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Dr. Pablo Diaz Benito (University of the Witwatersrand) TITLE: "Novel Charges in CFT's"

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Rene Kotze

The document describes novel charges Q`n and Q``n in conformal field theories (CFTs). It introduces notation and discusses restricted Schur polynomials and fR functions. Charges Q`n emerge from an infinite embedding chain of Lie algebras g(1) → g(2) → ... → g = u, so, sp. Charges Q``n are also defined. The eigenvalues of the charges involve fR functions.

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Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel Charges in CFT's P. Diaz, arXiv:1406.7671 Pablo Daz University of the Witwatersrand September 9, 2014 P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I and forcing hQ[O] O 0i = hOQ O 0i: P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I and forcing hQ[O] O 0i = hOQ O 0i: I The eigenvectors of the charges are restricted Schurs P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Table of contents Preliminaries Notation Restricted Schur polynomials Charges Q`n Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Charges Q``n Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Conclusion and future works P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
elds in a given operator. N or M refer to the rank of the gauge group. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
ned conformal dimension ON( ) = X 2Sn a()TrN( ) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
ned conformal dimension ON( ) = X 2Sn a()TrN( ) I = n1 1 n2 2 nr r $ = (n1; : : : nr ) ` n P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
ned conformal dimension ON( ) = X 2Sn a()TrN( ) I = n1 1 n2 2 nr r $ = (n1; : : : nr ) ` n I TrN( ) = I (I ); 2 Sn; I = i1 in; ir = 1; : : : ;N P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! I R ` n will be resolved by fQ`n NMg P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! I R ` n will be resolved by fQ`n NMg I is a collection of partitions. If = (n1; : : : ; nr ) then = (s1 ` n1; : : : ; sr ` nr ). Or, in other words, is an irrep of S Sn. Label will be resolved by fQ``n NM g P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R I where f U(N) R = Y (i ;j)2R (N + j i) f SO(N) R = Y (i ;j)2R (N + 2j i 1) f Sp(N) R = Y (i ;j)2R (N + j 2i + 1) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R I where f U(N) R = Y (i ;j)2R (N + j i) f SO(N) R = Y (i ;j)2R (N + 2j i 1) f Sp(N) R = Y (i ;j)2R (N + j 2i + 1) I They also appear in the eigenvalues of the charges P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras I Our charges emerge naturally from the in
nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras I Our charges emerge naturally from the in
nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I Many ways of performing the embedding but it doesn't matter which one we choose P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) I Symplectic algebras: i J + Jti = 0, J = 0N=2 IN=2 IN=2 0N=2 i = A B C D 2 sp(N) ! i = 0 B@ A B C D 1 CA 2 sp(M) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) I Symplectic algebras: i J + Jti = 0, J = 0N=2 IN=2 IN=2 0N=2 i = A B C D 2 sp(N) ! i = 0 B@ A B C D 1 CA 2 sp(M) I Embedding EMN is an injective map: vN 1 ; vN 2 ; : : : ; vN N2 # # # vM 1 ; vM 2 ; : : : ; vM N2 ; vM N2+1; : : : ; vM M2 P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: I ProjNM are surjective maps. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: I ProjNM are surjective maps. vM 1 ; vM 2 ; : : : ; vM N2 ; vM N2+1; : : : ; vM M2 # # # # # vN 1 ; vN 2 ; : : : ; vN N2 ; 0; : : : ; 0 I They have the obvious property: ProjNM EMN = IdN P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) I So, ProjNM is a map between GI operators built on i 2 g(M) and GI operators built on i 2 g(N) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM I In other words hAvMN[ON] O 0M i = hONProjNM O 0M i P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM I In other words hAvMN[ON] O 0M i = hONProjNM O 0M i Can we
nd an explicit expression for AvMN? P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action I Averaging operators are given by AvMNON = Z g2G(M) dg Adg (ON); G = U; SO or Sp P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action I Averaging operators are given by AvMNON = Z g2G(M) dg Adg (ON); G = U; SO or Sp I The adjoint action applies to every i in the GI operator AdgTrN(ZY ) = TrN(gZg1gYg1) = gi1j1Zj1j2g1 j2i2 gi3j3Yj3j4g1 j4i4 i2i3i4i1 ; where i = 1; : : : ;N j = 1; : : : ;M: Z; Y are embedded P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
