Permutation algebrasand Matrix/Tensor holography

Sanjaye Ramgoolam

Queen Mary, University of London

9’th Joburg workshop on String theory.Based on :

Ben Geloun, Ramgoolam,(arXiv-1307.6490 ; AIHPD ) - GR1307“Counting tensor model observables and branched covers of the 2-sphere”

Ben Geloun, Ramgoolam, (arxiv:1708.03524; JHEP ) - GR1708“Tensor models, Kronecker coefficients and permutation centralizer algebras,”

Permutation algebras : simple example from Sn

The symmetric group Sn is the group of rearrangements of{1,2, · · · ,n}. Order of group n!.

It has an associative product. Given σ1, σ2, σ3 ∈ Sn :

(σ1σ2)σ3 = σ1(σ2σ3)

The group algebra C(Sn) is the set of linear combinations

A =∑σ∈Sn

Aσσ with Aσ ∈ C

This is a vector space, of dimension n!. It has a product,inherited from the group product.

AB = (∑σ1

Aσ1σ1)(∑σ2

Bσ2σ2)

=∑σ1,σ2

Aσ1Bσ2(σ1σ2)

C(Sn) : associative, semisimple algebra

C(Sn) is a non-commutative associative algebra (vector spacewith associative product)

(AB)C = A(BC)

Another important property is that it is a semi-simple algebra.There is a non-degenerate bilinear pairing.

< A,B >=<∑σ1

Aσ1σ1,∑σ2

Bσ2σ2 >

=∑σ1

∑σ2

Aσ1Bσ2δ(σ1σ−12 )

δ(σ) is the delta-function on the group. 1 if σ = () and zerootherwise.

Wedderburn-Artin : Matrix Blocks

Semi-simple algebras are isomorphic to a direct sum of matrixalgebras (Wedderburn-Artin theorem).

The dimension of C(Sn) is a sum of squares. For everyirreducible representation R of Sn, of dimension dR, there is aset of d2

R elements in C(Sn).

n! =∑

R

d2R

Given any irrep VR, we have matrices DRij (σ). The DR

ij are d2R

functions, correspond to some elements in C(Sn).

QRij =

dR

n!

∑σ

DRij (σ)σ

Basis for matrices Eij multiply as EijEkl = δjkEkl .

QRij QS

kl = δRSδjkQRil

Wedderburn-Artin : Matrix Blocks

Semi-simple algebras are isomorphic to a direct sum of matrixalgebras (Wedderburn-Artin theorem).

The dimension of C(Sn) is a sum of squares. For everyirreducible representation R of Sn, of dimension dR, there is aset of d2

R elements in C(Sn).

n! =∑

R

d2R

Given any irrep VR, we have matrices DRij (σ). The DR

ij are d2R

functions, correspond to some elements in C(Sn).

QRij =

dR

n!

∑σ

DRij (σ)σ

Basis for matrices Eij multiply as EijEkl = δjkEkl .

QRij QS

kl = δRSδjkQRil

Another example : the centre of C(Sn)

The centre of C(Sn), denoted Z(C(Sn)), is the subspace whichcommutes with all elements A. This is a commutativeassociative algebra.

For z ∈ Z(C(Sn)), and any A ∈ C(Sn)

zA = Az

For any σ ∈ Sn,

σ =∑γ∈Sn

γσγ−1

is in Z(C(Sn)).

σ is a sum of permutations in the same conjugacy class as σ,with equal weight for each element of the conjugacy class.

Permutations can be described in cycle notation. Example S3 :

σ1 =

{1 2 31 2 3

}= (1)(2)(3) = ()

σ2 =

{1 2 32 1 3

}= (12)(3) = (12)

σ3 =

{1 2 32 3 1

}= (123)

A cycle structure in S3 corresponds to a partition of 3 intopositive integers.

[1,1,1] = [13]→ (1)(2)(3)[2,1]→ (12)(3), (13)(2), (1)(23)[3]→ (1,2,3), (1,3,2)

Fact : Conjugacy classes σ ∼ γσγ−1 in Sn correspond to cyclestructures.

So a basis for the algebra Z(C(S3)) is

T1 = ()T2 = (1,2) + (2,3) + (1,3)T3 = (1,2,3) + (1,3,2)

Products of these central elements can be expanded in termsof the central elements.

