arXiv:0704.1038v3  [hep-th]  2 Nov 2007
DIAS-STP-07-05
arXiv:0704.1038 [hep-th]
Counting BPS Operators in N = 4 SYM
F.A. Dolan
Institiúid Ard-léinn Bhaile Átha Cliath,
(Dublin Institute for Advanced Studies,)
10 Burlington Rd., Dublin 4, Ireland
The free field partition function for a generic U(N) gauge theory, where the fun-
damental fields transform in the adjoint representation, is analysed in terms of
symmetric polynomial techniques. It is shown by these means how this is related
to the cycle polynomial for the symmetric group and how the large N result may be
easily recovered. Higher order corrections for finite N are also discussed in terms of
symmetric group characters. For finite N , the partition function involving a single
bosonic fundamental field is recovered and explicit counting of multi-trace quarter
BPS operators in free N = 4 super Yang Mills discussed, including a general result
for large N . The partition function for quarter BPS operators in the chiral ring
of N = 4 super Yang Mills is analysed in terms of plane partitions. Asymptotic
counting of BPS primary operators with differing R-symmetry charges is discussed
in both free N = 4 super Yang Mills and in the chiral ring. Also, general and
explicit expressions are derived for SU(2) gauge theory partition functions, when
the fundamental fields transform in the adjoint, for free field theory.
Keywords: Characters, Partition Functions, Gauge Theory, N = 4 Super Yang
Mills
E-mail: fdolan@stp.dias.ie
http://arxiv.org/abs/0704.1038v3
1. Introduction
For the past while there has been intense interest in finite N partition functions for
Yang Mills theories, especially in super-symmetric ones, particularly with regard to their
construction for BPS states and the counting thereof [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].
Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it
being by now the archetypal example of a conformal field theory for which we have a
dual description in terms of string theory, by means of the AdS/CFT conjecture [16].
An example is in the assiduous efforts that have been made to explain the entropy of
certain BPS black holes in AdS5×S5 [17,18,19,20] in terms of microscopic counting of dual
operators in N = 4 super Yang Mills, with gauge group SU(N), by means of partition
functions [1]. While this problem remains unsolved to date, essentially due to the difficulty
of defining what is meant by these dual operators for finite N , there are other equally
interesting sectors of (super) Yang Mills theories where more progress with counting has
been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors
involving BPS operators.
For N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators
involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multi-
trace operators involving two bosonic operators Z, Y while the eighth consists of ones
involving three bosonic operators Z, Y,X and two fermionic operators λ, λ̄. (All these
operators are here assumed to belong to the Lie algebra of U(N).)
For the chiral ring, the commuting/anti-commuting of these operators is at the heart
of why we can write very concise and elegant generating functions for the finite N multi-
trace partition functions [1]. Analysis of these partition functions, in terms of the counting
of operators, has become a sophisticated industry where such approaches as the so-called
‘Plethystic Program’ have provided substantial results [3,9].
This paper is devoted largely to the issue of partition functions for free field theory
and particularly to the counting of gauge invariant multi-trace operators for the case of two
bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super
Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to
the chiral ring when they do.
The partition function for a free, massless quark-gluon gas was computed long ago [21].
This involved taking particle statistics into account using coherent state techniques and
then imposing the gauge singlet condition by integrating over the relevant gauge group.
With some modifications to the expression thus derived we may write the multi-trace
partition function for some generic bosonic/fermionic fundamental fields in terms of an
integral over the gauge group, involving the single particle partition function [22,23]. This
is the starting point here.
For U(N) gauge theories we may easily write down the integral, though, even for this
case, its evaluation is far from simple. One approach (which we adapt here for the SU(2)
case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in
principle be evaluated by summing the contributions from poles inside origin centred unit
discs, in each of the N complex planes - similar techniques have been used in [10]. Due to
the number of poles this becomes unfeasible for higher values of N . Another approach is
to use the fact that the complex integral that interests us provides for an inner product
for symmetric polynomials - see Macdonald [24], pp. 363 - 372, for a related discussion.
Taking this point more seriously reveals an alternative route to evaluating the free field
theory partition function which exposes not only the large N case in an almost trivial way,
but also how and where this differs from the finite N case.
This treatment also reveals an alternative interpretation of the free field partition
function at finite N - it is related to a gauge group average of the cycle polynomial for the
symmetric permutation group (after a certain identification of ‘letters’ with gauge group
valued variables). This point is not dwelt on further here though makes the connection
between the partition function and Polya enumeration explicit. For single trace operators
at large N this connection has already been made [22,25] for N = 4 super Yang Mills,
whereby the partition function for single trace operators is related to the cycle polynomial
for the cyclic permutation group.1
Another issue is how to use the expression for the free field partition function to give
explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with
gauge group U(N). The case for one bosonic fundamental field has been widely discussed
and, for finite N , the operators are counted by partition numbers of non-negative integers
into at most N parts (which is here denoted by pN (n) - no closed formula for these numbers
for arbitrary N, n exist, though they have a ‘nice’ generating function). Here, this result is
re-derived, from a symmetric function perspective, by employing the well known Cauchy-
Littlewood formula.
To proceed further with counting, for the quarter BPS sector of N = 4 super Yang
Mills, for instance, character methods prove to be both natural and indispensable. Char-
acters in relation to conformal field theories prove to be very convenient for encoding
the allowable representations [15,29,30,31,32,33] and for studying related partition func-
tions, [34,35,36,33]. For N = 4 super Yang Mills, it was shown in [33] that, if we are
1 For more recent applications of symmetric polynomials and Polya enumeration to super-
symmetric quantum mechanical models, analysed in terms of Fock space methods, see [26,27,28].
to distinguish among primary operators with differing conformal dimensions, spins and
R-symmetry charges, such counting is most easily achieved using reductions of the full
N = 4 superconformal characters, in certain limits that isolate corresponding sectors of
short/semi-short operators. (One such limit corresponds to the index constructed in [1].)
This point may be easily illustrated for quarter BPS primary operators in N = 4 su-
per Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit
construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the
counting of quarter BPS primary operators is complicated by the fact that, if we are to
keep track of differing R-symmetry representations, any partition function restricted to
this sector must be expanded in terms of U(2) characters (or two-variable Schur polyno-
mials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are
letters corresponding to the fields Z, Y , then by expanding
Z(t, u) =
N(n,m) s(n,m)(t, u) , s(n,m)(t, u) =
tn+1um − tmun+1
t− u , (1.1)
in terms of two-variable Schur polynomials s(n,m)(t, u), we obtain the numbers N(n,m) of
gauge invariant quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry
representation and having conformal dimension n + m (so that they are superconformal
primary highest weight states in the corresponding quarter BPS supermultiplets). (The
case m = 0 counts gauge invariant half BPS primary operators.)
Here, the free field partition function is thus expanded in terms of Schur polynomials,
depending on the same variables as the one particle partition function, the two boson
case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood
formula (and, if we include fermions, another formula due to Littlewood). Generally, we
may obtain a result that relates the counting numbers to a sum over Kronecker coefficients.
These arise naturally in the theory of the symmetric permutation group, though remain
somewhat mysterious from a combinatorial perspective.
Specialising to the two boson case, a recursive procedure is employed here for the
counting of multi-trace quarter BPS operators in free field theory at finite N . This issue
was given considerable discussion for the large N case in [33] - here the results of [33] are
generalised in terms of a generating function that may be employed to count quarter BPS
operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed
in the latter case for the numbers N(n,m) in (1.1) for large n and fixed m.
To complete the discussion, counting of quarter BPS operators in the chiral ring of
N = 4 super Yang Mills is investigated in terms of expanding over U(2) characters as in
(1.1). An explicit formula is given for the corresponding finite N partition function, with
a short combinatorial interpretation given in terms of plane partitions, and specialised to
large N . For the latter case, the exponential behaviour of the numbers N(n,m) in (1.1) is
found for n and m both comparably large. This behaviour is consistent with a special case
addressed in [3,9]. By way of completion, a similar discussion for an arbitrary number of
bosonic fundamental fields in the chiral ring is included.
Two appendices are included; the first establishes some notation used for partitions
and gives some standard results for the symmetric group and symmetric polynomials, the
second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang
Mills, with gauge group U(N), for which explicit formulae are given in the main text.
Footnotes contain further details and points of clarification.
2. Free Field Partition Functions
We start from the single particle partition function which is here denoted by f(t) for
some variables2 t = (t1, t2 . . .). The general form of f(t) is
f(t) =
aiti , (2.1)
where each ti is a letter corresponding to a fundamental field and ai are signs, being +1
for a bosonic field, or −1 for a fermionic field.
For compact gauge Lie group G, the multi-trace partition function is then given by
[21], (see [22,23] for refinements,)
ZG(t) =
dµG(g) exp
f(tn)χR(g
, (2.2)
involving the Haar (or G-invariant, or Hurwitz) measure dµG(g) for g ∈ G (so that∫
dµG(g)F (g) =
dµG(g)F (gh) =
dµG(g)F (hg) for all h ∈ G and
dµG(g) = 1)
and where χR(g) is the character for the R representation of G, assuming that the funda-
mental fields transform in identical gauge group representations R.
For G = U(N), so that for any matrix U ∈ U(N) we may write U = VΘV †, where V
is a unitary matrix and Θ = diag.(eiθ1 , . . . , eiθN ), 0 ≤ θi < 2π, and for some F (U) = F (Θ),
independent of V , then we may write
dµU(N)(U)F (U) =
(2π)NN !
1≤k<l≤N
|eiθk − eiθl |2 F (Θ) , (2.3)
2 In what follows roman letters are used to denote a collection of variables and, for x =
(x1, . . . , xi), y = (y1, . . . , yj) for example, the shorthand x
α is used to mean (x1
α, . . . , xi
α) and
z = xy to mean z = (z11, . . . , zij) where zrs = xrys. The latter convenient notation has been used
by Macdonald [24].
which is, of course, related to the Weyl parametrisation of U(N). Thus, for such F (U), the
left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χR(U) gen-
erally depends on linear combinations of tr(U j)ktr(U †l)m for various non-negative integers
j, k, l,m then any function of χR(U) is an example of such an F (U).
We are interested in the case where R = Adj. is the adjoint representation so that for
U(N) we have that χAdj.(U) = trU trU
† (while for SU(N) then χAdj.(U) = trU trU
†−1).
For U(N) we then find that, using (2.2) with (2.3),
ZU(N)(t) =
(2π)NN !
1≤k<l≤N
|eiθk − eiθl |2 exp
f(tn)
j,k=1
ein(θj−θk)
(2.4)
We may write (2.4) as an N -fold contour integral by first making the variable change
zj = e
iθj so that the integrals in (2.4) are around unit circles in each zj complex plane and
then we obtain
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1) exp
f(tn)pn(z)pn(z
, (2.5)
where ∆(z) =
1≤i<j≤N (zi − zj) is the Vandermonde determinant and pn(z) =
i=1 zi
is a power symmetric polynomial - see appendix A for a brief discussion of symmetric
polynomials. This integral may then in principle be evaluated by deforming the contours
so as to extract the residues at poles within the discs |zj | < 1, 1 ≤ j ≤ N .
A crucial observation is that, for some N variable symmetric polynomials g(z), h(z),
(2πi)NN !
∆(z)∆(z−1)g(z)h(z−1) , (2.6)
acts as an inner product - this is easy to see in terms of Schur polynomials which provide
an orthonormal basis for symmetric polynomials. The reader may now wish to peruse
appendix A where notation regarding partitions and a short discussion of symmetric poly-
nomials is included.
The General and Large N Cases for U(N)
For application of inner products to (2.5) we have that, in terms of power symmetric
polynomials pλ(z) for partitions λ,
f(tn)pn(z)pn(z
nanan!
f(tn)anpn(z)
anpn(z
−1)an ,
fλ(t)pλ(z)pλ(z
−1) ,
(2.7)
with the definitions of
nanan! , fλ(t) =
f(tn)an , (2.8)
being in terms of the frequency representation of λ, (1a1 , 2a2 , . . .) with
n≥1 n an = |λ|,
the weight of the partition λ (note that the frequency representation of λ is simply a
convenient re-ordering of the parts of λ).
In (2.7) the numbers zλ have a standard combinatorial interpretation - for a given
permutation σ ∈ Sm with a1 1-cycles, a2 2-cycles etc., so that
n≥1 n an = m = |λ|, then
n≥1 n
anan! is the size of the centraliser Zσ = {τ ; τ ∈ Sm, τστ−1 = σ} of σ ∈ Sm.
(This may be easily seen as under conjugation of σ by τ then τ can permute the cycles
of length n among themselves in an! ways and/or render a cyclic rotation on each of the
individual cycles in nan ways.) More details of the symmetric group are to be found in
appendix A.
We may immediately observe that (2.7) represents a sum over cycle polynomials of
the symmetric group Sm. This is given by, for letters u1, u2, . . . um,3
Cm(u) =
a1,...,am≥0
δa1+2a2+...+mam,m
nanan!
uλ , (2.9)
where ‘λ ⊢ m’ means that λ is any partition of m - see appendix A for notation - and
an , (2.10)
in terms of the frequency representation of λ above. Identifying un = f(t
n)pn(z)pn(z
then we may rewrite
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)Cm(u) , (2.11)
3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G
of Sm. This is given by
j1(g) · · ·um
jm(g) =
|Kg|u1
j1(g) · · ·um
jm(g) ,
where ji(g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles
and Kg denotes the conjugacy classes of G with class representatives g. The size of the conjugacy
class Kg is given by |Kg| = |G|/|Zg| where Zg is the centraliser of g ∈ G. For the present case
then, G = Sm, |Sm| = m!, and |Zσ| = zλ, where λ gives the cycle structure of σ ∈ Sm, and thus,
for the corresponding conjugacy class Kλ, |Kλ| = m!/zλ.
the sum of the U(N) group averages of each of the cycle polynomials Cm(u). (Physically,
the interpretation is that the cycle index for the symmetric permutation group accounts
for particle statistics while integration over the gauge group imposes the gauge singlet
condition. For purposes of clarity, the U(N) case has been focused upon here, though
from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby
the letters un = f(t
n)χR(g
n) for the fundamental fields transforming in identical gauge
group representations, R.)
Directly from (2.7), in terms of the inner product (2.6), then
ZU(N)(t) =
fλ(t)
pλ, pλ
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
, (2.12)
where on the right-hand side of (2.12) we have used an expression for the inner product
of two power symmetric polynomials expressed in terms of the characters of the symmet-
ric group, given in appendix A. (Here ‘ℓ(µ)’ means the number of non-zero parts of the
partition µ.)
Using a result of appendix A (essentially orthogonality relations for symmetric group
characters), (2.12) may be rewritten as
ZU(N)(t) =
|λ|≤N
fλ(t) +
|λ|>N
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
. (2.13)
In the large N limit, ZU(N)(t) simplifies considerably as only the first term in (2.13)
need be considered. Using the frequency representation of λ then
ZU(∞)(t) =
fλ(t) =
f(tn)an =
1− f(tn) , (2.14)
a result which has been obtained using Polya counting methods for single trace operators
and saddle point approximations [22,23].
Higher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained
by successive evaluation of
µ⊢|λ|
ℓ(µ)≤N
. One method is to employ the Murnaghan-
Nakayama Rule, used to compute χ
λ using skew hooks and Young diagrams. (A readable
account of the Murnaghan-Nakayama Rule may be found in [38], though of course it is
explained in many standard textbooks that discuss the symmetric group.)
For the case of |λ| = N+1 then we may observe that,
µ⊢N+1
ℓ(µ)≤N
µ⊢N+1
ℓ(µ)≤N+1
)2 − (χνλ)2 = zλ − (χνλ)2 , ν = (1N+1) , (2.15)
since the partition ν = (1N+1) is the only one excluded among those partitions µ of
N+1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for
ν = (1L),
(χνλ)
1 for |λ| = L
0 otherwise
, (2.16)
since χνλ in this case is just a sign. (This may be easily seen as there is only one possible
way to remove successive skew hooks, which in this case are just column Young diagrams
of length λi, from the (1
L) column Young diagram to leave one of normal shape, in this
case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain
ZU(N)(t) =
|λ|≤N+1
fλ(t)−
|λ|=N+1
fλ(t) +
|λ|>N+1
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
. (2.17)
By a similar line of argument we may do the same for the case of |λ| = N+2. We
have that, for ν1 = (1
N+2) and ν2 = (2, 1
N+1),
µ⊢N+2
ℓ(µ)≤N
µ⊢N+2
ℓ(µ)≤N+2
)2 − (χν1λ )
2 − (χν2λ )
2 = zλ − (χν1λ )
2 − (χν2λ )
2 . (2.18)
We may determine, for ν = (2, 1L),
(χνλ)
(a1 − 1)2 for |λ| =
n≥1 n an = L+2
0 otherwise
. (2.19)
Using (2.16), (2.19) with (2.18) in (2.17) then we obtain
ZU(N)(t) =
|λ|≤N+2
fλ(t)−
N+1≤|λ|≤N+2
fλ(t)−
|λ|=N+2
fλ(t)
|λ|>N+2
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
(2.20)
where, in the frequency representation of λ,
(a1−1)2
(a1 − 1)!
(a1 − 2)!