2Sn ()II 0 (
)J0 J WgU(M)(
): P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
2Sn ()II 0 (
)J0 J WgU(M)(
): I where WgU(N)() = 1 n! X R`n l(R)N dR f U(N) R R(); 2 Sn: [Collins '03] P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
2Sn ()II 0 (
)J0 J WgU(M)(
): I where WgU(N)() = 1 n! X R`n l(R)N dR f U(N) R R(); 2 Sn: [Collins '03] I Similar results are found for integrals over the orthogonal and the symplectic groups. [Collins, Matsumoto '09] [Matsumoto '13] P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: I They are self-adjoint by construction for all M N P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: I They are self-adjoint by construction for all M N What are their eigenvalues and eigenvectors? P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; I And their eigenvalues are all dierent for dierent R's, P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; I And their eigenvalues are all dierent for dierent R's, so I Restricted Schur polynomials must be orthogonal on the label R P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i I to obtain 1 f G(M) R hG(M) R;;m G(M) R;;m0i = 1 f G(N) R hG(N) R;;m G(N) R;;m0i; 8M N P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i I to obtain 1 f G(M) R hG(M) R;;m G(M) R;;m0i = 1 f G(N) R hG(N) R;;m G(N) R;;m0i; 8M N I So hG(N) R;;m G(N) S;;m0i = c(R; ; ;m;m0)f G(N) R RS P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. I We construct a self-adjoint set of charges Q`n NM ProjNM AvMN : VN ! VN: P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. I We construct a self-adjoint set of charges Q`n NM ProjNM AvMN : VN ! VN: I It turns out that their eigenvectors are restricted Schur polynomials and that we can partially reconstruct the free
eld two-point function. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) I But AvMN changes its Image accordingly Av(E) MNG(N) R;;m( ) = f G(N) R f G(M) R G(M) R;;m( (E)); 8M N P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) I But AvMN changes its Image accordingly Av(E) MNG(N) R;;m( ) = f G(N) R f G(M) R G(M) R;;m( (E)); 8M N I So NM Av(E) MN = Proj(E0) NM Av(E0) MN Q`n Proj(E) NM P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) Dim(R;N)Dim(R;N;S;M) Dim(S;M) P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) Dim(R;N)Dim(R;N;S;M) Dim(S;M) P R = 1 P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
nal irrep (R;N). P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. I It turns out that Dim(R;N; R;M) = 1, for all R;N;M. P. Diaz Novel Charges in CFT's
Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. I It turns out that Dim(R;N; R;M) = 1, for all R;N;M. I Our eigenvalues are f U(N) R f U(M) R = Dim(R;N) Dim(R;M) P. Diaz Novel Charges in CFT's

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احياء السادس العلمي - الفصل الثالث (التكاثر) منهج متميزين/كلية بغداد/موهوبين
S LS
 

Dr. Pablo Diaz Benito (University of the Witwatersrand) TITLE: "Novel Charges in CFT's"

  • 1. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel Charges in CFT's P. Diaz, arXiv:1406.7671 Pablo Daz University of the Witwatersrand September 9, 2014 P. Diaz Novel Charges in CFT's
  • 2. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges P. Diaz Novel Charges in CFT's
  • 3. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng P. Diaz Novel Charges in CFT's
  • 4. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
  • 5. nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp P. Diaz Novel Charges in CFT's
  • 6. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
  • 7. nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I and forcing hQ[O] O 0i = hOQ O 0i: P. Diaz Novel Charges in CFT's
  • 8. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Novel charges I We obtain fQ`n NM;M Ng and fQ``n NM ;M Ng I from the in
  • 9. nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I and forcing hQ[O] O 0i = hOQ O 0i: I The eigenvectors of the charges are restricted Schurs P. Diaz Novel Charges in CFT's
  • 10. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Table of contents Preliminaries Notation Restricted Schur polynomials Charges Q`n Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Charges Q``n Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Conclusion and future works P. Diaz Novel Charges in CFT's
