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

PreliminariesCharges Q`n

Charges Q``n

Conclusion and future works

Novel Charges in CFT’sP. Diaz, arXiv:1406.7671

Pablo Dıaz

University of the Witwatersrand

September 9, 2014

P. Diaz Novel Charges in CFT’s


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Novel charges

I We obtain

{Q`nNM ,M > N} and {Q``nNM ,M ≥ N}

I from the infinite embedding chain:

g(1) ↪→ g(2) ↪→ · · · g = u, so, sp

I and forcing〈Q[O]O′〉 = 〈OQO′〉.

I The eigenvectors of the charges are restricted Schurs



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Novel charges

I We obtain



g(1) ↪→ g(2) ↪→ · · · g = u, so, sp





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Novel charges

I We obtain



g(1) ↪→ g(2) ↪→ · · · g = u, so, sp





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Novel charges

I We obtain



g(1) ↪→ g(2) ↪→ · · · g = u, so, sp





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Novel charges

I We obtain



g(1) ↪→ g(2) ↪→ · · · g = u, so, sp





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Table of contents

PreliminariesNotationRestricted Schur polynomials

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Embedding chainProjection and averaging operators: Q`n

Recovering the two-point functionEigenvalues as probabilities

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Requirementsλ-adjoint actionEigenvalues and eigenvectors of Q``nNM




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NotationRestricted Schur polynomials

Notation

I Two infinite sets of charges {Q`nNM ,M > N}, {Q``nNM ,M ≥ N}

I n: Total number of fields in a given operator. N or M refer tothe rank of the gauge group.

I Generic operators with well-defined conformal dimension

ON(Ψ) =∑σ∈Sn

a(σ)TrN(σΨ)

I Ψ = φ⊗n11 ⊗ φ⊗n22 ⊗ · · · ⊗ φ⊗nrr ↔ λ = (n1, . . . nr ) ` n

I TrN(σΨ) = ΨIσ(I ), σ ∈ Sn, I = i1 · · · in, ir = 1, . . . ,N



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Notation

I Two infinite sets of charges {Q`nNM ,M > N}, {Q``nNM ,M ≥ N}I n: Total number of fields in a given operator. N or M refer to

the rank of the gauge group.


ON(Ψ) =∑σ∈Sn

a(σ)TrN(σΨ)





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Notation




ON(Ψ) =∑σ∈Sn

a(σ)TrN(σΨ)





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Notation




ON(Ψ) =∑σ∈Sn

a(σ)TrN(σΨ)





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Notation




ON(Ψ) =∑σ∈Sn

a(σ)TrN(σΨ)





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Restricted Schur polynomials

I Restricted Schur polynomials form a basis of GI operators anddiagonalize the free two-point function

χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn

χGR,µ,m(σ)TrG(N)(σΨ)

I Sλ = Sn1 × Sn2 × · · · × Snr , so |Sλ| = n1!n2! · · · nr !

I R ` n will be resolved by {Q`nNM}I µ is a collection of partitions. If λ = (n1, . . . , nr ) thenµ = (s1 ` n1, . . . , sr ` nr ). Or, in other words, µ is an irrep ofSλ ⊂ Sn. Label µ will be resolved by {Q``nNM }



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χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn






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χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn






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χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn






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χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn



I R ` n will be resolved by {Q`nNM}

I µ is a collection of partitions. If λ = (n1, . . . , nr ) thenµ = (s1 ` n1, . . . , sr ` nr ). Or, in other words, µ is an irrep ofSλ ⊂ Sn. Label µ will be resolved by {Q``nNM }



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χG(N)R,µ,m(Ψ) =

1

|Sλ|∑σ∈Sn






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fR functions

I fG(N)R are polynomials of N of degree n.They appear in the

correlators of Shurs as

〈χG(N)R,µ,m(Ψ)χ

G(N)S ,ν,m′(Ψ)〉 ∝ δRSδµνδmm′f

G(N)R

I where

fU(N)R =

∏(i,j)∈R

(N + j − i)

fSO(N)R =

∏(i,j)∈R

(N + 2j − i − 1)

fSp(N)R =

∏(i,j)∈R

(N + j − 2i + 1)

I They also appear in the eigenvalues of the charges



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fR functions

I fG(N)R are polynomials of N of degree n.