Example :

T1T2 = T2T2T2 = 3T1 + 2T3

Centre of C(Sn) and irreducible reps.A general partition of n, denoted [1p1 ,2p2 , ...] specified by{p1,p2, · · · ,pn}

n = p1 + 2p2 + · · ·+ npn

Number of these is a famous number sequence p(n) - innumber theory, combinatorics, string theory

Counting Fact : Number of conjugacy classes is equal tonumber of irreps.

Corresponding construction fact: There is a basis ofZ(C(Sn)) which comes from irreps. PR. Can be written in termsof characters

χR(σ) =∑

i

DRii (σ)

p(n) =∑

R

1 =∑

R

12

Key lessons about Permutation algebras

1. Related to interesting number sequences : n! ; p(n)

2. Have two types of bases : permutation basis ;Representation basis.

3. Have a matrix structure (Wedderburn-Artin) related torepresentation bases. Dimension is a sum of squares.

Rest of talk

1. Permutation algebras control the counting of observablesin matrix/tensor models. Both Finite N and large N.

2. Representation bases and Matrix structure of the algebrasare related to properties of correlators in Gaussian matrixand tensor models.

3. Interesting number sequences.4. Covering space geometry interpretation of

counting/correlators in the matrix/tensor models.

WHY ? - Invariant theory and Schur-Weyl dualityFor U(N) the tensor product V ⊗ V of the fundamental with theanti-fundamental contains one invariant (singlet).

∑i

ei ⊗ ei =∑i,j

ei ⊗ ej δji

For V⊗n ⊗ V⊗n, we have n! invariants.

vσ =∑

i1,··· ,in

ei1 ⊗ ei2 ⊗ · · · ⊗ ein ⊗ eiσ(1)⊗ eiσ(2)

⊗ · · · ⊗ eiσ(n)

= ei1 ⊗ ei2 ⊗ · · · ⊗ ein ⊗ ej1 ⊗ ej2 ⊗ · · · ⊗ ejn δj1iσ(1)· · · δjn

iσ(n)

OUTLINE

1. Permutation algebras and matrix/tensor invariants.I Holomorphic invariants of a complex matrix.I Parametrizing invariants with permutations and describing

redundancy.I A catalog of algebras and the matrix/tensor problems they

solve.

2. Complex tensor models.I Counting of invariants : permutation and representation

basisI Applications : thermodynamics, orthogonal bases.I Color-symmetrized counting: Perm and Rep basis.

Implications.

3. Covering space geometryI Complex matrix model.I Tensor models.

Part 1: One complex matrixConsider a 1-matrix model. Z a complex matrix. Transformingin the adjoint of a U(N) gauge symmetry.

Z → UZU†

Gauge invariant polynomial holomorphic functions of Z aretraces. e.g. tr Z3, (tr Z2)tr Z, (tr Z)3

Oσ(Z ) = Z i1iσ(1)

Z i2iσ(2)

Z i3iσ(3)

σ = (1)(2)(3) −→ (tr Z)3

σ = (1,2)(3) −→ (tr Z2)tr Zσ = (1,2,3) −→ tr Z3

For σ = (1,2,3), we have

Oσ(Z ) = Z i1i2

Z i2i3

Z i3i1

= trZ3

For σ = (1,3,2)

Oσ(Z ) = Z i1i3

Z i2i1

Z i3i2

= Z i1i3

Z i3i2

Z i2i1

= trZ3

This equivalence can be described in terms of a permutationgroup property

(1,2,3) = (2,3)(1,3,2)(2,3)

Conjugate permutations produce the same trace.

Oσ(Z ) = Oγσγ−1(Z )

Conjugacy Classes to Irreducible RepresentationsSn representation theory gives a nice basis for the centre ofC(Sn). It is the basis of projectors, one for each irrep,constructed from characters.

PR =dR

n!

∑σ∈Sn

χR(σ)σ

This basis allows an easy way to implement finite N constraints,related to the “stringy exclusion principle.”

In AdS5/CFT4, this is used in mapping gauge-invariantoperators to giant graviton branes. ( Corley, Jevicki,Ramgoolam 2001).

Schur-Weyl dualityYoung diagrams label symmetry types of tensors

Si1,··· ,inR ei1 ⊗ ei2 ⊗ · · · ⊗ ein

i.e. irreducible representations of U(N) appearing in V⊗nN .

