Operators, Correlators and Free Fermions for SO(N) and Sp(N)
aa r X i v : . [ h e p - t h ] M a r WITS-CTP-114
Operators, Correlators and Free Fermionsfor
S O ( N ) and S p ( N ) Pawel Caputa a , Robert de Mello Koch a,b and Pablo Diaz a a National Institute for Theoretical Physics,Department of Physics and Centre for Theoretical Physics,University of Witwatersrand, Wits, 2050,South Africa b Institute of Advanced Study,Durham UniversityDurham DH1 3RL, UK
ABSTRACT
Using the recently constructed basis for local operators in free SO ( N ) gauge theory we derivean exact formula for the correlation functions of multi trace operators. This formula is usedto obtain a simpler form and a simple product rule for the operators in the SO ( N ) basis.The coefficients of the product rule are the Littlewood-Richardson numbers which determinethe corresponding product rule in free U ( N ) gauge theory. SO ( N ) gauge theory is dual to anon-oriented string theory on the AdS × R P geometry. To explore the physics of this stringtheory we consider the limit of the gauge theory that, for the U ( N ) gauge theory, is dual tothe pp-wave limit of AdS × S . Non-planar unoriented ribbon diagrams do not survive thislimit. We give arguments that the number of operators in our basis matches counting usingthe exact free field partition function of free SO ( N ) gauge theory. We connect the basis wehave constructed to free fermions, which has a natural interpretation in terms of a class of -BPS bubbling geometries, which arise as orientifolds of type IIB string theory. Finally,we obtain a complete generalization of these results to Sp ( N ) gauge theory by provingthat the finite N physics of SO ( N ) and Sp ( N ) gauge theory are related by exchangingsymmetrizations and antisymmetrizations and replacing N by − N . [email protected] [email protected] [email protected] ontents SO ( N ) basis 84 Multi trace Correlators Again 105 Counting 146 Link to Free Fermions 177 Sp ( N ) gauge theory 198 Discussion 24A Simplifying the SO ( N ) basis 25B Correlation Functions 27C Jacobians 29 C.1 U ( N ) Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29C.2 SO ( N ) Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31C.3 Sp ( N ) Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 In a previous paper[1] we have initiated the study of local operators in SO ( N ) gauge theorywhich have a bare dimension that can depend parametrically on N . For these operators, oneneeds to sum more than just the planar diagrams to capture the large N limit. We dealtwith this problem by employing group representation theory to define local operators whichgeneralize the Schur polynomials of the theory with gauge group U ( N ). We found that thefree field two point function is diagonalized by our operators. In this article we will extendour understanding in a number of important ways.A basic result of [1] is the basis of local operators, given by O R ( Z ) = 1(2 n )! X σ ∈ S n χ R ( σ ) σ i i i i ··· i n − i n i n i n − j j ··· j n − j n Z j j Z j j · · · Z j n − j n (1.1)1or SO ( N ) with N even, we also need to include Q R ( Z ) = ǫ i i ··· i N ( N + 4)! X σ ∈ S N +2 p χ R ( σ ) σ i i ··· i N − i N i N +1 i N +2 i N +2 i N +1 ··· i N + p − i N + p i N + p i N + p − j j ··· j N +2 p − j N +2 p × Z j j Z j j · · · Z j N +2 p − j N +2 p − Z j N +2 p − j N +2 p (1.2)We will focus our discussion on the O R ( Z ) which we understand better than the Q R ( Z ).The operator label R for O R is a Young diagram with 2 n boxes, that is, an irreduciblerepresentation of the symmetric group S n . To obtain a non-zero operator O R ( Z ), n mustbe even. Thus, 2 n is divisible by 4. In fact, the only representations R which lead to anon-zero operator are built from the “basic block” as we now explain. Choose a partitionof n/
2, or equivalently a Young diagram r with n/ R are obtained by replacing each box in r by the “basic block” . To reflect the relationbetween r and R , we use the notation r = R/
4. Thus, the number of gauge invariantoperators of the type O R ( Z ) built using n fields is equal to the number of partitions of n .As an example, for n = 4 the allowed labels are R / , R / R = , R = (1.4)Given the form of the Young diagrams R we consider, we will list the row lengths of R as( r , r , r , r , · · · , r N , r N ). In section 2 we start by using the results of [1] to derive an exactformula for the free field theory correlation functions of multi trace operators. The result isgiven in equation (2.15). Using this formula we explain in section 3, how to obtain a simplerform for the O R ( Z ) of our SO ( N ) basis. The resulting operators, with a more convenientnormalization, are χ S ( Z ) = 1 (cid:0) n (cid:1) ! X ν ∈ S n − l ( ν ) χ S/ ( ν )Tr V ⊗ n ( ν ( Z ) ⊗ n ) (1.5)With the new normalization, the two point function of these operators is h χ R ( Z ) χ S ( ¯ Z ) i = δ RS Y i ∈ odd boxes in S c i (1.6) Recall that there are two invariant tensors for SO ( N ): the Kronecker delta δ ij and the ǫ i i ··· i N . We canuse either of these tensors when contracting indices to obtain gauge invariant operators. For SO ( N ) with N odd ǫ i i ··· i N has an odd number of indices, so that we can’t build a gauge invariant operator that uses onlya single ǫ i i ··· i N . c i in the above formula. A box appearing in column a and row b has factor N + a − b . Theright hand side is equal to the product of the factors of the boxes in every second row asshown below ∗ ∗ ∗ ∗∗ ∗ (1.7)We called these the odd boxes in [1] because they referred to boxes labeled with an oddinteger in a Young-Yamanouchi labeling of the states in the S n irreducible representation.With the new simplified form of the SO ( N ) operators, we are able, in section 3, to give aproduct rule for our operators. The product rule is ( S/ ⊢ n , R/ ⊢ n ) χ S ( Z ) χ R ( Z ) = X T/ ⊢ n n g R/ S/ T/ χ T ( Z ) (1.8)where g R/ S/ T/ is the Littlewood-Richardson coefficient.These results constitute a rather complete understanding of the local operators in SO ( N )gauge theory, comparable to what has been achieved for the U ( N ) theory. This programwas initiated in the context of U ( N ) gauge theory, by Corley, Jevicki and Ramgoolam in[2]. In particular, [2] showed that the half-BPS operators constructed using a single complexmatrix can be described using Schur polynomials and they demonstrated that the Schurpolynomials