  • 11. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
  • 12. nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng P. Diaz Novel Charges in CFT's
  • 13. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
  • 14. nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
  • 15. elds in a given operator. N or M refer to the rank of the gauge group. P. Diaz Novel Charges in CFT's
  • 16. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
  • 17. nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
  • 18. elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
  • 19. ned conformal dimension ON( ) = X 2Sn a()TrN( ) P. Diaz Novel Charges in CFT's
  • 20. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
  • 21. nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
  • 22. elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
  • 23. ned conformal dimension ON( ) = X 2Sn a()TrN( ) I = n1 1 n2 2 nr r $ = (n1; : : : nr ) ` n P. Diaz Novel Charges in CFT's
  • 24. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Notation I Two in
  • 25. nite sets of charges fQ`n NM;M Ng, fQ``n NM ;M Ng I n: Total number of
  • 26. elds in a given operator. N or M refer to the rank of the gauge group. I Generic operators with well-de
  • 27. ned conformal dimension ON( ) = X 2Sn a()TrN( ) I = n1 1 n2 2 nr r $ = (n1; : : : nr ) ` n I TrN( ) = I (I ); 2 Sn; I = i1 in; ir = 1; : : : ;N P. Diaz Novel Charges in CFT's
  • 28. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials P. Diaz Novel Charges in CFT's
  • 29. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function P. Diaz Novel Charges in CFT's
  • 30. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) P. Diaz Novel Charges in CFT's
  • 31. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! P. Diaz Novel Charges in CFT's
  • 32. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! I R ` n will be resolved by fQ`n NMg P. Diaz Novel Charges in CFT's
  • 33. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials Restricted Schur polynomials I Restricted Schur polynomials form a basis of GI operators and diagonalize the free two-point function G(N) R;;m( ) = 1 jSj X 2Sn GR ;;m()TrG(N)( ) I S = Sn1 Sn2 Snr , so jSj = n1!n2! nr ! I R ` n will be resolved by fQ`n NMg I is a collection of partitions. If = (n1; : : : ; nr ) then = (s1 ` n1; : : : ; sr ` nr ). Or, in other words, is an irrep of S Sn. Label will be resolved by fQ``n NM g P. Diaz Novel Charges in CFT's
  • 34. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions P. Diaz Novel Charges in CFT's
  • 35. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n. P. Diaz Novel Charges in CFT's
  • 36. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R P. Diaz Novel Charges in CFT's
  • 37. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R I where f U(N) R = Y (i ;j)2R (N + j i) f SO(N) R = Y (i ;j)2R (N + 2j i 1) f Sp(N) R = Y (i ;j)2R (N + j 2i + 1) P. Diaz Novel Charges in CFT's
  • 38. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Notation Restricted Schur polynomials fR functions I f G(N) R are polynomials of N of degree n.They appear in the correlators of Shurs as hG(N) R;;m( )G(N) S;;m0( )i / RSmm0f G(N) R I where f U(N) R = Y (i ;j)2R (N + j i) f SO(N) R = Y (i ;j)2R (N + 2j i 1) f Sp(N) R = Y (i ;j)2R (N + j 2i + 1) I They also appear in the eigenvalues of the charges P. Diaz Novel Charges in CFT's
  • 39. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras P. Diaz Novel Charges in CFT's
  • 40. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras I Our charges emerge naturally from the in
  • 41. nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp P. Diaz Novel Charges in CFT's
  • 42. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embedding chain of Lie algebras I Our charges emerge naturally from the in
  • 43. nite embedding chain: g(1) ,! g(2) ,! g = u; so; sp I Many ways of performing the embedding but it doesn't matter which one we choose P. Diaz Novel Charges in CFT's
  • 44. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings P. Diaz Novel Charges in CFT's
  • 45. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) P. Diaz Novel Charges in CFT's
  • 46. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) I Symplectic algebras: i J + Jti = 0, J = 0N=2 IN=2 IN=2 0N=2 i = A B C D 2 sp(N) ! i = 0 B@ A B C D 1 CA 2 sp(M) P. Diaz Novel Charges in CFT's