R has n boxes and height no bigger than N.

All Young diagrams R with n boxes correspond to irreps of Sn(standard tableaux, with boxes labelled 1,2, · · · n form a basisof an irrep).

The space V⊗nN has an action of U(N) as well as a commuting

action of Sn. The decomposition into irreps of U(N)× Sn is

V⊗nN =

⊕R:l(R)≤N

V U(N)R ⊗ V (Sn)

R

Permutations in matrix/tensor invariants : TWO JOBS

I In the above we used permutation σ to parametrizeinvariants.

I Then we observed a redundancy given by γ, related tobosonic symmetry.

I In the above σ ∈ Sn, γ ∈ Sn, and

σ ∼ γσγ−1

I More generally parametrizing invariants involves asequence of permutations living in a group G.

I Describing redundancy involves a subgroup H, and anaction of the subgroup H on G.

Perm equivalence classes fundamental to physics of gauge invariantsIdentifying the permutation equivalence classes for a givenproblem in classifying and studying correlators ofgauge-invariants opens up a standard box set of mathematicaltools ...

Perm equiv classes. to representation theory basis is ageneralization of the Fourier transform - in the context ofdiscrete algebras.

The rep theory basis exposes a matrix block structure,expected from Wedderburn-Artin theorem, which is useful inmulti-matrix and tensor problems.

Allows - via Schur-Weyl duality - to describe finite N effects.

Table: Matrix/Tensor combinatorics and Permutation Algebras

G H H-action

1-matrix Z Sn Sn Conjugationn copies of Z σ γ γσγ−1

2-matrix Z ,Y Sm+n Sm × Sn Conjugation(m,n) copies σ γ γσγ−1

Complex3-index tensor S×3

n Sn × Sn left and right action( n copies Φ and (σ1, σ2, σ3) (γ1, γ2) (γ1σ1γ2, γ1σ2γ2, γ1σ3γ2)n copies of Φ )

3-index Sn × Sn Sn Diag conjugationgauge-fixed (σ1, σ2) γ (γσ1γ

−1, γσ2γ−1)

The 2-matrix example generalizes to Quivers : a permutation in G forevery node ; a permutation in H for every edge ( Pasukonis,Ramgoolam - arXiv:1301.1980)

For rank d complex tensors Φi1···id the invariants build from ncopies of Φ and n copies of Φ, restricting to U(N)d gaugeinvariants, we have

G = S×dn

H = Sn the diagonally embedded subgroup

The gauge symmetry is H × H.The equivalence relation is

(σ1, · · · , σd ) ∼ (γ1σ1γ2, · · · , γ1σdγ2)

An equivalent formulation is

(1, τ1, τ2, · · · , τd−1) ∼ (1, γ−12 τ1γ2, · · · , γ−1

2 τd−1γ2)

So now we have

G = S×(d−1)n

H = Diag(Sn)

H acts by conjugation and generates equivalent permutations.This is related by a partial gauge-fixing to the previousformulation. Use γ1 to fix one permutation to 1, then γ2becomes a conjugation action :

(1, σ−11 σ2, · · · , σ−1

1 σd )

PART 2 : Permutation equivalences from pictures14 December 2017 23:44

New Section 105 Page 1

Φi1,j1,k1 · · ·Φi2,j2,k2 · · ·Φin,jn,kn

Φp1,q1,r1 · · ·Φp2,q2,r2 · · ·Φpn,qn,rn

Must contract in all possible ways :~i ↔ ~p~j ↔ ~q~k ↔ ~r

Multi-index tensor models with U(N)×n gauge invariance have been studied actively in the tensor model community(Gurau, Rivasseau and others, starting around 2009), in connection with double scaling limits leading to melonicgraphs.

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Permutation equivalences and Burnside LemmaWhenever a group H acts on a set S, the number of orbits ofthe H action (or equivalence classes under H action) is given interms of the fixed points of the H-action by Burnside Lemma.Applying this to the 3-index tensor models.

Z3(n) =1

(n!)2

∑γ1,γ2∈Sn

∑σ1,σ2,σ3

δ(γ1σ1γ2σ−11 )δ(γ1σ2γ2σ

−12 )δ(γ1σ3γ2σ

−13 )

Manipulating this leads to

Z3(n) =∑p`n

Symp

where

Sym p =n∏

i=1

ipi pi !