diagonalize the free field two point function. The study of the finite N physicsof U ( N ) gauge theories is by now well developed. There are a number of bases of localoperators that diagonalize the free field two point function[3, 4, 5, 6, 7, 8, 9, 10] and weknow how to diagonalize the one-loop dilatation operator[11, 12] for certain operators dualto giant graviton branes[13, 14, 15, 16, 17]. This diagonalization has provided new integrablesectors, with the spectrum of the dilatation operator reducing to that of decoupled harmonicoscillators which describe the excitations of the system[14, 15, 16]. Integrability in the planarlimit was discovered in [18, 19] and is reviewed in [20]. For a study of the SU ( N ) theory see[21].Given the results we have developed, we are now in a position to probe the finite N physics of SO ( N ) gauge theory. Recall that according to the AdS/CFT duality[22, 23,24], finite N physics of the gauge theory[25] corresponds to non-perturbative (in the stringcoupling) physics of objects such as giant graviton branes[26, 27, 28] and the stringy exclusionprinciple[29]. N = 4 super Yang-Mills with SO ( N ) or Sp ( N ) gauge group is dual to theAdS × R P geometry[30]. In this case one expects a non-oriented string theory so thatthe study of non-perturbative stringy physics, which is captured by the finite N physicsof the gauge theory, is likely to provide new insights extending what can be learned fromthe AdS × S example which involves oriented string. For studies in this direction see [31].3omputations in the gauge theory, must sum both the planar and the non-planar diagrams.At the non-planar level there are genuine differences between the U ( N ) and the SO ( N ) or Sp ( N ) gauge theories. Recall that matrix model Feynman diagrams in double line notationrepresent discrete triangulations of Riemann surfaces[32]. For Hermitian matrices we dealwith oriented triangulations whereas for symplectic or anti-symmetric matrices unorientedtriangulations[33]. In general, a Feynman diagram is weighted by λ g − b + c N − c − g +2 , (1.9)where N is the number of colors, λ = g Y M N is the ’t Hooft coupling, g is the numberof handles, b the number of boundaries and c the number of cross-caps on the surface.Thus, for the SO ( N ) or Sp ( N ) gauge theories, the leading non-planar corrections come fromribbon graphs that triangulate non-orientable Feynman diagrams with a single cross-cap.The large N limit of correlation functions of operators with a bare dimension that dependsparametrically on N are sensitive to this non-planar structure of the theory. With the goalof probing this structure, in section 4 we use our technology to compute free field theorycorrelation functions of multi trace operators, to all orders in 1 /N . Given these correlators,we can consider the double scaling limit defined by [34, 35] N → ∞ and J → ∞ with J N fixed , g Y M fixed (1.10)where J is the number of fields in the gauge theory operator. In this limit some non-planar diagrams (string interactions) survive, giving a non-trivial normalization of the twoand three-point correlators of single trace operators. This limit is particularly interestingbecause in the dual gravity picture it corresponds to taking a pp-wave limit of AdS × S ,a background in which the superstring theory can be quantized. We find that non-planarunoriented diagrams in SO ( N ) gauge theory do not survive this limit.We have argued that O R (and for N even, the Q R ) give a basis. A weak point in ourargument is that we have not demonstrated that these operators are a complete set. Thisissue is considered in section 5. We focus on N even, which is the more involved case, as aconsequence of the fact that we may use ǫ i i ··· i N when constructing gauge invariant operators.By counting the number of operators we have constructed, we are able to reproduce the exactfree field partition function of the SO ( N ) gauge theory[36] for N = 4 , N , in (5.3) and at anyodd N in (5.14).The basis that we have constructed allows us to study the dynamics of the gauge in-variant observables of a single matrix model. Of course, this problem can be reduced toeigenvalue dynamics which is itself equivalent to the dynamics of free fermions in an exter-nal potential[37]. The Schur polynomial basis for the U ( N ) gauge theory has a very directlink to free fermion dynamics. It is natural to ask if there is a similar connection betweenfree fermions and the basis we have constructed. We develop this link in some detail in4ection 6 and show that there is indeed a natural connection to free fermions. Our operatorscan be mapped to states of fermions moving in a harmonic oscillator potential, with definiteparity and maximum angular momentum for a given energy.Although we will not do so in this article, we have developed enough technology that itwould be natural to initiate a systematic study of the dilatation operator in non-planar large N limits of the SO ( N ) and Sp ( N ) gauge theories. A detailed study of the planar spectralproblem of N = 4 super Yang-Mills with gauge groups SO ( N ) and Sp ( N ) has been carriedout in [38]. The essential difference between the theories with gauge groups U ( N ) or SO ( N )is that in the SO ( N ) case certain states are projected out. Thus, the planar spectral problemof the SO ( N ) theory can again be mapped to an integrable spin chain[38]. It is interestingto ask if the new integrable sectors discovered in [14, 15, 16] are also present in large N butnon-planar limits of SO ( N ) and Sp ( N ) gauge theory.It is well known that there is a close relationship between SO ( N ) group theory and Sp ( N )group theory. These relations imply that the dimension of a given irreducible representationof Sp ( N ) is equal to that representation of SO ( N ) with symmetrizations exchanged withantisymmetrizations (i.e. transpose the Young tableau) and N replaced by − N [39]. TheQCD loop equations for SO ( N ) gauge theory and Sp ( N ) gauge theory in 3 + 1 dimensionsenjoy the same connection[40]. This same relation has been observed in two dimensionalYang-Mills theory[41]. Motivated by this background, we proceed to study the finite N physics of Sp ( N ) gauge theory. We are able to argue that precisely the same connectionrelates the finite N physics for the orthogonal and symplectic gauge theories. In this way, insection 7 we obtain a rather complete