Jack wavefunctions and W theories Benoit Estienne joint work with - PowerPoint PPT Presentation

Symmetric Polynomials Monomial basis { m λ } The monomial function m λ is a symmetric polynomial in n variables { z i , i = 1 , . . . , n } : n � z λ i m λ ( { z i } ) = S ( i ) i =1 Partitions λ = ( λ 1 , . . . , λ N ) λ i are positive integers λ i > λ i +1 For λ = (4 , 4 , 2 , 1 , 1) : � � z 4 1 z 4 2 z 2 m λ = S 3 z 4 z 5 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 5 / 33

Symmetric Polynomials Monomial basis { m λ } The monomial function m λ is a symmetric polynomial in n variables { z i , i = 1 , . . . , n } : n � z λ i m λ ( { z i } ) = S ( i ) i =1 Partitions λ = ( λ 1 , . . . , λ N ) λ 1 λ i are positive integers λ 2 λ i > λ i +1 λ 3 λ 4 For λ = (4 , 4 , 2 , 1 , 1) : λ 5 � � z 4 1 z 4 2 z 2 m λ = S 3 z 4 z 5 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 5 / 33

Jack Polynomials J α λ ( z 1 , · · · , z N ) symmetric and homogeneous polynomials of N variables indexed by partitions λ = ( λ 1 , λ 2 , . . . , λ N ) depend rationally on a parameter α : the expansion over the m λ basis takes the form � J α λ = m λ + u λµ ( α ) m µ . µ<λ The Jacks J α λ are eigenfunctions of the Calogero-Sutherland Hamiltonian : N ( z i ∂ i ) 2 + 1 z i + z j � � H CS( α ) = ( z i ∂ i − z j ∂ j ) α z i − z j i =1 i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 6 / 33

Jack Polynomials J α λ ( z 1 , · · · , z N ) symmetric and homogeneous polynomials of N variables indexed by partitions λ = ( λ 1 , λ 2 , . . . , λ N ) depend rationally on a parameter α : the expansion over the m λ basis takes the form � J α λ = m λ + u λµ ( α ) m µ . µ<λ The Jacks J α λ are eigenfunctions of the Calogero-Sutherland Hamiltonian : N ( z i ∂ i ) 2 + 1 z i + z j � � H CS( α ) = ( z i ∂ i − z j ∂ j ) α z i − z j i =1 i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 6 / 33

Jack Polynomials J α λ ( z 1 , · · · , z N ) symmetric and homogeneous polynomials of N variables indexed by partitions λ = ( λ 1 , λ 2 , . . . , λ N ) depend rationally on a parameter α : the expansion over the m λ basis takes the form � J α λ = m λ + u λµ ( α ) m µ . µ<λ The Jacks J α λ are eigenfunctions of the Calogero-Sutherland Hamiltonian : N ( z i ∂ i ) 2 + 1 z i + z j � � H CS( α ) = ( z i ∂ i − z j ∂ j ) α z i − z j i =1 i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 6 / 33

Jack Polynomials J α λ ( z 1 , · · · , z N ) symmetric and homogeneous polynomials of N variables indexed by partitions λ = ( λ 1 , λ 2 , . . . , λ N ) depend rationally on a parameter α : the expansion over the m λ basis takes the form � J α λ = m λ + u λµ ( α ) m µ . µ<λ The Jacks J α λ are eigenfunctions of the Calogero-Sutherland Hamiltonian : N ( z i ∂ i ) 2 + 1 z i + z j � � H CS( α ) = ( z i ∂ i − z j ∂ j ) α z i − z j i =1 i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 6 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ 1 ≥ λ 3 + 2 λ i − λ i + k ≥ r λ 3 Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r λ 2 ≥ λ 4 + 2 λ 4 Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jacks wavefunction ( k , r ) admissible partitions λ i − λ i + k ≥ r (2 , 2) admissible Jack Polynomials with ( k , r ) clustering properties for the special value α = − ( k + 1) / ( r − 1) and for a ( k , r ) admissible partition λ [Feigin et al (2001) ] These Jacks are well defined. They have generalized clustering properties : they vanish as r powers when k + 1 particles come to the same point. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 7 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Jack Polynomials at α = − ( k + 1) / ( r − 1) Densest ( k , r ) admissible partitions The root partition for the wavefunction with the highest density is given by the occupation numbers k λ = [ k 00 . . . 0 k 00 . . . 0 k . . . ] r � �� � � �� � r − 1 r − 1 Trial wavefunctions generalizing the Read-Rezayi states These Jacks have been considered as trial many-body wavefunctions for non-Ablian FQH states [ Bernevig and Haldane (2007) ] at (bosonic) filling fraction ν = k / r r = 2 boils down to the Read-Rezayi Z k state conjectured to be connected to W conformal field theories Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 8 / 33