  • 47. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Embeddings I Orthogonal and unitary algebras i 2 u(N) or so(N) ! i = 0 B@ i 0MN 1 CA 2 u(M) or so(M) I Symplectic algebras: i J + Jti = 0, J = 0N=2 IN=2 IN=2 0N=2 i = A B C D 2 sp(N) ! i = 0 B@ A B C D 1 CA 2 sp(M) I Embedding EMN is an injective map: vN 1 ; vN 2 ; : : : ; vN N2 # # # vM 1 ; vM 2 ; : : : ; vM N2 ; vM N2+1; : : : ; vM M2 P. Diaz Novel Charges in CFT's
  • 48. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra P. Diaz Novel Charges in CFT's
  • 49. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
  • 50. ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: P. Diaz Novel Charges in CFT's
  • 51. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
  • 52. ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: I ProjNM are surjective maps. P. Diaz Novel Charges in CFT's
  • 53. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on the algebra I We also de
  • 54. ne a set of projectors adapted to the chain ProjNM : g(M) ! g(N); M N: I ProjNM are surjective maps. vM 1 ; vM 2 ; : : : ; vM N2 ; vM N2+1; : : : ; vM M2 # # # # # vN 1 ; vN 2 ; : : : ; vN N2 ; 0; : : : ; 0 I They have the obvious property: ProjNM EMN = IdN P. Diaz Novel Charges in CFT's
  • 55. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI P. Diaz Novel Charges in CFT's
  • 56. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) P. Diaz Novel Charges in CFT's
  • 57. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
  • 58. elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 P. Diaz Novel Charges in CFT's
  • 59. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
  • 60. elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM P. Diaz Novel Charges in CFT's
  • 61. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
  • 62. elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) P. Diaz Novel Charges in CFT's
  • 63. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
  • 64. elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) P. Diaz Novel Charges in CFT's
  • 65. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Projection operators on GI I We make ProjNM act on composite operators by simply acting as before on every slot of in OM( ) I If any of the
  • 66. elds i ;M does not come from i ;N via the embedding then ProjNMOM( ) = 0 I To be consequent we should write Proj(E) NM (but we won't) I So, ProjNM is a map between GI operators built on i 2 g(M) and GI operators built on i 2 g(N) P. Diaz Novel Charges in CFT's
  • 67. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators P. Diaz Novel Charges in CFT's
  • 68. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN P. Diaz Novel Charges in CFT's
  • 69. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? P. Diaz Novel Charges in CFT's
  • 70. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM P. Diaz Novel Charges in CFT's
  • 71. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM I In other words hAvMN[ON] O 0M i = hONProjNM O 0M i P. Diaz Novel Charges in CFT's
  • 72. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators I Operator$ states I ON as vectors belonging to the vector space VN. I hON O 0N i inner product in VN. I ProjNM : VM ! VN I What is the adjoint of ProjNM? AvMN Proj NM; AvMN : VN ! VM I In other words hAvMN[ON] O 0M i = hONProjNM O 0M i Can we
  • 73. nd an explicit expression for AvMN? P. Diaz Novel Charges in CFT's
  • 74. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action P. Diaz Novel Charges in CFT's
  • 75. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action I Averaging operators are given by AvMNON = Z g2G(M) dg Adg (ON); G = U; SO or Sp P. Diaz Novel Charges in CFT's
  • 76. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging operators and adjoint action I Averaging operators are given by AvMNON = Z g2G(M) dg Adg (ON); G = U; SO or Sp I The adjoint action applies to every i in the GI operator AdgTrN(ZY ) = TrN(gZg1gYg1) = gi1j1Zj1j2g1 j2i2 gi3j3Yj3j4g1 j4i4 i2i3i4i1 ; where i = 1; : : : ;N j = 1; : : : ;M: Z; Y are embedded P. Diaz Novel Charges in CFT's
  • 77. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) P. Diaz Novel Charges in CFT's
  • 78. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
  • 81. ): P. Diaz Novel Charges in CFT's
  • 82. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
  • 85. ): I where WgU(N)() = 1 n! X R`n l(R)N dR f U(N) R R(); 2 Sn: [Collins '03] P. Diaz Novel Charges in CFT's