description of the finite N physics of the Sp ( N ) gaugetheory.In section 8 we outline some open problems that we find interesting. Our goal in this section is to give an exact formula for the free field theory correlationfunctions of multi trace operators in SO ( N ) gauge theory. In our discussion below n , n and2 n will enter at various points. The reader is encouraged to keep in mind that our operatorsare built using n fields.We start by choosing any partition ν ⊢ n . The partition is then translated into the cyclestructure of a permutation. For example, if n = 4 there are 5 possible choices for ν , namely(4), (2) , (2) (1) , (3) (1), or (1) . We use these partitions ν (see [1]) to construct σ ν ∈ S n with cycle structure given by multiplying each of the parts of ν by 4. For the above list ofpartitions ν the cycle structures of the corresponding permutations σ ν are (16), (8) , (8)(4) ,(12)(4) and (4) . We will now explain, by providing a few examples, how we associate toeach cycle structure a canonical permutation. The cycle structure (12) (4) is associated to5he cycle σ ν = (1 , , , , , , , , , , , , , ,
16) (2.1)while (8) (4) is associated to σ ν = (1 , , , , , , , , , , , , ,
16) (2.2)So, the rule for obtaining the canonical permutation is to populate the largest cycles first,starting from 1 and counting up, until the permutation is completely determined. Notice thatthe canonical permutations are composed of cycles with cycle lengths that are a multipleof 4 and further, they always take an even number to an odd number. These canonicalpermutations can be used to define a “contractor” as follows C σ ν J = C σ ν j j ··· j n = n Y p =1 δ j p j σ (2 p ) C Jσ ν = C j j ··· j n σ ν = n Y p =1 δ j p j σ (2 p ) (2.3)To see that all indices appear on the right hand side and no index appears more than once,it is useful to remember that σ ν always takes an even number to an odd number. Using thecontractors, we can define the operators O σ ν R ( Z ) = 1(2 n )! X β ∈ S n χ R ( β ) C σ ν j j ··· j n β j j ··· j n i i ··· i n − i n Z i i · · · Z i n − i n (2.4)These operators are particularly convenient for the question of correlation functions of multitrace operators. Indeed, using the identity δ ( σ ) = 1(2 n )! X σ ∈ S n d R χ R ( σ ) (2.5)we easily find ( Z ) i i ν (1) ( Z ) i i ν (2) · · · ( Z ) i n i ν ( n = X R d R O σ ν R (2.6)Thus, any multi trace operator can easily be written as a linear combination of the O σ ν R .Introduce the notationTr V ⊗ n ( ν ( Z ) ⊗ n ) ≡ ( Z ) i i ν (1) ( Z ) i i ν (2) · · · ( Z ) i n i ν ( n (2.7)Every multi trace operator can be written in this way for a suitable ν . Thus, to evaluate anarbitrary two point function of multi trace operators, all we need to do is to compute thecorrelator h O σ ν R ¯ O σ µ S i . Using the results in [1], it is straight forward to see that h O σ ν R ( Z ) ¯ O σ µ S ( Z ) i = δ RS n !2 n (2 n )! d R X ψ ∈ S n Tr( P [ A ] Γ R ( ψ )) C νI ( ψ ) IJ C J µ (2.8)6e can define a permutation σ νµ by the condition C νI ( σ νµ ) IJ = C µJ (2.9)By considering an example at this point, it will be clear that σ νµ defines a unique elementof the double coset H ν \ S n /H µ , where H µ ( H ν ) is a stabilizer of σ µ ( σ ν ) respectively.Indeed, for σ ν = (1 , , , , , , , , , ,
12) (2.10)and σ µ = (1 , , , , , , , , ,
12) (2.11)we have σ νµ = (1 ,
5) (2.12)See Figure 1 for an illustration of this example.Figure 1: Relating C νI and C µI with a permutation.With the use of σ νµ we can now write h O σ ν R ( Z ) ¯ O σ µ S ( Z ) i = δ RS n !2 n (2 n )! d R X ψ ∈ S n Tr( P [ A ] Γ R ( ψ )) C µI ( σ − µν ψ ) IJ C J µ = δ RS n !2 n (2 n )! d R X ψ ∈ S n Tr( P [ A ] Γ R ( σ µν ψ )) C µI ( ψ ) IJ C J µ (2.13)Since we are contracting row and column labels of ψ in the same way, the computationbecomes very similar to the computation of [1]. In the next formula we introduce an S n [ S ]subgroup that belongs to the stabilizer of C µJ . The reader should recall[1] that P [ A ] is definedusing the embedding of S n [ S ] that stabilizes (1 , , · · · (2 n − , n ). It is now straightforward to obtain h O σ ν R ( Z ) ¯ O σ µ S ( Z ) i = δ RS n !2 n (2 n )! d R X ψ ∈B n X ψ ∈ S n [ S ] Tr( P [ A ] Γ R ( σ µν ψ ψ )) C µI ( ψ ψ ) IJ C J µ δ RS n !2 n (2 n )! d R X ψ ∈B n X ψ ∈ S n [ S ] Tr( P [ A ] Γ R ( σ µν ψ ψ )) C µI ( ψ ) IJ C J µ = δ RS ( n !2 n ) (2 n )! d R X ψ ∈B n Tr( P [ A ] Γ R ( σ µν ψ ) ˆ P [ S ] ) C µI ( ψ ) IJ C J µ = δ RS ( n !2 n ) (2 n )! d R Tr( P [ A ] Γ R ( σ µν ) ˆ P [ S ] ) Y i ∈ odd boxes in R c i = δ RS l ( ν )+ l ( µ ) χ R/ ( µ ) χ R/ ( ν ) d R Y i ∈ odd boxes in R c i (2.14)which is a remarkably simple formula. To obtain the last line we have used the results of[42] as explained in Appendix C of [1]. Using this in a completely straight forward way wefind h Tr( µ ( Z ) ⊗ n )Tr( ν ( ¯ Z ) ⊗ n ) i = X R/ ⊢ n l ( ν )+ l ( µ ) χ R/ ( µ ) χ R/ ( ν ) Y i ∈ odd boxes in R c i (2.15) SO ( N ) basis In this section we will argue that the result (2.15) allows us to write a simpler description ofour basis, that is closely related to the Schur polynomial basis of the U ( N ) theory. Towardsthis end, we begin by computing the correlator h O R ¯ O σ ν S i . First, it already follows from theresults of [1] that h O R ¯ O σ ν S i ∝ δ RS (3.1)so that we only need to compute h O R ¯ O σ ν R i . Now, noting that O R = O σ ν R with ν = 1 n , wecan apply (2.14) to find h O R ¯ O σ ν S i = δ RS n + l ( µ ) d R/ d R χ R/ ( µ ) Y i ∈ odd boxes in R c i (3.2)Now, since the O R constitute a basis and since (3.2) gives the two point function of ¯ O σ ν S with any O R , it is clear that (3.2) can be used to determine ¯ O σ ν S as a linear combination ofthe O R . We find O νR = 2 l ( ν ) − n d R/ χ R/ ( ν ) O R (3.3)Thus, we can now writeTr V ⊗ n ( ν ( Z ) ⊗ n ) = X R/ ⊢ n d R O σ ν R = X R/ ⊢ n d R d R/ l ( ν ) − n χ R/ ( ν ) O R (3.4)8otice that, using character orthogonality, we can now invert this relation. Indeed X ν ∈ S n − l ( ν ) χ S/ ( ν )Tr V ⊗ n ( ν ( Z ) ⊗ n ) = X R/ ⊢ n d R d R/ − n h X ν ∈ S n χ S/ ( ν ) χ R/ ( ν ) i O R = X R/ ⊢ n d R d R/ − n h (cid:16) n (cid:17) ! δ RS i O R = d S d S/ − n h (cid:16) n (cid:17) ! i O S (3.5)Consequently O S ( Z ) = d S/ d S n (cid:0) n (cid:1) ! X ν ∈ S n − l ( ν ) χ S/ ( ν )Tr V ⊗ n ( ν ( Z ) ⊗ n ) (3.6)The normalization in (3.6) looks rather unnatural. From now on we will adopt a newnormalization, given by χ S ( Z ) = 1 (cid:0) n (cid:1) ! X ν ∈ S n − l ( ν ) χ S/ ( ν )Tr V ⊗ n ( ν ( Z ) ⊗ n ) (3.7)Notice that we continue to label our operators by S , not by S/
