Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

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by  Miguel Tierz

EXACT SOLUTION OF CHERN-SIMONS-MATTER MATRIX MODELS
WITH CHARACTERISTIC/ORTHOGONAL POLYNOMIALS
MIGUEL TIERZ
Abstract. WesolveforfiniteN thematrixmodelofsupersymmetricU(N)Chern-Simonsthe-
orycoupledtoN fundamentalandN anti-fundamentalchiralmultipletsofR-charge1/2and
f f
ofmassm,byidentifyingitwith an averageofinversecharacteristic polynomials inaStieltjes-
6 Wigertensemble. ThisrequiresthecomputationoftheCauchytransformoftheStieltjes-Wigert
1
polynomials, which we carry out, findinga relationship with Mordell integrals, and hencewith
0
previous analytical results on the matrix model. The semiclassical limit of the model is ex-
2
pressed, for arbitrary Nf, in terms of a single Hermite polynomial. This result also holds for
r
p more general matter content,involving matrix models with double-sinefunctions.
A

7 1. Introduction
2
The study of supersymmetric gauge theories has been rekindled in recent years with con-
]
h siderable progress mainly due to two reasons: on one hand, the development of localization of
t
- supersymmetric gauge theories [1] ([2, 3] for recent reviews) and, on the other hand, the avail-
p
ability of powerful analytical tools to explicitly handle matrix models, a long-standing area of
e
h mathematical and theoretical physics [4]. These two subjects are now intimately related since
[
the localization method manages to reduce the original functional integral describing a quan-

