USp Matrix Model Revisited 080304@KEK originally with A. Tokura - PowerPoint PPT Presentation

USp Matrix Model Revisited 080304@KEK originally with A. Tokura (’97) later with A. Tsuchiya, T. Matsuo, B. Chen, H. Kihara (Osaka U.) in recent years with H. Kihara (KIAS), R. Yoshioka (OCU) I). Introduction • ten years of developments in reduced matrix models • review not as a reflection but as an opportunity for a better perspective

Contents I). Introduction II). Criteria and construction [I-Tok] III). Semi-uniqueness and loop variables [I-Tok, I-Tsuch] IV). S-D equation [I-Tsuch] V). Spacetime fluctuations represented by nonabelian Berry phase [I-Mats, CIK] VI). Attempts in recent years [IY, IKY]

II). Criteria and construction M BFSS by ’84 − → type I ← → hetero SO (32) IIA IIB � W Ω projection ← → S 9 open added T 9 hetero E 8 × E 8 16 + 16 supercharges 8 + 8 supercharges closed + open, nonorientable closed, orientable large k ⇓ ⇓ reduced model IIB matrix model USp matrix model IKKT IT

Action of IIB matrix model and USp matrix model Let me tell you the definitions of the models first. S IIB ( v M , Ψ) = 1 � 1 4[ v M , v N ][ v M , v N ] − 1 � ¯ ΨΓ M [ v M , Ψ] g 2 tr 2 Ψ : Maj-Weyl , 0 ≤ M, N ≤ 9 objects with : U(2k) matrices m = 0 , · · · , 3 . We also write as v m , 1 Φ I = 2 ( v 3+ I + iv 6+ I ) , I = 1 , 2 , 3 √ Ψ = ( λ, 0 , ψ 1 , 0 , ψ 2 , 0 , ψ 3 , 0 , 0 , ¯ λ, 0 , ¯ ψ 1 , 0 , ¯ ψ 2 , 0 , ¯ ψ 3 ) t using 4d superfield �� � �� � 1 � + 1 I e 2 V Φ I d 2 θd 2 ¯ θ Φ † d 2 θW α W α + h.c. + 4 d 2 θW 0 + h.c. ∴ S IIB = 4 g 2 tr g 2 √ W 0 = 2tr(Φ 1 [Φ 2 , Φ 3 ]]) Requirements for the model descending from perturbative type I superstrings i) closed (projected) ii) nonorientable iii) 8 + 8 susy

• To find the matrix counterpart of the Ω projection, recall for both USp and SO. � 0 � I ր USp adj (= sym) F = U (2 k ) adj − I 0 ց USp asym � 0 � ր I SO adj (= asym) F = analog of Ω U (2 k ) adj 0 I ց SO sym 2 ( • ∓ F − 1 • t F ) ρ ∓ • = 1 The projector ˆ In fact � M � N X t F + FX = 0 for X ∈ usp (2 k ) Lie alg. ∴ X = N ∗ − M ∗ � A � B , B t = − B, C t = − C ∴ Y ≡ ( TF ) j Y t F − FY = 0 for Y ∈ antisym i = A t C ∗ We found out planar diagram analysis ⇒ Chan-Paton factor of open loop all lead to usp consistency with wv field theory • Need to add open string degrees of freedom, ( Q • , ˜ Q • , ψ Q • , ψ • keeping 8 + 8 susy ⇒ fundamental rep. Q ) , #( fund ) = n f ˜