4. This deserves a few com-ments. In the context of SO ( N ) gauge theory, the Wick contractions are a sum over elementsof S n . To construct our operators, we have constructed projectors[1] using representations S of S n . Orthogonality of our operators then follows as a consequence of the fact that theseprojectors commute with the Wick contractions and are mutually orthogonal. The two pointfunction is given in terms of a product of factors of boxes in S , which were obtained [1] byevaluating the action of Jucys-Murphy elements on states in the carrier space of S . Clearly, S summarizes information about the group theory used to construct our operators: it is therepresentation that organizes the 2 n indices of the Z ij fields that appear in O S . Thus forexample, cut offs due to the stringy exclusion principle [29, 26] cut S off at N rows. S/ S/ S/ S are allowed) in our basis. Note also that for the operators Q R ( Z ), since the rows in R allhave an odd length, there is no notion of the R/ h χ R ( Z ) χ S ( ¯ Z ) i = δ RS Y i ∈ odd boxes in S c i ≡ δ RS f R, odd (3.8)The formulas of this section indicate a very interesting interplay between Young diagrams S ⊢ n and S/ ⊢ n .One immediate application of the new formula (3.7) is in the derivation of a product rule.Indeed, for S ⊢ m and R ⊢ n we have χ S ( Z ) χ R ( Z ) = 1 (cid:0) n (cid:1) ! (cid:0) m (cid:1) ! X ν ∈ S m X µ ∈ S n − l ( ν ) − l ( µ ) χ S/ ( ν ) χ R/ ( µ )Tr V ⊗ n ( ν ( Z ) ⊗ n )Tr V ⊗ n ( µ ( Z ) ⊗ n )9 1 (cid:0) n (cid:1) ! (cid:0) m (cid:1) ! X ν ∈ S m X µ ∈ S n X σ ∈ S n + m − l ( ν ) − l ( µ ) χ S/ ( ν ) χ R/ ( µ ) δ ( σµ − ◦ ν − )Tr V ⊗ n + m ( σ ( Z ) ⊗ n )= 1 (cid:0) n (cid:1) ! (cid:0) m (cid:1) ! (cid:0) n + m (cid:1) ! X ν ∈ S m X µ ∈ S n X σ ∈ S n + m X T/ ⊢ n + m − l ( σ ) χ S/ ( ν ) χ R/ ( µ ) χ T ( µ − ◦ ν − ) χ T ( σ )Tr V ⊗ n + m ( σ ( Z ) ⊗ n + m )= X T/ ⊢ n + m g R/ S/ T/ χ T ( Z ) (3.9)where we have used the formula[43] g R/ S/ T/ = 1 (cid:0) n (cid:1) ! (cid:0) m (cid:1) ! X ν ∈ S m X µ ∈ S n χ S/ ( ν ) χ R/ ( µ ) χ T ( µ − ◦ ν − ) (3.10)for the Littlewood-Richardson coefficient. The answer (2.15) gives a complete description of correlators in the trace basis. To obtainexplicit answers we need to evaluate symmetric group characters. We explain how thisevaluation is carried out in this section.In the U ( N ) case, very similar formulas have been obtained, exploiting the relationbetween operators written in the trace basis and the Schur polynomials. We could alsofollow this route given the new form of our operators in (3.7). Following this route, wewould write multi-point correlators in terms of products of our operators and then evaluatethese products using the Littlewood-Richardson coefficients. This approach computes thegeneral multi-trace operators knowing nothing more than a character for an n cycle of S n in aYoung diagram labeled by a hook representation and the Littlewood-Richardson coefficients.For SO ( N ) we can employ the formula (2.15) which requires the computation of char-acters, beyond the character for an n cycle of S n in a Young diagram labeled by a hookrepresentation. As we explain below, the evaluation of these characters is straight forward.Given the values of the characters we obtain below, the current computation could also beused to give an alternative derivation of the known U ( N ) correlation functions.To start, consider the computation for correlators of the form h Tr( Z J )Tr( Z J )Tr( ¯ Z J ) i = 2 X R/ ⊢ J χ R/ ( µ ) χ R/ ( ν ) Y i ∈ odd boxes in R c i (4.1)We have J = J + J and µ is a J cycle while ν is the product of a J cycle and a J cycle.We know that χ R/ ( µ ) will only be non-zero when R/ R/ [ k, J − k ]. A formula that will be useful is Y i ∈ odd boxes in R with R/ k, J − k ] c i = ( N + 2 k − N − J + 2 k − χ [ k, J − k ] ( µ ) = ( − J − k (4.3)Now consider χ [ k, J − k ] ( ν ). According to the Murnaghan-Nakayama rule[43], this characteris equal to a sum over all ways of extracting a border strip tableau of length J . For eachpossible extraction we have to multiply by ( − h where h is the height (= number of rows)of the removed border strip multiplied by the character of a J cycle in the irreduciblerepresentation labeled by the Young diagram obtained by removing the border strip from[ k, J − k ]. Thus, (in the following ( J ) denotes a J cycle) χ [ k, J − k ] ( ν ) = − θ ( k > J ) χ [ k − J , J − k ] (( J )) + ( − J θ ( J − k ≥ J ) χ [ k, J − k − J ] (( J ))= − θ ( k > J ) χ [ k − J , J − k ] (( J )) + ( − J θ ( J ≥ k ) χ [ k, J − k ] (( J ))= θ ( k > J )( − J − k +1 + θ ( J ≥ k )( − J + J − k = θ ( k > J )( − J − k +1 + θ ( J ≥ k )( − J − k (4.4)where θ ( k > J ) = 1 if k > J = 0 otherwise (4.5) θ ( J − k ≥ J ) = 1 if J − k ≥ J = 0 otherwise (4.6)It is now clear that χ [ k, J − k ] ( µ ) χ [ k, J − k ] ( ν ) = θ ( J ≥ k ) − θ ( k > J ) (4.7)Below we will want to generalize this formula a bit. Towards this end it is worth lookingback and realizing that the negative sign above arose because we removed J boxes from thefirst row - giving a removed tableaux with height 1. Bear this in mind when considering thesubsequent character formulas we obtain, since we will again obtain θ functions with a signdetermined by how many border strip tableau were removed from the first row.Using this character formula we have h Tr( Z J )Tr( Z J )Tr( ¯ Z J ) i = 2 X R/ ⊢ J χ R/ ( µ ) χ R/ ( ν ) Y i ∈ odd boxes in R c i This notation for the Young diagram is listing row lengths, i.e. [ k, J − k ] has k boxes in its first row andthen it has J − k rows which each have a single box.