Conformal field theories as wavefunctions generators To describe a N particles quantum Hall ground state, a polynomial P N ( { z i } ) has to be a SU (2) spin singlet : � L − P N = i ∂ i P N ( { z i } ) = 0 � � � z i ∂ i − N φ L z P N = P N ( { z i } ) = 0 i 2 � � � L + P N = − z 2 i ∂ i + z i N φ P N ( { z i } ) = 0 i All these properties are automatically ensured by global conformal invariance for single channel correlators : � ( z i − z j ) γ � Φ( z 1 ) . . . Φ( z N ) � i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 9 / 33

Conformal field theories as wavefunctions generators To describe a N particles quantum Hall ground state, a polynomial P N ( { z i } ) has to be a SU (2) spin singlet : � L − P N = i ∂ i P N ( { z i } ) = 0 � � � z i ∂ i − N φ L z P N = P N ( { z i } ) = 0 i 2 � � � L + P N = − z 2 i ∂ i + z i N φ P N ( { z i } ) = 0 i All these properties are automatically ensured by global conformal invariance for single channel correlators : � ( z i − z j ) γ � Φ( z 1 ) . . . Φ( z N ) � i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 9 / 33

Conformal field theories as wavefunctions generators To describe a N particles quantum Hall ground state, a polynomial P N ( { z i } ) has to be a SU (2) spin singlet : � L − P N = i ∂ i P N ( { z i } ) = 0 � � � z i ∂ i − N φ L z P N = P N ( { z i } ) = 0 i 2 � � � L + P N = − z 2 i ∂ i + z i N φ P N ( { z i } ) = 0 i All these properties are automatically ensured by global conformal invariance for single channel correlators : � ( z i − z j ) γ � Φ( z 1 ) . . . Φ( z N ) � i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 9 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic chiral algebra • additional Z k symmetry encoded in the fusion rules of a set of chiral operators Ψ q ( z ) : [ Ψ n ] × [ Ψ m ] = [ Ψ n + m ] consistency (bootstrap) fixes the conformal dimensions : ∆ n = r n ( k − n ) 2 k • r ≥ 2 is an integer : r = 2 : FZ parafermions [Fateev, Zamolodchikov (1985)] ⇒ Read-Rezayi states r = 3 : (for k even) non unitary [Jacob, Mathieu (2002)] ⇒ Gaffnian r = 4 : second parafermionic serie [Dotsenko, Jacobsen, Santachiara (2003)] Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 10 / 33

Parafermionic correlators and clustering properties Parafermionic correlators Let’s consider a parafermionic CFT Z ( r ) k . The following function is a symmetric polynomial � P ( k , r ) ( z i − z j ) 2∆ 1 − ∆ 2 ( { z i } ) = ˆ � Ψ( z 1 ) . . . Ψ( z N ) � N i < j � ( z i − z j ) r / k . = � Ψ( z 1 ) . . . Ψ( z N ) � i < j and is a SU (2) singlet. Clustering properties More interestingly, this polynomial vanishes as r powers when k + 1 particles come to the same point ! Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 11 / 33