  • 86. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Weingarten calculus: integrals over G(M) I For U(N) we have (g1 = g+): Z g2U(M) dg gi1 j1 gin jn (g+)j 0 1 i 0 1 (g+)j 0 n i 0 n = X ;
  • 89. ): I where WgU(N)() = 1 n! X R`n l(R)N dR f U(N) R R(); 2 Sn: [Collins '03] I Similar results are found for integrals over the orthogonal and the symplectic groups. [Collins, Matsumoto '09] [Matsumoto '13] P. Diaz Novel Charges in CFT's
  • 90. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition P. Diaz Novel Charges in CFT's
  • 91. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: P. Diaz Novel Charges in CFT's
  • 92. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: I They are self-adjoint by construction for all M N P. Diaz Novel Charges in CFT's
  • 93. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: composition I Consider the set of operators Q`n NM ProjNM AvMN : VN ! VN: I They are self-adjoint by construction for all M N What are their eigenvalues and eigenvectors? P. Diaz Novel Charges in CFT's
  • 94. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs P. Diaz Novel Charges in CFT's
  • 95. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; P. Diaz Novel Charges in CFT's
  • 96. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; I And their eigenvalues are all dierent for dierent R's, P. Diaz Novel Charges in CFT's
  • 97. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Averaging and projection: orthogonality of Schurs I Their eigenvectors are restricted Schur polynomials Q`n NMG(N) R;;m( ) = f G(N) R f G(M) R G(N) R;;m( ); 8M N; I And their eigenvalues are all dierent for dierent R's, so I Restricted Schur polynomials must be orthogonal on the label R P. Diaz Novel Charges in CFT's
  • 98. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function P. Diaz Novel Charges in CFT's
  • 99. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N P. Diaz Novel Charges in CFT's
  • 100. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i P. Diaz Novel Charges in CFT's
  • 101. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i I to obtain 1 f G(M) R hG(M) R;;m G(M) R;;m0i = 1 f G(N) R hG(N) R;;m G(N) R;;m0i; 8M N P. Diaz Novel Charges in CFT's
  • 102. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Recovering the two-point function I Insert AvMNG(N) R;;m = f G(N) R f G(M) R G(M) R;;m; 8M N I into hAvMN[G(N) R;;m]G(M) R;;m0i = hG(N) R;;mProjNM G(M) R;;m0i I to obtain 1 f G(M) R hG(M) R;;m G(M) R;;m0i = 1 f G(N) R hG(N) R;;m G(N) R;;m0i; 8M N I So hG(N) R;;m G(N) S;;m0i = c(R; ; ;m;m0)f G(N) R RS P. Diaz Novel Charges in CFT's
  • 103. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology P. Diaz Novel Charges in CFT's
  • 104. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! P. Diaz Novel Charges in CFT's
  • 105. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. P. Diaz Novel Charges in CFT's
  • 106. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. P. Diaz Novel Charges in CFT's
  • 107. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. P. Diaz Novel Charges in CFT's
  • 108. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. I We construct a self-adjoint set of charges Q`n NM ProjNM AvMN : VN ! VN: P. Diaz Novel Charges in CFT's
  • 109. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Summary of the methodology I The starting point is g(1) ,! g(2) ,! I To implement this structure we create the set ProjNM; M N, which are maps between GI operators. I We look for the adjoint operators of this set with respect to the free two-point function. I We found an explicit form of AvMN = Proj NM in terms of integrals over the groups. I We construct a self-adjoint set of charges Q`n NM ProjNM AvMN : VN ! VN: I It turns out that their eigenvectors are restricted Schur polynomials and that we can partially reconstruct the free
  • 110. eld two-point function. P. Diaz Novel Charges in CFT's
  • 111. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings P. Diaz Novel Charges in CFT's
  • 112. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) P. Diaz Novel Charges in CFT's
  • 113. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) P. Diaz Novel Charges in CFT's
  • 114. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) I But AvMN changes its Image accordingly Av(E) MNG(N) R;;m( ) = f G(N) R f G(M) R G(M) R;;m( (E)); 8M N P. Diaz Novel Charges in CFT's