)/ ∞∏
nanan! . (2.21)
We may proceed in this manner to compute explicit higher order corrections though
this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the
corrections will always involve contributions from (2.16) and (2.19) as well as extra ones
coming from
µ⊢|λ|
ℓ(µ)≤|λ|
µ⊢|λ|
ℓ(µ)≤N
The One Boson Case for U(N)
For the case of one bosonic fundamental field (applicable to half BPS operators for
N = 4 super Yang Mills), we have f(t) = t in (2.5), so that we may write
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)
j,k=1
1− tzjzk−1
. (2.22)
To evaluate this integral we may use the Cauchy-Littlewood formula,
1− xiyj
ℓ(λ)≤min.{L,M}
sλ(x1, . . . , xL)sλ(y1, . . . , yM) , (2.23)
where the sum on the right-hand side is over all partitions λ such that the corresponding
Young diagrams have no more than min.{L,M} rows, ℓ(λ) ≤ min.{L,M}. With xi = tzi,
yi = zi
−1, i = 1, . . . , N , in (2.23), so that sλ(tz1, . . . , tzN ) = t
|λ|sλ(z), and employing also
(2.6) and the orthonormality of Schur polynomials, we may easily obtain,
ZU(N)(t) =
ℓ(λ)≤N
sλ, sλ
ℓ(λ)≤N
t|λ| . (2.24)
By changing summation variables so that λi − λi+1 = ai, i = 1, . . . , N−1, λN = aN then
we may write
ZU(N)(t) =
a1,...,aN=0
ta1+2a2+...+NaN = PN (t) , (2.25)
where4
PN (t) =
1− ti . (2.26)
Of course this is nothing other than the generating function for the number pN (n) of
partitions of n into no more than N parts since, by definition,
ℓ(λ)≤N
δ|λ|,n = pN (n) , (2.27)
so that by the above
pN (n)t
n = PN (t) . (2.28)
4 1/P∞(t) =
(1−tn) is commonly called the Euler function, denoted by Φ(t). P∞(t) =∑∞
p(n)tn acts as a generating function for the number of unordered partitions of n, p(n). Note
that pN (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the
same as the total number of partitions of n so long as n ≤ N .
This makes explicit the connection between ZU(N)(t) and the partition numbers pN (n).
The SU(2) Gauge Group Case
Here we first consider f(t) =
j=1 tj in (2.1) so that the variables 0 ≤ ti < 1
represent k bosons in the single particle partition function. For such fields transforming in
the adjoint representation of SU(2) then (2.2) simplifies significantly. For any U ∈ SU(2)
we may write U = VΘV †, where V is unitary and Θ = diag.(eiθ, e−iθ), for 0 ≤ θ < 2π, so
that for F (U) = F (θ), then in usual Weyl parametrisation,
SU(2)
dµSU(2)(U)F (U) =
dθ sin2 θ F (θ)
1− cos θ
F ( θ
1− cos θ
F ( θ
(2.29)
where F (θ) = F (θ + π) is assumed in writing the last line. In the present case, F (U) =
F (θ) =
n≥1 f(t
n)χAdj.(U
n)/n, where χAdj.(U) = tr(U)tr(U
†) − 1 = e2iθ + e−2iθ + 1 =
2 cos 2θ + 1, so that
ZSU(2)(t) =
dθ (1− cos θ)
(1− tj)(1− tjeiθ)(1− tje−iθ)
. (2.30)
Making the variable change z = eiθ, and using that F (θ) = F (−θ) is even, then
ZSU(2)(t) =
(1− z)
(1− tj)(1− tjz)(1− tjz−1)
, (2.31)
where the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily
computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = tj .
ZSU(2)(t) =
1− ti2
j 6=i
(ti − tj)(1− titj)(1− tj)
. (2.32)
This partition function has an interesting interpretation from a group theory perspec-
tive. We may write,
1 + t
(1− tz)(1− tz−1) =
χn(z)t
n , (2.33)
where
χj(z) =
2 − z−j− 12
2 − z− 12
, j ∈ 1
Z , (2.34)
is an SU(2) character, corresponding to the spin j irreducible representation, Rj. Now the
integral in (2.31) acts as an SU(2) inner product,
χj , χk
(1− z)χj(z)χk(z−1) = δjk , (2.35)
for j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35),
(1− tj2)ZSU(2)(t) =
n1,...,nk=0
χn1 · · ·χnk , 1
n1 · · · tknk , (2.36)
acts as a generating function for the number of singlets in the decomposition of the SU(2)
representation Rn1 ⊗· · ·⊗Rnk .5 By using that χn(z) =
j=−n z
j we may use the Cauchy
residue theorem to compute explicitly that
χn1 · · ·χnk , 1
· · ·
(δj1+···+jk,n1+···+nk − δj1+···+jk,n1+···+nk+1) . (2.37)
If we modify the one particle partition function to include k bosons and k̄ fermions
and hence consider (2.1) in the form f(t, t̄) =
j=1 tj −
̄=1 t̄̄ then we may similarly as
above evaluate
ZSU(2)(t, t̄) =
(1− z)
1≤j≤k
1≤̄≤k̄
(1− t̄̄)(1− t̄̄z)(1− t̄̄z−1)
(1− tj)(1− tjz)(1− tjz−1)
, (2.38)
where the contour is around the unit disc |z| = 1. So long as k > k̄ then (2.38) receives
contributions from only those simple poles at z = tj so that for this case we obtain
ZSU(2)(t, t̄)
k−k̄−1
1− ti2
1≤̄≤k̄
1≤j≤k,j 6=i
(ti − t̄̄)(1− tit̄̄)(1− t̄̄)
(ti − tj)(1− titj)(1− tj)
. (2.39)
For k ≤ k̄ then (2.38) also receives contributions from poles at z = 0. For instance
ZSU(2)(t, t̄)
ti(1− ti2)
1≤j,̄≤k
j 6=i
(ti − t̄̄)(1− tit̄̄)(1− t̄̄)
(ti − tj)(1− titj)(1− tj)
1≤j,̄≤k
t̄̄(1− t̄̄)
tj(1− tj)
(2.40)
5 Generating functions for products of Lie algebra representations have been considered
elsewhere, in [39] for instance. A generating function for the number of singlets in n products of
the fundamental times n products of the anti-fundamental representations for SU(N) was found
by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a
nice physics oriented discussion of similar issues. The special case of R 1
⊗ · · · ⊗R 1
(2n products
of the fundamental) for SU(2), contains a Catalan number, 1
, of singlet representations.
where the last term on the right-hand side of (2.40) comes from the simple pole at z = 0.
These formulae should be useful for computing the multi-trace partition functions,
for fundamental fields transforming in an SU(2) gauge group, in other sectors of N = 4
super Yang Mills. For instance, after a suitable identification of the variables tj , t̄̄ with
variables in single particle partition functions for semi-short sectors of N = 4 super Yang
Mills, described in detail in [33], then (2.40) should allow for an explicit expression for
corresponding multi-trace partition functions. They may also be useful for computing the
N = 4 superconformal index of [1] for SU(2) gauge group, or at least for restrictions of it
such as described in [33] or [42].
3. Counting Operators in Free N = 4 Super Yang Mills
In this section, the counting of half and quarter BPS operators for free N = 4 super
Yang Mills, when the fundamental fields transform in the adjoint representation of U(N),
is discussed in some detail.
Counting Operators Directly
We may, of course, proceed to count multi-trace half and quarter BPS primary oper-
ators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for
quarter BPS operators.
(Z, Y ) forms a U(2) doublet, where U(2) has generators given by a subset of the
SU(4)R generators, Hi, Ei±, 1 ≤ i ≤ 3, where Hi are the Cartan sub-algebra genera-
tors and Ei± are ladder operators satisfying (in the Chevalley-Serre basis) [Hi, Ej±] =
±KijEj±, with [Kij] being the usual SU(4) Cartan matrix. The U(2) generators consist
of the SU(2) generators H2, E2±, where explicitly [(H1, H2, H3), E2±] = ∓(1,−2, 1)E2±,
along with the generator H1+H2+H3, whose eigenvalues give the conformal dimensions
in this case, [H1+H2+H3, (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E2+, Z] = 0,
[E2−, Z] = Y , [(H1, H2, H3), Z] = (0, 1, 0)Z, [(H1, H2, H3), Y ] = (1,−1, 1)Y so that an
operator involving n Z’s and m Y ’s transforms in the [m,n−m,m] SU(4)R R-symmetry
representation.
For k-trace half BPS primary operators transforming in the [0, n, 0] SU(4)R R-
symmetry representation, with conformal dimension n, then in terms of the fundamental
field Z a basis is provided by,
tr(Zn1) · · · tr(Znk) ,
ni = n . (3.1)
We have that, due to trace identities for finite N , tr(Zn) for n > N is expressible in terms
of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis
for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with
every ni ≤ N , ordered so that n1 ≥ n2 ≥ . . . ≥ nk ≥ 0, i.e. so that (n1, . . . , nk) is a
partition of n where each part ni ≤ N . With this restriction, the number of multi-trace
half BPS primary operators for a given n is
N(n) = pN (n) , (3.2)
since the number pN (n), in (2.27), of partitions of n into ≤ N parts is the same as the
number of partitions of n in which each part is ≤ N - see [43] for a simple proof employing
generating functions.
For quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry repre-
sentation, a basis for k-trace operators is
Zn1jY m1j
· · · tr
ZnkjY mkj
nij = n ,
mij = m, (3.3)
where there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to
the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k,
then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace
and also trace identities for finite N must be accounted for. Assuming that this is done,
let M(n,m) denote the number of elements in this minimal basis for multi-trace operators
of the form (3.3). Then, to obtain the number N(n,m) of multi-trace quarter BPS primary
operators in the SU(4)R representation [m,n−m,m], the number of U(2) descendants, in
the SU(4)R representation [m,n−m,m], of multi-trace quarter BPS primary operators, in
SU(4)R representations [j, n+m−2j, j], 0 ≤ j ≤ m−1, must be subtracted from M(n,m).
(These descendants arise due to the relation [E2−, Z] = Y . Acting with (E2−)
m−j on the
highest weight state in the SU(4)R representation [j, n+m−2j, j] we obtain a descendant in
the SU(4)R representation [m,n−m,m].) The number of such U(2) descendants coincides
with N(n+m−j,j), the number of corresponding primary operators. In this way, we obtain
M(n,m) = N(n,m) +N(n+1,m−1) + . . .+N(n+m−1,1) +N(n+m) , (3.4)
so that N(n,m) = M(n,m) −M(n+1,m−1) may be obtained recursively for each m.
We may illustrate by counting all multi-trace quarter BPS primary operators in the
[1, n−1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is
provided by
tr(Zn1) · · · tr(Znk)tr(ZjY ) ,
ni = n− j . (3.5)
Cyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U(N)
trace identities imply, similarly as for the half BPS case, that a minimal basis for multi-
trace operators requires j < N and each ni ≤ N in (3.5) for every 1 ≤ k ≤ n−j, so
that (n1, . . . , nk) forms a partition of n−j, with every part ≤ N . Thus, by a similar
argument as for the half BPS case, M(n,1) =
j=0 pN (n−j). Finally, to ensure that only
primary operators are counted then we must subtract off contributions from descendants
of half BPS primary operators in the [0, n+1, 0] SU(4)R representation, of which there are
pN (n+1). Using (3.4) with (3.2) we then conclude that
N(n,1) =
pN (n− j)− pN (n+1) , (3.6)
gives the number of multi-trace quarter BPS primary operators in the [1, n−1, 1] R-
symmetry representation.
Counting in this fashion becomes more difficult for greater m and now a procedure is
described employing symmetric polynomials to find a generating function for the numbers
of multi-trace quarter BPS primary operators in the [m,n−m,m] SU(4)R representation,
for m = 0, 1, 2 at finite N and for any n,m at large N . This generating function is
subsequently used to provide asymptotic counting for fixed m, large n in the large N limit.
Counting Operators via Expansion of Partition Functions in Schur Polynomials
For k bosonic fundamental fields, we may take f(t) =
j=1 tj in (2.1) so that (2.5)
may be written as
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)
r,s=1
1− tjzrzs−1
. (3.7)
Often it is the case that such partition functions should be expanded in terms of sλ(t), the
k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for
counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula
(2.23) to expand in this way, to obtain
ZU(N)(t) =
ℓ(λ)≤min.{k,N2}
Nλ sλ(t) , (3.8)
where
(2πi)NN !
∆(z)∆(z−1) sλ(zz
−1) , (3.9)
where zz−1 has components zizj
−1, 1 ≤ i, j ≤ N .
From Macdonald [24] we have that
sλ(xy) =
µ,ν⊢|λ|
γλµνsµ(x)sν(y) , (3.10)
in terms of Kronecker coefficients,
γλµν =
σ∈S|λ|
χλ(σ)χµ(σ)χν(σ)
ρ⊢|λ|
χλρ χ
(3.11)
being a sum over irreducible S|λ| characters evaluated at σ ∈ S|λ|, related to a sum over
irreducible S|λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in
the second line. Using (2.6) along with the orthonormality property of Schur polynomials
we find that
µ⊢|λ|
ℓ(µ)≤N
γλµµ . (3.12)
The situation becomes much more involved if we include also k̄ fermionic fields, so
that (2.1) may be written in the form f(t, t̄) =
j=1 tj −
̄=1 t̄̄, and attempt to ex-
pand ZU(N)(t, t̄) in terms of products of Schur polynomials sλ(t)sµ(t̄). Such expansion
is required for counting, for instance, for the free field partition function in the eighth
BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded,
analogous to (1.1), in terms of SU(2|3) characters, which may be expressed in terms of a
linear combination of products of two-variable and three-variable Schur polynomials. (See
[33] for a discussion of counting for the eighth BPS sector along these lines.) Including
fermions, (3.7) becomes modified by
ZU(N)(t, t̄) =
(2πi)NN !
∆(z)∆(z−1)
1≤j≤k
1≤̄≤k̄
r,s=1
1− t̄̄zrzs−1
1− tjzrzs−1
. (3.13)
To achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with
another formula of Littlewood,
(1 + xiyj) =
ℓ(λ)≤L,ℓ(λ̃)≤M
sλ(x1, . . . , xL) sλ̃(y1, . . . , yM ) , (3.14)
where λ̃ is the partition conjugate to λ (where the rows and columns of the Young diagram
corresponding to λ are interchanged) and where the sum is restricted to those λ whereby
the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns,
ℓ(λ̃) ≤ M . We may thus write
ZU(N)(t, t̄) =
ℓ(λ)≤min.{k,N2}
ℓ(µ)≤k̄,ℓ(µ̃)≤N2
Nλ,µ sλ(t) sµ(t̄) , (3.15)
where
Nλ,µ =
(−1)|µ|
(2πi)NN !
∆(z)∆(z−1) sλ(zz
−1)sµ̃(zz
−1) . (3.16)
Obviously these numbers are considerably more involved than those in (3.9). We may of
course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients.
Counting Quarter BPS Operators by Symmetric Polynomial Methods
The two bosonic fundamental field case is now focused upon.6 In particular, the
numbers N(n,m) in (1.1) are evaluated using results of the last sub-section.
We may proceed to evaluate Nλ recursively. The simplest case is for Nλ = N(n),
whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and
(2.26), that
N(n)tn =
(2πi)NN !
∆(z)∆(z−1)
j,k=1
1− tzjzk−1
= PN (t) , (3.17)
so that, by (2.28), N(n) is given by (3.2).
More generally to evaluate N(n,m) from (3.9) we may use, for y = zz−1,
s(m)(y)s(n)(y) = s(n,m)(y) + s(n+1,m−1)(y) + . . .+ s(n+m−1,1)(y) + s(n+m)(y) ,
s(m)(zz
−1) =
ℓ(µ)≤N
sµ(z)sµ(z
−1) , (3.18)
6 The two boson case leads to an interesting generalisation of an identity in [44] involv-
ing Littlewood-Richardson coefficients cνλµ, the coefficients that appear in the decomposition
sλ(x)sµ(x) =
cνλµsν(x). With f(t) in (2.1) given by f(t) = t1 + t2, and expanding appro-
priately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of
Schur polynomials and the result (2.14), we obtain (note that cνλµ = 0 if |ν| 6= |λ|+ |µ|)
ZU(∞)(t1, t2) =
〈sλsµ, sλsµ〉∞ =
λ,µ,ν
ν⊢|λ|+|µ|
1− t1n − t2n
which reduces to Theorem 4.1 of [44] if we take t1 = t2 = t.
where the expression in the first line of (3.18) may be easily seen using Young tableaux
multiplication rules while (2.23) determines the expression in the second line. From (3.9)
with (3.18), we may find a useful generating function, in terms of a formal variable t, for
the numbers in (3.4) as follows,
F (m)N (t) =
M(n,m)tn
(2πi)NN !
∆(z)∆(z−1) s(m)(zz
s(n)(zz
−1)tn
(2πi)NN !
∆(z)∆(z−1)
ℓ(µ)≤N
sµ(z)sµ(z
j,k=1
1− tzjzk−1
ℓ(µ)≤N
sλsµ, sλsµ
(3.19)
so that we may write
N(n,m) =
F (m)N (t)−
F (m−1)N (t)
, (3.20)
which allows for recursive determination of N(n,m).
