Superconformal field theories A-Maximization classically the trace - PowerPoint PPT Presentation

Superconformal field theories

A-Maximization classically the trace of the energy–momentum tensor for a scale- invariant theory vanishes. trace anomaly. r: µν ) 2 − a ( ˜ R µνρσ ) 2 + . . . , g 3 ˜ 1 T µ β ( F b µ = central charge a ˜ β is the numerator of the exact NSVZ β function R µνρσ is the curvature tensor Cardy conjectured that a satisfies: a IR < a UV

A-Maximization in SCFT a can determined by ‘t Hooft anomalies of SC R -charge 3Tr R 3 − Tr R 3 � � a = 32 T µν and the R -current ( J Rµ ) in the same supermultiplet α ( x, θ, ¯ In superspace, super-energy–momentum tensor T α ˙ θ ) contains: θ components and T µν in the θ 2 component J Rµ , J αµ in the θ and ¯ superconformal R -charge: R = R 0 + � i c i Q i

A-Maximization superconformal symmetry relates different triangle anomalies � J R J R J i � related to � TTJ i � by 9 Tr R 2 Q i = Tr Q i two-point function 1 � J i ( x ) J k (0) � ∝ τ ik x 4 Unitarity ⇒ τ ik to have positive definite eigenvalues Superconformal symmetry ⇒ Tr RQ i Q k = − τ ik 3 ⇒ Tr RQ i Q k is negative definite

A-Maximization Intriligator and Wecht: correct choice of the R -charge R = R 0 + � i c i Q i maximizes a -charge: ∂a 3 � 9 Tr R 2 Q i − Tr Q I � = = 0 ∂c i 32 ∂ 2 a 27 = 16 Tr RQ i Q k < 0 ∂c i ∂c k

The simplest chiral SCFT SU ( N ) SU ( F ) SU ( F + N − 4) U (1) R Q R ( Q ) 1 Q R ( Q ) 1 A R (A) 1 1 F = 0, breaks SUSY F = 1 , 2, runaway vacua F = 3, quantum deformed moduli space F = 4, s-confining F = 5, splits into an IR free and IR fixed point sectors F > 5 ?

Moduli Space parameterized by mesons M = Q ¯ Q , H = ¯ QA ¯ Q and baryons: N even N odd ¯ ¯ Q N Q N QA N − 1 / 2 A N/ 2 Q 2 A ( N − 2) / 2 Q 3 A ( N − 3) / 2 . . . . . . Q k A ( N − k ) / 2 Q k A ( N − k ) / 2 where k ≤ min ( N, F )

R-charge anomaly cancellation for large F, N ⇒ � N R (A) = F � � � 2 − R ( Q ) − F + 1 R ( Q ) N In general 2 − 6 R ( Q ) = N + b ( N − F ) + c 6 F N + bF − c R ( Q ) = F + N − 4 − 12 R (A) = N − 2 bF 3( N 2 − 1 + FN ( R ( Q ) − 1) 3 + N ( F + N − 4)( R ( Q ) − 1) 3 a = + N ( N − 1) / 2( R ( A ) − 1) 3 ) − ( N 2 − 1) − NF ( R ( Q ) − 1) − N ( F + N − 4)( R ( Q ) − 1) − N ( N − 1) 1 2 ( R ( A ) − 1)

R-charge a-maximization gives for F, N : � 2 + 2 ( − 4+ N 12 − 9 ( N F ) ( N F ) F (73 N F − 4)) R ( Q ) = R ( Q ) = − 3( − 4+( N F − 4) N F ) even though theory is chiral

R-charge 3 _ _ Q 0.6 H=QAQ 0.5 2.5 A 0.4 2 0.3 _ Μ =QQ 0.2 1.5 0.1 0.5 0.75 1 1.25 1.5 1.75 2 0.5 0.75 1 1.25 1.5 1.75 2 (a) The R -charges of the fundamental fields, with R ( Q ) = R ( Q ) (b) The corresponding dimensions of the meson operators.