Parafermionic correlators and clustering properties Parafermionic correlators Let’s consider a parafermionic CFT Z ( r ) k . The following function is a symmetric polynomial � P ( k , r ) ( z i − z j ) 2∆ 1 − ∆ 2 ( { z i } ) = ˆ � Ψ( z 1 ) . . . Ψ( z N ) � N i < j � ( z i − z j ) r / k . = � Ψ( z 1 ) . . . Ψ( z N ) � i < j and is a SU (2) singlet. Clustering properties More interestingly, this polynomial vanishes as r powers when k + 1 particles come to the same point ! Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 11 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA k − 1 conformal field theories : some basic properties Extended conformal symmetry These theories have first been introduced in the case k = 3 by Fateev and Zamolodchikov (1987) : the so-called W 3 theory generalized to any k by Fateev and Lykyanov (1988) they are the prototype of CFT with extended symmetries : in addition to the stress-energy tensor T ( z ), the chiral algebra contains k − 2 currents W ( s ) ( z ) of integer spin s = 3 , . . . , k − 1. Minimal models For a discrete serie of values of the central charge, these CFT are minimal. The central charge of the WA k − 1 ( p , p ′ ) models is: � � 1 − k ( k + 1)( p − p ′ ) 2 c ( p , p ′ ) = ( k − 1) pp ′ p and p ′ are coprimes, and these models are unitary for p ′ = p + 1. Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 12 / 33

WA 1 theories : minimal models of Virasoro algebra Virasoro algebra The conformal symmetry is encoded in a single current : the stress-enery tensor T ( z ). Its mode obey the celebrated Virasoro algebra : [ L n , L m ] = ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 Primary fields Primary fields are anihilated by all positive modes L n : T ( z )Φ ∆ (0) = ∆ z 2 Φ ∆ (0) + 1 z ∂ Φ ∆ (0) + O (1) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 13 / 33

WA 1 theories : minimal models of Virasoro algebra Virasoro algebra The conformal symmetry is encoded in a single current : the stress-enery tensor T ( z ). Its mode obey the celebrated Virasoro algebra : [ L n , L m ] = ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 Primary fields Primary fields are anihilated by all positive modes L n : T ( z )Φ ∆ (0) = ∆ z 2 Φ ∆ (0) + 1 z ∂ Φ ∆ (0) + O (1) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 13 / 33

WA 1 theories : minimal models of Virasoro algebra Virasoro algebra The conformal symmetry is encoded in a single current : the stress-enery tensor T ( z ). Its mode obey the celebrated Virasoro algebra : [ L n , L m ] = ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 Primary fields Primary fields are anihilated by all positive modes L n : T ( z )Φ ∆ (0) = ∆ z 2 Φ ∆ (0) + 1 z ∂ Φ ∆ (0) + O (1) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 13 / 33

WA 1 theories : minimal models of Virasoro algebra Virasoro algebra The conformal symmetry is encoded in a single current : the stress-enery tensor T ( z ). Its mode obey the celebrated Virasoro algebra : [ L n , L m ] = ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 Primary fields Primary fields are anihilated by all positive modes L n : T ( z )Φ ∆ (0) = ∆ z 2 Φ ∆ (0) + 1 z ∂ Φ ∆ (0) + O (1) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 13 / 33

Minimal models of Virasoro algebra WA 1 ( p , p ′ ) central charge c = 1 − 6( p − p ′ ) 2 pp ′ finite number of primary fields Φ ( n | n ′ ) labeled by the Kac table : 1 ≤ n ≤ p ′ − 1 1 ≤ n ′ ≤ p − 1 with conformal dimension ∆ ( n , n ′ ) = ( np − n ′ p ′ ) 2 − ( p − p ′ ) 2 4 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 14 / 33

Minimal models of Virasoro algebra WA 1 ( p , p ′ ) central charge c = 1 − 6( p − p ′ ) 2 pp ′ finite number of primary fields Φ ( n | n ′ ) labeled by the Kac table : 1 ≤ n ≤ p ′ − 1 1 ≤ n ′ ≤ p − 1 with conformal dimension ∆ ( n , n ′ ) = ( np − n ′ p ′ ) 2 − ( p − p ′ ) 2 4 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 14 / 33

Minimal models of Virasoro algebra WA 1 ( p , p ′ ) central charge c = 1 − 6( p − p ′ ) 2 pp ′ finite number of primary fields Φ ( n | n ′ ) labeled by the Kac table : 1 ≤ n ≤ p ′ − 1 1 ≤ n ′ ≤ p − 1 with conformal dimension ∆ ( n , n ′ ) = ( np − n ′ p ′ ) 2 − ( p − p ′ ) 2 4 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 14 / 33

