 2, τ by 2. In these conventions, the Seiberg-Witten curve with no flavors is y 2 = x 3 ux Λ4 x uλ4. Taking the mass m of the one flavor m large the low energy scale becomes Λ 4 = mλ 3 1. Taking m with Λ held fixed, the curve must reduce to the no flavor case a = b = 1 4, c = 0. A large u corresponds to a VEV for a that is simply a mass term. When u = m 2 the masses cancel and there is indeed a singularity for large u m 2 /(64t) (the b term was absorbed in t) t = 1/64. In summary, the correct curve is y 2 = x 3 ux 2 + m 4 Λ3 1x 1 64 Λ6 1.

25 Massless Flavor The curve y 2 = x 3 ux 2 + m 4 Λ3 1x 1 64 Λ6 1. The coupling constant diverges when two roots coincide, which is when the discriminant vanishes. As m 0 vanishes when = (4A 3 27C)C, u = e 2πik/3 3 Λ /3. Interpretation: for F = 1 flavor there is a Z 3 symmetry on the moduli space. Monodromies at these points are conjugate to T.

26 More Flavors Curves for more flavors F are obtained similarly. For F flavors the monodromy at is determined by the β function as M = T F 4. The central charge Z of the N = 2 algebra is more complicated with F 0: it depends on the masses of the flavors as well as on global U(1) charges.

27 Massless Flavors The final result for singularities and the associated monodromies in the case of F massless flavors: F monodromies BPS charges (n m, n e ) 0 ST S 1, D 2 T D2 1 (1, 0), (1, 2) 1 ST S 1, D 1 T D1 1, D 2T D2 1 (1, 0), (1, 1), (1, 2) 2 ST 2 S 1, D 1 T 2 D1 1 (1, 0), (1, 1) 3 ST 4 S 1, (ST 2 S)T (ST 2 S) 1 (1, 0), (2, 1) The conjugation matrices are D n = T n S. The physical origin of monodromy D n T k Dn 1 : k massless dyons with charge (1, n). In each case the product of the monodromies multiply to M.