They appear in thecorrelators of Shurs as



G(N)R

I where

fU(N)R =

∏(i,j)∈R

(N + j − i)

fSO(N)R =

∏(i,j)∈R

(N + 2j − i − 1)

fSp(N)R =

∏(i,j)∈R

(N + j − 2i + 1)




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fR functions





G(N)R

I where

fU(N)R =

∏(i,j)∈R

(N + j − i)

fSO(N)R =

∏(i,j)∈R

(N + 2j − i − 1)

fSp(N)R =

∏(i,j)∈R

(N + j − 2i + 1)




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fR functions





G(N)R

I where

fU(N)R =

∏(i,j)∈R

(N + j − i)

fSO(N)R =

∏(i,j)∈R

(N + 2j − i − 1)

fSp(N)R =

∏(i,j)∈R

(N + j − 2i + 1)




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fR functions





G(N)R

I where

fU(N)R =

∏(i,j)∈R

(N + j − i)

fSO(N)R =

∏(i,j)∈R

(N + 2j − i − 1)

fSp(N)R =

∏(i,j)∈R

(N + j − 2i + 1)




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Embedding chain of Lie algebras

I Our charges emerge naturally from the infinite embeddingchain:

g(1) ↪→ g(2) ↪→ · · · g = u, so, sp

I Many ways of performing the embedding but it doesn’tmatter which one we choose



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g(1) ↪→ g(2) ↪→ · · · g = u, so, sp




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g(1) ↪→ g(2) ↪→ · · · g = u, so, sp




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Embeddings

I Orthogonal and unitary algebras

φi ∈ u(N) or so(N)→ φi =

φi

0M−N

∈ u(M) or so(M)

I Symplectic algebras: φiJ + Jφti = 0, J =

(0N/2 IN/2

−IN/2 0N/2

)

φi =

(A BC D

)∈ sp(N)→ φi =

A B

C D

∈ sp(M)

I Embedding EMN is an injective map:

vN1 , vN

2 , . . . , vNN2

↓ ↓ ↓vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2



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Embeddings



φi

0M−N

∈ u(M) or so(M)


(0N/2 IN/2

−IN/2 0N/2

)

φi =

(A BC D

)∈ sp(N)→ φi =

A B

C D

∈ sp(M)


vN1 , vN

2 , . . . , vNN2

↓ ↓ ↓vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2



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Embeddings



φi

0M−N

∈ u(M) or so(M)


(0N/2 IN/2

−IN/2 0N/2

)

φi =

(A BC D

)∈ sp(N)→ φi =

A B

C D

∈ sp(M)


vN1 , vN

2 , . . . , vNN2

↓ ↓ ↓vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2



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Embeddings



φi

0M−N

∈ u(M) or so(M)


(0N/2 IN/2

−IN/2 0N/2

)

φi =

(A BC D

)∈ sp(N)→ φi =

A B

C D

∈ sp(M)


vN1 , vN

2 , . . . , vNN2

↓ ↓ ↓vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2



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Projection operators on the algebra

I We also define a set of projectors adapted to the chain

ProjNM : g(M)→ g(N), M > N.

I ProjNM are surjective maps.

vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2

↓ ↓ ↓ ↓ ↓vN1 , vN

2 , . . . , vNN2 , 0, . . . , 0

I They have the obvious property: ProjNM ◦ EMN = IdN



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vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2

↓ ↓ ↓ ↓ ↓vN1 , vN

2 , . . . , vNN2 , 0, . . . , 0




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vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2

↓ ↓ ↓ ↓ ↓vN1 , vN

2 , . . . , vNN2 , 0, . . . , 0




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vM1 , vM

2 , . . . , vMN2 , vM

N2+1, . . . , vMM2

↓ ↓ ↓ ↓ ↓vN1 , vN

2 , . . . , vNN2 , 0, . . . , 0




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Projection operators on GI

I We make ProjNM act on composite operators by simply actingas before on every slot of Ψ in OM(Ψ)

I If any of the fields φi,M does not come from φi,N via theembedding then ProjNMOM(Ψ) = 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 φi ∈ g(M)and GI operators built on φi ∈ g(N)



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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 φi ∈ g(M)

and GI operators built on φi ∈ g(N)



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Averaging operators

I Operator↔ statesI ON as vectors belonging to the vector space VN .I 〈ONO′N〉 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

〈AvMN [ON ]O′M〉 = 〈ONProjNMO′M〉

Can we find an explicit expression for AvMN?