Applying this to the case of Nλ = N(n,1) we have, from (3.18)
s(1)(zz
−1) = s(1)(z)s(1)(z
−1) , (3.21)
so that, from (3.19),
F (1)N (t) =
N(n,1) +N(n+1)
ℓ(λ)≤N
s(1)sλ, s(1)sλ
. (3.22)
Using (again, this may be easily seen from Young tableaux multiplication rules)
s(1)(z)sλ(z) =
sλ+er (z) , (3.23)
for {er; 1 ≤ r ≤ N, er · es = δrs} being usual orthonormal vectors, we find that
F (1)N (t) =
N(n,1) +N(n+1)
ℓ(λ)≤N
r,s=1
sλ+er , sλ+es
. (3.24)
Now for any partition λ,
sλ+er , sλ+es
vanishes unless er = es for any r, s and λr−1−λr >
0 for r = 2, . . . , N , due to
s(λ1,...,λr−1,λr+1,...,λN )(z) = 0 for λr−1 = λr , r > 1 . (3.25)
Changing summation variables to those in (2.25) then we have, with the definition (2.26),
F (1)N (t) =
N(n,1) +N(n+1)
a1,...,aN≥0
ta1+...+NaN +
a1,...,aN≥0
ta1+...+NaN
a1,...,aN≥0
ta1+...+NaN =
1− tPN−1(t) .
(3.26)
Thus, using (2.28), (3.2) with (3.26),7
N(n,1) =
pN−1(j)−N(n+1) =
pN−1(j)− pN (n+ 1) . (3.27)
For the case of Nλ = N(n,2) we have that, from (3.18),
s(2)(zz
−1) = s(2)(z)s(2)(z
−1) + s(1,1)(z)s(1,1)(z
−1) , (3.28)
so that, from (3.19), we have
F (2)N (t) =
ℓ(λ)≤N
s(2)sλ, s(2)sλ
s(1,1)sλ, s(1,1)sλ
. (3.29)
Using
s(2)(z)sλ(z) =
sλ+2er (z) +
1≤r<s≤N
sλ+er+es(z) ,
s(1,1)(z)sλ(z) =
1≤r<s≤N
sλ+er+es(z) ,
(3.30)
7 This formula agrees with (3.6) due to
pN (n − j) =
pN−1(j) which follows
because the corresponding generating functions match,
pN (n− j)t
= (1 + t+ . . .+ t
)PN (t) =
PN−1(t) =
pN−1(j)t
along with (3.25) and
s(λ1,...,λr−1,λr+2,...,λN )(z) = −s(λ1,...,λr+1,λr−1+1,...,λN )(z) , (3.31)
for the cases where λr−1 = λr, we may obtain, with the definition (2.26),
F (2)N (t) =
1−tN+1
(1−t)(1−t2)PN−1(t) +
(1−t)(1−t2)PN−2(t) , (3.32)
where the first contribution comes from
|λ|〈s(2)sλ, s(2)sλ
while the second comes
|λ|〈s(1,1)sλ, s(1,1)sλ
. Since the partition number pk(−n) = 0 for n = 1, 2, . . .
we may write, using (2.28),8
(1− t)(1− t2)Pk(t) =
n,i,j=0
pk(n− i− 2j)tn . (3.33)
Thus, from (3.32) with (3.27),
N(n,2) = −
pN−1(j)
i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)) if n ≤ N ,∑∞
i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)− pN−1(N+1−n−i−2j)) if n ≥ N+1.
(3.34)
Tables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few
cases of n,N . Notice from these tables that the numbers N(n,m) below the diagonal line
N ≥ n+m for a given n are the same for all N . This is a general feature that derives from
values of N(n,m) for N ≥ n + m, which numbers may be obtained from a corresponding
generating function that is now constructed.
Using these techniques, we may provide a consistency check of (3.17), (3.26), (3.32)
along with a general result for N(n,m) for high enough values of N , N ≥ m + n. This
employs the orthogonality property of power symmetric polynomials pλ(z) (in the large N
8 This is a special case of the following: for any f(n), n ∈ Z, that satisfies f(−n) = 0,
n = 1, 2, . . ., then we may (at least formally) write
Pk(t)
f(n)t
n,i1,...,ik=0
f(n− i1 − 2i2 − . . .− kik)t
limit) along with
s(n)(z) =
pλ(z)
i1,...,in=0
i1!i2! · · · in!
δi1+2i2+...+nin,n p1(z)
p2(z)
)i2 · · ·
pn(z)
(3.35)
Using the trivial identity pλ(xy) = pλ(x)pλ(y) then from (2.6), (3.19) with (3.35) we have
F (m)∞ (t) =
pλpµ, pλpµ
pν , pν
(3.36)
where for (1a1 , 2a2 , . . .) being the frequency representation of λ and (1b1 , 2b2 , . . .) being
that of µ then ν has frequency representation (1a1+a2 , 2a2+b2 , . . .) so that |ν| = n + m.
This agrees with F (m)N (t) in a series expansion up to O(tN−m) (since the last equation in
(3.36) is also valid for finite N so long as |ν| = n +m ≤ N , by a result of appendix A).
Now, since
(aj+bj)!
aj !bj!
, (3.37)
we obtain from (3.36) that,
F (m)∞ (t) =
a1,...,an=0
b1,...,bm=0
δa1+···+nan,nδb1+···+mbm,m
(aj + bj)!
aj !bj!
b1,...,bm=0
δb1+···+mbm,m
(aj + bj)!
aj!bj !
1− tj
b1,...,bm=0
δb1+···+mbm,m
(1− tj)bj+1
1− tj .
(3.38)
For the first few cases we have that, with PN (t) as defined in (2.26),
F (m)∞ (t) =
P∞(t) for m = 0
1−tP∞(t) for m = 1
(1−t)(1−t2)P∞(t) for m = 2
, (3.39)
whose series expansion agrees with (3.17), (3.26), (3.32) up to O(tN−m) for, respectively,
m = 0, 1, 2. We may use (3.20) with (3.36) to determine N(n,m) exactly for N ≥ n+m.
Asymptotic Counting of Quarter BPS Operators at Large N
Asymptotic counting for the one boson case in the large N limit, for which, with PN (t)
as defined in (2.26), with p(n) being the total number of (unordered) partitions of n,
ZU(∞)(t) = P∞(t) =
p(n)tn , (3.40)
is the multi-trace partition function, entails finding an asymptotic value for the partition
number p(n) for ‘large’ n. This may be achieved by performing a saddle point approxima-
tion of p(n) = 1
dtP∞(t)t
−n−1. The function P∞(t) has a ‘large’ singularity at t = 1,
but in addition has singularities at all other roots of unity - see [45] on the validity of ignor-
ing these contributions asymptotically. This method was used by Hardy and Ramanujan
to find their celebrated formula, here given in a less detailed form as,
p(n) ∼ 1
, (3.41)
which was improved by Rademacher to give p(n) exactly. Their method relied crucially on
the modular properties of P∞(t).
Focusing now on the two bosonic fundamental field case for which, in the large N
limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for
the numbers Q(n,m, b) = Q(n,m, b1, . . . , bm), with constraint equation
j=1 jbj = m,
defined by
(1− tj)bj+1
1− tj = 1 +
Q(n,m, b)tn . (3.42)
Having found these we may then attempt to find the dominant contribution to (3.20) with
(3.38) for large N . In order to give asymptotic values for Q(n,m, b) we may follow [46] and
apply a formula due to Meinardus which gives a general result for the generating function
(1− tn)−an = 1 +
r(n)tn . (3.43)
A detailed version of Meinardus’ theorem may be found in [46] but for purposes of brevity
we may note that it implies that, as n → ∞,
r(n) ∼ C nκ exp
AΓ(α+ 1)ζ(α+ 1)nα
)1/(α+1)
(α+ 1)/α
, (3.44)
where ζ(s) =
j=1 j
−s is the Riemann zeta function and the constants C, κ, α, A are
determined by the auxiliary Dirichlet series,
D(s) =
, (3.45)
which must converge for Re(s) > α, a positive real number, and possess an analytic
continuation in the region Re(s) ≥ c, −1 < c < 0, such that, in this region, D(s) is
analytic except at a simple pole at s = α where it has residue A. In terms of α,A then
2π(1+α)
AΓ(α+ 1)ζ(α+ 1)
)(1−2D(0))/2(α+1)
expD′(0) ,
κ = (D(0)− 1− 1
α)/(α+ 1) .
(3.46)
Applying Meinardus’ theorem to the case of (3.42), clearly we have
D(s) =
+ ζ(s) , (3.47)
so that, assuming m =
j=1 jbj is fixed, D(s) has a simple pole at s = α = 1 where it
has residue A = 1. Using
D(0) = −1
bj , expD
′(0) =
, (3.48)
then, from (3.44) with (3.46), we may easily determine that, as n → ∞,
Q(n,m, b) ∼ 1
bj m∏
. (3.49)
This reduces to (3.41) when bj = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using
(3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞,
N(n,m) =
b1,...,bm≥0∑
jbj=m
Q(n,m, b)−
b1,...,bm−1≥0∑
jbj=m−1
Q(n+1, m−1, b)
(3.50)
since Q(n,m,m, 0, . . . , 0), for b1 = m, bj = 0, j > 1, dominates over all other terms in
(3.50). This gives asymptotic values for the numbers in (1.1), for counting quarter BPS
operators, transforming in [m,n−m,m] SU(4)R representations, in the large N limit of
free N = 4 super Yang Mills, as previously described.
4. Counting Operators in the Chiral Ring of N = 4 Super Yang Mills
For the purposes of counting operators in the chiral ring of N = 4 super Yang Mills,
we denote corresponding multi-trace partition functions by CU(N)(t).
The generating function for CU(N)(t) for the case of one bosonic fundamental field has
been written in the form [1,3,9]
C(ν, t) =
1− νtn =
νNCU(N)(t) , (4.1)
so that ν acts as a chemical potential for the rank of the gauge group U(N). The equiv-
alence CU(N)(t) = ZU(N)(t) = PN (t), with ZU(N)(t) as in (2.25), is actually a special case
of the q-Binomial theorem. Writing - see [43] for notation -
(a; q)k = (1− a)(1− aq) · · · (1− aqk−1) , (4.2)
then the q-Binomial theorem is, for |x|, |q| < 1,
(a; q)k
(q; q)k
(ax; q)∞
(x; q)∞
. (4.3)
(Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t)∞ = C(ν, t) above
and 1/(t; t)N = PN (t) in (2.26), then CU(N)(t) = ZU(N)(t) = PN (t) straightforwardly.
This special case of the q-Binomial theorem is due to Euler.)
For the two boson case, so that the single particle partition function is given by
f(t, u) = t + u for some t, u, then the generating function for the finite N chiral ring
partition function CU(N)(t, u) is given by [1,3,9]
C(ν, t, u) =
n,m=0
1− νtnum =
νNCU(N)(t, u) . (4.4)
This function is more difficult to analyse in terms of counting though has been investigated
by Stanley [47] in relation to partitions - there it has been dubbed the ‘double Eulerian’
generating function. Through use of the Cauchy-Littlewood formula, then we may expand
CU(N)(t, u) in terms of partitions of N as,
CU(N)(t, u) =
hλ(t)hλ(u) , (4.5)
where
hλ(t) = sλ(1, t, t
2, . . .) , (4.6)
so that using an identity for Schur polynomials to be found in [24,47] then
CU(N)(t, u) =
1≤i<j≤N (1− tλi−λj+j−i)(1− uλi−λj+j−i)∏N
i=1(t; t)λi+N−i(u; u)λi+N−i
(i−1)λi . (4.7)
(4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π
being all column-strict plane partitions of shape λ, |π| =
i,j πij ,
hλ(t) = sλ(1, t, t
2, . . .) =
t|π| . (4.8)
Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connec-
tion between the ‘double Eulerian’ generating function and major indices of permutations
see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2
whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corre-
sponding to a Young diagram with a single row of two boxes) π11 ≥ π12 ≥ 0 gives all
column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a
Young diagram with a single column of two boxes) then π11 > π21 ≥ 0 gives all column-
strict plane partitions of shape (1, 1). Thus,
h(2,0)(t) =
π11,π12≥0
π11≥π12
tπ11+π12 =
(1− t)(1− t2) ,
h(1,1)(t) =
π11,π21≥0
π11>π21
tπ11+π21 =
(1− t)(1− t2) ,
(4.9)
so that, from (4.5) for N = 2,
CU(2)(t, u) = h(2,0)(t)h(2,0)(u)+h(1,1)(t)h(1,1)(u) =
1 + ut
(1− t)(1− t2)(1− u)(1− u2) , (4.10)
which is the correct result as may be verified by extracting the ν2 coefficient in an expansion
of (4.4) up to O(ν2).
9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane parti-
tion of shape λ is an array π = (πij) of non-negative integers with finitely many non-zero entries,
that is arranged in a Young tableaux with shape λ - see appendix A - such that the numbers πij
are weakly decreasing along each row, πij ≥ πi j+1 ≥ 0, and strictly decreasing down each column,
πij > πi+1 j ≥ 0. The sum of the parts of π is given by |π| =
πij. (Note that in contrast to
the definition in [47], here we are allowing πij = 0, for some i, j, to be a part of the plane partition
π with shape λ.)
In the large N limit,
CU(∞)(t, u) =
n1,n2≥0
n1+n2>0
1− tn1un2 , (4.11)
upon which attention is shortly focused.
For the numbers N(n,m) → N̂(n,m) counting quarter BPS primary operators for the
chiral ring of N = 4 super Yang Mills, belonging to [m,n − m,m] SU(4)R R-symmetry
representations, as in (1.1), we have 10
N̂(n,m) =
dt du CU(N)(t, u) s(n,m)(t−1, u−1) (t−1 − u−1)2 . (4.12)
These may be more conveniently evaluated in terms of the numbers in (3.4) M(n,m) →
M̂(n,m), counting all chiral ring quarter BPS operators in the [m,n−m,m] SU(4)R repre-
sentation, given by
M̂(n,m) =
(2πi)2
dt du CU(N)(t, u) t−n−1u−m−1 , (4.13)
so that N̂(n,m) = M̂(n,m)−M̂(n+1,m−1). Defining Pλ(n) to be the number of column-strict
plane partitions π of shape λ so that |π| =
i,j πij = n, then, from (4.5) with (4.8) and
(4.13), M̂(n,m) =
λ⊢N Pλ(n)Pλ(m). Thus,
N̂(n,m) =
(Pλ(n)Pλ(m)− Pλ(n+1)Pλ(m−1)) , (4.14)
counts chiral ring quarter BPS primary operators in SU(4)R representations [m,n−m,m]
for any n,m at finite N .
Asymptotic Counting for Chiral Ring BPS Operators at Large N
For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills
at large N , a relatively crude method is employed here which nevertheless captures the
exponential behaviour of counting numbers of interest. This method is based on saddle
point approximations of functions near a dominant singularity - see [45] for a useful sum-
mary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we
are interested only in the exponential behaviour of such numbers anyhow.)
10 This formula employs the orthonormality relation of Schur polynomials described here
and has appeared in a similar context in [33], appendix B.
To illustrate, we consider the one boson case in the large N limit again. We first find
a convenient ‘approximating function’ as follows,
P∞(t) =
1− tn = exp
ln(1− tn)
∼ exp
ds ln(1− ts)
= exp
6 ln t
(4.15)
which has an ‘easier’ singularity structure. (The approximation in the second step may
be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using
(4.15) then for large enough n,
p(n) =
dt P∞(t)t
−n−1 ∼ 1
dt eg(t) , g(t) = − π
6 ln t
− n ln t . (4.16)
We may approximate the latter integral for large n by noting that the dominant contribu-
tion is at the saddle point t′ = e−π/
6n ∼ 1 for which
g(t′) = π
n , g′(t′) = 0 , g′′(t′) =
6n3 eπ
2/3n = α , (4.17)
so that, for t′′ = t− t′,
p(n) ∼ eπ
2n/3 1
dt′′ e
αt′′2 ∼ eπ
2n/3 1
ds e−
αs2 =
2n/3 .
(4.18)
Thus,
ln p(n) ∼ π
n , (4.19)
which captures the correct behaviour of ln p(n) for large n, according to (3.41).
We may proceed analogously for the quarter BPS chiral ring multi-trace partition
function at large N , (4.11), which we approximate by
CU(∞)(t, u) =
n1,n2≥0
n1+n2>0
1− tn1un2 = exp
n1,n2≥0
n1+n2>0
ln(1− tn1un2)
∼ exp
dv dw ln(1− tvuw)
= exp
( ζ(3)
ln t lnu
(4.20)
In this case we have, from (4.13),
M̂(n,m) ∼
(2πi)2
dt du eg(t,u) , (4.21)
where
g(t, u) =
ln t lnu
− n ln t−m lnu , (4.22)
for n,m large. The dominant contribution to the integral, for n,m both large and of the
same order, occurs about the point (t′, u′) ∼ (1, 1) where
t′ = e−(ζ(3)mn
−2)1/3 , u′ = e−(ζ(3)nm
−2)1/3 , (4.23)
for which
g(t′, u′) = 3 3
ζ(3)nm ,
g(t, u)
(t′,u′)
g(t, u)
(t′,u′)
= 0 ,
g(t, u)
(t′,u′)
= 2(ζ(3)−1n5m−1)
3 e2(ζ(3)mn
−2)1/3 = α ,
g(t, u)
(t′,u′)
= 2(ζ(3)−1m5n−1)
3 e2(ζ(3)nm
−2)1/3 = β ,
g(t, u)
(t′,u′)
= (ζ(3)−1n2m2)
3 e(ζ(3)mn
−2)1/3+(ζ(3)nm−2)1/3 = γ .