  • 115. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Dierent embeddings I Dierent embeddings (E)6= (E0) I Dierent embeddings changes the support of projectors NM6= Proj(E0) NM ; Proj(E) NMO( (E0)) = 0 Proj(E) I But AvMN changes its Image accordingly Av(E) MNG(N) R;;m( ) = f G(N) R f G(M) R G(M) R;;m( (E)); 8M N I So NM Av(E) MN = Proj(E0) NM Av(E0) MN Q`n Proj(E) NM P. Diaz Novel Charges in CFT's
  • 116. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I P. Diaz Novel Charges in CFT's
  • 117. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 P. Diaz Novel Charges in CFT's
  • 118. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ P. Diaz Novel Charges in CFT's
  • 119. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ P. Diaz Novel Charges in CFT's
  • 120. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) P. Diaz Novel Charges in CFT's
  • 121. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) Dim(R;N)Dim(R;N;S;M) Dim(S;M) P. Diaz Novel Charges in CFT's
  • 122. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues I N = 3 N = 2 PPPPPPP @ @ @ N = 1 ; @ @ @ J J JJ Dim( ;2) Dim( ;3) Dim(R;N)Dim(R;N;S;M) Dim(S;M) P R = 1 P. Diaz Novel Charges in CFT's
  • 123. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II P. Diaz Novel Charges in CFT's
  • 124. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
  • 125. nal irrep (R;N). P. Diaz Novel Charges in CFT's
  • 126. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
  • 127. nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. P. Diaz Novel Charges in CFT's
  • 128. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
  • 129. nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. I It turns out that Dim(R;N; R;M) = 1, for all R;N;M. P. Diaz Novel Charges in CFT's
  • 130. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
  • 131. nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. I It turns out that Dim(R;N; R;M) = 1, for all R;N;M. I Our eigenvalues are f U(N) R f U(M) R = Dim(R;N) Dim(R;M) P. Diaz Novel Charges in CFT's
  • 132. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Embedding chain Projection and averaging operators: Q`n Recovering the two-point function Eigenvalues as probabilities Probabilistic interpretation of the eigenvalues II I We can interpret Dim(R;N)Dim(R;N;S;M) Dim(S;M) as a natural probability associated to a Markov process, level by level, with initial irrep (S;M) and
  • 133. nal irrep (R;N). I Each step down from (T;M0) and (U;M0 1) has probability Dim(U;M01) Dim(T;M0) if T and U are linked and 0 otherwise. I It turns out that Dim(R;N; R;M) = 1, for all R;N;M. I Our eigenvalues are f U(N) R f U(M) R = Dim(R;N) Dim(R;M) I So, the eigenvalues are the probabilities of starting from irrep (R;M) and go down to (R;N) by means of a Markov process. P. Diaz Novel Charges in CFT's
  • 134. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n: Motivation P. Diaz Novel Charges in CFT's
  • 135. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n: Motivation I Charges Q`n NM resolve the label R ` n of R;;m which is related to the total number of
  • 136. elds n. P. Diaz Novel Charges in CFT's
  • 137. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n: Motivation I Charges Q`n NM resolve the label R ` n of R;;m which is related to the total number of
  • 138. elds n. I Operators built on have abundancies = (n1; : : : ; nr ) of
  • 139. elds 1; : : : ; r . P. Diaz Novel Charges in CFT's
  • 140. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n: Motivation I Charges Q`n NM resolve the label R ` n of R;;m which is related to the total number of
  • 141. elds n. I Operators built on have abundancies = (n1; : : : ; nr ) of
  • 142. elds 1; : : : ; r . I Charges Q`n NM, by construction, do not care about . P. Diaz Novel Charges in CFT's
  • 143. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n: Motivation I Charges Q`n NM resolve the label R ` n of R;;m which is related to the total number of
  • 144. elds n. I Operators built on have abundancies = (n1; : : : ; nr ) of
  • 145. elds 1; : : : ; r . I Charges Q`n NM, by construction, do not care about . I There must be other charges related to that resolve the small labels. P. Diaz Novel Charges in CFT's
  • 146. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n NM , requirements P. Diaz Novel Charges in CFT's
  • 147. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n NM , requirements I They should be related to the embedding chain and sensitive to , so we impose Q ProjNM Av MN Q``n NM = X `n 6=(n) Q NM P. Diaz Novel Charges in CFT's