Number sequencesEasy to work out - higher orders with Mathematica (GR-1307) :

1,4,11,43,161, · · ·

Connected invariants (analogs of single traces) by using thetechnology of Plethystic logarithms :

1,3,26,97,624...

Applications of Plethystic Logs in SUSY gauge theories (Feng,Hanany, He, hepth/0701063).

General Rank dThese methods generalize easily to higher rank.

Zd (n) =∑p`n

( Sym (p))d−2

Open questions

I Find generating functions

Zd (x) =∞∑

n=0

Zd (n)xn

(n!)a

some fixed a.I And asymptotics.

Similar sequences for operators in higher dimensional tensorfield theories have been studied in Beccaria-Tseytlin(1703.044660); some asymptotic estimates in Klebanov et. al. (arXiv:1707.09347 )

Rep. theory countingIrreducible representations of Sn correspond to Young diagramsR. In the tensor product

VR1 ⊗ VR2 ⊗ VR3

the one-dimensional irrep appears some number of timesC(R1,R2,R3). This is called the Kronecker coefficient.The counting for the 3-index case is

Z3(n) =∑

R1,R2,R3

(C(R1,R2,R3))2

Finite N cutoff easy : l(Ri) ≤ N ( follows from Schur-Weylduality).This was written in a number of recent papers : Mattioli-Ramgoolam-1601.06086Diaz-Rey : 1706.02667De Mello Koch, Grossman, Tribelhorn : 1707.01455Ben Geloun, Ramgoolam: 1708.03524

Rep. theory counting and Wedderburn ArtinThe permutation equivalence classes

(σ1, σ2, σ3) ∼ (γ1σ1γ2, γ1σ2γ2, γ1σ3γ2)

lead to a semisimple, associative algebra: the subspace ofC(Sn)⊗ C(Sn)× C(Sn) invariant under left and rightmultiplication by the diagonal Sn. Call it K(n).

This algebra is associative and semi-simple. Has a basis

QR1,R2,R3ij

which gives matrix decomposition, where1 ≤ i , j ≤ C(R1,R2,R3).Ben Geloun, Ramgoolam, 1708

Structure of Matrix multiplication in blocks labelled by triples(R1,R2,R3) with non-vanishing Kronecker coefficients :

QR1,R2,R3ij Q

R′1,R′2,R′3

kl = δjkδR1,R′1δR2,R′2δR3,R′3QR1,R2,R3

il

This rep basis for K(n) is a linear combination of permutations

QR1,R2,R3ij =

∑σ1,σ2,σ3

QR1,R2,R3ij;σ1,σ2,σ3

σ1 ⊗ σ2 ⊗ σ3

There are corresponding operators O~R;ij(Φ, Φ).

Representation Basis and Gaussian CorrelatorsThese operators forming the representation theoretic basis(matrix basis) have orthogonal two-point functions.Gaussian Model :

Z =

∫ ∏i,j,k

dΦijkdΦijke−12∑

i,j,k Φijk Φijk

〈: OR1,R2,R3µ1ν1

: : OS1,S2,S3µ2ν2

:〉

= δR1,S1δR2,S2δR3,S3δµ1,µ1δµ2,ν2C(R1,R2,R3)DimNR1DimNR2 DimNR3

Note : normal-ordered.

Color symmetrizationThese tensor invariants correspond to colored graphs. Forn = 2, we have

Sym([2]) + Sym([1,1]) = 2 + 2 = 4

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Color symmetrization using permutation triplesColor exchange involves rearranging (σ1, σ2, σ3) by 3!permutations.

Using this observation, along with Burnside Lemma, leads to aformula for the color-symmetrized counting ( where graphsrelated by color exchange are treated as equivalent).

Zcs(n) =16

(∑p

Sym p

)+

12

1n!

∑σ,γ∈Sn

δ(γ2σγ−2σ−1) +13

1n!

∑σ,γ

δ(γ3σ3)

The sequence :

1,2,5,15 · · ·

Last 2 terms : as sums over partitions ( see GR-1307 ).

A surprising integralityThe summands are separately integer sequences :∑

p

Sym p

1n!

∑σ,γ∈Sn

δ(γ2σγ−2σ−1)

1n!

∑σ,γ∈Sn

δ(γ3σ3)

The surprising integrality.