(4.24)
So long as m,n are both large and of the same order, the saddle point approximation is
justified and we obtain
M̂(n,m) ∼ e3
ζ(3)nm 1
dv dw e−
(αv2+βw2+2γvw) = h(α, β, γ)e3
ζ(3)nm ,
(4.25)
where
h(α, β, γ) =
αβ − γ2
(ζ(3)m−2n−2)
3 e−(ζ(3)mn
−2)1/3−(ζ(3)nm−2)1/3 . (4.26)
We thus have that for n,m both comparably large,
lnM̂(n,m) ∼ 3 3
ζ(3)nm , (4.27)
so that
ln N̂(n,m) = ln
M̂(n,m) − M̂(n+1,m−1)
∼ lnM̂(n,m) ∼ 3 3
ζ(3)nm . (4.28)
It is difficult to check the consistency of this result given the dearth of literature on
these types of multi-variable generating functions and their asymptotic behaviour, however,
we may consider the simpler function, also considered in [9],
CU(∞)(t, t) =
(1− tn)n+1 =
E(r) tr , (4.29)
where, in terms of the counting numbers N̂(n,m), from (1.1),
CU(∞)(t, t) =
(n−m+1)N̂(n,m) tn+m ⇒ E(r) =
(r−2m+1)N̂(r−m,m) .
(4.30)
Hence, E(r) counts quarter BPS primary operators in the chiral ring of N = 4 super
Yang Mills, transforming in [m,n−m,m] SU(4)R representations, with the same conformal
dimensions r = n+m. Extracting the dominant contribution to ln E(r) from (4.30), which
occurs at the maximum value of m, mM = [
r], and using (4.28), we obtain
ln E(r) ∼ ln N̂(r−mM,mM) ∼ 32
2ζ(3)r2 . (4.31)
By considering (4.29) directly, we may employ Meinardus’ theorem (described in the
third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus’
theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet
series (3.45), with aj = j+1, has two simple poles. To overcome this difficulty we split
(4.29) into a product of two functions, both separately amenable to application of Meinar-
dus’ theorem. One is the reciprocal of the Euler function, P∞(t) in (4.15). The other, the
MacMahon function, is given by
M(t) =
(1− tn)n =
q(r) tr , (4.32)
and has been considered in a similar context as here in [3].11 Writing
CU(∞)(t, t) = P∞(t)M(t) , (4.33)
with P∞(t) as in (4.15), so that, using (4.29),
E(r) =
r1,r2≥0
r1+r2=r
p(r1)q(r2) , (4.34)
we may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant
contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary
Dirichlet series for M(t) in (4.32) is, from (3.45) with aj = j,
D(s) = ζ(s− 1) , (4.35)
11 The relation of M(t) in (4.32) to plane partitions is given a description in [46]. Briefly,
q(r) gives the number of ordinary plane partitions π, so that πij ≥ πi+1 j > 0, πij ≥ πi j+1 > 0,
with |π| =
πij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions.
In fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula
first found by Wright [48] for the number of plane partitions q(r) of the number r.
which has a simple pole at s = α = 2, at which the residue is A = 1. Thus, from (3.44),
ln q(r) ∼ 3
2ζ(3)r2 . (4.36)
This is consistent with (4.31) as the dominant contribution to lnE(r) comes from the r1 = 0
term in (4.34) (since p(r) ≪ q(r) as r → ∞) so that lnE(r) ∼ ln q(r).
It has not escaped attention that the method used here, to capture the exponential
behaviour of asymptotic values for the numbers M̂(n,m), may be easily extended to chiral
ring sectors other than the quarter BPS one. Suppose, for simplicity, that Zj , 1 ≤ j ≤ k−1,
are commuting bosonic fundamental fields, in the U(N) Lie algebra, so that the single
particle partition function is given by f(t) =
j=1 tj , in terms of the corresponding letters
tj . Let M̂(m1,...,mk−1) denote the number of independent operators involving products of
m1 Z1’s, m2 Z2’s etc. in corresponding multi-trace operators. The multi-trace partition
function, in the large N limit, is given by,
CU(∞)(t) =
n1,...,nk−1≥0
n1+...+nk−1>0
1− t1n1 · · · tk−1nk−1
, (4.37)
which may be crudely approximated by, similarly as before,12
CU(∞)(t) ∼ exp
dvj ln(1−t1v1 · · · tk−1vk−1)
= exp
(−1)k+1ζ(k)
ln t1 · · · ln tk−1
. (4.38)
Thus, without going into as much detail, for the analogue of (4.27) we have, (assuming mj
are all comparably large,)
lnM̂(m1,...,mk−1) ∼ g(t
1, . . . , t
k−1) = k
ζ(k)m1 · · ·mk−1 , (4.39)
where g(t′1, . . . , t
k−1) is the value of
g(t1, . . . , tk−1) =
(−1)k+1ζ(k)
ln t1 · · · ln tk−1
−m1 ln t1 − . . .−mk−1 ln tk−1 , (4.40)
12 For Lin(x) =
xj/jn being the usual Polylogarithm, with Lin(1) = ζ(n), n > 1,
Lin(0) = 0, then with the convention Li1(x) = − ln(1− x), the following integral
Lin(zx) = Lin+1(z) ,
may be useful for showing this, after a suitable change of variables.
at the saddle point (t′1, . . . , t
k−1) ∼ (1, . . . , 1), where
(ln t′1, . . . , ln t
k−1) = − k
ζ(k)m1 · · ·mk−1 (1/m1, . . . , 1/mk−1) , (4.41)
so that
g(t1, . . . , tk−1)
,...,t′
= 0 , j = 1, . . . , k−1 . (4.42)
(4.39) is consistent with a result implied by Meinardus’ theorem. The function,13
CU(∞)(t, . . . , t) ∼
(1− tn)−n
k−2/(k−2)! =
c(k, r)tr , (4.43)
has auxiliary Dirichlet series, from (3.45) with aj = j
k−2/(k − 2)!,
D(s) =
(k − 2)! ζ(s+ 2− k) , (4.44)
which has a simple pole at s = α = k−1 at which the residue is A = 1/(k − 2)!, so that,
from (3.44),
ln c(k, r) ∼ k
(k−1) ζ(k) rk−1 . (4.45)
(4.45) is precisely the result that may be obtained from (4.39) if we maximise the product
m1 · · ·mk−1, subject to the constraint
j=1 mj = r, for which the solution is mj = mM =
r/(k − 1) (relaxing the constraint that mj be non-negative integers, which is irrelevant
asymptotically), so that lnM̂(mM,...,mM) ∼ ln c(k, r).
This is applicable to counting multi-trace operators in the eighth BPS chiral ring sec-
tor for N = 4 super Yang Mills with fundamental fields Z, Y,X involving m1 Z’s, m2 Y ’s,
m3 X ’s. Expanding the corresponding partition function (4.37), with k = 4, in terms of
Schur polynomials s(m1,m2,m3)(t), m1 ≥ m2 ≥ m3 ≥ 0, similar to (1.1), the expansion
coefficients N̂(m1,m2,m3) count spinless multi-trace primary operators transforming in the
[m2 +m3, m1 −m2, m2 −m3] SU(4)R R-symmetry representation, with conformal dimen-
sionsm1+m2+m3 [33]. Just as in (4.28), asymptotically lnN(m1,m2,m3) ∼ lnM(m1,m2,m3).
This counting, however, ignores contributions of the fermionic fields λ, λ̄, which it may be
important to include in order to give correct counting of eighth BPS chiral ring operators.
13 This may be easily seen from (4.37), as the number of solutions to
mj = n, where
mj are non-negative integers, is the binomial number
n+k−2
which, to leading order in large n,
behaves like nk−2/(k−2)!. More properly, we should split the product CU(∞)(t, . . . , t) into pieces
separately amenable to Meinardus’ theorem, as for the prior case for k = 3, however, just as for
that case, the numbers c(k, r) dominate, and so other contributions are ignored here.
5. Conclusions
There are some obvious questions not answered by this work. The first is whether
or not the approach in the second section using symmetric polynomials can give insight
into thermodynamics at finite N , such as for the Hagedorn transition, for example. While
it gives the large N expression (2.14) in an elementary way, its wider applicability or
usefulness to such questions is unclear. The approach is undoubtedly useful for finding
exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS
operators) and (3.9), (3.16) may be useful for analysing counting for more complicated
sectors of N = 4 super Yang Mills, with gauge group U(N).
The second question is how the arguments employing symmetric polynomial tech-
niques here may be extended to other gauge groups, the most pertinent being perhaps
SU(N). Arguments here employing (2.6) and the orthonormality property of Schur poly-
nomials should remain largely unaffected for SU(N). Exact values for counting numbers
obtained here should require some modification for SU(N), though asymptotic values may
be unchanged.
The third question concerns asymptotic values for counting numbers and how these
may be improved. The asymptotic counting formulae given in such papers as [3,9] for
chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan,
Meinardus, etc., all of which derive from single variable generating functions. It is hoped
that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS
operators, that distinguishes between differing R-symmetry charges, represents a serious
attempt at going beyond consideration of single variable generating functions.14 Improving
upon these formulae will require more sophisticated techniques, perhaps along the lines
used to find those of Hardy and Ramanujan or Meinardus and employing any modular
properties of the multi-variable functions involved. This issue may also be important for
microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4
superconformal symmetry, depend on special values of R-symmetry charges [17,18,19,20]
- see [50] for a related detailed discussion.
Thus far, the elegant results for finite N partition functions for chiral ring sectors have
been interpreted from a largely geometric perspective - it may be interesting to investigate
more how such results are related to the theory of random matrices and/or symmetric
polynomials.
14 After submission of the first version of this paper to the electronic archive, I received
an e-mail from Hai Lin pointing out an interesting comparison between (4.28) here and (2.14) of
[49], obtained in quite a different context. The two formulae are essentially the same given the
numerical value 3 3
ζ(3) = 3.189 . . ., correct to three decimal places.
Acknowledgements
I warmly thank Yang-Hui He, Paul Heslop, Hugh Osborn, Christian Romelsberger and
Christian Saemann for useful comments and discussions. This work is supported by an
IRCSET (Irish Research Council for Science, Engineering and Technology) Post-doctoral
Fellowship.
Appendix A. Partitions, symmetric group characters, symmetric polynomials
and inner products
A generic partition λ is any finite or infinite sequence λ = (λ1, λ2, . . .) of non-negative
integers in decreasing order λ1 ≥ λ2 ≥ . . . ≥ 0 containing only finitely many non-zero
terms. Often it is convenient to omit zero entries. The non-zero entries are called the
parts of λ the number of which we denote by ℓ(λ). The sum of the parts of λ is called
the weight of λ which we denote by |λ| =
i λi. If |λ| = L then λ is a partition of L
and we write λ ⊢ L. For convenience we sometimes write λ in its frequency representation
which is a reordering of the entries in λ, indicating the number of times each successive
non-negative integer occurs, (1a1 , 2a2 , . . .) so that exactly an of the parts of λ equal n and
|λ| =
n≥1 n an.
In terms of standard Young diagrams, λ corresponds to a Young diagram of shape λ,
with λ1 boxes in the first row, λ2 boxes in the second row etc.; the number of parts ℓ(λ)
is simply the number of rows and the weight |λ| is the total number of boxes.
For the symmetric group, SN , the irreducible representations are labelled by par-
titions λ ⊢ N - see [38] for a useful summary - so that, for Xλ(σ), σ ∈ SN , being a
corresponding matrix representation, then the character of σ ∈ SN in the representation
Xλ is χλ(σ) = tr(Xλ(σ)). The characters are class functions so that they take a constant
value on conjugacy classes and, recalling that for SN the conjugacy classes Kµ are labelled
by partitions µ ⊢ N , corresponding to the cycle structure of a class representative, then
χλ(σ) = χλµ for all σ ∈ Kµ. With zλ as defined in (2.8), a crucial property of SN charac-
ters is the orthogonality of the matrix [zµ
−1/2χλµ]λµ. This gives rise to the orthogonality
relations, for λ, µ ⊢ N , (see also Ch. IV of [51] for a related discussion,)
χλ(σ)χµ(σ) =
χλν χ
ν = δλµ , (A.1)
and ∑
χνλ χ
µ = zλδλµ . (A.2)
A convenient basis for N variable symmetric polynomials are Schur polynomials
sλ(z) = sλ(z1, . . . , zN ) labelled by λ = (λ1, . . . , λN ). They may be expressed in a number
of ways [24,47]. For convenience we write them as
sλ(z) = aλ+ρ(z)/aρ(z) , (A.3)
where ρ, the Weyl vector, is given by ρ = (N − 1, N − 2, . . . , 1, 0) and
aλ+ρ(z) =
sgn(σ) zσ(1)
λ1+N−1 · · · zσ(j)λj+N−j · · · zσ(N)λN = det[ziλj+N−j ] , (A.4)
aρ(z) = det[zi
N−j ] =
1≤i<j≤N
(zi − zj) = ∆(x) , (A.5)
being the Vandermonde determinant. Schur polynomials sλ(z) have a standard interpre-
tation as corresponding to the characters of irreducible U(N) (or, for
i zi = 1, SU(N))
Lie algebra representations. Here, λ gives the shape of the Young tableaux for the corre-
sponding U(N) Lie algebra representation.
For λ = (λ1, . . . , λN ) and µ = (µ1, . . . , µN ) where λi, µi ∈ Z then, from the definition
of (2.6) along with (A.3),
sλ, sµ
sgn(σ)δλσµ =
sgn(σ)δλµσ , (A.6)
where, for any λ′ = (λ′1, . . . , λ
N ), λ
′σ = σ(λ′+ρ)−ρ is the shifted Weyl reflection of λ′ by σ,
with the action of SN on λ′ being given by σ(λ′1, . . . , λ′N ) = (λ′σ(1), . . . , λ′σ(N)). (Equation
(A.6) is a reflection of sλ(x) = sgn(σ)sλσ(x) for any partition λ and σ ∈ SN - note that this
property is useful for showing (3.25), (3.31). λσ has a standard interpretation in terms of
U(N) Lie algebra representations - for the Verma module with dominant integral highest
weight having orthonormal basis labels λ, λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0, then λσ, for σ 6= idSN ,
are the orthonormal basis labels for the highest weights of all invariant sub-modules. This
fact may be exploited to derive the Weyl character formula (A.3) for the irreducible U(N)
Lie algebra representation with dominant integral highest weight having orthonormal basis
labels λ, or, alternatively, Young tableaux of shape λ.)
When λ, µ are partitions so that λ1 ≥ . . . ≥ λN ≥ 0 and µ1 ≥ . . . ≥ µN ≥ 0 then
(A.6) reduces to a well defined inner product,
sλ, sµ
= δλµ , (A.7)
so that in this case the Schur polynomials are orthonormal. Note that in order that sλ(x)
be non-zero for some arbitrary partition λ then ℓ(λ) ≤ N , so that (A.7) is zero for ℓ(λ) > N
or ℓ(µ) > N .
Another basis for symmetric polynomials are the power symmetric polynomials, pλ(z),
for λ = (λ1, . . . , λL) ⊢ L, which are defined by
pλ(z) = pλ1(z)pλ2(z) · · ·pλL(z) , pn(z) =
n . (A.8)
Note that there is no longer the restriction that ℓ(λ) ≤ N as for Schur polynomials.
Symmetric group characters may be used to relate the two bases for symmetric poly-
nomials [24,47] so that, with the definition of zλ in (2.8),
sλ(z) =
χλµ pµ(z) , (A.9)
(a theorem of Frobenius) and, for λ ⊢ L,
pλ(z) =
ℓ(µ)≤N
λ sµ(z) . (A.10)
((A.9) with χ
λ = 1 for all λ ⊢ N is useful for obtaining (3.35).)
Regarding the inner product (2.6), then using (A.10) along with (A.7), we then have
that, for λ ⊢ L, µ ⊢ M , 〈
pλ, pµ
= δLM
ℓ(ν)≤N
µ . (A.11)
Orthogonality of symmetric group characters implies, from (A.2), that for |λ|, |µ| ≤ N
then (A.11) simplifies to, with the definition of zλ in (2.8),
pλ, pµ
= zλδλµ . (A.12)
Appendix B. Tables
N n 2 3 4 5 6 7 8 9 10 11
1 1 1 1 1 1 1 1 1 1 1
2 2 2 3 3 4 4 5 5 6 6
3 2 3 4 5 7 8 10 12 14 16
4 2 3 5 6 9 11 15 18 23 27
5 2 3 5 7 10 13 18 23 30 37
6 2 3 5 7 11 14 20 26 35 44
Numbers of multi-trace half BPS primary operators, with conformal dimension n
and belonging to [0, n, 0] R-symmetry representations, for free N = 4 SYM with
U(N) gauge group. (For every N there is one [0, 0, 0] and [0, 1, 0] representation -
these are omitted above.)
N(n,1)
N n 2 3 4 5 6 7 8 9 10 11
2 1 1 2 2 3 3 4 4 5 5
3 1 2 4 5 8 10 13 16 20 23
4 1 2 5 7 12 16 23 30 40 49
5 1 2 5 8 14 20 30 41 57 74
6 1 2 5 8 15 22 34 48 69 92
7 1 2 5 8 15 23 36 52 76 104
Numbers of multi-trace quarter BPS primary operators, with conformal dimension
n+1 and belonging to [1, n−1, 1] R-symmetry representations, for free N = 4 SYM
with U(N) gauge group. (n = 0, 1 cases are all zero.)
N(n,2)
N n 2 3 4 5 6 7 8 9 10 11
3 3 5 10 14 21 27 36 44 55 65
4 3 6 14 21 36 50 73 96 130 163
5 3 6 15 25 44 66 101 142 200 267
6 3 6 15 26 48 74 118 171 251 346
7 3 6 15 26 49 78 126 188 281 398
8 3 6 15 26 49 79 130 196 298 428
Numbers of multi-trace quarter BPS primary operators, with conformal dimension
n+2 and belonging to [2, n−2, 2] R-symmetry representations, for free N = 4 SYM
with U(N) gauge group. (n = 0, 1 cases are all zero.)
