PoS(CORFU2015)107 http://pos.sissa.it/ gauge invariants using permutations, sub- ) N ( U N ∗ gauge theories. I review the use of permutations in classifying gauge invariants in one- N [email protected] Speaker. Group algebras of permutations havelarge proved highly useful in solving a number of problems in matrix and multi-matrix models andplicable to computing tensor their models correlators. andbranched have covers. These revealed The methods a key are idea link is also between to ap- tensor parametrize models and the counting of ject to equivalences. Correlatorsclasses. are related Fourier to transformation group onnice theoretic symmetric bases properties groups of of by functions these on means these equivalence oftifying equivalence CFT representation classes. duals theory of This offers giant has gravitons applicationsquiver and in gauge their theory AdS/CFT perturbations. correlators, in It uncovering iden- links has to alsothe two lead combinatorics dimensional to of topological general trace field results monoids. theory on and ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Centre for Research in string theory, School of Physics and Astronomy, Queen Mary, University of London E-mail: Sanjaye Ramgoolam Permutations and the combinatorics of gauge invariants for general Proceedings of the Corfu Summer Instituteand 2015 Gravity" "School and Workshops on Elementary1-27 Particle September Physics 2015 Corfu, Greece PoS(CORFU2015)107 (2.1) (1.1) global . Aside ) n 6 ( SO . In the simplest ) . Gauge invariant 4 2 | Sanjaye Ramgoolam 2 iX , 2 + ( 1 X PSU = , for natural numbers Z ) n S ( . The gauge group of the CFT4 is C 5 S 3 ) 4 SYM, the canonical example of gauge × ) Z n = 5 tr S ( ( , C 4 SYM, define N AdS 0 Z = ∞ = 2 n M superconformal algebra. tr 2 N ) ≡ Z 4 | ) tr 2 ∞ , , S 2 ( ( 3 C ] relates ten dimensional string theory to four dimensional Z 1 tr SYM: 1-complex matrix model PSU . They transform in the vector representation of an 6 4 X , = ··· , N 1 X . For the less supersymmetric quarter and eighth-BPS sectors, we encounter n S . The fields transform in the adjoint. In addition to gauge fields and fermions, we have six ) A very useful tool for thinking about the matrix invariant theory problems we encounter, and Using two of the six hermitian scalars in The AdS/CFT correspondence [ The physical context of what I will describe is Another algebra, more precisely a sequence of algebras, will play a crucial role in this talk. N ( N-dependent Combinatorics of gauge theories. from the obvious Lagrangian symmetries, the direct sum of symmetric group algebras, the role of permutationsthese in tensor these spaces. problems, Diagrams will fora be representing powerful linear tensor way operators, spaces to their uncover and products thewill and linear hidden be traces, simplicity operators provide illustrated of acting multi-index as on tensor wematrix manipulations. proceed problems These from tools encountered the in simplesttensor example AdS/CFT models. of and 1-matrix to to problems more complex of multi- tensor invariants encountered2. in Half-BPS sector in string duality, where the dual is string theory in a multi-matrix problem and a generalization of conjugacy class algebras will play a role. This is the sequence of group algebras of symmetric groups 1. Introduction conformal quantum field theory. In CFT,to the local operator-state correspondence operators, relates which quantumenumeration states are of gauge these composites invariant and composites theThe built computation emergence of from from their these the correlators algebraic is elementaryof structures an the fields. algebraic of strings problem. the and The branes extra in dimensionsgeometry these of emerging backgrounds, from the which algebra. string we expect background, Thealgebras from underlies idea the work that duality in non-commutative is space-time geometry, an a geometry subject exampleat is of of the interest to to Corfu2015 be many in conference. reconstructed the from structures audience In emerge from this