Two Free Mesons reduce F from the Banks–Zaks fixed point at F ∼ 2 N meson M = Q ¯ Q goes free at 9 N F = F 1 = 7) ≈ 0 . 3386 N √ 4(4+ meson H = ¯ QA ¯ Q is still interacting Kutasov: assume only one accidental U (1) for the free meson M then a int = a − a ( R ( M )) a − 3 3( R ( Q ) + R ( Q ) − 1) 3 � 32 F ( F + N − 4) = − ( R ( Q ) + R ( Q ) − 1)) meson H = ¯ QA ¯ Q goes free at F = F 2 ≈ 0 . 2445 N

Two Free Mesons R 1.5 0.6 1.4 0.5 _ Q 0.4 1.3 H 0.3 1.2 Q M 0.2 1.1 0.1 A F/N 0.25 0.3 0.35 0.4 0.45 0.25 0.3 0.35 0.4 0.45 0.5 (a) The R -charges of the fundamental fields (b) The corresponding dimensions of the meson operators

Dual Description for N odd and F ≥ 5: SU ( F − 3) Sp (2 F − 8) SU ( N + F − 4) SU ( F ) y ˜ 1 1 p 1 1 1 q 1 1 a 1 1 1 l 1 1 B 1 1 1 1 M 1 1 H 1 1 1 with a superpotential W = c 1 Mql ˜ y + c 2 Hll + B 1 qp + a ˜ y ˜ y M = Q ¯ Q and H = ¯ QA ¯ Q are mapped to elementary fields

Dual Description even N and F ≥ 5: SU ( F − 3) Sp (2 F − 8) SU ( N + F − 4) SU ( F ) SU (2) y ˜ 1 1 1 p 1 1 1 q 1 1 1 a 1 1 1 1 l 1 1 1 S 1 1 1 B 0 1 1 1 1 1 M 1 1 1 H 1 1 1 1 with a superpotential y + B 0 ap 2 W = c 1 Mql ˜ y + c 2 Hll + Sqp + a ˜ y ˜

Dual Description M goes free at F = F 1 , c 1 → 0 chiral operator ql ˜ y has dimension 2 H goes free at F = F 2 , c 2 → 0 chiral operator ll has dimension 2 corresponding to R -charge 4 / 3 l has R -charge 2 / 3 and dimension 1? is l free? self-consistent if l is a gauge-invariant operator recall SUSY QCD: W = c Mφφ when M goes free, the coupling c → 0 the chiral operator φφ has dimen- sion 2 a -maximization ⇒ φ , φ are free if we assume accidental axial symmetry for dual quarks accidental axial symmetry only if dual gauge group is IR-free dual β function ⇒ dual looses asymptotic freedom for F < 3 N/ 2 dual quarks are free

Dual β function with c 1 and c 2 set to zero W = B 1 qp + a ˜ y ˜ y Sp (2 F − 8) has ˜ y and l with gauge interactions − g 3 � � β ( g ) = 3(2 F − 6) − ( F − 3)(1 − γ ˜ y | g =0 ) − ( N + F − 4) 16 π 2 + O ( g 5 ) , nonperturbative SU ( F − 3) and superpotential corrections through anoma- lous dimension γ ˜ y Sp (2 F − 8) is IR free if N − 4 F + 11 − ( F − 3) γ ˜ y | g =0 > 0

Dual β function superpotential has dimension 3 ⇒ ( ∗ ) γ a + 2 γ ˜ y = 0 a ( F − 3) / 2 is a gauge-invariant operator ⇒ F − 3 + F − 3 4 γ a ≥ 1 ( ∗∗ ) 2 y ≤ 1 Combining eqns (*) and (**) we have that for large F , γ ˜ large F and large N limit with F < N/ 5: β ∝ N − 4 F + 11 − ( F − 3) γ ˜ y > 0 , Sp (2 F − 8) is IR free assuming l is free for F < F 2 we can check that Sp (2 F − 8) becomes IR free at F = F 2

