RSOS paths, quasi-particles and fermionic characters Pierre Mathieu - PowerPoint PPT Presentation

RSOS paths, quasi-particles and fermionic characters Pierre Mathieu Universit´ e Laval (collaboration with Patrick Jacob)

Minimal model M ( p ′ , p ) ◮ Central charge c = 1 − 6 ( p − p ′ ) 2 pp ′ (with p ′ , p coprime and say p > p ′ ) ◮ Conformal dimensions: h r , s = ( pr − p ′ s ) 2 −( p − p ′ ) 2 = h p ′ − r , p − s 4 pp ′ 1 ≤ r ≤ p ′ − 1 1 ≤ s ≤ p − 1 and ◮ Highest-weight modules are completely degenerate

b b b b b Embedding pattern of singular vectors ( r , s ) ∼ ( p ′ − r , p − s )

Character of the irreducible modules ◮ Character: − q rs − q ( p ′ − r )( p − s ) 1 χ ( p ′ , p ) ( q ) = r , s ( q ) ∞ ( q ) ∞ ( q ) ∞ + q rs +( p ′ + r )( p − s ) + q ( p ′ − r )( p − s )+ r ( 2 p − s ) − ··· ( q ) ∞ ( q ) ∞ where (Verma character) � 1 1 p ( n ) q n ≡ � n ≥ 1 ( 1 − q n ) = ( q ) ∞ n ≥ 0 ◮ This formula is thus an alternating sign expression ... ◮ obtained by representation theory... ◮ but not very physical !

Fermionic character formula ◮ Every character in minimal models has a representation in terms of a positive multiple-sum ◮ This is called a fermionic character ◮ It reflects the filling of the space of states with quasi-particles subject to restrictions and without singular vectors

Fermionic character formula ◮ Every character in minimal models has a representation in terms of a positive multiple-sum ◮ This is called a fermionic character ◮ It reflects the filling of the space of states with quasi-particles subject to restrictions and without singular vectors ◮ Example: Ising vacuum: q 2 m 2 ∞ � χ ( 3 , 4 ) 1 , 1 ( q ) = ( q ) 2 m m = 0 where m � ( 1 − q i ) ( q ) m ≡ i = 1 manifestly positive: 1 ( 1 − q j ) = 1 + q j + q 2 j + ...

The M ( 2 , 2 k + 1 ) model [Feigin-Nakanashi-Ooguri ’91] q mBm + Cm ∞ � χ ( 2 , 2 k + 1 ) ( q ) = 1 , s ( q ) m 1 ··· ( q ) m k − 1 m 1 , ··· , m k − 1 = 0 where B ij = min ( i , j ) C j = max ( j − s + 1 )

The M ( 2 , 2 k + 1 ) model [Feigin-Nakanashi-Ooguri ’91] q mBm + Cm ∞ � χ ( 2 , 2 k + 1 ) ( q ) = 1 , s ( q ) m 1 ··· ( q ) m k − 1 m 1 , ··· , m k − 1 = 0 where B ij = min ( i , j ) C j = max ( j − s + 1 ) Basis of states: L − n 1 ··· L − n N | h 1 , s � ( n i > 0 ) with n i ≥ n i + k − 1 + 2 n p − s + 1 ≥ 2 and (sort of generalized exclusion principle plus a boundary condition that selects the module) Origin in CFT: These constraints come from the non-trivial vacuum singular vector

The M ( 2 , 2 k + 1 ) model [Feigin-Nakanashi-Ooguri ’91] q mBm + Cm ∞ � χ ( 2 , 2 k + 1 ) ( q ) = 1 , s ( q ) m 1 ··· ( q ) m k − 1 m 1 , ··· , m k − 1 = 0 where B ij = min ( i , j ) C j = max ( j − s + 1 ) Questions ◮ What is the CFT interpretation of the m j ? ◮ If a fermionic form is related to an integrable perturbation: this is the φ 1 , 3 one: how does this enters in the structure?