11 2 J X k =1 ( θ ( J ≥ k ) − θ ( k > J )) ( N + 2 k − N − J + 2 k − J X k =1 − J X k = J +1 ! ( N + 2 k − N − J + 2 k − h Tr( Z J )Tr( Z J )Tr( Z J )Tr( ¯ Z J ) i = 2 X R/ ⊢ J χ R/ ( µ ) χ R/ ( ν ) Y i ∈ odd boxes in R c i (4.9)We have J = J + J + J and µ is a J cycle while ν is the product of a J cycle, a J cycleand a J cycle. We know that χ R/ ( µ ) will only be non-zero when R/ R/ k, J − k ]. Arguing exactly as we didabove, a simple application of the Murnaghan-Nakayama rule gives χ [ k, J − k ] ( µ ) χ [ k, J − k ] ( ν ) = θ ( k ≤ J ) − θ ( J < k ≤ J + J ) − θ ( J < k ≤ J + J ) + θ ( J + J < k ≤ J ) (4.10)The first term on the right hand side comes from removing both cycles ( J ) and ( J ) fromthe column. The second term on the right hand side comes from removing cycle ( J ) fromrow 1 and cycle ( J ) from the column. The third term on the right hand side comes fromremoving cycle ( J ) from row 1 and cycle ( J ) from the column. The fourth term on theright hand side comes from removing both cycles ( J ) and ( J ) from the first row. Thus h Tr( Z J ) Tr( Z J )Tr( Z J )Tr( ¯ Z J ) i = 2 X R/ ⊢ J χ R/ ( µ ) χ R/ ( ν ) Y i ∈ odd boxes in R c i = 2 J X k =1 ( θ ( k ≤ J ) − θ ( J < k ≤ J + J ) − θ ( J < k ≤ J + J )+ θ ( J + J + 1 < k ≤ J )) ( N + 2 k − N − J + 2 k − J X k =1 − J + J X k = J +1 − J + J X k = J +1 + J X k = J + J +1 ! ( N + 2 k − N − J + 2 k − J n = J + J + · · · + J n − ) h n − Y i =1 Tr( Z J i )Tr( ¯ Z J n ) i = 2 n J X k =1 − J + J X k = J +1 − · · · − J + J n − X k = J n − +1 + J + J + J X k = J + J +1 + · · · + J n X k = J + ··· + J n − × ( N + 2 k − N − J n + 2 k − U ( N ) result which was derived in [44]. Seealso[35, 45, 46, 47].There is a particularly interesting double scaling limit of N = 4 super Yang-Millstheory[34, 35] defined by N → ∞ and J → ∞ with J N fixed , g Y M fixed (4.13)where J is the number of fields in the trace. In this limit some non-planar diagrams survive,leading to a new renormalized genus counting parameter J N . This limit is AdS/CFT dualto a pp-wave limit of AdS × S , in which the superstring theory can be quantized. Sincewe have computed two and three point correlators we can explore this limit in the SO ( N )gauge theory.In order to extract the double scaling limit of the two and three point correlators we willneed the following identity [48]Γ( N + p + 1)Γ( N − p ) = ( N + p )!( N − p − p + p Y l =0 ( N + p − l )= N p + p +1 exp " p + p X l =0 ln(1 + p − lN ) . (4.14)Expanding for large N and summing over l yieldsΓ( N + p + 1)Γ( N − p ) ∼ N p + p +1 exp (cid:20) N ( p − p )( p + p + 1) + O (1 /N ) (cid:21) . (4.15)Applying it to our SO(N) correlators gives h Tr ( ¯ Z J ) T r ( Z J ) i = 4 J X k =1 Γ( N + 2( J − k ) + 1)Γ( N − k + 1) ∼ N J J X k =1 exp (cid:18) J (2 J − k + 1) N (cid:19) ∼ J N J sinh J N J N . (4.16)and h Tr ( ¯ Z J )Tr ( Z J )Tr ( Z J ) i = 8 J X k =1 − J X k = J +1 ! Γ( N + 2( J − k ) + 1)Γ( N − k + 1) ∼ × N J e J N sinh J J N sinh J J N e J N − ∼ × J N J sinh J J N sinh J J NJ N (4.17)Compare these results with U(N) correlators in the double scaling limit [35] h Tr ( ¯ Z J )Tr ( Z J ) i = J X k =1 Γ( N + k )Γ( N − J + k ) ∼ J N J sinh J N J N (4.18)13 Tr ( ¯ Z J )Tr ( Z J )Tr ( Z J ) i = J X k =2 J +1 − J X k =1 ! Γ( N + k )Γ( N − J + k ) ∼ N J e J N (coth J N −
1) sinh 2 J J N sinh 2 J J N ∼ J N J sinh J J N sinh J J NJ N (4.19)It is clear that non-planar unoriented diagrams in SO ( N ) gauge theory do not survive thislimit.Our result is similar to earlier results obtained for the double scaling limit of the matrixmodel relevant for the c = 1 string[49]. From this point of view, the ribbon graphs of thematrix model are identified as a triangulation of the string worldsheet. These double scalinglimits take N → ∞ simultaneously with the world sheet continuum limit in such a way thatthe string coupling is held finite, so that sums over continuum surfaces of any topology arecaptured. This limit for antisymmetric matrices has been discussed in [50]. In this doublescaling regime too, only orientable surfaces survive. When constructing gauge invariant operators, it is possible to contract indices using any in-variant tensors. For SO ( N ) there are two such tensors: the Kronecker delta (which contractspairs of indices) and the ǫ i i ··· i N tensor. In [1], taking both invariant tensors into account,we made a precise conjecture for the basis that can be constructed. In this section we wouldlike to count the number of operators we proposed and thereby verify that it is indeed acomplete set. We will focus on the case that N is even. We will end this section with a fewcomments on the odd N case.To start, recall the discussion of [1]: operators of the form (1.1) are the complete set ofoperators that can be built without using ǫ i i ··· i N . They correspond to the set of operatorsthat can be written as a product of traces of even powers of Z . Further, they are labeled byYoung diagrams with an even number of boxes in each column and row. As explained in [1],these Young diagrams R built using 2 n boxes with the restriction that R has not more than N rows, can be indexed by partitions of n that have no more than N parts. Consequently,we can write the partition function for the number of operators of the form (1.1) as F ( x ) = N Y i =1 − x i (5.1)The coefficient of the x p in the expansion of F ( x ) tells us how many operators can beconstructed using p Z fields. Next we need to count the number of operators (1.2) built using ǫ i i ··· i N . These operators all have a dimension ≥ N . They are constructed by contracting14he indices of N Z fields with ǫ i i ··· i N and contracting the remaining indices in pairs. Thesecan also be constructed in the form (1.1), with labels R that consist of a single column of N boxes with a second Young diagram stacked to the right. To get a non-zero operator, thissecond Young diagram must again have an even number of columns and rows. Clearly then,the partition function for the operators constructed using one ǫ i i ··· i N is F ( x ) = x N F ( x ) = x N N Y i =1 − x i (5.2)Operators constructed using any even number of ǫ i i ··· i N s lead to Young diagrams that haveboth an even number of rows and columns, and hence are included in (5.1). Similarly,operators constructed using any odd number of ǫ i i ··· i N s are included in (5.2). Consequently,the complete partition function for the operators in our basis is F ( x ) = (1 + x N ) N Y i =1 − x i (5.3)The partition for free Yang Mills theory on a compact space has been computed in [36].The result is G ( x ) = Z [ dO ] ∞ X n =0 x nE χ Sym n ( R ) ( O ) = Z [ dO ] e P ∞ m =1 xmEm χ R ( O m ) (5.4)Here we take R to be the adjoint representation (since our field Z transforms in the adjoint)and Sym n ( R ) is the representation obtained by taking the symmetric product of n copiesof the adjoint. Set E = 1 and then expand to get a polynomial in x . The coefficient of x n counts the number of operators that can be built using n Z s. Consequently, if our basis iscomplete, we should find G ( x ) = F ( x ). To evaluate (5.4) we need the