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  The free field partition function for a generic U(N) gauge theory, where the
fundamental fields transform in the adjoint representation, is analysed in
terms of symmetric polynomial techniques. It is shown by these means how this
is related to the cycle polynomial for the symmetric group and how the large N
result may be easily recovered. Higher order corrections for finite N are also
discussed in terms of symmetric group characters. For finite N, the partition
function involving a single bosonic fundamental field is recovered and explicit
counting of multi-trace quarter BPS operators in free \N=4 super Yang Mills
discussed, including a general result for large N. The partition function for
BPS operators in the chiral ring of \N=4 super Yang Mills is analysed in terms
of plane partitions. Asymptotic counting of BPS primary operators with
differing R-symmetry charges is discussed in both free \N=4 super Yang Mills
and in the chiral ring. Also, general and explicit expressions are derived for
SU(2) gauge theory partition functions, when the fundamental fields transform
in the adjoint, for free field theory.

Introduction
For the past while there has been intense interest in finite N partition functions for
Yang Mills theories, especially in super-symmetric ones, particularly with regard to their
construction for BPS states and the counting thereof [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].
Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it
being by now the archetypal example of a conformal field theory for which we have a
dual description in terms of string theory, by means of the AdS/CFT conjecture [16].
An example is in the assiduous efforts that have been made to explain the entropy of
certain BPS black holes in AdS5×S5 [17,18,19,20] in terms of microscopic counting of dual
operators in N = 4 super Yang Mills, with gauge group SU(N), by means of partition
functions [1]. While this problem remains unsolved to date, essentially due to the difficulty
of defining what is meant by these dual operators for finite N , there are other equally
interesting sectors of (super) Yang Mills theories where more progress with counting has
been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors
involving BPS operators.
For N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators
involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multi-
trace operators involving two bosonic operators Z, Y while the eighth consists of ones
involving three bosonic operators Z, Y,X and two fermionic operators λ, λ̄. (All these
operators are here assumed to belong to the Lie algebra of U(N).)
For the chiral ring, the commuting/anti-commuting of these operators is at the heart
of why we can write very concise and elegant generating functions for the finite N multi-
trace partition functions [1]. Analysis of these partition functions, in terms of the counting
of operators, has become a sophisticated industry where such approaches as the so-called
‘Plethystic Program’ have provided substantial results [3,9].
This paper is devoted largely to the issue of partition functions for free field theory
and particularly to the counting of gauge invariant multi-trace operators for the case of two
bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super
Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to
the chiral ring when they do.
The partition function for a free, massless quark-gluon gas was computed long ago [21].
This involved taking particle statistics into account using coherent state techniques and
then imposing the gauge singlet condition by integrating over the relevant gauge group.
With some modifications to the expression thus derived we may write the multi-trace
partition function for some generic bosonic/fermionic fundamental fields in terms of an
integral over the gauge group, involving the single particle partition function [22,23]. This
is the starting point here.
For U(N) gauge theories we may easily write down the integral, though, even for this
case, its evaluation is far from simple. One approach (which we adapt here for the SU(2)
case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in
principle be evaluated by summing the contributions from poles inside origin centred unit
discs, in each of the N complex planes - similar techniques have been used in [10]. Due to
the number of poles this becomes unfeasible for higher values of N . Another approach is
to use the fact that the complex integral that interests us provides for an inner product
for symmetric polynomials - see Macdonald [24], pp. 363 - 372, for a related discussion.
Taking this point more seriously reveals an alternative route to evaluating the free field
theory partition function which exposes not only the large N case in an almost trivial way,
but also how and where this differs from the finite N case.
This treatment also reveals an alternative interpretation of the free field partition
function at finite N - it is related to a gauge group average of the cycle polynomial for the
symmetric permutation group (after a certain identification of ‘letters’ with gauge group
valued variables). This point is not dwelt on further here though makes the connection
between the partition function and Polya enumeration explicit. For single trace operators
at large N this connection has already been made [22,25] for N = 4 super Yang Mills,
whereby the partition function for single trace operators is related to the cycle polynomial
for the cyclic permutation group.1
Another issue is how to use the expression for the free field partition function to give
explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with
gauge group U(N). The case for one bosonic fundamental field has been widely discussed
and, for finite N , the operators are counted by partition numbers of non-negative integers
into at most N parts (which is here denoted by pN (n) - no closed formula for these numbers
for arbitrary N, n exist, though they have a ‘nice’ generating function). Here, this result is
re-derived, from a symmetric function perspective, by employing the well known Cauchy-
Littlewood formula.
To proceed further with counting, for the quarter BPS sector of N = 4 super Yang
Mills, for instance, character methods prove to be both natural and indispensable. Char-
acters in relation to conformal field theories prove to be very convenient for encoding
the allowable representations [15,29,30,31,32,33] and for studying related partition func-
tions, [34,35,36,33]. For N = 4 super Yang Mills, it was shown in [33] that, if we are
1 For more recent applications of symmetric polynomials and Polya enumeration to super-
symmetric quantum mechanical models, analysed in terms of Fock space methods, see [26,27,28].
to distinguish among primary operators with differing conformal dimensions, spins and
R-symmetry charges, such counting is most easily achieved using reductions of the full
N = 4 superconformal characters, in certain limits that isolate corresponding sectors of
short/semi-short operators. (One such limit corresponds to the index constructed in [1].)
This point may be easily illustrated for quarter BPS primary operators in N = 4 su-
per Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit
construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the
counting of quarter BPS primary operators is complicated by the fact that, if we are to
keep track of differing R-symmetry representations, any partition function restricted to
this sector must be expanded in terms of U(2) characters (or two-variable Schur polyno-
mials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are
letters corresponding to the fields Z, Y , then by expanding
Z(t, u) =
N(n,m) s(n,m)(t, u) , s(n,m)(t, u) =
tn+1um − tmun+1
t− u , (1.1)
in terms of two-variable Schur polynomials s(n,m)(t, u), we obtain the numbers N(n,m) of
gauge invariant quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry
representation and having conformal dimension n + m (so that they are superconformal
primary highest weight states in the corresponding quarter BPS supermultiplets). (The
case m = 0 counts gauge invariant half BPS primary operators.)
Here, the free field partition function is thus expanded in terms of Schur polynomials,
depending on the same variables as the one particle partition function, the two boson
case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood
formula (and, if we include fermions, another formula due to Littlewood). Generally, we
may obtain a result that relates the counting numbers to a sum over Kronecker coefficients.
These arise naturally in the theory of the symmetric permutation group, though remain
somewhat mysterious from a combinatorial perspective.
Specialising to the two boson case, a recursive procedure is employed here for the
counting of multi-trace quarter BPS operators in free field theory at finite N . This issue
was given considerable discussion for the large N case in [33] - here the results of [33] are
generalised in terms of a generating function that may be employed to count quarter BPS
operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed
in the latter case for the numbers N(n,m) in (1.1) for large n and fixed m.
To complete the discussion, counting of quarter BPS operators in the chiral ring of
N = 4 super Yang Mills is investigated in terms of expanding over U(2) characters as in
(1.1). An explicit formula is given for the corresponding finite N partition function, with
a short combinatorial interpretation given in terms of plane partitions, and specialised to
large N . For the latter case, the exponential behaviour of the numbers N(n,m) in (1.1) is
found for n and m both comparably large. This behaviour is consistent with a special case
addressed in [3,9]. By way of completion, a similar discussion for an arbitrary number of
bosonic fundamental fields in the chiral ring is included.
Two appendices are included; the first establishes some notation used for partitions
and gives some standard results for the symmetric group and symmetric polynomials, the
second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang
Mills, with gauge group U(N), for which explicit formulae are given in the main text.
Footnotes contain further details and points of clarification.
2. Free Field Partition Functions
We start from the single particle partition function which is here denoted by f(t) for
some variables2 t = (t1, t2 . . .). The general form of f(t) is
f(t) =
aiti , (2.1)
where each ti is a letter corresponding to a fundamental field and ai are signs, being +1
for a bosonic field, or −1 for a fermionic field.
For compact gauge Lie group G, the multi-trace partition function is then given by
[21], (see [22,23] for refinements,)
ZG(t) =
dµG(g) exp
f(tn)χR(g
, (2.2)
involving the Haar (or G-invariant, or Hurwitz) measure dµG(g) for g ∈ G (so that∫
dµG(g)F (g) =
dµG(g)F (gh) =
dµG(g)F (hg) for all h ∈ G and
dµG(g) = 1)
and where χR(g) is the character for the R representation of G, assuming that the funda-
mental fields transform in identical gauge group representations R.
For G = U(N), so that for any matrix U ∈ U(N) we may write U = VΘV †, where V
is a unitary matrix and Θ = diag.(eiθ1 , . . . , eiθN ), 0 ≤ θi < 2π, and for some F (U) = F (Θ),
independent of V , then we may write
dµU(N)(U)F (U) =
(2π)NN !
1≤k<l≤N
|eiθk − eiθl |2 F (Θ) , (2.3)
2 In what follows roman letters are used to denote a collection of variables and, for x =
(x1, . . . , xi), y = (y1, . . . , yj) for example, the shorthand x
α is used to mean (x1
α, . . . , xi
α) and
z = xy to mean z = (z11, . . . , zij) where zrs = xrys. The latter convenient notation has been used
by Macdonald [24].
which is, of course, related to the Weyl parametrisation of U(N). Thus, for such F (U), the
left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χR(U) gen-
erally depends on linear combinations of tr(U j)ktr(U †l)m for various non-negative integers
j, k, l,m then any function of χR(U) is an example of such an F (U).
We are interested in the case where R = Adj. is the adjoint representation so that for
U(N) we have that χAdj.(U) = trU trU
† (while for SU(N) then χAdj.(U) = trU trU
†−1).
For U(N) we then find that, using (2.2) with (2.3),
ZU(N)(t) =
(2π)NN !
1≤k<l≤N
|eiθk − eiθl |2 exp
f(tn)
j,k=1
ein(θj−θk)
(2.4)
We may write (2.4) as an N -fold contour integral by first making the variable change
zj = e
iθj so that the integrals in (2.4) are around unit circles in each zj complex plane and
then we obtain
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1) exp
f(tn)pn(z)pn(z
, (2.5)
where ∆(z) =
1≤i<j≤N (zi − zj) is the Vandermonde determinant and pn(z) =
i=1 zi
is a power symmetric polynomial - see appendix A for a brief discussion of symmetric
polynomials. This integral may then in principle be evaluated by deforming the contours
so as to extract the residues at poles within the discs |zj | < 1, 1 ≤ j ≤ N .
A crucial observation is that, for some N variable symmetric polynomials g(z), h(z),
(2πi)NN !
∆(z)∆(z−1)g(z)h(z−1) , (2.6)
acts as an inner product - this is easy to see in terms of Schur polynomials which provide
an orthonormal basis for symmetric polynomials. The reader may now wish to peruse
appendix A where notation regarding partitions and a short discussion of symmetric poly-
nomials is included.
The General and Large N Cases for U(N)
For application of inner products to (2.5) we have that, in terms of power symmetric
polynomials pλ(z) for partitions λ,
f(tn)pn(z)pn(z
nanan!
f(tn)anpn(z)
anpn(z
−1)an ,
fλ(t)pλ(z)pλ(z
−1) ,
(2.7)
with the definitions of
nanan! , fλ(t) =
f(tn)an , (2.8)
being in terms of the frequency representation of λ, (1a1 , 2a2 , . . .) with
n≥1 n an = |λ|,
the weight of the partition λ (note that the frequency representation of λ is simply a
convenient re-ordering of the parts of λ).
In (2.7) the numbers zλ have a standard combinatorial interpretation - for a given
permutation σ ∈ Sm with a1 1-cycles, a2 2-cycles etc., so that
n≥1 n an = m = |λ|, then
n≥1 n
anan! is the size of the centraliser Zσ = {τ ; τ ∈ Sm, τστ−1 = σ} of σ ∈ Sm.
(This may be easily seen as under conjugation of σ by τ then τ can permute the cycles
of length n among themselves in an! ways and/or render a cyclic rotation on each of the
individual cycles in nan ways.) More details of the symmetric group are to be found in
appendix A.
We may immediately observe that (2.7) represents a sum over cycle polynomials of
the symmetric group Sm. This is given by, for letters u1, u2, . . . um,3
Cm(u) =
a1,...,am≥0
δa1+2a2+...+mam,m
nanan!
uλ , (2.9)
where ‘λ ⊢ m’ means that λ is any partition of m - see appendix A for notation - and
an , (2.10)
in terms of the frequency representation of λ above. Identifying un = f(t
n)pn(z)pn(z
then we may rewrite
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)Cm(u) , (2.11)
3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G
of Sm. This is given by
j1(g) · · ·um
jm(g) =
|Kg|u1
j1(g) · · ·um
jm(g) ,
where ji(g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles
and Kg denotes the conjugacy classes of G with class representatives g. The size of the conjugacy
class Kg is given by |Kg| = |G|/|Zg| where Zg is the centraliser of g ∈ G. For the present case
then, G = Sm, |Sm| = m!, and |Zσ| = zλ, where λ gives the cycle structure of σ ∈ Sm, and thus,
for the corresponding conjugacy class Kλ, |Kλ| = m!/zλ.
the sum of the U(N) group averages of each of the cycle polynomials Cm(u). (Physically,
the interpretation is that the cycle index for the symmetric permutation group accounts
for particle statistics while integration over the gauge group imposes the gauge singlet
condition. For purposes of clarity, the U(N) case has been focused upon here, though
from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby
the letters un = f(t
n)χR(g
n) for the fundamental fields transforming in identical gauge
group representations, R.)
Directly from (2.7), in terms of the inner product (2.6), then
ZU(N)(t) =
fλ(t)
pλ, pλ
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
, (2.12)
where on the right-hand side of (2.12) we have used an expression for the inner product
of two power symmetric polynomials expressed in terms of the characters of the symmet-
ric group, given in appendix A. (Here ‘ℓ(µ)’ means the number of non-zero parts of the
partition µ.)
Using a result of appendix A (essentially orthogonality relations for symmetric group
characters), (2.12) may be rewritten as
ZU(N)(t) =
|λ|≤N
fλ(t) +
|λ|>N
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
. (2.13)
In the large N limit, ZU(N)(t) simplifies considerably as only the first term in (2.13)
need be considered. Using the frequency representation of λ then
ZU(∞)(t) =
fλ(t) =
f(tn)an =
1− f(tn) , (2.14)
a result which has been obtained using Polya counting methods for single trace operators
and saddle point approximations [22,23].
Higher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained
by successive evaluation of
µ⊢|λ|
ℓ(µ)≤N
. One method is to employ the Murnaghan-
Nakayama Rule, used to compute χ
λ using skew hooks and Young diagrams. (A readable
account of the Murnaghan-Nakayama Rule may be found in [38], though of course it is
explained in many standard textbooks that discuss the symmetric group.)
For the case of |λ| = N+1 then we may observe that,
µ⊢N+1
ℓ(µ)≤N
µ⊢N+1
ℓ(µ)≤N+1
)2 − (χνλ)2 = zλ − (χνλ)2 , ν = (1N+1) , (2.15)
since the partition ν = (1N+1) is the only one excluded among those partitions µ of
N+1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for
ν = (1L),
(χνλ)
1 for |λ| = L
0 otherwise
, (2.16)
since χνλ in this case is just a sign. (This may be easily seen as there is only one possible
way to remove successive skew hooks, which in this case are just column Young diagrams
of length λi, from the (1
L) column Young diagram to leave one of normal shape, in this
case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain
ZU(N)(t) =
|λ|≤N+1
fλ(t)−
|λ|=N+1
fλ(t) +
|λ|>N+1
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
. (2.17)
By a similar line of argument we may do the same for the case of |λ| = N+2. We
have that, for ν1 = (1
N+2) and ν2 = (2, 1
N+1),
µ⊢N+2
ℓ(µ)≤N
µ⊢N+2
ℓ(µ)≤N+2
)2 − (χν1λ )
2 − (χν2λ )
2 = zλ − (χν1λ )
2 − (χν2λ )
2 . (2.18)
We may determine, for ν = (2, 1L),
(χνλ)
(a1 − 1)2 for |λ| =
n≥1 n an = L+2
0 otherwise
. (2.19)
Using (2.16), (2.19) with (2.18) in (2.17) then we obtain
ZU(N)(t) =
|λ|≤N+2
fλ(t)−
N+1≤|λ|≤N+2
fλ(t)−
|λ|=N+2
fλ(t)
|λ|>N+2
fλ(t)
µ⊢|λ|
ℓ(µ)≤N
(2.20)
where, in the frequency representation of λ,
(a1−1)2
(a1 − 1)!
(a1 − 2)!