Minimal models of Virasoro algebra WA 1 ( p , p ′ ) central charge c = 1 − 6( p − p ′ ) 2 pp ′ finite number of primary fields Φ ( n | n ′ ) labeled by the Kac table : 1 ≤ n ≤ p ′ − 1 1 ≤ n ′ ≤ p − 1 with conformal dimension ∆ ( n , n ′ ) = ( np − n ′ p ′ ) 2 − ( p − p ′ ) 2 4 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 14 / 33

WA 1 (3 , 2 + r ) theories and parafermions λ Φ ( n | n ′ +1) Φ (1 | 2) × Φ ( n | n ′ ) Φ ( n | n ′ − 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions λ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions λ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions Φ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Ψ × Ψ = I Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions Φ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Ψ × Ψ = I Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions Φ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Ψ × Ψ = I Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions Φ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Ψ × Ψ = I Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

WA 1 (3 , 2 + r ) theories and parafermions Φ Φ (1 | 3) Φ (1 | 2) × Φ (1 | 2) Ψ × Ψ = I Φ (1 | 1) Fermionic field Φ (1 | 2) In the theory WA 1 (3 , 2 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ × Ψ = I and its conformal dimension is ∆ (1 | 2) = r 4 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 2 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 15 / 33

Null vector at level 2 for the field Ψ = Φ (1 | 2) The following field � � 3 r + 2 L 2 χ 2 = L − 2 − Ψ − 1 This degeneracy translates into a PDE for correlators: ∂ 2 � Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = r + 2 � L − 2 Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � 3 Virasoro modes have a geometric interpretation � ˆ � ( L − 2 Φ( z ))Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = D j � Φ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � j where D j are differential operators acting on the j th field: 1 1 ˆ D j = ( z − w j ) 2 ∆ j + ( z − w j ) ∂ w j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 16 / 33

Null vector at level 2 for the field Ψ = Φ (1 | 2) The following field � � 3 r + 2 L 2 χ 2 = L − 2 − Ψ − 1 This degeneracy translates into a PDE for correlators: ∂ 2 � Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = r + 2 � L − 2 Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � 3 Virasoro modes have a geometric interpretation � ˆ � ( L − 2 Φ( z ))Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = D j � Φ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � j where D j are differential operators acting on the j th field: 1 1 ˆ D j = ( z − w j ) 2 ∆ j + ( z − w j ) ∂ w j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 16 / 33

Null vector at level 2 for the field Ψ = Φ (1 | 2) The following field � � 3 r + 2 L 2 χ 2 = L − 2 − Ψ − 1 This degeneracy translates into a PDE for correlators: ∂ 2 � Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = r + 2 � L − 2 Ψ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � 3 Virasoro modes have a geometric interpretation � ˆ � ( L − 2 Φ( z ))Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � = D j � Φ( z )Φ 1 ( w 1 )Φ 2 ( w 2 ) · · · � j where D j are differential operators acting on the j th field: 1 1 ˆ D j = ( z − w j ) 2 ∆ j + ( z − w j ) ∂ w j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 16 / 33

WA 1 (3 , 2 + r ) theories and PDE Null vector at level 2 N N i � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = r + 2 � � i L ( i ) z 2 i ∂ 2 z 2 − 2 � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � 3 i =1 i =1 translates into the following PDE : H WA 1 ( r ) � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = 0 H WA 1 is a differential operator of order 2: z 2 z i z j ( ∂ j − ∂ i ) � � � ( z i ∂ i ) 2 + γ 1 ( r ) j ( z j − z i ) 2 + γ 2 ( r ) + N γ 3 ( r ) ( z j − z i ) i i � = j i � = j γ 1 = − r ( r + 2) γ 2 = r + 2 γ 3 = − r ( r − 1) , et 12 6 12 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 17 / 33