adjoint character (wefocus on SO (2 n )) χ R (x) = X ≤ i 11 + x 11 + x + x (5.12)so that for N = 2 n = 6 we have G ( x ) = (1 + x ) 11 − x − x − x (5.13)This proves that for N = 4 , G ( x ) = F ( x ) which provides strong support that we haveindeed constructed a basis. Given the results of this section, we conjecture that the partitionfunction for SO ( N ), with N even is given by (5.3). Our results also lead us to conjecturethat for SO ( N ), with N odd, the partition function is G ( x ) = N − Y i =1 − x i (5.14)16 Link to Free Fermions The dynamics of gauge invariant operators of a single Hermitian matrix model reduces toeigenvalue dynamics[37]. If one restricts to the gauge invariant and purely holomorphic orantiholomorphic observables of complex matrix models, one can again reduce to eigenvaluedynamics[52]. This eigenvalue dynamics can then be mapped to the dynamics of N freefermions in an external potential[2]. For the free complex matrix model, the basis providedby the Schur polynomials has a particularly close relationship to free fermion dynamics: theSchur polynomials can be mapped to free fermion wave functions[2]. In this section our goalis to argue that our operators (3.7) also describe free fermion wave functions.Before considering the eigenvalue dynamics it is useful to recall a few facts about thewave functions of the single particle Hamiltonian H = − ∂∂z ∂∂ ¯ z + z ¯ z (6.1)which describes a single particle moving in a harmonic oscillator potential, in two dimensions.The ground state wave function (not normalized) is ψ ( z, ¯ z ) = e − z ¯ z (6.2)Further, at any given energy level, the state with largest angular momentum (again, notnormalized) is given by ψ l ( z, ¯ z ) = z l e − z ¯ z (6.3)The parity of this wave function is given by ( − l so that (for example) if l is even we havean even parity state. These wave functions will play a prominent role below.To start, return to the two point function (3.8) and rewrite it in terms of eigenvalues δ RS f R, odd = h χ R ( Z ) χ S ( ¯ Z ) i = Z [ dzd ¯ z ] | ∆( z ) | χ R ( z ) χ S (¯ z ) e − P N i =1 z i ¯ z i (6.4)where the required Jacobian ∆( z ) = Q ≤ i 1. The power p of the leading20erm is determined rather simply by the sign of the permutations β and τ . To compute thesign of a permutation decompose it into a product of transpositions. This decompositionis not unique. The sign of the permutation sgn( σ ) = ( − m where m is the number oftranspositions in the product. sgn( σ ) is well defined, i.e. it does not depend on the specificdecomposition of σ into transpositions. Our final result is h T σ ν σ µ i Sp ( N ) = sgn( β )sgn( τ ) (cid:20) h T σ ν σ µ i SO ( N ) (cid:12)(cid:12)(cid:12) N →− N (cid:21) (7.9)This result will play an important role below.Figure 2: A comparison of the free field two point function of the SO ( N ) gauge theory(shown in (A) above) and of the Sp ( N ) gauge theory (shown in (B) above).Introduce the operators O R ( Z ) = 1(2 n )! X σ ∈ S n χ R ( σ ) σ i i i i ··· i n − i n i n i n − j j ··· j n − j n Z j j · · · Z j n − j n (7.10)¯ O R ( Z ) = 1(2 n )! X σ ∈ S n χ R ( σ ) σ j j ··· j n − j n i i i i ··· i n − i n i n i n − Z j j · · · Z j n − j n (7.11)To compute the two point function of these operators we will be using (7.9). For this reasonwe will need to spell out the gauge group we are using to compute the correlator. We willneed one more result from group theory. Recall that to get the conjugate (or transpose) R c of a Young diagram R we need to swap rows and columns. For example R = R c = (7.12)The characters of R and R c are related by χ R ( σ ) = sgn( σ ) χ R c ( σ ) (7.13)21rom the point of view of the projectors used in defining O R , taking the conjugate of R corresponds to swapping symmetrization and antisymmetrization of indices. We are nowready to compute the correlation functions of the O R . For N even, R ⊢ n and S ⊢ m wefind h O R ( Z ) ¯ O S ( Z ) i Sp ( N ) = 1(2 n )! 1(2 m )! X σ ∈ S n X τ ∈ S m χ R ( σ ) χ S ( τ ) ×h σ i i i i ··· i n − i n i n i n − j j ··· j n − j n Z j j · · · Z j n − j n τ j j ··· j m − j m i i i i ··· i m − i m i m i m − Z j j · · · Z j m − j m i Sp ( N ) = 1(2 n )! 1(2 m )! X σ ∈ S n X τ ∈ S m χ R ( σ ) χ S ( τ )sgn( σ )sgn( τ ) ×h σ i i i i ··· i n − i n i n i n − j j ··· j n − j n Z j j Z j j · · · Z j n − j n τ j j ··· j m − j m i i i i ··· i m − i m i m i m − Z j j · · · Z j m − j m i SO ( N ) (cid:12)(cid:12)(cid:12) N →− N = 1(2 n )! 1(2 m )! X σ ∈ S n X τ ∈ S m χ R c ( σ ) χ S c ( τ ) ×h σ i i i i ··· i n − i n i n i n − j j ··· j n − j n Z j j · · · Z j n − j n τ j j ··· j m − j m i i i i ··· i m − i m i m i m − Z j j · · · Z j m − j m i SO ( N ) (cid:12)(cid:12)(cid:12) N →− N = (cid:2) h O R c ( Z ) ¯ O S c ( Z ) i SO ( N ) i N →− N (7.14)Using the results of [1] we now immediately find h O R ( Z ) ¯ O S ( Z ) i Sp ( N ) = δ RS n (cid:18) d R/ d R (cid:19) Y i ∈ odd boxes in R c c i (cid:12)(cid:12)(cid:12) N →− N (7.15)It is straight forward to verify that Y i ∈ odd boxes in R c c i (cid:12)(cid:12)(cid:12) N →− N = Y i ∈ even boxes in R c i (7.16)so that we finally obtain h O R ( Z ) ¯ O S ( Z ) i Sp ( N ) = δ RS n (cid:18) d R/ d R (cid:19) Y i ∈ even boxes in R c i (7.17)Recall that the even boxes will occupy every second row, including the top most row. As anexample we have filled the even boxes below ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ (7.18)We can now repeat many of the arguments we developed for the SO ( N ) theory. Sincemany of the details are basically the same, we will for the most part simply quote the results. Recall that as symmetric group representations we have d R = d R c . We have used this when we wrote(7.15). h Tr (cid:0) µ ( ¯ Z ) ⊗ n (cid:1) Tr (cid:0) ν ( Z ) ⊗ n (cid:1) i Sp ( N ) = 2 l ( µ )+ l ( ν ) X R/ ⊢ n/ χ R/ ( µ ) χ R/ ( ν ) Y i ∈ even in R c i (7.19)Using this result we can again give a simpler form for our operators χ S ( Z ) = 1 (cid:0) n (cid:1) ! X ν ∈ S n − l ( ν ) χ S/ ( ν )Tr V ⊗ n ( ν ( Z ) ⊗ n ) (7.20)The two point function of these operators is h χ R ( Z ) χ S ( ¯ Z ) i Sp ( N ) = δ RS Y i ∈ even boxes in S c i (7.21)These operators again enjoy a simple product rule ( S/ ⊢ n , R/ ⊢ n ) χ S ( Z ) χ R ( Z ) = X T/ ⊢ n n g R/ S/ T/ χ T ( Z ) (7.22)where g R/ S/ T/ is the Littlewood-Richardson coefficient.Using (7.19) we can again give a formula for general extremal correlation functions h