)/ ∞∏
nanan! . (2.21)
We may proceed in this manner to compute explicit higher order corrections though
this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the
corrections will always involve contributions from (2.16) and (2.19) as well as extra ones
coming from
µ⊢|λ|
ℓ(µ)≤|λ|
µ⊢|λ|
ℓ(µ)≤N
The One Boson Case for U(N)
For the case of one bosonic fundamental field (applicable to half BPS operators for
N = 4 super Yang Mills), we have f(t) = t in (2.5), so that we may write
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)
j,k=1
1− tzjzk−1
. (2.22)
To evaluate this integral we may use the Cauchy-Littlewood formula,
1− xiyj
ℓ(λ)≤min.{L,M}
sλ(x1, . . . , xL)sλ(y1, . . . , yM) , (2.23)
where the sum on the right-hand side is over all partitions λ such that the corresponding
Young diagrams have no more than min.{L,M} rows, ℓ(λ) ≤ min.{L,M}. With xi = tzi,
yi = zi
−1, i = 1, . . . , N , in (2.23), so that sλ(tz1, . . . , tzN ) = t
|λ|sλ(z), and employing also
(2.6) and the orthonormality of Schur polynomials, we may easily obtain,
ZU(N)(t) =
ℓ(λ)≤N
sλ, sλ
ℓ(λ)≤N
t|λ| . (2.24)
By changing summation variables so that λi − λi+1 = ai, i = 1, . . . , N−1, λN = aN then
we may write
ZU(N)(t) =
a1,...,aN=0
ta1+2a2+...+NaN = PN (t) , (2.25)
where4
PN (t) =
1− ti . (2.26)
Of course this is nothing other than the generating function for the number pN (n) of
partitions of n into no more than N parts since, by definition,
ℓ(λ)≤N
δ|λ|,n = pN (n) , (2.27)
so that by the above
pN (n)t
n = PN (t) . (2.28)
4 1/P∞(t) =
(1−tn) is commonly called the Euler function, denoted by Φ(t). P∞(t) =∑∞
p(n)tn acts as a generating function for the number of unordered partitions of n, p(n). Note
that pN (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the
same as the total number of partitions of n so long as n ≤ N .
This makes explicit the connection between ZU(N)(t) and the partition numbers pN (n).
The SU(2) Gauge Group Case
Here we first consider f(t) =
j=1 tj in (2.1) so that the variables 0 ≤ ti < 1
represent k bosons in the single particle partition function. For such fields transforming in
the adjoint representation of SU(2) then (2.2) simplifies significantly. For any U ∈ SU(2)
we may write U = VΘV †, where V is unitary and Θ = diag.(eiθ, e−iθ), for 0 ≤ θ < 2π, so
that for F (U) = F (θ), then in usual Weyl parametrisation,
SU(2)
dµSU(2)(U)F (U) =
dθ sin2 θ F (θ)
1− cos θ
F ( θ
1− cos θ
F ( θ
(2.29)
where F (θ) = F (θ + π) is assumed in writing the last line. In the present case, F (U) =
F (θ) =
n≥1 f(t
n)χAdj.(U
n)/n, where χAdj.(U) = tr(U)tr(U
†) − 1 = e2iθ + e−2iθ + 1 =
2 cos 2θ + 1, so that
ZSU(2)(t) =
dθ (1− cos θ)
(1− tj)(1− tjeiθ)(1− tje−iθ)
. (2.30)
Making the variable change z = eiθ, and using that F (θ) = F (−θ) is even, then
ZSU(2)(t) =
(1− z)
(1− tj)(1− tjz)(1− tjz−1)
, (2.31)
where the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily
computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = tj .
ZSU(2)(t) =
1− ti2
j 6=i
(ti − tj)(1− titj)(1− tj)
. (2.32)
This partition function has an interesting interpretation from a group theory perspec-
tive. We may write,
1 + t
(1− tz)(1− tz−1) =
χn(z)t
n , (2.33)
where
χj(z) =
2 − z−j− 12
2 − z− 12
, j ∈ 1
Z , (2.34)
is an SU(2) character, corresponding to the spin j irreducible representation, Rj. Now the
integral in (2.31) acts as an SU(2) inner product,
χj , χk
(1− z)χj(z)χk(z−1) = δjk , (2.35)
for j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35),
(1− tj2)ZSU(2)(t) =
n1,...,nk=0
χn1 · · ·χnk , 1
n1 · · · tknk , (2.36)
acts as a generating function for the number of singlets in the decomposition of the SU(2)
representation Rn1 ⊗· · ·⊗Rnk .5 By using that χn(z) =
j=−n z
j we may use the Cauchy
residue theorem to compute explicitly that
χn1 · · ·χnk , 1
· · ·
(δj1+···+jk,n1+···+nk − δj1+···+jk,n1+···+nk+1) . (2.37)
If we modify the one particle partition function to include k bosons and k̄ fermions
and hence consider (2.1) in the form f(t, t̄) =
j=1 tj −
̄=1 t̄̄ then we may similarly as
above evaluate
ZSU(2)(t, t̄) =
(1− z)
1≤j≤k
1≤̄≤k̄
(1− t̄̄)(1− t̄̄z)(1− t̄̄z−1)
(1− tj)(1− tjz)(1− tjz−1)
, (2.38)
where the contour is around the unit disc |z| = 1. So long as k > k̄ then (2.38) receives
contributions from only those simple poles at z = tj so that for this case we obtain
ZSU(2)(t, t̄)
k−k̄−1
1− ti2
1≤̄≤k̄
1≤j≤k,j 6=i
(ti − t̄̄)(1− tit̄̄)(1− t̄̄)
(ti − tj)(1− titj)(1− tj)
. (2.39)
For k ≤ k̄ then (2.38) also receives contributions from poles at z = 0. For instance
ZSU(2)(t, t̄)
ti(1− ti2)
1≤j,̄≤k
j 6=i
(ti − t̄̄)(1− tit̄̄)(1− t̄̄)
(ti − tj)(1− titj)(1− tj)
1≤j,̄≤k
t̄̄(1− t̄̄)
tj(1− tj)
(2.40)
5 Generating functions for products of Lie algebra representations have been considered
elsewhere, in [39] for instance. A generating function for the number of singlets in n products of
the fundamental times n products of the anti-fundamental representations for SU(N) was found
by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a
nice physics oriented discussion of similar issues. The special case of R 1
⊗ · · · ⊗R 1
(2n products
of the fundamental) for SU(2), contains a Catalan number, 1
, of singlet representations.
where the last term on the right-hand side of (2.40) comes from the simple pole at z = 0.
These formulae should be useful for computing the multi-trace partition functions,
for fundamental fields transforming in an SU(2) gauge group, in other sectors of N = 4
super Yang Mills. For instance, after a suitable identification of the variables tj , t̄̄ with
variables in single particle partition functions for semi-short sectors of N = 4 super Yang
Mills, described in detail in [33], then (2.40) should allow for an explicit expression for
corresponding multi-trace partition functions. They may also be useful for computing the
N = 4 superconformal index of [1] for SU(2) gauge group, or at least for restrictions of it
such as described in [33] or [42].
3. Counting Operators in Free N = 4 Super Yang Mills
In this section, the counting of half and quarter BPS operators for free N = 4 super
Yang Mills, when the fundamental fields transform in the adjoint representation of U(N),
is discussed in some detail.
Counting Operators Directly
We may, of course, proceed to count multi-trace half and quarter BPS primary oper-
ators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for
quarter BPS operators.
(Z, Y ) forms a U(2) doublet, where U(2) has generators given by a subset of the
SU(4)R generators, Hi, Ei±, 1 ≤ i ≤ 3, where Hi are the Cartan sub-algebra genera-
tors and Ei± are ladder operators satisfying (in the Chevalley-Serre basis) [Hi, Ej±] =
±KijEj±, with [Kij] being the usual SU(4) Cartan matrix. The U(2) generators consist
of the SU(2) generators H2, E2±, where explicitly [(H1, H2, H3), E2±] = ∓(1,−2, 1)E2±,
along with the generator H1+H2+H3, whose eigenvalues give the conformal dimensions
in this case, [H1+H2+H3, (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E2+, Z] = 0,
[E2−, Z] = Y , [(H1, H2, H3), Z] = (0, 1, 0)Z, [(H1, H2, H3), Y ] = (1,−1, 1)Y so that an
operator involving n Z’s and m Y ’s transforms in the [m,n−m,m] SU(4)R R-symmetry
representation.
For k-trace half BPS primary operators transforming in the [0, n, 0] SU(4)R R-
symmetry representation, with conformal dimension n, then in terms of the fundamental
field Z a basis is provided by,
tr(Zn1) · · · tr(Znk) ,
ni = n . (3.1)
We have that, due to trace identities for finite N , tr(Zn) for n > N is expressible in terms
of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis
for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with
every ni ≤ N , ordered so that n1 ≥ n2 ≥ . . . ≥ nk ≥ 0, i.e. so that (n1, . . . , nk) is a
partition of n where each part ni ≤ N . With this restriction, the number of multi-trace
half BPS primary operators for a given n is
N(n) = pN (n) , (3.2)
since the number pN (n), in (2.27), of partitions of n into ≤ N parts is the same as the
number of partitions of n in which each part is ≤ N - see [43] for a simple proof employing
generating functions.
For quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry repre-
sentation, a basis for k-trace operators is
Zn1jY m1j
· · · tr
ZnkjY mkj
nij = n ,
mij = m, (3.3)
where there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to
the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k,
then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace
and also trace identities for finite N must be accounted for. Assuming that this is done,
let M(n,m) denote the number of elements in this minimal basis for multi-trace operators
of the form (3.3). Then, to obtain the number N(n,m) of multi-trace quarter BPS primary
operators in the SU(4)R representation [m,n−m,m], the number of U(2) descendants, in
the SU(4)R representation [m,n−m,m], of multi-trace quarter BPS primary operators, in
SU(4)R representations [j, n+m−2j, j], 0 ≤ j ≤ m−1, must be subtracted from M(n,m).
(These descendants arise due to the relation [E2−, Z] = Y . Acting with (E2−)
m−j on the
highest weight state in the SU(4)R representation [j, n+m−2j, j] we obtain a descendant in
the SU(4)R representation [m,n−m,m].) The number of such U(2) descendants coincides
with N(n+m−j,j), the number of corresponding primary operators. In this way, we obtain
M(n,m) = N(n,m) +N(n+1,m−1) + . . .+N(n+m−1,1) +N(n+m) , (3.4)
so that N(n,m) = M(n,m) −M(n+1,m−1) may be obtained recursively for each m.
We may illustrate by counting all multi-trace quarter BPS primary operators in the
[1, n−1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is
provided by
tr(Zn1) · · · tr(Znk)tr(ZjY ) ,
ni = n− j . (3.5)
Cyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U(N)
trace identities imply, similarly as for the half BPS case, that a minimal basis for multi-
trace operators requires j < N and each ni ≤ N in (3.5) for every 1 ≤ k ≤ n−j, so
that (n1, . . . , nk) forms a partition of n−j, with every part ≤ N . Thus, by a similar
argument as for the half BPS case, M(n,1) =
j=0 pN (n−j). Finally, to ensure that only
primary operators are counted then we must subtract off contributions from descendants
of half BPS primary operators in the [0, n+1, 0] SU(4)R representation, of which there are
pN (n+1). Using (3.4) with (3.2) we then conclude that
N(n,1) =
pN (n− j)− pN (n+1) , (3.6)
gives the number of multi-trace quarter BPS primary operators in the [1, n−1, 1] R-
symmetry representation.
Counting in this fashion becomes more difficult for greater m and now a procedure is
described employing symmetric polynomials to find a generating function for the numbers
of multi-trace quarter BPS primary operators in the [m,n−m,m] SU(4)R representation,
for m = 0, 1, 2 at finite N and for any n,m at large N . This generating function is
subsequently used to provide asymptotic counting for fixed m, large n in the large N limit.
Counting Operators via Expansion of Partition Functions in Schur Polynomials
For k bosonic fundamental fields, we may take f(t) =
j=1 tj in (2.1) so that (2.5)
may be written as
ZU(N)(t) =
(2πi)NN !
∆(z)∆(z−1)
r,s=1
1− tjzrzs−1
. (3.7)
Often it is the case that such partition functions should be expanded in terms of sλ(t), the
k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for
counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula
(2.23) to expand in this way, to obtain
ZU(N)(t) =
ℓ(λ)≤min.{k,N2}
Nλ sλ(t) , (3.8)
where
(2πi)NN !
∆(z)∆(z−1) sλ(zz
−1) , (3.9)
where zz−1 has components zizj
−1, 1 ≤ i, j ≤ N .
From Macdonald [24] we have that
sλ(xy) =
µ,ν⊢|λ|
γλµνsµ(x)sν(y) , (3.10)
in terms of Kronecker coefficients,
γλµν =
σ∈S|λ|
χλ(σ)χµ(σ)χν(σ)
ρ⊢|λ|
χλρ χ
(3.11)
being a sum over irreducible S|λ| characters evaluated at σ ∈ S|λ|, related to a sum over
irreducible S|λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in
the second line. Using (2.6) along with the orthonormality property of Schur polynomials
we find that
µ⊢|λ|
ℓ(µ)≤N
γλµµ . (3.12)
The situation becomes much more involved if we include also k̄ fermionic fields, so
that (2.1) may be written in the form f(t, t̄) =
j=1 tj −
̄=1 t̄̄, and attempt to ex-
pand ZU(N)(t, t̄) in terms of products of Schur polynomials sλ(t)sµ(t̄). Such expansion
is required for counting, for instance, for the free field partition function in the eighth
BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded,
analogous to (1.1), in terms of SU(2|3) characters, which may be expressed in terms of a
linear combination of products of two-variable and three-variable Schur polynomials. (See
[33] for a discussion of counting for the eighth BPS sector along these lines.) Including
fermions, (3.7) becomes modified by
ZU(N)(t, t̄) =
(2πi)NN !
∆(z)∆(z−1)
1≤j≤k
1≤̄≤k̄
r,s=1
1− t̄̄zrzs−1
1− tjzrzs−1
. (3.13)
To achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with
another formula of Littlewood,
(1 + xiyj) =
ℓ(λ)≤L,ℓ(λ̃)≤M
sλ(x1, . . . , xL) sλ̃(y1, . . . , yM ) , (3.14)
where λ̃ is the partition conjugate to λ (where the rows and columns of the Young diagram
corresponding to λ are interchanged) and where the sum is restricted to those λ whereby
the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns,
ℓ(λ̃) ≤ M . We may thus write
ZU(N)(t, t̄) =
ℓ(λ)≤min.{k,N2}
ℓ(µ)≤k̄,ℓ(µ̃)≤N2
Nλ,µ sλ(t) sµ(t̄) , (3.15)
where
Nλ,µ =
(−1)|µ|
(2πi)NN !
∆(z)∆(z−1) sλ(zz
−1)sµ̃(zz
−1) . (3.16)
Obviously these numbers are considerably more involved than those in (3.9). We may of
course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients.
Counting Quarter BPS Operators by Symmetric Polynomial Methods
The two bosonic fundamental field case is now focused upon.6 In particular, the
numbers N(n,m) in (1.1) are evaluated using results of the last sub-section.
We may proceed to evaluate Nλ recursively. The simplest case is for Nλ = N(n),
whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and
(2.26), that
N(n)tn =
(2πi)NN !
∆(z)∆(z−1)
j,k=1
1− tzjzk−1
= PN (t) , (3.17)
so that, by (2.28), N(n) is given by (3.2).
More generally to evaluate N(n,m) from (3.9) we may use, for y = zz−1,
s(m)(y)s(n)(y) = s(n,m)(y) + s(n+1,m−1)(y) + . . .+ s(n+m−1,1)(y) + s(n+m)(y) ,
s(m)(zz
−1) =
ℓ(µ)≤N
sµ(z)sµ(z
−1) , (3.18)
6 The two boson case leads to an interesting generalisation of an identity in [44] involv-
ing Littlewood-Richardson coefficients cνλµ, the coefficients that appear in the decomposition
sλ(x)sµ(x) =
cνλµsν(x). With f(t) in (2.1) given by f(t) = t1 + t2, and expanding appro-
priately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of
Schur polynomials and the result (2.14), we obtain (note that cνλµ = 0 if |ν| 6= |λ|+ |µ|)
ZU(∞)(t1, t2) =
〈sλsµ, sλsµ〉∞ =
λ,µ,ν
ν⊢|λ|+|µ|
1− t1n − t2n
which reduces to Theorem 4.1 of [44] if we take t1 = t2 = t.
where the expression in the first line of (3.18) may be easily seen using Young tableaux
multiplication rules while (2.23) determines the expression in the second line. From (3.9)
with (3.18), we may find a useful generating function, in terms of a formal variable t, for
the numbers in (3.4) as follows,
F (m)N (t) =
M(n,m)tn
(2πi)NN !
∆(z)∆(z−1) s(m)(zz
s(n)(zz
−1)tn
(2πi)NN !
∆(z)∆(z−1)
ℓ(µ)≤N
sµ(z)sµ(z
j,k=1
1− tzjzk−1
ℓ(µ)≤N
sλsµ, sλsµ
(3.19)
so that we may write
N(n,m) =
F (m)N (t)−
F (m−1)N (t)
, (3.20)
which allows for recursive determination of N(n,m).
Applying this to the case of Nλ = N(n,1) we have, from (3.18)
s(1)(zz
−1) = s(1)(z)s(1)(z
−1) , (3.21)
so that, from (3.19),
F (1)N (t) =
N(n,1) +N(n+1)
ℓ(λ)≤N
s(1)sλ, s(1)sλ
. (3.22)
Using (again, this may be easily seen from Young tableaux multiplication rules)
s(1)(z)sλ(z) =
sλ+er (z) , (3.23)
for {er; 1 ≤ r ≤ N, er · es = δrs} being usual orthonormal vectors, we find that
F (1)N (t) =
N(n,1) +N(n+1)
ℓ(λ)≤N
r,s=1
sλ+er , sλ+es
. (3.24)
Now for any partition λ,
sλ+er , sλ+es
vanishes unless er = es for any r, s and λr−1−λr >
0 for r = 2, . . . , N , due to
s(λ1,...,λr−1,λr+1,...,λN )(z) = 0 for λr−1 = λr , r > 1 . (3.25)
Changing summation variables to those in (2.25) then we have, with the definition (2.26),
F (1)N (t) =
N(n,1) +N(n+1)
a1,...,aN≥0
ta1+...+NaN +
a1,...,aN≥0
ta1+...+NaN
a1,...,aN≥0
ta1+...+NaN =
1− tPN−1(t) .