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Averaging operators




I In other words





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Averaging operators




I In other words





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Averaging operators




I In other words





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Averaging operators




I In other words





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Averaging operators




I In other words





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Averaging operators and adjoint action

I Averaging operators are given by

AvMNON =

∫g∈G(M)

dg Adg (ON), G = U,SO or Sp

I The adjoint action applies to every φi in the GI operator

AdgTrN(ZY ) = TrN(gZg−1gYg−1)

= gi1j1Zj1j2g−1j2i2

gi3j3Yj3j4g−1j4i4δi2i3δi4i1 ,

where

i = 1, . . . ,N j = 1, . . . ,M. Z ,Y are embedded



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AvMNON =

∫g∈G(M)

dg Adg (ON), G = U, SO or Sp





where




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AvMNON =

∫g∈G(M)

dg Adg (ON), G = U, SO or Sp





where




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Weingarten calculus: integrals over G (M)

I For U(N) we have (g−1 = g+):∫g∈U(M)

dg g i1j1· · · g in

jn(g+)

j′1i ′1· · · (g+)

j′ni ′n

=∑

α,β∈Sn

(α)II ′(β)J′

J WgU(M)(αβ).

I where

WgU(N)(σ) =1

n!

∑R`n

l(R)≤N

dR

fU(N)R

χR(σ), σ ∈ Sn. [Collins ’03]

I Similar results are found for integrals over the orthogonal and thesymplectic groups. [Collins, Matsumoto ’09] [Matsumoto ’13]



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jn(g+)

j′1i ′1· · · (g+)

j′ni ′n

=∑

α,β∈Sn

(α)II ′(β)J′

J WgU(M)(αβ).

I where

WgU(N)(σ) =1

n!

∑R`n

l(R)≤N

dR

fU(N)R





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jn(g+)

j′1i ′1· · · (g+)

j′ni ′n

=∑

α,β∈Sn

(α)II ′(β)J′

J WgU(M)(αβ).

I where

WgU(N)(σ) =1

n!

∑R`n

l(R)≤N

dR

fU(N)R





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jn(g+)

j′1i ′1· · · (g+)

j′ni ′n

=∑

α,β∈Sn

(α)II ′(β)J′

J WgU(M)(αβ).

I where

WgU(N)(σ) =1

n!

∑R`n

l(R)≤N

dR

fU(N)R





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Averaging and projection: composition

I Consider the set of operators

Q`nNM ≡ ProjNM ◦ AvMN : VN → VN .

I They are self-adjoint by construction for all M > N

What are their eigenvalues and eigenvectors?



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Averaging and projection: orthogonality of Schurs

I Their eigenvectors are restricted Schur polynomials

Q`nNMχG(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R

χG(N)R,µ,m(Ψ), ∀M > N,

I And their eigenvalues are all different for different R’s, so

I Restricted Schur polynomials must be orthogonal on the labelR



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fG(N)R

fG(M)R






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fG(N)R

fG(M)R


I And their eigenvalues are all different for different R’s,

so




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fG(N)R

fG(M)R






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Recovering the two-point function

I Insert

AvMNχG(N)R,µ,m =

fG(N)R

fG(M)R

χG(M)R,µ,m, ∀M > N

I into

〈AvMN [χG(N)R,µ,m]χ

G(M)R,ν,m′〉 = 〈χG(N)

R,µ,mProjNM χG(M)R,ν,m′〉

I to obtain1

fG(M)R

〈χG(M)R,µ,mχ

G(M)R,ν,m′〉 =

1

fG(N)R

〈χG(N)R,µ,mχ

G(N)R,ν,m′〉, ∀M > N

I So〈χG(N)

R,µ,mχG(N)S ,ν,m′〉 = c(R, µ, ν,m,m′)f

G(N)R δRS



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I Insert

AvMNχG(N)R,µ,m =

fG(N)R

fG(M)R


I into




I to obtain1

fG(M)R

〈χG(M)R,µ,mχ

G(M)R,ν,m′〉 =

1

fG(N)R

〈χG(N)R,µ,mχ

G(N)R,ν,m′〉, ∀M > N

I So〈χG(N)


G(N)R δRS



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I Insert

AvMNχG(N)R,µ,m =

fG(N)R

fG(M)R


I into




I to obtain1

fG(M)R

〈χG(M)R,µ,mχ

G(M)R,ν,m′〉 =

1

fG(N)R

〈χG(N)R,µ,mχ

G(N)R,ν,m′〉, ∀M > N

I So〈χG(N)