  • 148. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n NM , requirements I They should be related to the embedding chain and sensitive to , so we impose Q ProjNM Av MN Q``n NM = X `n 6=(n) Q NM I They reduce to Q`n NM when = (n) P. Diaz Novel Charges in CFT's
  • 149. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Charges Q``n NM , requirements I They should be related to the embedding chain and sensitive to , so we impose Q ProjNM Av MN Q``n NM = X `n 6=(n) Q NM I They reduce to Q`n NM when = (n) I They are self-adjoint with respect to the free two-point function P. Diaz Novel Charges in CFT's
  • 150. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN P. Diaz Novel Charges in CFT's
  • 151. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN I Operators Av MN are de
  • 152. ned as Av MN[O( )] Z g1;:::;gr2G(M) g d Ad[ g][O( )]; 1 g gr where [g] = g n1 r P. Diaz Novel Charges in CFT's
  • 153. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN I Operators Av MN are de
  • 154. ned as Av MN[O( )] Z g1;:::;gr2G(M) g d Ad[ g][O( )]; 1 g gr where [g] = g n1 r I One has to de
  • 155. ne Ad P. Diaz Novel Charges in CFT's
  • 156. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN I Operators Av MN are de
  • 157. ned as Av MN[O( )] Z g1;:::;gr2G(M) g d Ad[ g][O( )]; 1 g gr where [g] = g n1 r I One has to de
  • 158. ne Ad I First attemp, for = (2; 1) we have Adg ;hTrN(ZZY ) = TrN(gZg1gZg1hYh1) P. Diaz Novel Charges in CFT's
  • 159. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN I Operators Av MN are de
  • 160. ned as Av MN[O( )] Z g1;:::;gr2G(M) g d Ad[ g][O( )]; 1 g gr where [g] = g n1 r I One has to de
  • 161. ne Ad I First attemp, for = (2; 1) we have Adg ;hTrN(ZZY ) = TrN(gZg1gZg1hYh1) I But then Q are not self-adjoint... P. Diaz Novel Charges in CFT's
  • 162. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Av MN I Operators Av MN are de
  • 163. ned as Av MN[O( )] Z g1;:::;gr2G(M) g d Ad[ g][O( )]; 1 g gr where [g] = g n1 r I One has to de
  • 164. ne Ad I First attemp, for = (2; 1) we have Adg ;hTrN(ZZY ) = TrN(gZg1gZg1hYh1) I But then Q are not self-adjoint... Can we cure it? P. Diaz Novel Charges in CFT's
  • 165. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM -adjoint action P. Diaz Novel Charges in CFT's
  • 166. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM -adjoint action I We can, if we shue the slots of left (or right) action of the group elements P. Diaz Novel Charges in CFT's
  • 167. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM -adjoint action I We can, if we shue the slots of left (or right) action of the group elements I In the former example Adg ;hTrN(ZZY ) = 1 3! TrN(gZg1gZg1hYh1) + TrN(gZg1hZg1gYh1) + : : : P. Diaz Novel Charges in CFT's
  • 168. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM -adjoint action I We can, if we shue the slots of left (or right) action of the group elements I In the former example Adg ;hTrN(ZZY ) = 1 3! TrN(gZg1gZg1hYh1) + TrN(gZg1hZg1gYh1) + : : : I For unitary groups the -adjoint action over multitraces is Ad[ g][TrU(N)( )] = 1 jSnj X 2Sn g I (J) J J0 g I 0 J0(1)I 0 I : P. Diaz Novel Charges in CFT's
  • 169. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM P. Diaz Novel Charges in CFT's
  • 170. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM I For unitary groups we
  • 171. nd the equation Q NM(U(N) R;;ij( )) = jSj jSnj f U(N) R f U(M) U(N) R;;ij( ); f = fs1 fs2 fsr P. Diaz Novel Charges in CFT's
  • 172. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM I For unitary groups we
  • 173. nd the equation Q NM(U(N) R;;ij( )) = jSj jSnj f U(N) R f U(M) U(N) R;;ij( ); f = fs1 fs2 fsr I If 0 and 06= then Q NM(U(N) R;;ij( 0)) = 0 P. Diaz Novel Charges in CFT's
  • 174. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM I For unitary groups we
  • 175. nd the equation Q NM(U(N) R;;ij( )) = jSj jSnj f U(N) R f U(M) U(N) R;;ij( ); f = fs1 fs2 fsr I If 0 and 06= then Q NM(U(N) R;;ij( 0)) = 0 I So, we de
  • 176. ne Q``n NM = X `n 6=(n) Q NM; M N which acts nontrivially on any GI non 12 -BPS operator P. Diaz Novel Charges in CFT's
  • 177. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM for other groups P. Diaz Novel Charges in CFT's