2
tum field theory into a much simpler matrix integral. Thus, it enormously reduces the task of
v
7 computing observables in a supersymmetric gauge theory, but there still remains the issue of
7 explicitly computing N integrations, which is typically not an straightforward task and requires
2
the use of specific matrix model techniques.
6
0 The theory we shall focus on is = 2 supersymmetric U(N) Chern-Simons (CS) on the
1. three-sphere S3 with N fundamentNal and N antifundamental chiral multiplets of mass m.
f f
0 Indeed the partition function on S3 can be determined by the localization techniques of Pestun,
6
1 which were adapted to the 3d case in [5, 6, 7, 8]. In the case of the partition function for U(N)
: = 3 Chern-Simons theory at level k coupled to N fundamental and N¯ anti-fundamental
v f f
N
i chiral multiplets of R-charge q the matrix model is [8]1
X
(1.1)
r
a N N
Z = 1 dNσ eiπkσ2j (sb=1(i iq σj))Nf (sb=1(i iq+σj))N¯f (2sinhπ(σi σj))2,
N! − − − −
Z j=1 i<j
Y Y
where s (σ) denotes the double sine function [7, 8] (and references therein). This matrix
b=1
model corresponds to the case where the matter chiral multiplets have R-charge q and belong
to the representation R of the gauge group. The fact that for = 3 theories the R-symmetry is
N
non-abelian allows us to fix an R-charge which is not altered under the RG flow2. In this paper,
we focus on a comprehensive analyzing of the case where q = 1/2 and R = r r, in which case,
⊕
1Notice that we have changed the sign of the Chern-Simons level with respect to that in [8] in order to make
contact with our conventions.
2Our formalism will implicitly allow for imaginary masses and thus one can view the matrix model as corre-
sponding to a N =2 theory,for which the Z-extremization process [7] has not been carried out.
1
2 MIGUELTIERZ
due to a basic property of the double sine function, the matter contribution simplifies in the
following way [8]
1
(1.2) s (i ρ σˆ ) s (i +ρ σˆ ) = ,
b=1 2 − i i · b=1 2 i i 2coshπρ σˆ
ρ∈r ρ∈r i i
Y Y
which renders the matrix model equal to
(1.3) ZU(N) = 1 dNµ i<j4sinh2(12(µi −µj)) e−21gPiµ2i+iηPiµi ,
Nf (2π)N N! Z Q i 2cosh(21(µi +m)) Nf
where g = 2πi with k Z the Chern-Simons lQeve(cid:0)l and µ /2π repres(cid:1)ent the eigenvalues of the
k ∈ i
scalar field σ belonging to the three dimensional vector multiplet. In (1.3) the radius R of the
three-sphere has been set to one. It can be restored by rescaling m mR, µ µ R. The
→ i → i
partition function is periodic in imaginary shifts of the mass, Z(m+i2πn)= Z(m), for integer
n. The addition of a a Fayet-Iliopoulos term (FI) in the Lagrangian adds a linear term in the
potential of the matrix model [5, 6, 8]. Thus η is a real parameter denoting our FI parameter.
Notice that the variables in (1.3) are rescaled with a 2π factor with regards to those in [5, 6, 8]
and with regard to the ones in (1.1). That is, µ = 2πσ .
i i
A variant of the matrix model (1.3), with 2N hypermultiplets has been previously analyzed
f
in [9, 10]. In [10], the approach is to express (1.3) as a Hankel determinant whose entries are
(combinations of) Mordell integrals, which are integrals of the type [11]
∞ e(l+1)µ+m
(1.4) I(l,m) = dµ e−µ2/2g.
1+eµ+m
−∞
Z
This integral I (1.4) was computed by Mordell [11] for general parameters. Typically, it is given
in terms of infinite sums of the theta-function type. However, in specific cases it assumes the
form of a Gauss’s finite sum [11, 10]. These specific cases precisely contain the one which is
physically relevant: g = 2πi/k with k Z. This works very well for N = 1 and has been
f
∈
more recently extended to higher flavour in [12] by studying parametric derivatives of Mordell
integrals. A large number of analytic results and explicit tests of Giveon-Kutasov duality can
be obtained in that way.
Analternative analytical approach isputforwardinthispaper. Inparticular, instead ofusing
the Mordell integral [11] method developed in [10, 12], we shall employ here the powerful for-
malism of characteristic polynomials in random matrix ensembles, which has been considerably
developed over the last decade [13, 14].
Thepaper is organized as follows. In the next Section we introduce the necessary background
on random matrices and averages of characteristic polynomials on random matrix ensembles,
together with its connections with orthogonal polynomials. We shall then show, in Section 3,
that the matrix model (1.3) above can be solved in terms of such averages in a Stieltjes-Wigert
ensemble. Determinantal formulas for the model in [10] and for the model (1.3) with different
masses are given as well.
In Section 4, we characterize analytically the Stieltjes transform of the Stieltjes-Wigert poly-
nomials, which is the fundamental object in Section 2 and also the building block in the de-
terminantal formulas that emerge in the N > 1 case. The polynomials are obtained explicitly
f
and shown to be related to Mordell integrals (1.4). A number of particular cases for both low
and large N are presented and consistency checks are carried out. In particular, using modular
properties of the Mordell integral [15] we also establish the relationship with the method of
Hankel determinants of Mordell integrals used in [10] and extended in [12].
3
In Section 5, we study the semiclassical limit of the matrix model, finding it to be given by
a single Hermite polynomial. This holds for N 1, because the number of flavours appears
f
≥
in the spectral parameter of the characteristic polynomial average (hence, in the variable of the
Hermite polynomial) and, therefore, no determinantal expression is required in this limit for
N > 1. The same behavior is shown to hold also for the model (1.1) and not just (1.3). Hence
f
the solution of the modelin the semiclassical limit is still a single Hermite polynomial, only with
a more involved specialization of the variable, which becomes complex, of the polynomial. We
finally conclude with a summary and avenues for further research.
2. Random matrices and characteristic polynomials, definitions.
In random matrix theory, ensembles of N N Hermitian matrices H are described by a
× { }
measure dα with finite moments x kdα(x) < , k = 0,1,2, , and the probability dis-
R
| | ∞ ···
tribution function for the eigenvalues x = x (H) of matrices H in the ensembles is of the
R { i i }
form
1
(2.1) Prob (x) = ∆(x)2dα(x)
α,N
Z
N
where dα(x) = N dα(x ), ∆(x) = (x x ) is the Vandermonde determinant for
i=1 i N≥i>j≥1 i − j
the x ’s, and Z = ∆(x)2dα(x) is the normalization constant. Therefore the integrals
i NQ ··· Q
that emerge from localization and after the suitable change of variables, are precisely of the
R R
same form as Z . Choosing the measure dα(x) to be of the form dα(x) = e−V(x)dx where V(x)
N
is known as the potential of the matrix model, one can also write (2.1) alternatively as
N
1
(2.2) det(f (x ))N det(g (x ))N dx
Z i j i,j=1 i j i,j=1 j
N
j=1
Y
where
(2.3) f (x) = xi−1, g (x) = xi−1e−V(x).
i i
Indeed, this follows upon recognizing (2.1) to be basically a product of two Vandermonde de-
terminants. In the last decade or so, the mathematical apparatus of random matrix theory has
been developed considerably [4] and, in particular, there has been much progress in the study
of averages of products and ratios of the characteristic polynomials
N
(2.4) D [µ,H] = (µ x (H)),
N i
−
i=1
Y
of random matrices with respect to a number of ensembles [13, 14]3. The variable µ in (2.4) is
a spectral parameter, which will be further characterized below.
Notice now that, as happens in the case of pure Chern-Simons theory [16], the change of
variables
(2.5) zi = ceµi, c= eg(cid:16)N+N2f−iη(cid:17),
brings the matrix model (1.3) in standard random-matrix form [10, 12]
(2.6)
ZU(N) = e−g2N(N+N2f−iη)(N−5N2f+iη)e−3mNNf/2 dNz (z z )2 e−21gPi(lnzi)2 .
Nf (2π)NN! Z[0,∞)N i<j i − j i(ce−m +zi)Nf
Y
Q
3The literature on characteristic polynomials in random matrix theory is much larger and we only quote the
works that specifically treat thecases which we will show describe Chern-Simons-matter.
4 MIGUELTIERZ
We recall that the Stieltjes-Wigert ensemble is characterized by V(z) = 1 ln2z and solves
2g
exactly pure U(N) Chern-Simons theory on S3, as was first shown in [16]. One possible inter-
pretation of the model (2.6) is as the normalization constant of the random matrix ensemble
with potential V(z) = 1 ln2z +N ln 1+zem , where z (0, ). However, this logarithmic
2g f c ∈ ∞
deformation of the Stieltjes-Wigert weight does not have a known closed system of orthogonal
(cid:0) (cid:1)
polynomials. Therefore, in principle a full analysis of (2.6) can not be carried out with the same
level of detail as with the much simpler Stieltjes-Wigert ensemble.
Remark. The main idea in this paper is that a more detailed analytical characterization of the
Chern-Simons-matter matrix model can be actually obtained by interpreting (2.6) as the average
of the inverse of a characteristic polynomial (to be defined in what follows) in a Stieltjes-Wigert
ensemble.
Let us then begin by introducing the simplest and most basic result on characteristic poly-
nomials. Notice that one can associate the average characteristic polynomial to the probability
distribution (2.2)–(2.3)
1 ∞ ∞ n n
(2.7) P (z) = ... (z x ) det(f (x ))n det(g (x ))n dx .
n Z˜n Z−∞ Z−∞j=1 − j  i j i,j=1 i j i,j=1j=1 j
Y Y
 