algebraic talk, data I in the will context be of describing CFT correlators. some ways in which geometrical hermitian adjoint scalars symmetry, which forms part of the will be found to play angebras important are role useful in in the description organizing of thesector the multiplicities of state of space representations interest, of of the CFT. Permutation maximallytions al- supersymmetric will half-BPS be sector, related the toconjugacy multiplicity classes problems of of of representa- invariant theory of one matrix, and will be organized by the operators such as U PoS(CORFU2015)107 as n (2.5) (2.6) (2.7) (2.3) (2.4) (2.2) ⊗ Z relating J and -fold tensor = n N ∆ V are annihilated by Sanjaye Ramgoolam j i ) on the Z Z n ] and refs. therein to ( n i S 2 j σ n , i e n ∈ e O S ]. σ = ∈ as a sum of positive integers. 5
··· ⊗ , γ n ··· ⊗ 2 4 ) j . In this picture a line represents 2 n i ( γσ ]. e 1 e σ n 3 i ⊗ e ⊗ , permutations act as ⊗ . Acting with the generators of the 1 Z N j ) 1 i 1 4 e V e | − | ) 2 n which determine the order in which the γ n j ) σ , i n for -dimensional vector space n n ( 2 Z n ⊗ ··· ⊗ n n σ S ( i ⊗ } ⊗ i N ) N ⊗ 1 V ( Z Z ··· N ∈ | Z σ tr i 2 n 2 j i i e ··· ≤ PSU σ e = Z = ) i 3 R-symmetry. The enumeration of these BPS oper- 1 1 1 ( 1 j
i ) 1 σ − i ) = 1 i ≤ , the number of gauge invariant polynomials is equal Z 6 . It follows that n γ i − Z ( n ) 1 n N e , ⊗ ⊗ i σ N ≡ ⊗ ··· ⊗ < n e SO Z V ··· 2 ) = ( γσγ { σ i ⊗ γ n n 2 n i e i Z . There is a standard action of O e ⊗ e ( n n ⊗ Z ⊗ ⊗ N ⊗ 1 N i V 1 V n e i ⊗ tr h ··· ⊗ N e ( V 2 i , i.e. the number of ways of writing = tr e n σ ⊗ ) = 1 ) = Z i as an operator acting on an ( e Z charge inside the ( ( σ -operators acts in the standard way as Z 1 n ) Z O − ⊗ 1 , and restricting to ( n Z γσγ U = O ∆ , a box is a linear operator and a contraction is an identification of an upper line n ⊗ N V It is plausible and very easy to check that, for any permutation For a fixed N-dependent Combinatorics of gauge theories. Using the standard connection between linearfor operators example, and we can diagrams, represent such this as diagrammatically used as in in knot Figure theory This is an equality of operators acting on lower indices are contracted with the upper indices with a lower line. Onright the picture left of the the upper picture, indexhorizontal this line line is ends and represented the on by two a joining horizontalof horizontal top ines thinking and line, are about bottom. the understood operators and to On lower correlators be the index is identified. line explained The ends further diagrammatic on in way [ another super-algebra produces an ultra-short representation.in All this ultra-short representations way. are generated Their correlators have non-renormalization properties (see [ This is also the number of conjugacyinvariants classes can of be the symmetric constructed group from and this permutations is no accident. Gauge an operator on the tensor product The tensor product of One checks that a state in to the number of partitions of which are holomorphic gauge-invariant polynomialshalf in of the matrix the elements 32 supercharges in the super-algebra earlier work) which allows comparison betweenpergravity. free These conformal holomorphic field gauge theory invariant operators computations satisfy and the su- BPS condition It is useful to think of product. Choosing a set of basis vectors the dimension to a ators using a Young diagram basis, the computationof of Young correlators diagram in states this with basis giant and gravitons the identification was developed in [ PoS(CORFU2015)107 (2.9) (2.8) (2.10) (2.11) . Thus n S Sanjaye Ramgoolam ) n ( 1 ) ). The calculation can also − ) 1 n