WA 1 (3 , 2 + r ) theories and PDE Null vector at level 2 N N i � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = r + 2 � � i L ( i ) z 2 i ∂ 2 z 2 − 2 � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � 3 i =1 i =1 translates into the following PDE : H WA 1 ( r ) � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = 0 H WA 1 is a differential operator of order 2: z 2 z i z j ( ∂ j − ∂ i ) � � � ( z i ∂ i ) 2 + γ 1 ( r ) j ( z j − z i ) 2 + γ 2 ( r ) + N γ 3 ( r ) ( z j − z i ) i i � = j i � = j γ 1 = − r ( r + 2) γ 2 = r + 2 γ 3 = − r ( r − 1) , et 12 6 12 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 17 / 33

WA 1 (3 , 2 + r ) theories and PDE Null vector at level 2 N N i � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = r + 2 � � i L ( i ) z 2 i ∂ 2 z 2 − 2 � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � 3 i =1 i =1 translates into the following PDE : H WA 1 ( r ) � Ψ( z 1 )Ψ( z 2 ) · · · Ψ( z N ) � = 0 H WA 1 is a differential operator of order 2: z 2 z i z j ( ∂ j − ∂ i ) � � � ( z i ∂ i ) 2 + γ 1 ( r ) j ( z j − z i ) 2 + γ 2 ( r ) + N γ 3 ( r ) ( z j − z i ) i i � = j i � = j γ 1 = − r ( r + 2) γ 2 = r + 2 γ 3 = − r ( r − 1) , et 12 6 12 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 17 / 33

WA 1 (3 , 2 + r ) theories and Jacks [Cardy (2004)] Jack polynomial By restauring the charge part, we consider the following polynomial wavefunction : � ( z i − z j ) r / 2 . P N ˆ = � Ψ( z 1 ) . . . Ψ( z N ) � i < j ⇒ It is an eigenvalue of the Calogero-Sutherland Hamiltonian for α = − 2+1 r − 1 , corresponding to the densest (2 , r ) admissible partition ! This proves the following relation : � ( z i − z j ) r / 2 . = J − 3 / ( r − 1) � Ψ( z 1 ) . . . Ψ( z N ) � [20 r − 1 20 r − 1 ··· 2] i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 18 / 33

WA 1 (3 , 2 + r ) theories and Jacks [Cardy (2004)] Jack polynomial By restauring the charge part, we consider the following polynomial wavefunction : � ( z i − z j ) r / 2 . P N ˆ = � Ψ( z 1 ) . . . Ψ( z N ) � i < j ⇒ It is an eigenvalue of the Calogero-Sutherland Hamiltonian for α = − 2+1 r − 1 , corresponding to the densest (2 , r ) admissible partition ! This proves the following relation : � ( z i − z j ) r / 2 . = J − 3 / ( r − 1) � Ψ( z 1 ) . . . Ψ( z N ) � [20 r − 1 20 r − 1 ··· 2] i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 18 / 33

WA 1 (3 , 2 + r ) theories and Jacks [Cardy (2004)] Jack polynomial By restauring the charge part, we consider the following polynomial wavefunction : � ( z i − z j ) r / 2 . P N ˆ = � Ψ( z 1 ) . . . Ψ( z N ) � i < j ⇒ It is an eigenvalue of the Calogero-Sutherland Hamiltonian for α = − 2+1 r − 1 , corresponding to the densest (2 , r ) admissible partition ! This proves the following relation : � ( z i − z j ) r / 2 . = J − 3 / ( r − 1) � Ψ( z 1 ) . . . Ψ( z N ) � [20 r − 1 20 r − 1 ··· 2] i < j Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 18 / 33

WA 2 algebra The algebra is generated by two currents T ( z ) and W ( z ) : ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 [ L n , L m ] = [ L n , W m ] = (2 n − m ) W n + m 16 c 360 n ( n 2 − 1)( n 2 − 4) δ n + m , 0 [ W n , W m ] = 22 + 5 c ( n − m )Λ n + m + � ( n + m + 2)( n + m + 3) � − ( n + 2)( m + 2) + ( n − m ) L n + m 15 6 Primary fields Φ ∆ ,ω ∆Φ(0) + ∂ Φ(0) T ( z )Φ ∆ ,ω (0) = + . . . z 2 z ω Φ(0) + W − 1 Φ(0) + W − 2 Φ(0) W ( z )Φ ∆ ,ω (0) = + . . . z 3 z 2 z Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 19 / 33