Tr ( ¯ Z J n )Tr ( Z J ) ... Tr ( Z J n − ) i Sp ( N ) =( − n n J X k =1 − ... + ... − ... + J n X k = J + ...J n − +1 Γ( N + 2 k )Γ( N − J n − k ) (7.23)where J n = P n − i =1 J i . In particular, for two point functions we find h Tr (cid:0) ¯ Z J (cid:1) Tr (cid:0) Z J (cid:1) i Sp ( N ) = 4 J X k =1 Γ( N + 2 k )Γ( N − J − k ) (7.24)We have also considered the double scaling limit of the two-point functions in the Sp ( N )gauge theory. We find h Tr (cid:0) ¯ Z J (cid:1) Tr (cid:0) Z J (cid:1) i ∼ N J J X k =1 exp (cid:20) JN (4 k − J − (cid:21) = 4 N J e J (2 J +3) N − e − J (2 J − N e JN − ∼ J N J sinh J N J N (7.25)Similarly for three-point functions h Tr ( ¯ Z J )Tr ( Z J )Tr ( Z J ) i = − J X k =1 − J X k = J +1 ! Γ( N + 2 k )Γ( N − J − k )) ∼ N J e J N − e J N sinh 2 J J N sinh 2 J J N ∼ J N J sinh J J N sinh J J N J /N (7.26)23ince this is the same as the SO ( N ) result, we again see that in this double scaling limitonly orientable higher genus surfaces contribute.Our operators can again be related to free fermion wave functions. Indeed, start from δ RS f R, even = h χ R ( Z ) χ S ( ¯ Z ) i = Z [ dzd ¯ z ] | ∆( z ) | χ R ( z ) χ S (¯ z ) e − P N i =1 z i ¯ z i (7.27)where ∆( z ) = Q N k =1 z k Q ≤ i This work is based upon research supported by the South African Re-search Chairs Initiative of the Department of Science and Technology and National ResearchFoundation. Any opinion, findings and conclusions or recommendations expressed in thismaterial are those of the authors and therefore the NRF and DST do not accept any liabilitywith regard thereto. RdMK would like to thank Collingwood College, Durham for theirsupport. PC would like to thank Vikram Vyas for hospitality during the last stages of theproject. Finally, the work of PD is supported in part by a Claude Leon Fellowship. A Simplifying the SO ( N ) basis In this Appendix we will give an alternative derivation of (3.6). Our starting point is O σ µ R = 12 n ! X β ∈ S n χ R ( β ) C σ µ j j ··· j n (cid:0) β (cid:1) j j ··· j n i i ··· i n Z i i · · · Z i n − i n (A.1)Recall our shorthand notation for indices C σ µ J = C σ µ j j ··· j n , (cid:0) β (cid:1) JI = (cid:0) β (cid:1) j j ··· j n i i ··· i n , Z I = Z i i · · · Z i n − i n . (A.2)First, the contractor C σ µ J can be written as C σ µ J = δ j j σ µ (2) · · · δ j n j σ µ (2 n ) = δ j σ µ (1) j σ µ (2) δ j σ µ (3) j σ µ (4) · · · δ j σ µ (2 n − j σ µ (2 n ) = δ k k δ k k · · · δ k n − k n (cid:0) σ µ (cid:1) KJ , (A.3)25he second line follows because σ µ ( r ) = r + 1 for r odd. Consequently, (A.1) becomes O σ µ R = 12 n ! X β ∈ S n χ R ( β ) C σ µ J (cid:0) β (cid:1) JI Z I = 12 n ! X β ∈ S n χ R ( β ) δ k k · · · δ k n − k n (cid:0) σ µ (cid:1) KJ (cid:0) β (cid:1) JI Z I = 12 n ! X β ∈ S n χ R ( β ) δ k k · · · δ k n − k n (cid:0) βσ µ (cid:1) KI Z I = 12 n ! X β ∈ S n χ R ( βσ − µ ) δ k k · · · δ k n − k n (cid:0) β (cid:1) KI Z I . (A.4)Since Z is antisymmetric, for any η ∈ S n [ S ] we have Z η ( I ) = Z i η (1) i η (2) · · · Z i η (2 n − i η (2 n ) = sgn( η ) Z I , (A.5)It is now straight forward to see that δ k k · · · δ k n − k n (cid:0) ξβη (cid:1) KI Z I = δ k k · · · δ k n − k n Z η − β − ξ − ( K ) = δ k k · · · δ k n − k n Z η − β − ( K ) sgn( ξ )= δ k k · · · δ k n − k n (cid:0) βη (cid:1) KI Z I sgn( ξ )= δ k k · · · δ k n − k n (cid:0) β (cid:1) KI Z I sgn( ξ ) . (A.6)Consequently O σ µ R = 12 n ! X β ∈ S n χ R ( βσ − µ ) δ k k · · · δ k n − k n (cid:0) β (cid:1) KI Z I . = 12 n ! 1(2 n n !) X β ∈ S n X ξ,η ∈ S n [ S ] χ R ( ξβησ − µ ) δ k k · · · δ k n − k n (cid:0) ξβη (cid:1) KI Z I . = 12 n ! 1(2 n n !) X β ∈ S n h X ξ,η ∈ S n [ S ] χ R ( ξβησ − µ )sgn( ξ ) δ k k · · · δ k n − k n (cid:0) β (cid:1) KI Z I i . Now, (A.6) implies that the expression within brackets defines a function on the doublecoset S n [ S ] \ S n /S n [ S ], i.e. it takes the same value for different σ ∈ S n that represent thesame double coset element. Thus, we can trade the sum over β ∈ S n for a sum over cosetrepresentatives which is a sum over partitions O σ µ R = 12 n ! X ν ⊢ n/ X ξ,η ∈ S n [ S ] z ν χ R ( ξβ ν ησ − µ )sgn( ξ ) δ k k · · · δ k n − k n (cid:0) β ν (cid:1) KI Z I . = 1 d R X ν ⊢ n/ z ν l ( µ )+ l ( ν ) χ R/ ( µ ) χ R/ ( ν ) δ k k · · · δ k n − k n (cid:0) β ν (cid:1) KI Z I , (A.7)26o obtain the last line above we have used the mathematical identity[42] d R n ! X ξ,η ∈ S n [ S ] χ R ( ξβ ν ησ − µ )sgn( ξ ) = 2 l ( µ )+ l ( ν ) χ R/ ( µ ) χ R/ ( ν ) . (A.8)Thus, our operators become O σ µ R = 2 l ( µ ) χ R/ ( µ ) d R X ν ⊢ n/ z ν l ( ν ) χ R/ ( ν ) δ k k · · · δ k n − k n (cid:0) β ν (cid:1) KI Z I = 2 l ( µ ) χ R/ ( µ ) d R X ν ⊢ n/ z ν − l ( ν ) χ R/ ( ν ) C β ν I Z I = 2 l ( µ ) χ R/ ( µ ) d R n/ X σ ∈ S n/ − l ( σ ) χ R/ ( σ )Tr (cid:0) σ ( Z ) ⊗ n (cid:1) . (A.9)For the special case µ = (1 n/ ) we find O R = 2 n/ d R/ d R n/ X σ ∈ S n/ − l ( σ ) χ R/ ( σ )Tr (cid:0) σ ( Z ) ⊗ n (cid:1) . (A.10)which completes the demonstration. B Correlation Functions To test the result (2.15) we have studied a number of correlation functions using Mathematicaand analytic techniques. Consider first correlators of the form h Tr( Z ) p Tr( ¯ Z ) p i . Introducethe notation A p = h Tr( Z ) p Tr( ¯ Z ) p i (B.1)By studying Wick contractions (or equivalently Schwinger-Dyson equations) it is not hardto obtain the following recursion relation A p = (16 p ( p − 1) + 4 pN ( N − A p − (B.2)Since A = 1 we can easily generate explicit expression for A p . For example A = 122880 N − N + 6144000 N − N + 98918400 N − N +604569600 N − N + 1148190720 N − N (B.3)Introduce the short hand Y i ∈ odd boxes in R c i = f odd R (B.4)27ur formula (2.15) says A = 2 f odd + 16 f odd + 25 f odd + 36 f odd + 25 f odd + 16 f odd + f odd (B.5)which is indeed correct.Next, consider correlators of the form h Tr( X ) p Tr( X )Tr( ¯ X ) p +2 i . Introduce the notation B p = h Tr( X ) p Tr( X )Tr( ¯ X ) p +2 i (B.6)Again by studying Wick contractions (or equivalently Schwinger-Dyson equations) it is nothard to obtain the following recursion relation B p = 16(2 p + 4)(2 p + 2)(2 p )(2 p − B p − + 2(2 p + 4)( N − A p +1 +[4 N ( N − p + 4)(2 p + 2) + (2 p + 4)(2 p + 2)(2 p )8(3 N − A p (B.7)We now easily find, for example, B = 245760 N − N + 10813440 N − N + 142786560 N − N + 586383360 N − N + 188743680 N (B.8)Our formula (2.15) says B = 2 f odd + 8 f odd + 5 f odd + 0 − f odd − f odd − f odd (B.9)which is indeed correct.Finally, we have also performed a complete check in Mathematica for the full set ofoperators that can be built using 8 fields. The Mathematica and analytic results from (2.15)are again in complete agreement. A study of the Sp ( N ) coorelators has also been performedto confirm (7.19) using Mathematica. 