(3.26)
Thus, using (2.28), (3.2) with (3.26),7
N(n,1) =
pN−1(j)−N(n+1) =
pN−1(j)− pN (n+ 1) . (3.27)
For the case of Nλ = N(n,2) we have that, from (3.18),
s(2)(zz
−1) = s(2)(z)s(2)(z
−1) + s(1,1)(z)s(1,1)(z
−1) , (3.28)
so that, from (3.19), we have
F (2)N (t) =
ℓ(λ)≤N
s(2)sλ, s(2)sλ
s(1,1)sλ, s(1,1)sλ
. (3.29)
Using
s(2)(z)sλ(z) =
sλ+2er (z) +
1≤r<s≤N
sλ+er+es(z) ,
s(1,1)(z)sλ(z) =
1≤r<s≤N
sλ+er+es(z) ,
(3.30)
7 This formula agrees with (3.6) due to
pN (n − j) =
pN−1(j) which follows
because the corresponding generating functions match,
pN (n− j)t
= (1 + t+ . . .+ t
)PN (t) =
PN−1(t) =
pN−1(j)t
along with (3.25) and
s(λ1,...,λr−1,λr+2,...,λN )(z) = −s(λ1,...,λr+1,λr−1+1,...,λN )(z) , (3.31)
for the cases where λr−1 = λr, we may obtain, with the definition (2.26),
F (2)N (t) =
1−tN+1
(1−t)(1−t2)PN−1(t) +
(1−t)(1−t2)PN−2(t) , (3.32)
where the first contribution comes from
|λ|〈s(2)sλ, s(2)sλ
while the second comes
|λ|〈s(1,1)sλ, s(1,1)sλ
. Since the partition number pk(−n) = 0 for n = 1, 2, . . .
we may write, using (2.28),8
(1− t)(1− t2)Pk(t) =
n,i,j=0
pk(n− i− 2j)tn . (3.33)
Thus, from (3.32) with (3.27),
N(n,2) = −
pN−1(j)
i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)) if n ≤ N ,∑∞
i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)− pN−1(N+1−n−i−2j)) if n ≥ N+1.
(3.34)
Tables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few
cases of n,N . Notice from these tables that the numbers N(n,m) below the diagonal line
N ≥ n+m for a given n are the same for all N . This is a general feature that derives from
values of N(n,m) for N ≥ n + m, which numbers may be obtained from a corresponding
generating function that is now constructed.
Using these techniques, we may provide a consistency check of (3.17), (3.26), (3.32)
along with a general result for N(n,m) for high enough values of N , N ≥ m + n. This
employs the orthogonality property of power symmetric polynomials pλ(z) (in the large N
8 This is a special case of the following: for any f(n), n ∈ Z, that satisfies f(−n) = 0,
n = 1, 2, . . ., then we may (at least formally) write
Pk(t)
f(n)t
n,i1,...,ik=0
f(n− i1 − 2i2 − . . .− kik)t
limit) along with
s(n)(z) =
pλ(z)
i1,...,in=0
i1!i2! · · · in!
δi1+2i2+...+nin,n p1(z)
p2(z)
)i2 · · ·
pn(z)
(3.35)
Using the trivial identity pλ(xy) = pλ(x)pλ(y) then from (2.6), (3.19) with (3.35) we have
F (m)∞ (t) =
pλpµ, pλpµ
pν , pν
(3.36)
where for (1a1 , 2a2 , . . .) being the frequency representation of λ and (1b1 , 2b2 , . . .) being
that of µ then ν has frequency representation (1a1+a2 , 2a2+b2 , . . .) so that |ν| = n + m.
This agrees with F (m)N (t) in a series expansion up to O(tN−m) (since the last equation in
(3.36) is also valid for finite N so long as |ν| = n +m ≤ N , by a result of appendix A).
Now, since
(aj+bj)!
aj !bj!
, (3.37)
we obtain from (3.36) that,
F (m)∞ (t) =
a1,...,an=0
b1,...,bm=0
δa1+···+nan,nδb1+···+mbm,m
(aj + bj)!
aj !bj!
b1,...,bm=0
δb1+···+mbm,m
(aj + bj)!
aj!bj !
1− tj
b1,...,bm=0
δb1+···+mbm,m
(1− tj)bj+1
1− tj .
(3.38)
For the first few cases we have that, with PN (t) as defined in (2.26),
F (m)∞ (t) =
P∞(t) for m = 0
1−tP∞(t) for m = 1
(1−t)(1−t2)P∞(t) for m = 2
, (3.39)
whose series expansion agrees with (3.17), (3.26), (3.32) up to O(tN−m) for, respectively,
m = 0, 1, 2. We may use (3.20) with (3.36) to determine N(n,m) exactly for N ≥ n+m.
Asymptotic Counting of Quarter BPS Operators at Large N
Asymptotic counting for the one boson case in the large N limit, for which, with PN (t)
as defined in (2.26), with p(n) being the total number of (unordered) partitions of n,
ZU(∞)(t) = P∞(t) =
p(n)tn , (3.40)
is the multi-trace partition function, entails finding an asymptotic value for the partition
number p(n) for ‘large’ n. This may be achieved by performing a saddle point approxima-
tion of p(n) = 1
dtP∞(t)t
−n−1. The function P∞(t) has a ‘large’ singularity at t = 1,
but in addition has singularities at all other roots of unity - see [45] on the validity of ignor-
ing these contributions asymptotically. This method was used by Hardy and Ramanujan
to find their celebrated formula, here given in a less detailed form as,
p(n) ∼ 1
, (3.41)
which was improved by Rademacher to give p(n) exactly. Their method relied crucially on
the modular properties of P∞(t).
Focusing now on the two bosonic fundamental field case for which, in the large N
limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for
the numbers Q(n,m, b) = Q(n,m, b1, . . . , bm), with constraint equation
j=1 jbj = m,
defined by
(1− tj)bj+1
1− tj = 1 +
Q(n,m, b)tn . (3.42)
Having found these we may then attempt to find the dominant contribution to (3.20) with
(3.38) for large N . In order to give asymptotic values for Q(n,m, b) we may follow [46] and
apply a formula due to Meinardus which gives a general result for the generating function
(1− tn)−an = 1 +
r(n)tn . (3.43)
A detailed version of Meinardus’ theorem may be found in [46] but for purposes of brevity
we may note that it implies that, as n → ∞,
r(n) ∼ C nκ exp
AΓ(α+ 1)ζ(α+ 1)nα
)1/(α+1)
(α+ 1)/α
, (3.44)
where ζ(s) =
j=1 j
−s is the Riemann zeta function and the constants C, κ, α, A are
determined by the auxiliary Dirichlet series,
D(s) =
, (3.45)
which must converge for Re(s) > α, a positive real number, and possess an analytic
continuation in the region Re(s) ≥ c, −1 < c < 0, such that, in this region, D(s) is
analytic except at a simple pole at s = α where it has residue A. In terms of α,A then
2π(1+α)
AΓ(α+ 1)ζ(α+ 1)
)(1−2D(0))/2(α+1)
expD′(0) ,
κ = (D(0)− 1− 1
α)/(α+ 1) .
(3.46)
Applying Meinardus’ theorem to the case of (3.42), clearly we have
D(s) =
+ ζ(s) , (3.47)
so that, assuming m =
j=1 jbj is fixed, D(s) has a simple pole at s = α = 1 where it
has residue A = 1. Using
D(0) = −1
bj , expD
′(0) =
, (3.48)
then, from (3.44) with (3.46), we may easily determine that, as n → ∞,
Q(n,m, b) ∼ 1
bj m∏
. (3.49)
This reduces to (3.41) when bj = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using
(3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞,
N(n,m) =
b1,...,bm≥0∑
jbj=m
Q(n,m, b)−
b1,...,bm−1≥0∑
jbj=m−1
Q(n+1, m−1, b)
(3.50)
since Q(n,m,m, 0, . . . , 0), for b1 = m, bj = 0, j > 1, dominates over all other terms in
(3.50). This gives asymptotic values for the numbers in (1.1), for counting quarter BPS
operators, transforming in [m,n−m,m] SU(4)R representations, in the large N limit of
free N = 4 super Yang Mills, as previously described.
4. Counting Operators in the Chiral Ring of N = 4 Super Yang Mills
For the purposes of counting operators in the chiral ring of N = 4 super Yang Mills,
we denote corresponding multi-trace partition functions by CU(N)(t).
The generating function for CU(N)(t) for the case of one bosonic fundamental field has
been written in the form [1,3,9]
C(ν, t) =
1− νtn =
νNCU(N)(t) , (4.1)
so that ν acts as a chemical potential for the rank of the gauge group U(N). The equiv-
alence CU(N)(t) = ZU(N)(t) = PN (t), with ZU(N)(t) as in (2.25), is actually a special case
of the q-Binomial theorem. Writing - see [43] for notation -
(a; q)k = (1− a)(1− aq) · · · (1− aqk−1) , (4.2)
then the q-Binomial theorem is, for |x|, |q| < 1,
(a; q)k
(q; q)k
(ax; q)∞
(x; q)∞
. (4.3)
(Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t)∞ = C(ν, t) above
and 1/(t; t)N = PN (t) in (2.26), then CU(N)(t) = ZU(N)(t) = PN (t) straightforwardly.
This special case of the q-Binomial theorem is due to Euler.)
For the two boson case, so that the single particle partition function is given by
f(t, u) = t + u for some t, u, then the generating function for the finite N chiral ring
partition function CU(N)(t, u) is given by [1,3,9]
C(ν, t, u) =
n,m=0
1− νtnum =
νNCU(N)(t, u) . (4.4)
This function is more difficult to analyse in terms of counting though has been investigated
by Stanley [47] in relation to partitions - there it has been dubbed the ‘double Eulerian’
generating function. Through use of the Cauchy-Littlewood formula, then we may expand
CU(N)(t, u) in terms of partitions of N as,
CU(N)(t, u) =
hλ(t)hλ(u) , (4.5)
where
hλ(t) = sλ(1, t, t
2, . . .) , (4.6)
so that using an identity for Schur polynomials to be found in [24,47] then
CU(N)(t, u) =
1≤i<j≤N (1− tλi−λj+j−i)(1− uλi−λj+j−i)∏N
i=1(t; t)λi+N−i(u; u)λi+N−i
(i−1)λi . (4.7)
(4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π
being all column-strict plane partitions of shape λ, |π| =
i,j πij ,
hλ(t) = sλ(1, t, t
2, . . .) =
t|π| . (4.8)
Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connec-
tion between the ‘double Eulerian’ generating function and major indices of permutations
see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2
whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corre-
sponding to a Young diagram with a single row of two boxes) π11 ≥ π12 ≥ 0 gives all
column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a
Young diagram with a single column of two boxes) then π11 > π21 ≥ 0 gives all column-
strict plane partitions of shape (1, 1). Thus,
h(2,0)(t) =
π11,π12≥0
π11≥π12
tπ11+π12 =
(1− t)(1− t2) ,
h(1,1)(t) =
π11,π21≥0
π11>π21
tπ11+π21 =
(1− t)(1− t2) ,
(4.9)
so that, from (4.5) for N = 2,
CU(2)(t, u) = h(2,0)(t)h(2,0)(u)+h(1,1)(t)h(1,1)(u) =
1 + ut
(1− t)(1− t2)(1− u)(1− u2) , (4.10)
which is the correct result as may be verified by extracting the ν2 coefficient in an expansion
of (4.4) up to O(ν2).
9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane parti-
tion of shape λ is an array π = (πij) of non-negative integers with finitely many non-zero entries,
that is arranged in a Young tableaux with shape λ - see appendix A - such that the numbers πij
are weakly decreasing along each row, πij ≥ πi j+1 ≥ 0, and strictly decreasing down each column,
πij > πi+1 j ≥ 0. The sum of the parts of π is given by |π| =
πij. (Note that in contrast to
the definition in [47], here we are allowing πij = 0, for some i, j, to be a part of the plane partition
π with shape λ.)
In the large N limit,
CU(∞)(t, u) =
n1,n2≥0
n1+n2>0
1− tn1un2 , (4.11)
upon which attention is shortly focused.
For the numbers N(n,m) → N̂(n,m) counting quarter BPS primary operators for the
chiral ring of N = 4 super Yang Mills, belonging to [m,n − m,m] SU(4)R R-symmetry
representations, as in (1.1), we have 10
N̂(n,m) =
dt du CU(N)(t, u) s(n,m)(t−1, u−1) (t−1 − u−1)2 . (4.12)
These may be more conveniently evaluated in terms of the numbers in (3.4) M(n,m) →
M̂(n,m), counting all chiral ring quarter BPS operators in the [m,n−m,m] SU(4)R repre-
sentation, given by
M̂(n,m) =
(2πi)2
dt du CU(N)(t, u) t−n−1u−m−1 , (4.13)
so that N̂(n,m) = M̂(n,m)−M̂(n+1,m−1). Defining Pλ(n) to be the number of column-strict
plane partitions π of shape λ so that |π| =
i,j πij = n, then, from (4.5) with (4.8) and
(4.13), M̂(n,m) =
λ⊢N Pλ(n)Pλ(m). Thus,
N̂(n,m) =
(Pλ(n)Pλ(m)− Pλ(n+1)Pλ(m−1)) , (4.14)
counts chiral ring quarter BPS primary operators in SU(4)R representations [m,n−m,m]
for any n,m at finite N .
Asymptotic Counting for Chiral Ring BPS Operators at Large N
For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills
at large N , a relatively crude method is employed here which nevertheless captures the
exponential behaviour of counting numbers of interest. This method is based on saddle
point approximations of functions near a dominant singularity - see [45] for a useful sum-
mary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we
are interested only in the exponential behaviour of such numbers anyhow.)
10 This formula employs the orthonormality relation of Schur polynomials described here
and has appeared in a similar context in [33], appendix B.
To illustrate, we consider the one boson case in the large N limit again. We first find
a convenient ‘approximating function’ as follows,
P∞(t) =
1− tn = exp
ln(1− tn)
∼ exp
ds ln(1− ts)
= exp
6 ln t
(4.15)
which has an ‘easier’ singularity structure. (The approximation in the second step may
be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using
(4.15) then for large enough n,
p(n) =
dt P∞(t)t
−n−1 ∼ 1
dt eg(t) , g(t) = − π
6 ln t
− n ln t . (4.16)
We may approximate the latter integral for large n by noting that the dominant contribu-
tion is at the saddle point t′ = e−π/
6n ∼ 1 for which
g(t′) = π
n , g′(t′) = 0 , g′′(t′) =
6n3 eπ
2/3n = α , (4.17)
so that, for t′′ = t− t′,
p(n) ∼ eπ
2n/3 1
dt′′ e
αt′′2 ∼ eπ
2n/3 1
ds e−
αs2 =
2n/3 .
(4.18)
Thus,
ln p(n) ∼ π
n , (4.19)
which captures the correct behaviour of ln p(n) for large n, according to (3.41).
We may proceed analogously for the quarter BPS chiral ring multi-trace partition
function at large N , (4.11), which we approximate by
CU(∞)(t, u) =
n1,n2≥0
n1+n2>0
1− tn1un2 = exp
n1,n2≥0
n1+n2>0
ln(1− tn1un2)
∼ exp
dv dw ln(1− tvuw)
= exp
( ζ(3)
ln t lnu
(4.20)
In this case we have, from (4.13),
M̂(n,m) ∼
(2πi)2
dt du eg(t,u) , (4.21)
where
g(t, u) =
ln t lnu
− n ln t−m lnu , (4.22)
for n,m large. The dominant contribution to the integral, for n,m both large and of the
same order, occurs about the point (t′, u′) ∼ (1, 1) where
t′ = e−(ζ(3)mn
−2)1/3 , u′ = e−(ζ(3)nm
−2)1/3 , (4.23)
for which
g(t′, u′) = 3 3
ζ(3)nm ,
g(t, u)
(t′,u′)
g(t, u)
(t′,u′)
= 0 ,
g(t, u)
(t′,u′)
= 2(ζ(3)−1n5m−1)
3 e2(ζ(3)mn
−2)1/3 = α ,
g(t, u)
(t′,u′)
= 2(ζ(3)−1m5n−1)
3 e2(ζ(3)nm
−2)1/3 = β ,
g(t, u)
(t′,u′)
= (ζ(3)−1n2m2)
3 e(ζ(3)mn
−2)1/3+(ζ(3)nm−2)1/3 = γ .
(4.24)
So long as m,n are both large and of the same order, the saddle point approximation is
justified and we obtain
M̂(n,m) ∼ e3
ζ(3)nm 1
dv dw e−
(αv2+βw2+2γvw) = h(α, β, γ)e3
ζ(3)nm ,
(4.25)
where
h(α, β, γ) =
αβ − γ2
(ζ(3)m−2n−2)
3 e−(ζ(3)mn
−2)1/3−(ζ(3)nm−2)1/3 . (4.26)
We thus have that for n,m both comparably large,
lnM̂(n,m) ∼ 3 3
ζ(3)nm , (4.27)
so that
ln N̂(n,m) = ln
M̂(n,m) − M̂(n+1,m−1)
∼ lnM̂(n,m) ∼ 3 3
ζ(3)nm . (4.28)
It is difficult to check the consistency of this result given the dearth of literature on
these types of multi-variable generating functions and their asymptotic behaviour, however,
we may consider the simpler function, also considered in [9],
CU(∞)(t, t) =
(1− tn)n+1 =
E(r) tr , (4.29)
where, in terms of the counting numbers N̂(n,m), from (1.1),
CU(∞)(t, t) =
(n−m+1)N̂(n,m) tn+m ⇒ E(r) =
(r−2m+1)N̂(r−m,m) .