WA 2 algebra The algebra is generated by two currents T ( z ) and W ( z ) : ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 [ L n , L m ] = [ L n , W m ] = (2 n − m ) W n + m 16 c 360 n ( n 2 − 1)( n 2 − 4) δ n + m , 0 [ W n , W m ] = 22 + 5 c ( n − m )Λ n + m + � ( n + m + 2)( n + m + 3) � − ( n + 2)( m + 2) + ( n − m ) L n + m 15 6 Primary fields Φ ∆ ,ω ∆Φ(0) + ∂ Φ(0) T ( z )Φ ∆ ,ω (0) = + . . . z 2 z ω Φ(0) + W − 1 Φ(0) + W − 2 Φ(0) W ( z )Φ ∆ ,ω (0) = + . . . z 3 z 2 z Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 19 / 33

WA 2 algebra The algebra is generated by two currents T ( z ) and W ( z ) : ( n − m ) L n + m + c 12 n ( n 2 − 1) δ n + m , 0 [ L n , L m ] = [ L n , W m ] = (2 n − m ) W n + m 16 c 360 n ( n 2 − 1)( n 2 − 4) δ n + m , 0 [ W n , W m ] = 22 + 5 c ( n − m )Λ n + m + � ( n + m + 2)( n + m + 3) � − ( n + 2)( m + 2) + ( n − m ) L n + m 15 6 Primary fields Φ ∆ ,ω ∆Φ(0) + ∂ Φ(0) T ( z )Φ ∆ ,ω (0) = + . . . z 2 z ω Φ(0) + W − 1 Φ(0) + W − 2 Φ(0) W ( z )Φ ∆ ,ω (0) = + . . . z 3 z 2 z Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 19 / 33

WA 2 ( p , p ′ ) minimal models central charge � � 1 − 12( p − p ′ ) 2 c = 2 pp ′ finite number of primary fields Φ ( n 1 , n 2 | n ′ 2 ) labeled by the Kac table : 1 , n ′ n 1 + n 2 ≤ p ′ − 1 n ′ 1 + n ′ 2 ≤ p − 1 with conformal dimension n ′ p ′ ) 2 − � ρ 2 ( p − p ′ ) 2 2 ) = ( � np − � ∆ ( n 1 , n 2 | n ′ 1 , n ′ 2 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 20 / 33

WA 2 ( p , p ′ ) minimal models central charge � � 1 − 12( p − p ′ ) 2 c = 2 pp ′ finite number of primary fields Φ ( n 1 , n 2 | n ′ 2 ) labeled by the Kac table : 1 , n ′ n 1 + n 2 ≤ p ′ − 1 n ′ 1 + n ′ 2 ≤ p − 1 with conformal dimension n ′ p ′ ) 2 − � ρ 2 ( p − p ′ ) 2 2 ) = ( � np − � ∆ ( n 1 , n 2 | n ′ 1 , n ′ 2 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 20 / 33

WA 2 ( p , p ′ ) minimal models central charge � � 1 − 12( p − p ′ ) 2 c = 2 pp ′ finite number of primary fields Φ ( n 1 , n 2 | n ′ 2 ) labeled by the Kac table : 1 , n ′ n 1 + n 2 ≤ p ′ − 1 n ′ 1 + n ′ 2 ≤ p − 1 with conformal dimension n ′ p ′ ) 2 − � ρ 2 ( p − p ′ ) 2 2 ) = ( � np − � ∆ ( n 1 , n 2 | n ′ 1 , n ′ 2 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 20 / 33

