On the correlation numbers in Minimal Gravity and Matrix Models - PowerPoint PPT Presentation

On the correlation numbers in Minimal Gravity and Matrix Models A.Belavin, A.Zamolodchikov

Two approaches to 2D quantum geometry Continuous approach Discret approach ↓ ↓ ”Liouville Gravity” ”Matrix Models” Impressive body of evidence that the two describe the same reality: • Operators O LG and O MM have identical scale dimensions k k • Some correlation numbers coincide: � O LG ...O LG � = � O MM ...O MM � 1 n 1 n But with ”naive” identification many correlation numbers are not in agreement. Resolution [ Moore, Seiberg, Staudacher, 1991 ]: Resonance re- lations : [ O k ] = [ τ k 1 ][ O k 2 ] ↓ + B k 1 k 2 O MM = O LG τ k 1 O LG Umbiguity k k k 2 k • In many cases the disagreement can be fixed by adjusting the parameters (e.g. B k 1 k 2 above). k 1

• This work: Trying to find exact map for special class of models: ”p − criticality” in ”Minimal Gravity” MG 2 / 2 p +1 ↔ One − Matrix Model ◦ The problem is rather ”rigid” (more constraints then the pa- rameters). ◦ Nonetheless, the map exists up to the level of four point corr. numbers. ◦ The resulting 1-, 2-, 3-, and 4-point correlation numbers are in perfect agreement. 2

1. Minimal Gravity 1.1. Quantum Geometry � � D [ g ] D [ φ ] e − S [ g,φ ] topologies g ( x ) - Riemannian metric on 2D manifold M (assume sphere), φ - ”matter” fields Invariant correlation functions (”correlation numbers”) : � O k N e − S [ g,φ ] D [ g, φ ] O k N � = Z − 1 � ˜ O k 1 ... ˜ O k 1 ... ˜ ˜ with � ˜ O k = M O k ( x ) dµ g ( x ) O k ( x ) - local fields (built from φ and g ). Generating function : { τ } = { τ 1 , ..., τ n } � D [ g, φ ] e − S τ [ g,φ ] , W ( { τ } ) = Z ( { τ } ) /Z ( { 0 } ) , Z ( { τ } ) = � τ k ˜ S τ [ g, φ ] = S 0 [ g, φ ] + O k k 3

so that � O k N � = ∂ N W ( { τ } ) � � ˜ O k 1 ... ˜ � � ∂τ k 1 ...∂τ k N � τ =0 The parameters { τ } may be regarded as the coordinates in the ”theory space” Σ.

1.2. Conformal Matter, and Liouville Gravity = − c g µν T matter 12 R µν Conformal Gauge g µν = e 2 bϕ ˆ g µν : Decoupling ⇒ S [ g, φ ] → S L [ ϕ ] + S Ghost [ B, C ] + S Matter [ φ ] with � � � S L [ φ ] = 1 � g µν ∂ µ ϕ∂ ν ϕ + Q ˆ R ϕ + 4 πµ e 2 b ϕ d 2 x , g ˆ ˆ 4 π � S Ghost [ B, C ] = 1 � g B µν ∇ µ C ν d 2 x , ˆ 2 π � � g µν B µν = 0 B µν = B νµ , ˆ , 26 − c = 1 + 6 Q 2 Q = b + 1 /b . ( S Matter [ φ ] is conformally invariant, with the central charge c ). 4

Correlation numbers � ˜ O k 1 ... ˜ O k N � with � V k ( x ) Φ k ( x ) d 2 x ˜ O k = Φ k ( x ) - (spinless) primary fields of the matter CFT, with the conformal dimensions (∆ k , ∆ k ) V k ( x ) - ”gravitational dressings”, V k ( x ) = e 2 a k ϕ ( x ) , a k ( Q − a k ) + ∆ k = 1 Gravitational dimensions of ˜ O k control the scale dependence of the corr. functions: δ k = − a k O k ∼ µ δ k , ˜ b 1.3. Correlation numbers O k n � = | ( x 1 − x 2 )( x 2 − x 3 )( x 3 − x 1 ) | 2 × � ˜ O k 1 ... ˜ � d 2 x 4 ...d 2 x n � O k 1 ( x 1 ) O k 2 ( x 2 ) O k 3 ( x 3 ) O k 4 ( x 4 ) ...O k n ( x n ) � � �� � ↓ � V k 1 ( x 1 ) ...V k n ( x n ) � Liouville � Φ k 1 ( x 1 ) ... Φ k n ( x n ) � Matter 5