γ n i l ( − γ n δ i l γ ) 1 n δ 2.11 ( ) n γ − n 2 k j ( n γ δ k j 2 γσ δ ) 1 2 1 ··· 2 i i j ) and ( σ ) x † ··· 1 ( δ ( ) n 1 2 1 )) 1 i j − 2.2 ⊗ ( N − 2 γ 1 ) 1 γ 1 δ i i V l l x n x tr δ δ S ( ) ) )( n ( 1 1 ( ( S Z 1 1 γ γ C = k k j j ∈ ( ∑ γ 2 i δ δ σ n ) 4 = 2 M O = = x . The result is ( )( i 3 2 2 i 1 j i † ) = n x ) k ∞ n † † l )( ))) S Z Z 2 ( Z . x ( )( Operator as trace in tensor space C ( 2 1 1 ··· σ Z x 1 ( ( k O 1 2 1 † 1 l j h i σ New New Section Page4 1 Z Z n n O j i h operators in tensor space by permutation operators. This can be Z † Figure 1: ))( 1 Z ··· x , 1 1 ( j i Z Z Z ( h 1 σ 15:52 O h . The correlator is calculated using the basic correlator n n 2 ) S 2 x ∈ 2 − σ 1 , x

22 March 2016 1 ( σ The correlator can be computed by putting together ( It turns out that permutations are important, not just for enumerating gauge invariants, but N-dependent Combinatorics of gauge theories. where and Wick’s theorem, which implies that be understood diagrammatically as in Figure also for computing their correlators. Considerinvariant a polynomial two of point function fixed involving degree afree holomorphic and field gauge limit. an This anti-holomorphic is polynomialtwo-point non-zero of functions if are fixed the thus degree two of polynomials in the have form the the same degree. The non-vanishing The enumeration of gauge invariants is the same as enumerating conjugacy classes in is a hidden algebra which organizes gauge invariants. We have dropped the space-time dependence which isamounts trivial. to Performing the relacing free field the pathexpressed integral diagrammatically in Figure PoS(CORFU2015)107 by 2 σ (2.12) (2.13) , 1 3 σ σ ! since the C n N ) 3 σ σ 0 2 C σ 0 Sanjaye Ramgoolam 1 ) σ ( 3 σ δ 1 n S − ∈ γ 3 2 σ , 2 , denoted by T γσ

1 ∈ σ 0 ∑ 2 1 is the identity permutation and 0 σ , dividing out by the sizes of these σ , − ( 2 γ 1 as a way to parametrize the gauge 1 T δ σ T sum can be replaced by , − 3 2 ∈ n σ 0 1 1 γ γσ S C T σ 1 σ | N C 2 n

T S N σ ! ∈ || n C n 3 1 S ∑ σ N T ∈ , ∑ branched over three points on the sphere. These | γ γ 1 5 = = = P ) = σ ( n ⊗ N V is defined to be 1 if tr ) Correlator as trace in tensor space σ ( New New Section Page5 1 δ ) to the sphere Σ Wick contraction sum as operator in tensor space New New Section Page6 1 only depends on the conjugacy class, we have replaced Figure 3: σ . With these sums in place, the | O 2 19:29 T | , . | Figure 2: 19:31 γ 1 T |

22 March 2016

22 March 2016 . n S of a permutation is given in terms of the number of cycles in This shows that the introduction of permutations in The final formula has an interpretation in terms of counting of branched covering maps from n ⊗ N N-dependent Combinatorics of gauge theories. The first step comes from the diagrammatic manipulation. In the second we use that, the trace in In the third line, we havegroup introduced to an write extra permutation a and neater a formula. delta function on the permutation Riemann surfaces (worldsheets otherwise. Finally since sums commute with invariant operators is extremelytheory useful, of since correlators can be given neatly in terms of group sums of permutations in their respectiveconjugacy conjugacy classes classes V PoS(CORFU2015)107 (2.14) (2.15) (2.16) 4 SYM, = N Sanjaye Ramgoolam boxes. We can define n can be written in terms of , i.e. the trace of the matrix 1 R + N trZ ) Z ], subsequently shown to diagonalize R ( ] to argue that giant gravitons are not d . In this case, there are relations among DimR σ 9 13 ! N in the irrep , O n ) > is to use the Fourier transform relation from σ 12 RS σ which are in the inverse image of three points , with conjugation equivalence, form a good n ( , 1 δ ) N Σ R P Z χ 11 = ( n , i 6 σ S → † ] and understood in terms of enhanced symmetries ∈ O ∑ ]. These types of holomorphic maps are called Belyi 10 Σ were identified with subdeterminant operators. With σ )) 7 14 : 5 ! ] these operators have diagonal 2-point function. Z , 1 f S ( n 3 6 S background of string theory, are related directly to Belyi [ 5 O } group element S ) = , correspond to Young diagrams with ∞ )( n n , Z Z × S S ( 1 ( 5 , R R 0 of O O { R h AdS this is not the case for N effects. While the ≤ N . It is shown that [ n R ]. 3 ]. 