G(N)R δRS



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I Insert

AvMNχG(N)R,µ,m =

fG(N)R

fG(M)R


I into




I to obtain1

fG(M)R

〈χG(M)R,µ,mχ

G(M)R,ν,m′〉 =

1

fG(N)R

〈χG(N)R,µ,mχ

G(N)R,ν,m′〉, ∀M > N

I So〈χG(N)


G(N)R δRS



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I Insert

AvMNχG(N)R,µ,m =

fG(N)R

fG(M)R


I into




I to obtain1

fG(M)R

〈χG(M)R,µ,mχ

G(M)R,ν,m′〉 =

1

fG(N)R

〈χG(N)R,µ,mχ

G(N)R,ν,m′〉, ∀M > N

I So〈χG(N)


G(N)R δRS



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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 tothe free two-point function.

I We found an explicit form of AvMN = Proj∗NM in terms ofintegrals over the groups.

I We construct a self-adjoint set of chargesQ`nNM ≡ ProjNM ◦ AvMN : VN → VN .

I It turns out that their eigenvectors are restricted Schurpolynomials and that we can partially reconstruct the freefield two-point function.



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I The starting point is g(1) ↪→ g(2) ↪→

I To implement this structure we create the setProjNM , M > N, which are maps between GI operators.







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Different embeddings

I Different embeddings Ψ(E) 6= Ψ(E ′)

I Different embeddings changes the support of projectors

Proj(E)NM 6= Proj

(E ′)NM , Proj

(E)NMO(Ψ(E ′)) = 0

I But AvMN changes its Image accordingly

Av(E)MNχ

G(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R

χG(M)R,µ,m(Ψ(E)), ∀M > N

I SoProj

(E)NM ◦ Av

(E)MN = Proj

(E ′)NM ◦ Av

(E ′)MN ≡ Q`nNM



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Proj(E)NM 6= Proj

(E ′)NM , Proj

(E)NMO(Ψ(E ′)) = 0


Av(E)MNχ

G(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R


I SoProj

(E)NM ◦ Av

(E)MN = Proj

(E ′)NM ◦ Av

(E ′)MN ≡ Q`nNM



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Proj(E)NM 6= Proj

(E ′)NM , Proj

(E)NMO(Ψ(E ′)) = 0


Av(E)MNχ

G(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R


I SoProj

(E)NM ◦ Av

(E)MN = Proj

(E ′)NM ◦ Av

(E ′)MN ≡ Q`nNM



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Proj(E)NM 6= Proj

(E ′)NM , Proj

(E)NMO(Ψ(E ′)) = 0


Av(E)MNχ

G(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R


I SoProj

(E)NM ◦ Av

(E)MN = Proj

(E ′)NM ◦ Av

(E ′)MN ≡ Q`nNM



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Proj(E)NM 6= Proj

(E ′)NM , Proj

(E)NMO(Ψ(E ′)) = 0


Av(E)MNχ

G(N)R,µ,m(Ψ) =

fG(N)R

fG(M)R


I SoProj

(E)NM ◦ Av

(E)MN = Proj

(E ′)NM ◦ Av

(E ′)MN ≡ Q`nNM



Charges Q``n




Probabilistic interpretation of the eigenvalues I

N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)

Dim(R,N)Dim(R,N;S,M)Dim(S,M)

∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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N = 3

N = 2

��

@@@

PPPPPPP

N = 1 ∅��

@@@@

��

��

��

JJJJ

��

Dim( ,2)

Dim( ,3)


∑R = 1



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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 final irrep (R,N).

I Each step down from (T ,M ′) and (U,M ′ − 1) has probabilityDim(U,M′−1)Dim(T ,M′) 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

fU(N)R

fU(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.



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fU(N)R

fU(M)R

=Dim(R,N)

Dim(R,M)




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fU(N)R

fU(M)R

=Dim(R,N)

Dim(R,M)




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fU(N)R

fU(M)R

=Dim(R,N)

Dim(R,M)




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fU(N)R

fU(M)R

=Dim(R,N)

Dim(R,M)




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fU(N)R

fU(M)R

=Dim(R,N)

Dim(R,M)




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Charges Q``n: Motivation

I Charges Q`nNM resolve the label R ` n of χR,µ,m which isrelated to the total number of fields n.

I Operators built on Ψ have abundancies λ = (n1, . . . , nr ) offields φ1, . . . , φr .

I Charges Q`nNM , by construction, do not care about λ.

I There must be other charges related to λ that resolve thesmall labels.