• The Liouville correlation functions are expressed in terms of the ”Conformal Blocks”, e.g. � V k 1 ( x 1 ) ...V k 4 ( x 4 ) � Liouville = � dP � � � � 4 π C L a k 1 , a k 2 , Q/ 2 + iP C L Q/ 2 − iP, a k 3 , a k 4 × |F ∆( P ) (1 − ∆ i | x i ) | 2 with ∆( P ) = Q 2 / 4 + P 2 , and the ”Liouville Structure Constants” 3 Υ b ( b ) Υ b (2 a i ) � ( Q − a ) /b � � πµ γ ( b 2 ) C L ( a 1 , a 2 , a 3 ) = Υ b ( a − Q ) Υ b ( a − a i ) i =1 where a = a 1 + a 2 + a 3 , � ∞ � � ( Q − 2 x ) 2 e − 2 t − sinh 2 (( Q/ 2 − x ) t ) dt log Υ b ( x ) = 4 sinh( bt ) sinh( t/b ) t 0 • Integration over the moduli x 4 , ..., x n is to be performed. 6

1.4. Matter CFT: ”Minimal Models” c = 1 − 6 ( p − q ) 2 M p/q pq Finite number of primary fields Φ ( n,m ) ( n = 1 , ..., p − 1 , m = 1 , ..., q − 1 , n ≤ m ) , with (in principle) computable correlation functions, e.g. � Φ ( n 1 ,m 1 ) ( x 1 ) ... Φ ( n 4 ,m 4 ) ( x 4 ) � MM = � C ( n,m ) ( n 1 ,m 1 )( n 2 ,m 2 ) C ( n,m ) ( n 3 ,m 3 )( n 4 ,m 4 ) |F ( n,m ) (∆ i | x ) | 2 ( n,m ) Fusion rules: N M � � Φ ( n 1 ,m 1 ) Φ ( n 2 ,m 2 ) = [Φ ( n,m ) ] , n = | n 1 − n 2 | +1 m = | m 1 − m 2 | +1 with N = min( n 1 + n 2 − 1 , 2 p − n 1 − n 2 − 1) , M = min( m 1 + m 2 − 1 , 2 q − m 1 − m 2 − 1) 7

1.5. ”Yang-Lee series” of the Minimal Models M 2 / 2 p +1 • M 2 / 2 p +1 has p primary fields Φ k ≡ Φ (1 ,k +1) , k = 0 , 1 , ..., p − 1 ( p, p + 1 , ..., 2 p − 1) Fusion rules k 1 + k 2 � [Φ k 1 ][Φ k 2 ] = [Φ k ] , [Φ k ] = [Φ 2 p − k − 1 ] k = | k 1 − k 2 | : 2 � + for even k ”Parity” : Φ k − for odd k Φ k = Φ 2 p − k − 1 → ”Parity violation” • Correlation functions: � Φ k � = δ k, 0 , � Φ k Φ k ′ � ∼ δ k,k ′ � Φ k 1 Φ k 2 Φ k 3 � = 0 � k 1 + k 2 < k 3 , etc , for k 1 + k 2 + k 3 even if k 1 + k 2 + k 3 < 2 p − 1 for k 1 + k 2 + k 3 odd 8

� Φ k 1 ... Φ k n � = 0 � k 1 + ... + k n − 1 < k n , for k 1 + ... + k n even if k 1 + ... + k n < 2 p − 1 for k 1 + ... + k n odd • Interpretations: M 2 / 3 - ”empty” theory (has only identity operator) M 2 / 5 - Yang-Lee edge criticality [ Cardy, 1985 ] M 2 / 2 p +1 - Yang-Lee multi-criticality?