15 in irrep is the character of the σ ) σ ( , which can be taken as R 1 χ P The above approach to matrix invariants, based on permutations and diagrammatic tensor Irreducible representations It is rather amazing that correlators in the distinguished half-BPS sector of The power of the permutation approach to gauge-invariants becomes particularly manifest ]. 8 N-dependent Combinatorics of gauge theories. the 2-point functions in the 2-matrix sector [ in gauge theory [ traces due to the Cayley Hamilton theorem which implies that operators corresponding to Young diagrams as 3. Multi-matrix model space methods, extends to multi-matrixproaches models. to 1-matrix This models, should e.g.matrix be generalization. contrasted reduction with to The many eigenvalue enumerationplications dynamics, other and to which ap- correlators quarter do of not and multi-matrixdescribed admit invariants eighth above has multi- generalize BPS direct to sectors ap- give of restrictedof Schur SYM. open operators. string I These excitations will were of explain introduced giant in how gravitons the permutation [ study methods with vanishing holomorphic derivatives at points on basis of operators when multi-traces. An elegant way to givegroup a theory, basis which at relates finite conjugacy classes to irreducible representations where representing Failure of diagonality oftraces. the Single trace giants basis which are had largethe been in Young the diagram used basis [ ingiant hand, graviton there states are [ natural candidates for CFT duals of single and multi- maps and have deep implications for[ number theory, in connection with the absolute Galois group branched covers are holomorphic maps on the equivalently in a distinguished when we consider finite maps, of central importance inof number Mathematics. theory, a subject which Gauss famously called the Queen PoS(CORFU2015)107 (3.1) (3.5) (3.3) (3.2) (3.4) is the n ) acting on 1 − γ and σ 1 Z ) ) ) − i n y n + Y m , + ( m − σ i Z γσγ i Sanjaye Ramgoolam i 1 ( z Y δ − n ··· + 1 ) ) m permutations 1 ( 1 S 1 + ∑ n − 1 ∈ m + ( S ∞ γ . There are linear operators m = σ copies of σ i i ∏ 2 i n R S × Y n ) is the number of = , × γσ m m ( m 1 m S m ∑ σ m S i i σ n ∈ ( Z γ S n β ! + C ) ) × m ··· n in irreps † ! N 1 ⊗ ) N Y m 2 ) ( n V , m S 2 σ , where β i + i tr , these are the only equivalences. Counting Z n 1 ( m Z ∈ = N γ 1 + ) − S UYU 1 ∑ − γ ( m , γ 1 σ < † 1 Y S i i ∈ = , − n γσγ Z 2 i Z σ 7 can be taken to run over an orthonormal basis for † O + = γσ for J )) UZU 1 m
( , σ Y I ) = σ , ( n 1 Y → Z δ , ⊗ − ( n ) 2 Z + Y copies of σ Y ( m , where the gauge symmetry acts as Number of operators with , S ⊗ σ n where ∑ O γσγ Y counting and correlators can be obtained by going to a Fourier ( ∈ Z , , n m ) O ( β )( m y Z ∼ N n ⊗ σ Y S m ( Z , z σ × Z R IJ 0 m subgroup. The free field 2-point function is n ∑ ( limit, i.e. when S = D + 1 n ∈ ∞ n m σ , ∑ S γ N ⊗ N m O V × h = is in the symmetric group tr by conjugation : m S n σ + ) = m Y S , Z ∈ ( . In the large σ in the gauge-invariant operator. In this two-matrix case, the equivalence takes the σ n S O Y gauge group on a two-torus, with a defect inserted on a circle, which constrains the × n m Number of operators with + which give functions S m ) S ∈ Consider the two-matrix problem, of enumerating gauge invariants built from two matrices. In As in the 1-matrix case, finite σ γ ( R N-dependent Combinatorics of gauge theories. As in the 1-matrixwhich case, determine we how the observe lower that indices are gauge contracted invariants with can the upper be indices parametrized by permutations This formula can be used to obtain the generating function The expression also has an interpretation aswith a partition function for topological lattice gauge theory holonomy to be in the the context of the quarter-BPS sectorinvariants of built SYM, from we two are matrices interested in holomorphic polynomial gauge for basis using matrix elements of the permutation number of form gauge invariant operators is equivalent to enumerating orbits of D Here the permutation permutations From the Burnside Lemma,points, we so know that that the number of orbits is equal to the number of