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Charges Q``n










Charges Q``n










Charges Q``n










Charges Q``n



Charges Q``nNM , requirements

I They should be related to the embedding chain and sensitiveto λ, so we impose

Qλ ≡ ProjNM ◦ AvλMN

(Q``nNM =

∑λ`nλ6=(n)

QλNM

)

I They reduce to Q`nNM when λ = (n)

I They are self-adjoint with respect to the free two-pointfunction



Charges Q``n






(Q``nNM =

∑λ`nλ 6=(n)

QλNM

)





Charges Q``n






(Q``nNM =

∑λ`nλ 6=(n)

QλNM

)





Charges Q``n






(Q``nNM =

∑λ`nλ 6=(n)

QλNM

)





Charges Q``n



AvλMN

I Operators AvλMN are defined as

AvλMN [O(Ψ)] ≡∫g1,...,gr∈G(M)

d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr

I One has to define Adλ

I First attemp, for λ = (2, 1) we have

Adλg ,hTrN(ZZY ) = TrN(gZg−1gZg−1hYh−1)

I But then Qλ are not self-adjoint... Can we cure it?



Charges Q``n



AvλMN



d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr







Charges Q``n



AvλMN



d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr







Charges Q``n



AvλMN



d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr







Charges Q``n



AvλMN



d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr




I But then Qλ are not self-adjoint...

Can we cure it?



Charges Q``n



AvλMN



d[g]

Adλ[g ][O(Ψ)],

where [g ] = g⊗n11 ⊗ · · · ⊗ g⊗grr







Charges Q``n



λ-adjoint action

I We can, if we shuffle the slots of left (or right) action of thegroup elements

I In the former example

Adλg ,hTrN(ZZY ) =1

3!

(TrN(gZg−1gZg−1hYh−1)

+ TrN(gZg−1hZg−1gYh−1) + . . .)

I For unitary groups the λ-adjoint action over multitraces is

Adλ[g ][TrU(N)(σΨ)] =1

|Sn|∑α∈Sn

[g]Iα(J)

[Ψ]JJ′

[g]I ′J′

(α−1σ)I′I .



Charges Q``n



λ-adjoint action




3!


+ TrN(gZg−1hZg−1gYh−1) + . . .)



|Sn|∑α∈Sn

[g]Iα(J)

[Ψ]JJ′

[g]I ′J′

(α−1σ)I′I .



Charges Q``n



λ-adjoint action




3!


+ TrN(gZg−1hZg−1gYh−1) + . . .)



|Sn|∑α∈Sn

[g]Iα(J)

[Ψ]JJ′

[g]I ′J′

(α−1σ)I′I .



Charges Q``n



λ-adjoint action




3!


+ TrN(gZg−1hZg−1gYh−1) + . . .)



|Sn|∑α∈Sn

[g]Iα(J)

[Ψ]JJ′

[g]I ′J′

(α−1σ)I′I .



Charges Q``n



Eigenvalues and eigenvectors of Q``nNM

I For unitary groups we find the equation

QλNM(χ

U(N)R,µ,ij(Ψ)) =

|Sλ||Sn|

fU(N)R

fU(M)µ

χU(N)R,µ,ij(Ψ), fµ = fs1fs2 · · · fsr

I If Ψλ′ and λ′ 6= λ then

QλNM(χ

U(N)R,µ,ij(Ψλ′)) = 0

I So, we define

Q``nNM ≡=∑λ`nλ6=(n)

QλNM , M ≥ N

which acts nontrivially on any GI non 12 -BPS operator



Charges Q``n





QλNM(χ

U(N)R,µ,ij(Ψ)) =

|Sλ||Sn|

fU(N)R

fU(M)µ



QλNM(χ


I So, we define


QλNM , M ≥ N




Charges Q``n





QλNM(χ

U(N)R,µ,ij(Ψ)) =

|Sλ||Sn|

fU(N)R

fU(M)µ



QλNM(χ


I So, we define


QλNM , M ≥ N




Charges Q``n





QλNM(χ

U(N)R,µ,ij(Ψ)) =

|Sλ||Sn|

fU(N)R

fU(M)µ



QλNM(χ


I So, we define

Q``nNM ≡=∑λ`nλ 6=(n)

QλNM , M ≥ N




Charges Q``n



Eigenvalues and eigenvectors of Q``nNM for other groups

I Let us write the eigenvector equation for all classical gaugegroups

Q``nNM (χU(N)R,µ,ij(Ψ)) =

|Sλ||Sn|

fU(N)R

fU(M)µ

χU(N)R,µ,ij(Ψ)