Definition of the model V = ˆ Φ 1 = ˆ ρ − Φ 1 , Φ I = ˆ ρ + Φ I , I = 2 , 3 ρ − V , adj adj asym �� � S USp = 1 � I e 2 V Φ I e − 2 V d 2 θd 2 ¯ θ Φ † d 2 θW α W α + 4 4 g 2 tr n f �� � �� � + 1 + 1 ( f ) e 2 V Q ( f ) + ˜ Q ( f ) e − 2 V ˜ � d 2 θd 2 ¯ θQ ∗ Q ∗ d 2 θW ( θ ) + h.c. ( f ) g 2 g 2 f =1 √ √ 2tr(Φ 1 [Φ 2 , Φ 3 ]) + � n f f =1 ( m ( f ) ˜ 2 ˜ W ( θ ) = Q ( f ) Q ( f ) + Q ( f ) Φ 1 Q ( f ) ) ↑ again using 4d superfield notation also can write S 0 : with Q ( f ) , ˜ S USp = S 0 + ∆ S Q ( f ) set to zero S 0 = S IIB (ˆ ρ b ± v m , ˆ ρ f ± Ψ A ) specific projection g 2 , Parameters (2 k ) , n ( f ) = 16 (by 6d gauge anomaly cancel.) m ( f ) , scaling : g 2 (2 k ) # = const. k → ∞ , # is difficult to determine.

Why transparency • Why matrices are strings • Why F ∼ Ω • USp not SO • Why fundamentals needed • n f =? • m ( f ) =?

• III). Semi-uniqueness and loop variables • [susy, projector] = 0 in the USp case • S IIB ; possesses 16 + 16 susy ∃ 8 + 8 susy • S USp ; need to check ; Is ρ b ∓ and ρ f ∓ unique?

Projector and susy how to have both consistently: • susy transf.  δ (1) v M = i ¯ ǫ Γ M Ψ     δ (1) Ψ = i 2 [ v M , v N ]Γ MN ǫ   16 + 16 IIB case δ (2) v M = 0     δ (2) Ψ = ξ   • We have examined the conditions: � ρ b ∓ , δ (1)(2) ] v M = 0 [ˆ ( ∗ ) ρ b ∓ , δ (1)(2) ] v M = 0 [ˆ To be more explicit, let ρ ( M ) b ∓ = Θ( M ∈ M − )ˆ ρ − + Θ( M ∈ M + )ˆ ˆ ρ + ρ ( A ) f ∓ = Θ( A ∈ A − )ˆ ρ − + Θ( A ∈ A + )ˆ ˆ ρ + M − ∪ M + = {{ 0 , 1 , 2 , · · · , 9 }} , M − ∩ M + = ∅ , etc ( ∗ ) yield eqs. w.r.t. ǫ, ξ, M − , M + , A − , A + demand 8 + 8 susy

• Solutions: our cases 6 adj + 4asym  ρ b ∓ = diag ( − , − , − , − , − , + , + , − , + , +) ˆ          I (2) 0         0 I (2)     ρ f ∓ = ˆ ˆ ρ − I (4) ⊗ + ˆ ρ + I (4) ⊗           0 I (2)              0 I (2)   M, hetero  ρ b ∓ = diag (+ , + , + , + , − , + , + , − , + , +) ˆ          0 I (2)         0 I (2)     ρ − I (4) ⊗ ρ + I (4) ⊗ ρ f ∓ = ˆ ˆ + ˆ           0 I (2)              0 I (2)   S. Rey, D. Lowe, ...

Gauge anomalies cancellation still somewhat mysterious • take matrix T dual ala W. Taylor albeit being against our spirit ⇒ (zero volume limit) of 6-d. wv. gauge theory • chirality Γ 6 = Γ 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 7     λ ψ 2 0 0      , − 1 , fund , − 1  , +1 ,      ψ 1   ψ 3    0 0 ∝ tr adj F 4 − tr asym F 4 − n f tr F 4 = (16 − n f )tr F 4 nonabelian anomaly ∴ n f = 16 • should be interpreted as a force balance. The cases n f � = 16 and IIB would imply an existence of residual interactions ⇒ gauge sym. breaking ??