fixed PoS(CORFU2015)107 , ) N of σ R ( (3.9) (3.8) (3.7) (3.6) 2 (3.10) ν , 1 ν , 2 2 R )) , 1 R R ) R , n . These can be χ 2 , n R S m , ] for the counting ; 1 × Sanjaye Ramgoolam R Λ 20 ( ( m g S M ( ) to have no more than i 1 Λ , it is useful to use group ) R , ν n ; Y R S , 2 , boxes Z × R m , which appear with multiplici- . The subgroup basis states can n ( ( , 2 ) m σ 1 ∑ 2 R C S R R , m O , , 1 1 , ) into irreps of 2 R with R 2 R R σ n V 2 R , ( + , 2 1 R boxes m 1 ⊗ ν . These restricted characters are used to , boxes respectively. A representation R ]. Another formula [ ) S 1 R i n ( n n | ν n 1 , S S 2 g is a multiplicity label taking values between I , ( 2 19 ν of + R , 1 R ; ∑ × , m is V R ν 2 1 R m ] from the group integral formula for counting boxes can be used to define functions 2 R m R ⊗ ih m R S , i χ I , ) 18 m 1 1 , 8 n 1 m and R with R ∑ ∈ + S 1 ν R with 2 ( m m | V ; 2 R γ , S 2 R ∑ V R 2 Λ I ∈ , with 2 , ∑ m 1 R σ 1 R , , 1 , R 1 1 R 1 = M R R m R i ) = | rows , 1 = 2 Y ν ) N , ; R n is , 2 rows Z constraints are easy to implement and they follow from Schur- + boxes 1 ( m m 2 Λ S R N , n N ν | ( , R 1 I 1 boxes V + , ν m , ∑ , R n 2 m 2 h R , + 1 R R , R ∑ m 1 counting of 2-matrix invariants is given by χ counting of 2-matrix gauge invariants with R | N N ]. They amount to restricting the Young diagram R counting of 2-matrix gauge invariants with has no more than 17 . we may write the relevant decomposition as representations N , ) R R equal to the Littlewood-Richardson coefficients. Thus there is a subgroup-adapted ) R ) has no more than Finite 16 , 2 = are state labels for the irreps 2 ( R R , 2 R 2 U . Since the functions of interest are invariant under , Finite , with states of the form m = 1 R R R , , 1 R V 1 ( m R g can be decomposed into a direct sum of representations ( In this Fourier basis, finite g n + m N-dependent Combinatorics of gauge theories. Weyl duality [ called restricted characters, labelled by threefunctions Young are diagrams invariant and under two conjugation multiplicitydefine by indices. restricted Schur These Polynomials be expanded in terms of a general orthonormal basis The branching coefficients where the dimension of the multiplicity space rows. Thus the finite This counting formula can begauge obtained invariants directly which [ was introduced by Sundborg [ in terms of the irrep theoretic data associated with reduction of the irrep 1 and parametrized by a pair of Young diagram where basis in S ties PoS(CORFU2015)107 ) ) n , N ( m , U (3.12) (3.14) (3.11) (3.13) Λ ] ( × ) 14 M lead to the N ( in the decom- U 3.9 , and n R S ⊗ × Sanjaye Ramgoolam R m S in 2 symmetry. The Noether µ , Λ of ) 1 2 ] µ global symmetry quantum n δ N [ ) 2 ( 2 ν , ⊗ ( 1 U in terms of the expressions ] ν U ) σ δ m n [ S to define a non-commutative asso- , , ) R ) m σ δ n ( ( 2 + ∝ ν m , A i 1 S † ν ( , 2 )) C R , VY Y 1 , , R R Z χ ( n 9 2 UZ + µ m ]. For the two-matrix case, we have a , 1 S ]. Brauer algebras provide another useful free field → ∑ µ ∈ , 15 2 σ Y 22 S , , , 1 = Z S S 2 χ 20 ν , 1 )( ν , . The equivalence of the two counting formulae was shown in Y 2 n , R S , Z 1 ( R R 2 × ν , Q m 1 ]. The starting point of this discussion is that the averaging over the S ν , ) and using the product in 2 24 R ]. Links to two dimensional topological field theory have been described , on these equivalence classes. The restricted characters in 1 3.3 R ) R 26 n χ , , h Kronecker coefficient (Clebsch-series multiplicity) for m 25 ( n , S A 18 into irreps of Λ is the ]. The operators in these different bases diagonalise commuting sets of Casimirs for ) Λ 23 , R Diagonal bases in the 2-matrix problem, with well-defined Recently these results have been generalized to the counting and correlators of quiver gauge The restricted Schur operators have diagonal 2-point function in the free field limit [ , ]. R ( 21 N-dependent Combinatorics of gauge theories. numbers have also been constructed [ theories, where the gauge group is a productfundamentals of [ unitary groups and the matter