Integrality from permutation algebraK(n) is the subspace of

C(Sn)⊗ C(Sn)⊗mC(Sn)

which is invariant under left and right action by Diag(Sn). Bothactions commute with S3 of exchanging the factors. So we candecompose K(n) into irreps of S3.

K(n) = K ⊕K ⊕K

The multiplicity of the three S3 irreps are integers. This - alongwith integrality of S3 characters, can be used to prove theintegrality above. ( GR1708 )This generalizes to rank d tensors.

Part 3 : Geometry of CorrelatorsRecall for complex matrix model :

Oσ(Z ) = Z i1iσ(1)· · ·Z in

iσ(n)

Conjugation equivalence :

Oσ(Z ) = Oγσγ−1(Z )

Gaussian model correlator :

< Oσ1(Z )Oσ2(Z †) >=∑γ∈Sn

∑σ3∈Sn

δ(γσ1γ−1σ2σ3)NCσ3

=Symp1 Symp2

n!

∑σ1∈Tp1

∑σ2∈Tp2

∑σ3∈Sn

δ(σ1σ2σ3)NCσ3

Geometry of Correlators : complex matrix modelAfter normalizing the observables

< Oσ1Oσ2 >=1n!

∑σ1∈Tp1 ,σ2∈Tp2

σ3∈Sn

δ(σ1σ2σ3)N2n−Bσ1−Bσ2−Bσ3

=∑

f :Σh→S2

{0,1,∞;p1,p2}

1Aut f

N2−2h

B(σ) = n − Cσ

This corresponds to counting branched covers of S2 with 3branch points. Power of N is the genus of the covering space.

There is an emergent target S2 from the combinatorics of thecorrelators, with gst ∼ 1/N.

Similar mathematics of branched covers was used to establisha string picture for 2D YM gauge theory by Gross and Taylor (92/93).

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Geometry of Counting tensor model observables.Counting of 3-index tensor model observables can bere-expressed as ( GR1307)

Z3(n) =1n!

∑γ∈Sn

∑τ1,τ2,τ3∈Sn

δ(τ1τ2τ3)δ(τ1γτ1γ−1)δ(γτ2γ

−1τ−12 )

We are counting the holomorphic maps which contribute in thecomplex matrix model, to correlators with arbitrary holo andanti-holo observables and taking N → 1 !

Question Complex matrix model : half-BPS sector ofAdS5/CFT 4Tensor model : potentially AdS2AdS Interpretation ?

Rank d tensors : Count branched covers of S2 with d − 1branch points.

Tensor model correlators

< Oσ1,σ2,σ3 >=∑γ∈Sn

N Euler character of covering 2D complex

δ(γσ1α1)δ(γσ2α2)δ(γσ3α3)

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Tensor model correlatorsThis 2-complex cannot be part of the cell-decomposition of asmooth 2-manifold. But can be part of the cell-decomposition ofa 3-manifold.

Higher rank tensors. More triangles join along an edge. Targetstill 3D.

Suggests a 2D topological string with 3D target underlies thecombinatorics of counting/correlators for rank d tensor models.2D target captures some aspects but not all.

c.f. discussions in quantum gravity community - higherdimensional morivations for higher rank tensor models.

Summary and Outlook

1. Permutation algebras provide a unifying framework formatrix and tensor models - counting and combinatorics ofgeneral observables in Gaussian model.

2. To make contact between these algebras and SYK -consider observable to be exponentiated quartic interaction; and study implications of the algebras for double scalinglimit - a new perspective on melons and beyond.

3. These Matrix algebras - associative, semisimple - arerelated to 2D topological field theories. This structure hasbeen studied in the context of CFT2 and branes ( Fuchs,Runkel, Schweiget 2003). So they are suggestive of acommon stringy structure in matrix models and tensormodels.

4. Another common stringy structure comes from theconnection between permutations and branched covers.

5. Interesting links to combinatorics.

CombinatoricsThe second term in the color symmetrized counting :

1n!

∑γ,σ

δ(γ2σγ−2σ−1) =∑

R

∑p`n

χRp

This term has a tensor model interpretation.

Is there a tensor model intepretation of the refinement ?∑p`n

χRp

This is also known to be a natural number from rep theoryarguments. An open problem of Stanley is to give aconstructive combinatoric interpretation of this number.

Possibly an avenue is to interpret this refinement in terms oftensor models.