  • 178. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM for other groups I Let us write the eigenvector equation for all classical gauge groups NM (U(N) Q``n R;;ij( )) = jSj jSnj f U(N) R f U(M) U(N) R;;ij( ) NM (SO(N) Q``n R;;i ( )) = jSj jS[S2]j2jS2nj f SO(N) R f SO(M) SO(N) R;;i ( ) NM (Sp(N) Q``n R;;i ( )) = jSj jS[S2]j2jS2nj f Sp(N) R f Sp(M) Sp(N) R;;i ( ); P. Diaz Novel Charges in CFT's
  • 179. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Eigenvalues and eigenvectors of Q``n NM for other groups I Let us write the eigenvector equation for all classical gauge groups NM (U(N) Q``n R;;ij( )) = jSj jSnj f U(N) R f U(M) U(N) R;;ij( ) NM (SO(N) Q``n R;;i ( )) = jSj jS[S2]j2jS2nj f SO(N) R f SO(M) SO(N) R;;i ( ) NM (Sp(N) Q``n R;;i ( )) = jSj jS[S2]j2jS2nj f Sp(N) R f Sp(M) Sp(N) R;;i ( ); I The value of the eigenvalues between 0 and 1 suggests that they should also have a probabilistic interpretation in terms of branching graphs paths P. Diaz Novel Charges in CFT's
  • 180. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Go back to the two-point function P. Diaz Novel Charges in CFT's
  • 181. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Go back to the two-point function I Remember that because of fQ`n NMg we know that hG(N) R;;m; G(N) S;;m0i = c(R; ; ;m;m0)f G(N) R RS; P. Diaz Novel Charges in CFT's
  • 182. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Requirements -adjoint action Eigenvalues and eigenvectors of Q``n NM Go back to the two-point function I Remember that because of fQ`n NMg we know that hG(N) R;;m; G(N) S;;m0i = c(R; ; ;m;m0)f G(N) R RS; I Now, using the same arguments, and because fQ``n NM g commutes with fQ`n NMg, we can narrow the result to hG(N) R;;m; G(N) S;;m0i = c(R; ;m;m0)f G(N) R RS; P. Diaz Novel Charges in CFT's
  • 183. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works P. Diaz Novel Charges in CFT's
  • 184. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 185. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. P. Diaz Novel Charges in CFT's
  • 186. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 187. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works P. Diaz Novel Charges in CFT's
  • 188. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 189. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg P. Diaz Novel Charges in CFT's
  • 190. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 191. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg I Find probabilistic interpretation of eigenvalues of fQ``n NM g P. Diaz Novel Charges in CFT's
  • 192. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 193. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg I Find probabilistic interpretation of eigenvalues of fQ``n NM g I See if we can drop matter in the adjoint P. Diaz Novel Charges in CFT's
  • 194. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 195. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg I Find probabilistic interpretation of eigenvalues of fQ``n NM g I See if we can drop matter in the adjoint I Connection with other charges studied before. Concretely the charges that come from the global U(N) U(N) symmetry of the free theory [Kimura, Ramgoolam]. P. Diaz Novel Charges in CFT's
  • 196. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 197. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg I Find probabilistic interpretation of eigenvalues of fQ``n NM g I See if we can drop matter in the adjoint I Connection with other charges studied before. Concretely the charges that come from the global U(N) U(N) symmetry of the free theory [Kimura, Ramgoolam]. I Exploiting Weingarten calculus. P. Diaz Novel Charges in CFT's
  • 198. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Conclusions and future works I We have constructed two in
  • 199. nite sets of charges which emerge naturally from the embedding chain. Their eigenvectors are restricted Schur polynomials for classical gauge groups. I For future works I Construction of charges fQm NMg I Find probabilistic interpretation of eigenvalues of fQ``n NM g I See if we can drop matter in the adjoint I Connection with other charges studied before. Concretely the charges that come from the global U(N) U(N) symmetry of the free theory [Kimura, Ramgoolam]. I Exploiting Weingarten calculus. I Conjecture: For an interacting correlator it will be possible to construct a set of commuting charges fDNM;M Ng such that DNN = D is the full dilatation operator. P. Diaz Novel Charges in CFT's
  • 200. Preliminaries Charges Q`n Charges Q``n Conclusion and future works Thanks! P. Diaz Novel Charges in CFT's