isand in bifundamentals some or surprising connectionshave to been described. trace This monoids arises because which theis have counting function an applications for infinite large in ranks product of computer the offrom science Unitary inverse groups an determinants. alphabet The consisting inverserelations determinant of between counts letters the words letters. corresponding constructed to simple loops, with partial commutation Matrix (Wedderburn-Artin) decomposition of the algebra This gives gives the intrinsic permutation meaningmatrices. of the The restricted structure characters, of without reference theseand to algebras can is closely be related viewed toan as Littlewood-Richardson appropriately coefficients, a defined sort Cartan (theimportant of role diagonals categorification in of the thereof. definition the of Wedderburn minimal The Artin sets decomposition) relations of charges. play between3.1 an the centre Quivers and enhanced symmetries in the free fieldsymmetry limit of [ left multiplication by unitary matrices We can also multiply on the right.charges For for each these matrix different there symmetries isresentation in can fact theory be a used labels to used constructdetermining to Casimirs the construct which labels the measure of different the thetion restricted bases. rep- centralizer Schur algebras basis Finding leads [ minimal toequivalence sets the classes of study in of charges ( the structureciative of algebra permuta- is the multiplicity of the irrep of the symmetric representations basis [ [ position of C PoS(CORFU2015)107 ⊗ ) n (4.1) (4.4) (4.2) (4.3) S which ( branch ) C d ’s. Thus 3 ¯ σ Φ , 2 effects can be σ , ) n 1 ( N ’s. The upshot is and its conjugate 3 σ γ σ ( k N ]. Explicit generat- , Sanjaye Ramgoolam ) V ) n used to construct the ( 28 1 2 ⊗ σ − 3 ¯ j Φ , N σ ) n 2 V ( 1 γ σ 3 ⊗ i and σ ) . The product of these three N Φ 1 , and lifting paths going round 1 ) Φ V ]. As in the use of permutations γ ) 1 } − 2 ( 2 ··· n − τ 28 γ δ ) , ) 1 1 3 ) ( 2 − 1 3 τ σ σ γ ··· 1 − 1 , where the equivalence is generated ) and taking the products in 2 k , 2 , ) γ τ . ) σ , ( 2 1 1 symmetry of permuting the n ( 2 , 2 τ γτ 4.2 { 2 S γ 2 γ , ( σ n j 2 2 τ 1 S , δ ∈ , ) τ σ σ ) 1 1 is equivalent to counting permutation triples ( 1 ( 2 1 1 1 τ γ γ γ σ − n ( 1 ( , i , τ 1 2 . This in turn is the same as counting branched δ 1 Φ γ γ ) γ n 1 − 1 10 k and an , γ σ − n which transforms as 1 j 1 1 , Φ σ -indices. k γ n , i 2 j ( k γτ is equivalent to counting branched covers with , γ ( ¯ i Φ . The three ranks can, more generally, be chosen to be 1 ∼ δ d Φ N σ n ) ¯ 1 V ··· S 3 γ 1 ∈ ( σ k 2 ⊗ , τ , ∑ 1 , δ j 2 N 1 3 , ¯ τ 1 σ V σ which determines the number of i n , , 2 S ¯ 1 n ⊗ Φ σ ∈ ∑ ∑ , σ γ 1 N ( σ ¯ ! V 1 n n ) = -indices and the S j ¯ ∈ , forming a sort of categorification thereof. Φ 2 , n γ ∑ , S 1 Φ γ ( 3 2 ! σ from Riemann surfaces to a sphere with three branch points. After choosing a 1 , n 2 n σ , symmetry of permuting the 1 leads to another important example of the permutation centralizer algebras defined σ n ) -indices, the n O S i subject to equivalences generated by S ]. ) ( 3 C 27 σ , ⊗ 2 ] (see concluding section there). The structure of this algebra is closely related to Kronecker , which transforms as ) Choose a positive integer Averaging over the equivalence classes defined by ( Permutation methods, of the type above which have been useful in enumerating multi-matrix Consider a complex 3-index tensor σ k n , , 24 j S 1 , i ( σ N-dependent Combinatorics of gauge theories. ¯ base point on the sphere, labelling the inverse images as product coefficients in Φ unequal. Polynomial invariants areindices constructed [ by contracting the upper indices with the lower There is an the counting of 3-index tensor invariants of degree This is the countingby of simulataneous equivalence conjugation classes by of acovers pairs permutation of degree ( ing functions for the counting,expressions new for connections correlators with were found. branched covers, as well as group