WA 2 ( p , p ′ ) minimal models central charge � � 1 − 12( p − p ′ ) 2 c = 2 pp ′ finite number of primary fields Φ ( n 1 , n 2 | n ′ 2 ) labeled by the Kac table : 1 , n ′ n 1 + n 2 ≤ p ′ − 1 n ′ 1 + n ′ 2 ≤ p − 1 with conformal dimension n ′ p ′ ) 2 − � ρ 2 ( p − p ′ ) 2 2 ) = ( � np − � ∆ ( n 1 , n 2 | n ′ 1 , n ′ 2 pp ′ Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 20 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 λ Φ ( n 1 ,n 2 | n ′ 1 +1 ,n ′ 2 ) Φ (11 | 21) × Φ ( n 1 ,n 2 | n ′ Φ ( n 1 ,n 2 | n ′ 1 ,n ′ 1 − 1 ,n ′ 2 ) 2 +1) Φ ( n 1 ,n 2 | n ′ 1 ,n ′ 2 − 1) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 21 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Φ (1 , 1 | 3 , 1) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 0) Φ (1 , 1 | 2 , 2) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 0 , 3) Φ (1 , 1 | 1 , 1) Φ (1 , 1 | 1 , 3) Φ (1 , 1 | 1 , 2) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 0 , 2) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 21 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Φ (1 , 1 | 3 , 1) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 0) Φ (1 , 1 | 2 , 2) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 0 , 3) Φ (1 , 1 | 1 , 1) Φ (1 , 1 | 1 , 3) Φ (1 , 1 | 1 , 2) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 0 , 2) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 21 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Φ (1 , 1 | 3 , 1) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 0) Φ (1 , 1 | 2 , 2) Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 0 , 3) Φ (1 , 1 | 1 , 1) Φ (1 , 1 | 1 , 3) Φ (1 , 1 | 1 , 2) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 0 , 2) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 21 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Φ (1 , 1 | 3 , 1) Ψ × Ψ = Ψ † Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 0) Φ (1 , 1 | 2 , 2) Ψ × Ψ † = I Φ (1 , 1 | 2 , 1) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 0 , 3) Φ (1 , 1 | 1 , 1) Φ (1 , 1 | 1 , 3) Ψ † × Ψ † = Ψ Φ (1 , 1 | 1 , 2) × Φ (1 , 1 | 1 , 2) Φ (1 , 1 | 2 , 1) Φ (1 , 1 | 0 , 2) Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 21 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Parafermionic fields Ψ = Φ (1 , 1 | 2 , 1) and Ψ † = Φ (1 , 1 | 1 , 2) In the theory WA 2 (4 , 3 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ † Ψ × Ψ = Ψ × Ψ † = I Ψ † × Ψ † = Ψ and their conformal dimension is ∆ (1 , 1 | 2 , 1) = ∆ (1 , 1 | 1 , 2) = r 3 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 3 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 22 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Parafermionic fields Ψ = Φ (1 , 1 | 2 , 1) and Ψ † = Φ (1 , 1 | 1 , 2) In the theory WA 2 (4 , 3 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ † Ψ × Ψ = Ψ × Ψ † = I Ψ † × Ψ † = Ψ and their conformal dimension is ∆ (1 , 1 | 2 , 1) = ∆ (1 , 1 | 1 , 2) = r 3 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 3 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 22 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Parafermionic fields Ψ = Φ (1 , 1 | 2 , 1) and Ψ † = Φ (1 , 1 | 1 , 2) In the theory WA 2 (4 , 3 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ † Ψ × Ψ = Ψ × Ψ † = I Ψ † × Ψ † = Ψ and their conformal dimension is ∆ (1 , 1 | 2 , 1) = ∆ (1 , 1 | 1 , 2) = r 3 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 3 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 22 / 33

WA 2 (4 , 3 + r ) CFT : parafermionic Z ( r ) theories 3 Parafermionic fields Ψ = Φ (1 , 1 | 2 , 1) and Ψ † = Φ (1 , 1 | 1 , 2) In the theory WA 2 (4 , 3 + r ) the field Ψ = Φ (1 | 2) obey the fusion rules : Ψ † Ψ × Ψ = Ψ × Ψ † = I Ψ † × Ψ † = Ψ and their conformal dimension is ∆ (1 , 1 | 2 , 1) = ∆ (1 , 1 | 1 , 2) = r 3 ⇒ This is a particular realization of a Z ( r ) parafermionic field theory 3 Benoit Estienne (LPTHE) Jack wavefunctions and W theories 08/20/2009 22 / 33