The M ( 3 , p ) models [Jacob, M ’06; Feigin et al ’06] Extended algebra construction: φ 2 , 1 × φ 2 , 1 = φ 1 , 1 + φ 3 , 1 = φ 1 , 1 e.g., with h ≡ h 2 , 1 = p − 2 φ ≡ φ 2 , 1 and 4 � � I +( z − w ) 2 2 h 1 φ ( z ) φ ( w ) = c T ( w )+ ··· S ( z − w ) 2 h and S = (− 1 ) p F where F counts the number of φ modes

Basis of states Generalized commutation relations + singular vector of φ : φ − s 1 φ − s 2 ··· φ − s N − 1 φ − s N | σ ℓ � , with s i ≥ s i + 1 − p s i ≥ s i + 2 + 1 2 + 3 , and the boundary conditions: s N − 1 ≥ − h + ℓ s N ≥ h − ℓ 2 + 1 , 2 where s N − 2 i ∈ Z + h + ℓ s N − 2 i − 1 ∈ Z − h + ℓ and 2 2 The spectrum is fixed by associativity

The M ( 3 , p ) character formula q mB ′ m + C ′ m � χ ( 3 , p ) ( q ) = , 1 , s ( q ) m 1 ··· ( q ) m k − 1 ( q ) 2 m k m 1 , m 2 , ··· m k ≥ 0 where k is defined via p = 3 k + 2 − ǫ ( ǫ = 0 , 1 ) with 1 ≤ i , j ≤ k − 1 kj = j kk = k + ǫ B ′ ij = min ( i , j ) , B ′ jk = B ′ B ′ 2 , , 4 and C ′ reads k = k − ǫ − s + 1 C ′ j = max ( j − s + 1 , 0 ) , C ′ , 2

The M ( 3 , p ) character formula q mB ′ m + C ′ m � χ ( 3 , p ) ( q ) = , 1 , s ( q ) m 1 ··· ( q ) m k − 1 ( q ) 2 m k m 1 , m 2 , ··· m k ≥ 0 where k is defined via p = 3 k + 2 − ǫ ( ǫ = 0 , 1 ) Questions ◮ What is the CFT interpretation of the m j ? ◮ If a fermionic form is related to an integrable perturbation: this is the φ 1 , 3 one: how does this fit with a formulation in terms of the φ 2 , 1 modes ?

On the CFT derivations of fermionic characters in M ( p ′ , p ) models ◮ The M ( 2 , p ) and the M ( 3 , p ) models are the only ones for which there is a ‘complete’ CFT derivation of the fermionic characters ◮ Generalization of the M ( 3 , p ) case to M ( p ′ , p ) : 1- Replace φ 2 , 1 by φ p ′ − 1 , 1 [M,Ridout ’07] 2- Treat φ 2 , 1 with its 2 channels [Feigen-Jimbo-Miwa-Mutkhin-Takayema ’04,’06] ◮ Monomial bases have been derived/conjectured for all models but the corresponding formula is not written ◮ These bases can be reexpressed in terms of RSOS-type configuration sums

Origins of Fermionic forms ◮ Bases in affine Lie algebras (parafermions) [Lepowski-Primc ’85] ◮ Bases for the M ( 2 , p ) models [Feigin-Ooguri-Nakanishi ’91] ◮ Dilogarithmic identities [Nahm et al ’92; Kuniba-Nakanishi-Suzuki (generalized parafermions)] ◮ Many conjectured expressions for the minimal models [Kedem-Klassen-McCoy-Melzer ’93] ◮ Counting of states in XXZ: truncation and q -deformation [Berkovich-McCoy-Schilling: ’94-’95] su ( 2 ) k [Bernard et al, Bouwknegt et al ’94] ◮ Spinon bases for � ◮ Mathematical transformations of identities (Bayley and Burge transforms) [Foda-Quano, Berkovich-McCoy,...]