theoretic the three branch points,permutations we is recover the permutations identity, ashomotopic it to should the be, tivial since path,that the counting path with of surrounding tensor trivial invariants all monodromy. of three rank points. Generalizing branch This is the points used is above to arguments constructfor explicit shows generating multi-matrix functions theories, in the [ applicationcaptured to by tensor performing the models Fourier also transform shows to that representations. finite C in [ invariants as outlined above, were used to approach the tensor invariants in [ 4. Tensor models invariants. These invariants can be parametrized by a triple of permutations contract the Using the Burnside Lemma, the number of these invariants is This expression can be easily simplified.that we The can first write step the is counting to as solve for one of the PoS(CORFU2015)107 n ]. S / 35 n ], the , ) 4 47 ]). This 34 T , ( 44 , 46 , 43 , 45 -dependent com- 42 n , Sanjaye Ramgoolam 41 , 40 , 1 superconformal theory on the orbifold (1998) 231] doi:10.1023/A:1026654312961 ) 2 4 11 , ]. Generalizations beyond unitary groups have been ]. 4 37 52 , in particular its centre, controls the , = ( ) , n S 36 51 ] and references therein). Some progress has been made in Acknowledgments ( N knows about the interactions of gravitons in AdS space, via ], the systematic classification of light-cone string diagrams C 33 , ) 49 ∞ , 32 S remains elusive. One would like to construct the BPS operators in ( , 48 C N 31 , 30 , , along with its representation theory and relations to Lie groups via Schur- ) 29 ∞ (some papers on the relevant combinatorics are [ S 3 ( S C 2d Yang Mills theory [ × ]. N 3 (1999) 1113 [Adv. Theor. Math. Phys. 39 , 38 AdS 38 Phys. [hep-th/9711200]. The algebra I thank the organizers of the Corfu2015 workshop on Non-commutative Field Theory and It is noteworthy that the algebra I have described a body of techniques based on permutations and diagrams to approach prob- ] and large [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. 50 N-dependent Combinatorics of gauge theories. Weyl duality, contains a lotexamples of and a information deeper about understanding holography of this and phenomenon quantum would be field fascinating. theory. Further holography. Elsewhere it has also been used in the classification of brane-tilings [ Gravity for the opportunity totension present allowed this to talk complete inquestions September the 2015 and proceedings and feedback. contribution. the generous Idraw I deadline on have thank ex- papers taken the that the have audience liberty appearedgrateful for since, to to but stimulating add all which my a fit collaborators few wellStefan on with small Cordes, the the Steve points research content Corley, in presented of David this the here: Garner,Heslop, Laurent talk. article Robert Joseph Freidel, de I which Ben Amihay Mello am Geloun, Hanany, Koch, also Tom Yang Antalhail Brown, Hui Jevicki, Mihailescu, He, Vishnu Greg Jejjala, Paul Moore, Yusuke Jurgis Kimura, Pasukonis, Paolo Diegoresearch Mattioli, Rodriguez-Gomez, is Rak Mi- supported Kyeong by Seong. STFC My consolidatedDuality” grant ST/L000415/1 “String Theory, Gauge Theory & References finding the general quarter or eighth BPS groundtion states, operator which acting are ground on states holomorphic of theThe invariants 1-loop built general dilata- from solution two at finite or threecorrespondence complex with matrices states [ of azation harmonic of oscillator giant in graviton moduli two spacesstudied dimensions, [ [ as expected from quanti- gives another instance where [ lems of enumerating gauge invariants andmodels, computing quiver correlators gauge in theories 1-matrix and models, tensorof multi-matrix models. loop-corrected They dilatation are operators, also particularly usefulgiant in in gravitons the computing ( the sector see spectrum [ of perturbations around half-BPS 5. Conclusions and Outlook binatorics of the orbifold theories. 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