1.6. Minimal gravity MG p/q : M p/q coupled to the Liouville Gravity • Early computations of the correlation numbers: [ Goulian & Li, 1991; Di Francesco & Kutasov, 1991; ... ] • Systematic approach [ Alexei Zamolodchikov, 2004; Belavin &Al.Zamolodchikov, 2006 ]: ”Higher Liouville Equations of Motion” � � moduli [ ... ] = moduli [total derivative] → Boundary terms • Results for MG 2 / 2 p +1 : � V k ( x ) Φ k ( x ) d 2 x , V k ( x ) = e ( k +2) b ϕ ( x ) ˜ O k = with � b = 2 / (2 p + 1) 9

⋆ One-point correlation numbers � O k � = 0 , ⋆⋆ Two-point numbers 1 O k ′ � = δ kk ′ 2 p − 2 k − 1 Leg 2 � ˜ O k ˜ L ( k ) , Z p with Z p = [(2 p − 1)(2 p + 1)(2 p + 3)] − 1 and �� − k +2 � � � � � 2 p +1 � π 2 γ 2 2 � 1 / 2 2 � πµ γ γ 2 p +1 2 p +1 2 Leg L ( k ) = � 2 p − 2 k − 1 � � 2 p − 2 k − 1 � 2 p − 1 γ γ 2 p +1 2 ⋆ ⋆ ⋆ Three-point correlation numbers: 3 O k 3 � = N k 1 k 2 k 3 � � ˜ O k 1 ˜ O k 2 ˜ Leg L ( k i ) Z p i =1 where N k 1 k 2 k 3 enforces the ”fusion rules” � 1 if the fusion rules of M 2 / 2 p +1 are satisfied N k 1 k 2 k 3 = 0 otherwise 10

⋆ ⋆ ⋆⋆ Four-point correlation numbers: 4 O k 4 � = Σ k 1 k 2 k 3 k 4 � � ˜ O k 1 ˜ O k 2 ˜ O k 3 ˜ Leg L ( k i ) Z p i =1 k 1 4 � � Σ k 1 ...k 4 = ( k 1 + 1)( p + k 1 + 3 / 2) − | p − 1 / 2 − k i − s | i =2 s = − k 1 : 2 Applies when the number of conformal blocks in � Φ k 1 ... Φ k 4 � is exactly k 1 . This holds for instance if and k 1 + k 4 ≤ k 2 + k 3 . k 1 ≤ k 2 ≤ k 3 ≤ k 4 , k 1 4 4 � p − 1 − 2 p +3 � � � � � � � ˜ � � 2 − k i − s � = ( k 1 +1) 2+ k i + F p ( k 1 + k i ) , s = − k 1 : 2 i =1 i =2 � F p ( k ) = ( p − k − 1)( p − k − 2) 1 for k ≥ 0 ˜ Θ( k − p ) , Θ( k ) = 2 0 for k < 0 ⋆...⋆ Higher-point functions are (in principle) computable [ Belavin, Al.Zamolodchikov, unpublished ]

• Generating function: { τ } = { τ 1 , τ 2 , ..., τ p − 1 } p − 1 � � � � � τ i ˜ W MG ( µ, { τ } ) = exp − O i i =1 MG 2 / 2 p +1 The cosmological constant µ may be treated as µ = τ 0 � e 2 bϕ ( x ) d 2 x S [ MG ] = ... + µ + ... � �� � � V 0 ( x ) Φ 0 ( x ) d 2 x , ˜ O 0 = Φ 0 = I Dimensions: k +2 τ k ∼ µ , k = 0 , 1 , ..., p − 1 2 By the definition � O k n � = ∂ n W MG ( µ, { τ i } ) � � ˜ O k 1 ... ˜ � { τ i } = { τ 1 , ..., τ n } , � ∂τ k 1 ...∂τ k n � { τ i } =0 11

2. Matrix Models Continuous (scaling) limit of the ensemble of planar graphs � �� � ↓ Quantum Geometry 2.1. One-matrix Model The planar graphs = Feynmann dia- grams associated with the perturbative evaluation of the matrix integral � � 2 M 2 − � � 1 αn n ! M n dM e − N tr n =3 Z = log M - Hermitian N × N matrix, N being the device for sorting out the topologies Z = N 2 Z 0 + Z 1 + ... + N 2 − 2 g Z g + ... Each term Z g generates discretized surfaces, of the topology g , made of triangles and higher polygons, with the weights deter- mined by α i . • We concentrate on g = 0 (sphere) Σ -space of the ”poten- n =3 α n tials” V ( M ) = � n ! M n . 12

• The sum of the planar graphs exhibits critical behavior , in the vicinities of certain critical hyper-surfaces in Σ: ... ⊂ Σ p ⊂ ... ⊂ Σ 2 ⊂ Σ 1 ⊂ Σ ↑ ” p -criticality”