(4.30)
Hence, E(r) counts quarter BPS primary operators in the chiral ring of N = 4 super
Yang Mills, transforming in [m,n−m,m] SU(4)R representations, with the same conformal
dimensions r = n+m. Extracting the dominant contribution to ln E(r) from (4.30), which
occurs at the maximum value of m, mM = [
r], and using (4.28), we obtain
ln E(r) ∼ ln N̂(r−mM,mM) ∼ 32
2ζ(3)r2 . (4.31)
By considering (4.29) directly, we may employ Meinardus’ theorem (described in the
third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus’
theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet
series (3.45), with aj = j+1, has two simple poles. To overcome this difficulty we split
(4.29) into a product of two functions, both separately amenable to application of Meinar-
dus’ theorem. One is the reciprocal of the Euler function, P∞(t) in (4.15). The other, the
MacMahon function, is given by
M(t) =
(1− tn)n =
q(r) tr , (4.32)
and has been considered in a similar context as here in [3].11 Writing
CU(∞)(t, t) = P∞(t)M(t) , (4.33)
with P∞(t) as in (4.15), so that, using (4.29),
E(r) =
r1,r2≥0
r1+r2=r
p(r1)q(r2) , (4.34)
we may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant
contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary
Dirichlet series for M(t) in (4.32) is, from (3.45) with aj = j,
D(s) = ζ(s− 1) , (4.35)
11 The relation of M(t) in (4.32) to plane partitions is given a description in [46]. Briefly,
q(r) gives the number of ordinary plane partitions π, so that πij ≥ πi+1 j > 0, πij ≥ πi j+1 > 0,
with |π| =
πij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions.
In fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula
first found by Wright [48] for the number of plane partitions q(r) of the number r.
which has a simple pole at s = α = 2, at which the residue is A = 1. Thus, from (3.44),
ln q(r) ∼ 3
2ζ(3)r2 . (4.36)
This is consistent with (4.31) as the dominant contribution to lnE(r) comes from the r1 = 0
term in (4.34) (since p(r) ≪ q(r) as r → ∞) so that lnE(r) ∼ ln q(r).
It has not escaped attention that the method used here, to capture the exponential
behaviour of asymptotic values for the numbers M̂(n,m), may be easily extended to chiral
ring sectors other than the quarter BPS one. Suppose, for simplicity, that Zj , 1 ≤ j ≤ k−1,
are commuting bosonic fundamental fields, in the U(N) Lie algebra, so that the single
particle partition function is given by f(t) =
j=1 tj , in terms of the corresponding letters
tj . Let M̂(m1,...,mk−1) denote the number of independent operators involving products of
m1 Z1’s, m2 Z2’s etc. in corresponding multi-trace operators. The multi-trace partition
function, in the large N limit, is given by,
CU(∞)(t) =
n1,...,nk−1≥0
n1+...+nk−1>0
1− t1n1 · · · tk−1nk−1
, (4.37)
which may be crudely approximated by, similarly as before,12
CU(∞)(t) ∼ exp
dvj ln(1−t1v1 · · · tk−1vk−1)
= exp
(−1)k+1ζ(k)
ln t1 · · · ln tk−1
. (4.38)
Thus, without going into as much detail, for the analogue of (4.27) we have, (assuming mj
are all comparably large,)
lnM̂(m1,...,mk−1) ∼ g(t
1, . . . , t
k−1) = k
ζ(k)m1 · · ·mk−1 , (4.39)
where g(t′1, . . . , t
k−1) is the value of
g(t1, . . . , tk−1) =
(−1)k+1ζ(k)
ln t1 · · · ln tk−1
−m1 ln t1 − . . .−mk−1 ln tk−1 , (4.40)
12 For Lin(x) =
xj/jn being the usual Polylogarithm, with Lin(1) = ζ(n), n > 1,
Lin(0) = 0, then with the convention Li1(x) = − ln(1− x), the following integral
Lin(zx) = Lin+1(z) ,
may be useful for showing this, after a suitable change of variables.
at the saddle point (t′1, . . . , t
k−1) ∼ (1, . . . , 1), where
(ln t′1, . . . , ln t
k−1) = − k
ζ(k)m1 · · ·mk−1 (1/m1, . . . , 1/mk−1) , (4.41)
so that
g(t1, . . . , tk−1)
,...,t′
= 0 , j = 1, . . . , k−1 . (4.42)
(4.39) is consistent with a result implied by Meinardus’ theorem. The function,13
CU(∞)(t, . . . , t) ∼
(1− tn)−n
k−2/(k−2)! =
c(k, r)tr , (4.43)
has auxiliary Dirichlet series, from (3.45) with aj = j
k−2/(k − 2)!,
D(s) =
(k − 2)! ζ(s+ 2− k) , (4.44)
which has a simple pole at s = α = k−1 at which the residue is A = 1/(k − 2)!, so that,
from (3.44),
ln c(k, r) ∼ k
(k−1) ζ(k) rk−1 . (4.45)
(4.45) is precisely the result that may be obtained from (4.39) if we maximise the product
m1 · · ·mk−1, subject to the constraint
j=1 mj = r, for which the solution is mj = mM =
r/(k − 1) (relaxing the constraint that mj be non-negative integers, which is irrelevant
asymptotically), so that lnM̂(mM,...,mM) ∼ ln c(k, r).
This is applicable to counting multi-trace operators in the eighth BPS chiral ring sec-
tor for N = 4 super Yang Mills with fundamental fields Z, Y,X involving m1 Z’s, m2 Y ’s,
m3 X ’s. Expanding the corresponding partition function (4.37), with k = 4, in terms of
Schur polynomials s(m1,m2,m3)(t), m1 ≥ m2 ≥ m3 ≥ 0, similar to (1.1), the expansion
coefficients N̂(m1,m2,m3) count spinless multi-trace primary operators transforming in the
[m2 +m3, m1 −m2, m2 −m3] SU(4)R R-symmetry representation, with conformal dimen-
sionsm1+m2+m3 [33]. Just as in (4.28), asymptotically lnN(m1,m2,m3) ∼ lnM(m1,m2,m3).
This counting, however, ignores contributions of the fermionic fields λ, λ̄, which it may be
important to include in order to give correct counting of eighth BPS chiral ring operators.
13 This may be easily seen from (4.37), as the number of solutions to
mj = n, where
mj are non-negative integers, is the binomial number
n+k−2
which, to leading order in large n,
behaves like nk−2/(k−2)!. More properly, we should split the product CU(∞)(t, . . . , t) into pieces
separately amenable to Meinardus’ theorem, as for the prior case for k = 3, however, just as for
that case, the numbers c(k, r) dominate, and so other contributions are ignored here.
5. Conclusions
There are some obvious questions not answered by this work. The first is whether
or not the approach in the second section using symmetric polynomials can give insight
into thermodynamics at finite N , such as for the Hagedorn transition, for example. While
it gives the large N expression (2.14) in an elementary way, its wider applicability or
usefulness to such questions is unclear. The approach is undoubtedly useful for finding
exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS
operators) and (3.9), (3.16) may be useful for analysing counting for more complicated
sectors of N = 4 super Yang Mills, with gauge group U(N).
The second question is how the arguments employing symmetric polynomial tech-
niques here may be extended to other gauge groups, the most pertinent being perhaps
SU(N). Arguments here employing (2.6) and the orthonormality property of Schur poly-
nomials should remain largely unaffected for SU(N). Exact values for counting numbers
obtained here should require some modification for SU(N), though asymptotic values may
be unchanged.
The third question concerns asymptotic values for counting numbers and how these
may be improved. The asymptotic counting formulae given in such papers as [3,9] for
chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan,
Meinardus, etc., all of which derive from single variable generating functions. It is hoped
that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS
operators, that distinguishes between differing R-symmetry charges, represents a serious
attempt at going beyond consideration of single variable generating functions.14 Improving
upon these formulae will require more sophisticated techniques, perhaps along the lines
used to find those of Hardy and Ramanujan or Meinardus and employing any modular
properties of the multi-variable functions involved. This issue may also be important for
microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4
superconformal symmetry, depend on special values of R-symmetry charges [17,18,19,20]
- see [50] for a related detailed discussion.
Thus far, the elegant results for finite N partition functions for chiral ring sectors have
been interpreted from a largely geometric perspective - it may be interesting to investigate
more how such results are related to the theory of random matrices and/or symmetric
polynomials.
14 After submission of the first version of this paper to the electronic archive, I received
an e-mail from Hai Lin pointing out an interesting comparison between (4.28) here and (2.14) of
[49], obtained in quite a different context. The two formulae are essentially the same given the
numerical value 3 3
ζ(3) = 3.189 . . ., correct to three decimal places.
Acknowledgements
I warmly thank Yang-Hui He, Paul Heslop, Hugh Osborn, Christian Romelsberger and
Christian Saemann for useful comments and discussions. This work is supported by an
IRCSET (Irish Research Council for Science, Engineering and Technology) Post-doctoral
Fellowship.
Appendix A. Partitions, symmetric group characters, symmetric polynomials
and inner products
A generic partition λ is any finite or infinite sequence λ = (λ1, λ2, . . .) of non-negative
integers in decreasing order λ1 ≥ λ2 ≥ . . . ≥ 0 containing only finitely many non-zero
terms. Often it is convenient to omit zero entries. The non-zero entries are called the
parts of λ the number of which we denote by ℓ(λ). The sum of the parts of λ is called
the weight of λ which we denote by |λ| =
i λi. If |λ| = L then λ is a partition of L
and we write λ ⊢ L. For convenience we sometimes write λ in its frequency representation
which is a reordering of the entries in λ, indicating the number of times each successive
non-negative integer occurs, (1a1 , 2a2 , . . .) so that exactly an of the parts of λ equal n and
|λ| =
n≥1 n an.
In terms of standard Young diagrams, λ corresponds to a Young diagram of shape λ,
with λ1 boxes in the first row, λ2 boxes in the second row etc.; the number of parts ℓ(λ)
is simply the number of rows and the weight |λ| is the total number of boxes.
For the symmetric group, SN , the irreducible representations are labelled by par-
titions λ ⊢ N - see [38] for a useful summary - so that, for Xλ(σ), σ ∈ SN , being a
corresponding matrix representation, then the character of σ ∈ SN in the representation
Xλ is χλ(σ) = tr(Xλ(σ)). The characters are class functions so that they take a constant
value on conjugacy classes and, recalling that for SN the conjugacy classes Kµ are labelled
by partitions µ ⊢ N , corresponding to the cycle structure of a class representative, then
χλ(σ) = χλµ for all σ ∈ Kµ. With zλ as defined in (2.8), a crucial property of SN charac-
ters is the orthogonality of the matrix [zµ
−1/2χλµ]λµ. This gives rise to the orthogonality
relations, for λ, µ ⊢ N , (see also Ch. IV of [51] for a related discussion,)
χλ(σ)χµ(σ) =
χλν χ
ν = δλµ , (A.1)
and ∑
χνλ χ
µ = zλδλµ . (A.2)
A convenient basis for N variable symmetric polynomials are Schur polynomials
sλ(z) = sλ(z1, . . . , zN ) labelled by λ = (λ1, . . . , λN ). They may be expressed in a number
of ways [24,47]. For convenience we write them as
sλ(z) = aλ+ρ(z)/aρ(z) , (A.3)
where ρ, the Weyl vector, is given by ρ = (N − 1, N − 2, . . . , 1, 0) and
aλ+ρ(z) =
sgn(σ) zσ(1)
λ1+N−1 · · · zσ(j)λj+N−j · · · zσ(N)λN = det[ziλj+N−j ] , (A.4)
aρ(z) = det[zi
N−j ] =
1≤i<j≤N
(zi − zj) = ∆(x) , (A.5)
being the Vandermonde determinant. Schur polynomials sλ(z) have a standard interpre-
tation as corresponding to the characters of irreducible U(N) (or, for
i zi = 1, SU(N))
Lie algebra representations. Here, λ gives the shape of the Young tableaux for the corre-
sponding U(N) Lie algebra representation.
For λ = (λ1, . . . , λN ) and µ = (µ1, . . . , µN ) where λi, µi ∈ Z then, from the definition
of (2.6) along with (A.3),
sλ, sµ
sgn(σ)δλσµ =
sgn(σ)δλµσ , (A.6)
where, for any λ′ = (λ′1, . . . , λ
N ), λ
′σ = σ(λ′+ρ)−ρ is the shifted Weyl reflection of λ′ by σ,
with the action of SN on λ′ being given by σ(λ′1, . . . , λ′N ) = (λ′σ(1), . . . , λ′σ(N)). (Equation
(A.6) is a reflection of sλ(x) = sgn(σ)sλσ(x) for any partition λ and σ ∈ SN - note that this
property is useful for showing (3.25), (3.31). λσ has a standard interpretation in terms of
U(N) Lie algebra representations - for the Verma module with dominant integral highest
weight having orthonormal basis labels λ, λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0, then λσ, for σ 6= idSN ,
are the orthonormal basis labels for the highest weights of all invariant sub-modules. This
fact may be exploited to derive the Weyl character formula (A.3) for the irreducible U(N)
Lie algebra representation with dominant integral highest weight having orthonormal basis
labels λ, or, alternatively, Young tableaux of shape λ.)
When λ, µ are partitions so that λ1 ≥ . . . ≥ λN ≥ 0 and µ1 ≥ . . . ≥ µN ≥ 0 then
(A.6) reduces to a well defined inner product,
sλ, sµ
= δλµ , (A.7)
so that in this case the Schur polynomials are orthonormal. Note that in order that sλ(x)
be non-zero for some arbitrary partition λ then ℓ(λ) ≤ N , so that (A.7) is zero for ℓ(λ) > N
or ℓ(µ) > N .
Another basis for symmetric polynomials are the power symmetric polynomials, pλ(z),
for λ = (λ1, . . . , λL) ⊢ L, which are defined by
pλ(z) = pλ1(z)pλ2(z) · · ·pλL(z) , pn(z) =
n . (A.8)
Note that there is no longer the restriction that ℓ(λ) ≤ N as for Schur polynomials.
Symmetric group characters may be used to relate the two bases for symmetric poly-
nomials [24,47] so that, with the definition of zλ in (2.8),
sλ(z) =
χλµ pµ(z) , (A.9)
(a theorem of Frobenius) and, for λ ⊢ L,
pλ(z) =
ℓ(µ)≤N
λ sµ(z) . (A.10)
((A.9) with χ
λ = 1 for all λ ⊢ N is useful for obtaining (3.35).)
Regarding the inner product (2.6), then using (A.10) along with (A.7), we then have
that, for λ ⊢ L, µ ⊢ M , 〈
pλ, pµ
= δLM
ℓ(ν)≤N
µ . (A.11)
Orthogonality of symmetric group characters implies, from (A.2), that for |λ|, |µ| ≤ N
then (A.11) simplifies to, with the definition of zλ in (2.8),
pλ, pµ
= zλδλµ . (A.12)
Appendix B. Tables
N n 2 3 4 5 6 7 8 9 10 11
1 1 1 1 1 1 1 1 1 1 1
2 2 2 3 3 4 4 5 5 6 6
3 2 3 4 5 7 8 10 12 14 16
4 2 3 5 6 9 11 15 18 23 27
5 2 3 5 7 10 13 18 23 30 37
6 2 3 5 7 11 14 20 26 35 44
Numbers of multi-trace half BPS primary operators, with conformal dimension n
and belonging to [0, n, 0] R-symmetry representations, for free N = 4 SYM with
U(N) gauge group. (For every N there is one [0, 0, 0] and [0, 1, 0] representation -
these are omitted above.)
N(n,1)
N n 2 3 4 5 6 7 8 9 10 11
2 1 1 2 2 3 3 4 4 5 5
3 1 2 4 5 8 10 13 16 20 23
4 1 2 5 7 12 16 23 30 40 49
5 1 2 5 8 14 20 30 41 57 74
6 1 2 5 8 15 22 34 48 69 92
7 1 2 5 8 15 23 36 52 76 104
Numbers of multi-trace quarter BPS primary operators, with conformal dimension
n+1 and belonging to [1, n−1, 1] R-symmetry representations, for free N = 4 SYM
with U(N) gauge group. (n = 0, 1 cases are all zero.)
N(n,2)
N n 2 3 4 5 6 7 8 9 10 11
3 3 5 10 14 21 27 36 44 55 65
4 3 6 14 21 36 50 73 96 130 163
5 3 6 15 25 44 66 101 142 200 267
6 3 6 15 26 48 74 118 171 251 346
7 3 6 15 26 49 78 126 188 281 398
8 3 6 15 26 49 79 130 196 298 428
Numbers of multi-trace quarter BPS primary operators, with conformal dimension
n+2 and belonging to [2, n−2, 2] R-symmetry representations, for free N = 4 SYM
with U(N) gauge group. (n = 0, 1 cases are all zero.)
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http://arxiv.org/abs/hep-th/0509235
http://arxiv.org/abs/hep-th/0611351