F < F 2 : Mixed Phase theory splits into two sectors in the IR, free magnetic sector: Sp (2 F − 8) SU ( N + F − 4) SU ( F ) l 1 M 1 H 1 1 interacting superconformal sector: is SU ( F − 3) Sp (2 F − 8) SU ( F ) y ˜ 1 p 1 1 q 1 a 1 1 B 1 1 1 W = B 1 qp + a ˜ y ˜ y

N = 1: Open Questioins • how nonperturbative effects make γ Q � = γ ¯ Q only for F < F 1 • mixed-phase first conjectured in theories with an adjoint, still not proven • SO with spinors

N = 2 SU ( N ) with N = 2 SUSY and F hypermultiplets in β ( g ) = − g 3 (3 N − N (1 − 2 β ( g ) /g ) − F ) 16 π 2 1 − Ng 2 / 8 π 2 adjoint in supermultiplet with gluon and gluino ⇒ Z = 1 /g 2 ⇒ γ ( g ) = 2 β ( g ) /g γ Q = 0, non-renormalization of the superpotential ⇒ non-renormalization of the K¨ ahler function, both related to a prepotential solving for β ( g ) β ( g ) = − g 3 16 π 2 (2 N − F ) exact at one-loop

N = 2 SCFT β ( g ) = − g 3 16 π 2 (2 N − F ) vanishes for F = 2 N β vanishes independent of g ⇒ line of fixed points Seiberg–Witten analysis ⇒ a i D have no logarithmic corrections classical relations between a i and a j D are exact theory with F = 2 N hypermultiplets is nonperturbatively conformal

Argyres–Douglas fixed points massless electrically and magnetically charged particles at the same point in the moduli space electric charge: g IR → 0, magnetic charge: g IR → ∞ IR fixed point? Argyres and Douglas: yes! N = 2 SU (2) with one flavor adjust mass and VEV so monopole and dyon points coincide for m = 3Λ 1 / 4 and u = 3Λ 2 1 / 4 � 3 � x − Λ 2 y 2 = 1 4 all three roots coincide Seiberg–Witten analysis shows that a D has no logarithmic corrections, theory is conformal

Argyres–Douglas fixed points charges in U (1) theories with IR fixed points do not produce long- range fields using d ≥ 1 2 [ C 2 ( r ) + C 2 ( V ) − C 2 ( r ′ )] . F µν , is in a (1 , 0) + (0 , 1) of SO (4), has a scaling dimension d ≥ 2 at an interacting IR fixed point generically d > 2 conformal symmetry and dimensional analysis ⇒ fields fall off as 1 /x d

Other SCFTs have several different interactions and are superconformal lines (or manifolds) of fixed points if there are n interactions and only p independent β functions then n − p dimensional manifold of fixed points moving in manifold ↔ changing coupling of an exactly marginal operator operator in L has scaling dimension 4, independent of couplings can also happen in N = 1 theories

N = 4 SUSY gauge theory N = 1 SUSY gauge theory with three chiral supermultiplets in the adjoint with a particular superpotential ≡ N = 2 SUSY gauge theory with an adjoint hypermultiplet In general N = 4 theories have a global SU (4) R × U (1) R R -symmetry restrictted to vector supermultiplet does not transform under the U (1) R λ , and the three adjoint fermions, ψ , transform as a 4 of the SU (4) R real adjoint scalars φ transform as a 6 of SU (4) R in terms of N = 1 fields, the SU (4) R symmetry is not manifest only SU (3) × U (1) subgroup is apparent for canonically normalized N = 1 superfields the superpotential is √ 2 ǫ ijk f abc Φ c Y i Φ a j Φ b W N =4 = − i 2 Y T r Φ 1 [Φ 2 , Φ 3 ] = √ k 3 where a, . . . , e = 1 , . . . , N 2 − 1 are the adjoint gauge indices i, . . . , m = 1 , 2 , 3 are SU (3) flavor indices, and Φ i = T a Φ a i for N = 4 SUSY, Y = g