Origins of Fermionic forms: The RSOS side ◮ RSOS models: Andrews-Baxter-Forrester (1984) (unitary case) Forrester-Baxter (1985) (non-unitary case) ◮ Key observation [Date, Jimbo, Kuniba, Miwa, Okado]: 1D configuration sums (obtained by CTM) in regime III (a lattice realization of the φ 1 , 3 perturbation) are the M ( p ′ , p ) irreducible characters ◮ Configurations sums leads to fermionic character in a systematic way [Melzer, Warnaar, Foda, Welsh]

Origins of Fermionic forms: The RSOS side ◮ RSOS models: Andrews-Baxter-Forrester (1984) (unitary case) Forrester-Baxter (1985) (non-unitary case) ◮ Key observation [Date, Jimbo, Kuniba, Miwa, Okado]: 1D configuration sums (obtained by CTM) in regime III (a lattice realization of the φ 1 , 3 perturbation) are the M ( p ′ , p ) irreducible characters ◮ Configurations sums leads to fermionic character in a systematic way [Melzer, Warnaar, Foda, Welsh] ...but the derivation is not constructive and the underlying quasi-particle structure remains unclear (except in the unitary case)

RSOS ( p ′ , p ) paths (regime-III) States in the finitized M ( p ′ , p ) minimal models (with p > p ′ ) are described by RSOS ( p ′ , p ) configurations Configurations ◮ Configuration = sequence of values of the height variables ℓ i ∈ { 1 , 2 , ··· , p − 1 } (0 ≤ i ≤ L ) ◮ with the admissibility condition: | ℓ i − ℓ i + 1 | = 1 ◮ and the boundary conditions: ℓ 0 , ℓ L − 1 and ℓ L fixed

RSOS ( p ′ , p ) paths (regime-III) States in the finitized M ( p ′ , p ) minimal models (with p > p ′ ) are described by RSOS ( p ′ , p ) configurations Configurations Paths ◮ A path is the contour of a ◮ Configuration = sequence of configuration. values of the height variables ℓ i ∈ { 1 , 2 , ··· , p − 1 } ◮ Path = sequence of NE or SE (0 ≤ i ≤ L ) edges joining the adjacent vertices ( i ,ℓ i ) and ( i + 1 ,ℓ i + 1 ) ◮ with the admissibility of the configuration within the condition: | ℓ i − ℓ i + 1 | = 1 rectangle 1 ≤ y ≤ p − 1 and ◮ and the boundary conditions: 0 ≤ x ≤ L ℓ 0 , ℓ L − 1 and ℓ L fixed ◮ with ℓ 0 and ℓ L fixed ◮ and fixed last edge: SE

b b b b b b b b b b b b b b b b b b b b b A typical configuration for the M ( p ′ , 7 ) model: ℓ 0 = 1, ℓ 19 = 4 ,ℓ 20 = 3 ( p − 1 =) 6 5 4 3 2 1 1 3 5 7 9 11 13 15 17 19

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b A typical configuration for the M ( p ′ , 7 ) model: ℓ 0 = 1, ℓ 19 = 4 ,ℓ 20 = 3 ( p − 1 =) 6 5 4 3 2 1 1 3 5 7 9 11 13 15 17 19 and the corresponding path (with ℓ 20 = 3) ( p − 1 =) 6 5 4 3 2 1 1 3 5 7 9 11 13 15 17 19

A typical path for the M ( p ′ , 7 ) model: ℓ 0 = 1 and ℓ 20 = 3 and final SE 6 5 4 3 2 1 5 1 3 7 9 11 13 15 17 19

A typical path for the M ( p ′ , 7 ) model: ℓ 0 = 1 and ℓ 20 = 3 and final SE 6 5 4 3 2 1 1 3 5 7 9 11 13 15 17 19 ◮ But this corresponds to a state for which model ? (value of p ′ ?) ◮ ...and to which module ( r , s ) ? ◮ ...and what is its conformal dimension?

b b b b Weighting the path The dependence of the path upon the parameter p ′ is via the weight function: L − 1 � w = w i ˜ ˜ i = 1 w i w i ˜ ˜ Vertex Vertex � � h + 1 h ( p − p ′ ) i − i h p 2 i i � � h ( p − p ′ ) i i h h − 1 p 2 i i