28 Jacobians For single matrix models, since we are interested in the dynamics of gauge invariant observ-ables, we can employ an eigenvalue description. The idea is to write the path integral asan integral over the eigenvalues and some angles. We then integrate out the angles. Theresulting measure is nontrivial. With a slight abuse of language, we refer to the measure asa Jacobian. In this Appendix we would like to compute the Jacobian for a single complex SO ( N ) matrix and for a single complex Sp ( N ) matrix. To the best of our knowledge, theseare new results.The approach we employ for determining the measure, is first to compute it for a realmatrix and then use this result to guess the answer for the complex matrix. We check thisguess by verifying that we get the correct answer for any correlation function we compute.To determine the Jacobian for a real (or Hermitian) matrix, we require that the Schwinger-Dyson equations in the original (matrix) variables agree with the Schwinger-Dyson equationsin the eigenvalue variables. This implies a differential equation for the Jacobian which wesolve. Although the Jacobians we compute in this way are all known, we have given ourderivation since it seems to be new and is simpler than existing derivations. To start, weillustrate the method with the U ( N ) matrix model and then move on to SO ( N ) and Sp ( N ).The use of Schwinger-Dyson equations to determine a Jacobian in this way was pioneeredin collective field theory[62]. See [63] for applications to multi matrix models and [64] forapplications to vector models. C.1 U ( N ) Matrix Models The main goal of this subsection is to illustrate how we compute the Jacobian using Schwinger-Dyson equations and also to illustrate the close connection between the Jacobian for the real(or Hermitian) matrix and the complex version.Consider the matrix model for a single matrix X living in the Lie algebra u( N ), i.e. X is Hermitian. Rewrite the Schwinger-Dyson equation0 = Z [ dX ] ddX ij (cid:16) [ X n − ] ij e − Tr( X ) (cid:17) = Z [ dX ] n − X r =0 Tr[ X n − r − ]Tr[ X r ] − Tr[ X n ] ! e − Tr( X ) (C.1)in terms of eigenvalue variables to obtain0 = Z [ dλ ] J ( λ ) n − X r =0 N X i,j =1 λ n − r − i λ rj − N X i =1 λ ni ! e − P Nl =1 λ l (C.2)29here J ( λ ) is the Jacobian we want to determine. After performing the sum over r we have0 = Z [ dλ ] J ( λ ) N X i,j =1 i = j λ n − i λ i − λ j + ( n − N X i =1 λ n − i − N X i =1 λ ni ! e − P Nl =1 λ l (C.3)Now, work directly in the eigenvalue variables0 = Z [ dλ ] N X i =1 ∂∂λ i (cid:16) λ n − i J ( λ ) e − P Nl =1 λ l (cid:17) = Z [ dλ ] J ( λ ) N X i =1 λ n − i ∂ log J ( λ ) ∂λ i + ( n − N X i =1 λ n − i − N X i =1 λ ni ! e − P Nl =1 λ l (C.4)Comparing (C.3) and the second line of (C.4) we learn that ∂ log J ( λ ) ∂λ i = 2 N X j =1 i = j λ i − λ j (C.5)This clearly implies that J ( λ ) = Y i We will now consider the matrix model relevant for SO ( N ) gauge theory. We consider thecase that N is even and start with a single real N × N antisymmetric matrix X . In thiscase, we can, with a unitary transformation, bring X into a block diagonal form X = x − x · · · x − x (C.9)Using this explicit form for the matrix X we easily find Tr( X J +1 ) = 0 andTr( X J ) = 2( − J N X i =1 x Ji (C.10)Since X is an antisymmetric matrix, elements above and below the diagonal are related bya sign. Consequently dX kl dX ij = δ ik δ jl − δ il δ jk (C.11)To get some practice with this derivative and since we need it in what follows, consider ddX ij ( X J − ) ij = J − X r =0 ( X r ) ik dX kl dX ij ( X J − r − ) lj = J − X r =0 ( X r ) ik ( δ ik δ jl − δ il δ kj )( X J − r − ) lj = J − X r =0 (cid:2) Tr( X r )Tr( X J − r − ) − Tr(( X T ) r X J − r − ) (cid:3) = J − X r =0 Tr( X r )Tr( X J − r − ) − J − X r =0 ( − r Tr( X J − )31 J − X r =0 Tr( X r )Tr( X J − r − ) − Tr( X J − ) (C.12)Now, consider the Schwinger-Dyson equation0 = Z [ dX ] ddX ij (cid:16) ( X J − ) ij e Tr( X ) (cid:17) (C.13)which implies h J − X r =0 Tr( X r )Tr( X J − r − ) − Tr( X J − ) i = − h Tr( X J ) i (C.14)Writing this in terms of eigenvalues gives h − J +1 J − X r =0 N X i,j =1 x ri x J − r − j − − J +1 N X i =1 x J − i i = − − J h N X i =1 x Ji i (C.15)or, after summing over r h N X i,j =1 ,i = j x Ji x i − x j + 2(2 J − N X i =1 x J − i i = 4 h N X i =1 x Ji i (C.16)Now, working in terms of the eigenvalue variables we have0 = Z [ dx ] N X i =1 ddx i (cid:18) J x J − i e − P N j =1 x j (cid:19) (C.17)which becomes0 = Z [ dx ] J e − P N j =1 x j N X i =1 (cid:18) x J − i d log Jdx i + (2 J − x J − i − x Ji (cid:19) (C.18)Comparing (C.37) and (C.39) we have d log Jdx i = N X i,j =1 ,j = i x i x i − x j (C.19)which is solved by J ( x ) = N Y i Finally, consider the matrix model relevant for Sp ( N ) gauge theory. In this case, N is even.Start with a single real N × N matrix X in the Lie algebra of Sp ( N ) which hence obeys X T = Ω X Ω (C.30)It is easy to see that0 = det( λ − X ) = det( λ − X T ) = det( λ − Ω X Ω) = det( λ + X ) (C.31)so that the eigenvalues of X come in pairs ± x i , i = 1 , , ..., N . Consequently, we findTr( X J +1 ) = 0 and Tr( X J ) = 2 N X i =1 x Ji (C.32)The relation (C.30) implies dX kl dX ij = δ ik δ jl + Ω il Ω kj (C.33)It is now straight forward to verify that ddX ij ( X J − ) ij = J − X r =0 Tr( X r )Tr( X J − r − ) + Tr( X J − ) (C.34)34ow, consider the Schwinger-Dyson equation0 = Z [ dX ] ddX ij (cid:16) ( X J − ) ij e − Tr( X ) (cid:17) (C.35)which implies h J − X r =0 Tr( X r )Tr( X J − r − ) + Tr( X J − ) i = 2 h Tr( X J ) i (C.36)Writing this in terms of eigenvalues and summing over r gives h N X i,j =1 ,i = j x Ji x i − x j + 2(2 J + 1) N X i =1 x J − i i = 4 h N X i =1 x Ji i (C.37)Now, working in terms of the eigenvalue variables we have0 = Z [ dx ] N X i =1 ddx i (cid:18) J x J − i e − P N j =1 x j (cid:19) (C.38)which becomes0 = Z [ dx ] J e − P N j =1 x j N X i =1 (cid:18) x J − i d log Jdx i + (2 J − x J − i − x Ji (cid:19) (C.39)Comparing (C.37) and (C.39) we have d log Jdx i = N X i,j =1 ,j = i x i x i − x j + 2 x i (C.40)which is solved by J ( x i ) = N Y k =1 x k N Y i