Q``nNM (χSO(N)R,µ,i (Ψ)) =

|Sλ||Sλ[S2]|2|S2n|

fSO(N)R

fSO(M)µ

χSO(N)R,µ,i (Ψ)

Q``nNM (χSp(N)R,µ,i (Ψ)) =

|Sλ||Sλ[S2]|2|S2n|

fSp(N)R

fSp(M)µ

χSp(N)R,µ,i (Ψ),

I The value of the eigenvalues between 0 and 1 suggests thatthey should also have a probabilistic interpretation in terms ofbranching graphs paths



Charges Q``n






|Sλ||Sn|

fU(N)R

fU(M)µ

χU(N)R,µ,ij(Ψ)


|Sλ||Sλ[S2]|2|S2n|

fSO(N)R

fSO(M)µ

χSO(N)R,µ,i (Ψ)


|Sλ||Sλ[S2]|2|S2n|

fSp(N)R

fSp(M)µ

χSp(N)R,µ,i (Ψ),




Charges Q``n






|Sλ||Sn|

fU(N)R

fU(M)µ

χU(N)R,µ,ij(Ψ)


|Sλ||Sλ[S2]|2|S2n|

fSO(N)R

fSO(M)µ

χSO(N)R,µ,i (Ψ)


|Sλ||Sλ[S2]|2|S2n|

fSp(N)R

fSp(M)µ

χSp(N)R,µ,i (Ψ),




Charges Q``n



Go back to the two-point function

I Remember that because of {Q`nNM} we know that

〈χG(N)R,µ,m, χ

G(N)S ,ν,m′〉 = c(R, µ, ν,m,m′)f

G(N)R δRS ,

I Now, using the same arguments, and because {Q``nNM }commutes with {Q`nNM}, we can narrow the result to

〈χG(N)R,µ,m, χ

G(N)S,ν,m′〉 = c(R, µ,m,m′)f

G(N)R δRSδµν ,



Charges Q``n





〈χG(N)R,µ,m, χ

G(N)S ,ν,m′〉 = c(R, µ, ν,m,m′)f

G(N)R δRS ,


〈χG(N)R,µ,m, χ


G(N)R δRSδµν ,



Charges Q``n





〈χG(N)R,µ,m, χ

G(N)S ,ν,m′〉 = c(R, µ, ν,m,m′)f

G(N)R δRS ,


〈χG(N)R,µ,m, χ


G(N)R δRSδµν ,



Charges Q``n


Conclusions and future works

I We have constructed two infinite sets of charges which emergenaturally from the embedding chain. Their eigenvectors arerestricted Schur polynomials for classical gauge groups.

I For future worksI Construction of charges {Qm

NM}I Find probabilistic interpretation of eigenvalues of {Q``nNM }I See if we can drop matter in the adjointI Connection with other charges studied before. Concretely the

charges that come from the global U(N)× U(N) symmetry ofthe 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 {DNM ,M ≥ N} suchthat DNN = D is the full dilatation operator.



Charges Q``n











Charges Q``n




I For future works

I Construction of charges {QmNM}

I Find probabilistic interpretation of eigenvalues of {Q``nNM }I See if we can drop matter in the adjointI Connection with other charges studied before. Concretely the






Charges Q``n





NM}

I Find probabilistic interpretation of eigenvalues of {Q``nNM }I See if we can drop matter in the adjointI Connection with other charges studied before. Concretely the






Charges Q``n





NM}I Find probabilistic interpretation of eigenvalues of {Q``nNM }

I See if we can drop matter in the adjointI Connection with other charges studied before. Concretely the






Charges Q``n





NM}I Find probabilistic interpretation of eigenvalues of {Q``nNM }I See if we can drop matter in the adjoint

I Connection with other charges studied before. Concretely thecharges that come from the global U(N)× U(N) symmetry ofthe free theory [Kimura, Ramgoolam].





Charges Q``n











Charges Q``n







I Exploiting Weingarten calculus.

I Conjecture: For an interacting correlator it will be possible toconstruct a set of commuting charges {DNM ,M ≥ N} suchthat DNN = D is the full dilatation operator.



Charges Q``n











Charges Q``n


Thanks!


Date post:	17-Jul-2015
Category:	Education
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Dr. Pablo Diaz Benito (University of the Witwatersrand) TITLE: "Novel Charges in CFT's"

Education