Closed and open loops • Recall we have added the open string deg. of freedom, ⇒ fundamental rep. ( Q, ˜ Q ) , #( fund ) = n f Q, ψ Q , ψ ˜ • To make flavor symmetry ( ≈ gauge sym. of strings) manifest,  � ψ Q ( f ) Q ( f )  Q ( f ) = , ψ Q ( f ) = F − 1 ˜ F − 1 ψ ˜ Q ( f − n f ) Q ( f − nf )  tr e lM • basic operators (observables): cf. one-matrix model ≡ tr ← − Π n 1 n = n 0 exp( − ip M Φ [ p M • , η ; n 1 , n 2 ] n v M − i ¯ η n Ψ) = = Φ [ ∓ p M • , η ; n 0 , n 1 ] i.e. nonorientable ( f ) ) + ( θψ Q ( f ) + F − 1 ¯ · Λ · Π ( f ) ≡ ( ξ Q ( f ) + F − 1 ξ ∗ Q ∗ θψ ∗ Q ( f ) ) f ’ f ≡ Λ ′ · Π ( f ′ ) FU [ · · · ]Λ · Π ( f ) = - Ψ f ′ f [ p M • , η • ; n 0 , n 1 ; Λ ′ , Λ] f f ’ • , ∓ η • ; n 0 , n 1 ; Λ ′ , Λ] = ∓ Ψ ff ′ [ ∓ p M

original(worldsheet) C-P factor Lie algebra − : ⇔ ⇐ our choice so (2 n f ) usp (2 k ) + : usp (2 n f ) ⇔ so (2 k )

IV). S-D equation Schwinger-Dyson eq. as before f ’ Φ [( i )] ; i-th closed loop f Ψ f ′ f [( i )] ; i-th open loop Consider � dµ ∂ ∂X r tr( U ([1]) T r U [(1)]) Φ [(2)] · · · Φ [( N )] Ψ [(1)] · · · Φ [( L )] e − S 0 = � dµ ∂ ∂X r Λ (1) ′ · Π f (1) ′ FU [(1)] T r U [(1)]Λ (1) · Π f (1) Φ [(1)] · · · Φ [( N )] Ψ [(2)] · · · Φ [( L )] e − S 0 = X r = v r M or Ψ r � ∂ U [(1)]Λ (1) · Π f (1) Φ [(1)] · · · Φ [( N )] Ψ [(2)] · · · Φ [( L )] e − S 0 = dµ ∂ Z ( f ) i Z ( f ) i = Q ( f ) i or ψ Q ( f ) i

• We have shown that eqs. are closed w.r.t. the loops eq. of motion acting on the loop → deformation of the loop complete set of nonorientable interactions among loops consistent with two kinds of elementary local moves. � dµ ∂ • ; n (1) 2 , n (1) • ; n (1) 1 , n (1) ∂X r { tr( U [ p (1) • , η (1) + 1] T r U [ p (1) • , η (1) 0 = 0 ]) 1 Φ [(2)] · · · Φ [( N )] Ψ [(1)] · · · Φ [( L )] e − S } T r : generator of usp(2k), X r : A r M or Ψ r α 0 = + � � 1 + + 2 1 � � + + 2 1 � � + + 2

� 2 k 2 ± k ( T r ) j i δ j k ∓ F − 1 k = 1 i ( T r ) l 2 ( δ l ik F lj ) r =1 � 2 k 2 ± k ( T r ) j k • = 1 2 ( • ∓ F − 1 • t F ) = ˆ i ( T r ) l ρ ∓ r =1 I) ⇒ 1 g 2 � δ X Φ [(1) : X r ] Φ [(2)] · · · Φ [( N )] Ψ [(1)] · · · � 0 = (1) kinetic term +(2) splitting and twisting term #( loops ) increases by 1 +(3) joining with a closed string #( loops ) decreases by 1 +(4) joining with an open string #( loops ) decreases by 1

V). Spacetime fluctuations represented by nonabelian Berry phase • variables  x (1)  M ...     x ( k )   v M = u ( M ) u ( M ) − 1   M   ∓ x (1)   ↑ M   ...     ∓ x ( k ) M ≡ u ( M ) X M u ( M ) − 1 ↑ spacetime pts. → Ψ integrate → s to give → ψ f dynamics to the spacetime pts. → Q f For simplicity, set u ( M ) = 1 , Q f = 0 note: spacetime pts are dynamical variables

t x quantum mechanics parameter d.v. QFT parameter parameter reduced matrix model d.v. d.v. • Rather than ln Z eff [ x ( i ) M ] , we measure for a given path Γ in x ( k ) Σ �� one-particle projector �� Γ M • �� 1P-projector �� = nonabelian Berry phase S fermion = S MM + S gf + S Yukawa