It then follows from a classical calculation of Heine, see e.g. [17], that P can be characterized
n
as the nth monic orthogonal polynomial with respect to the weight function e−V(x). This means
that the polynomials P satisfy the conditions
n
P (x) = xn+O(xn−1)
n
for all n N and
∈
∞
(2.8) P (x)P (x)e−V(x) dx = c c δ
n m n m m,n
−∞
Z
for all n,m N, for certain c R.
n
∈ ∈
Likewise, one can define the average inverse characteristic polynomial corresponding to the
probability distribution (2.2)–(2.3) as [13]
1 ∞ ∞ n n
(2.9) Q (z) = ... (z x )−1 det(f (x ))n det(g (x ))n dx ,
n Z˜n Z−∞ Z−∞j=1 − j  i j i,j=1 i j i,j=1j=1 j
Y Y
 
for z C R. Then it holds that
∈ \
∞ P (x)e−V(x)
n−1
(2.10) Q (z) = dx.
n
z x
−∞
Z −
Thus Q (z) is the Stieltjes (or Cauchy) transform of the P (x) polynomials multiplied by
n n−1
their weight function e−V(x). The normalizing partition function in (2.9) is, in terms of the
orthogonal polynomials Z˜ = N! N−1c2 .
n k=0 k
Q
5
3. Chern-Simons-matter models as correlation functions of characteristic
polynomials
The first step is to notice that already (2.9) can be immediately identified with (2.6) in the
case N = 1. More precisely, we have that
f
ZU(N) = 1 dNz (z z )2 e−21gPNi=1(lnzi)2
Nf=1 ZNU(fN=)0 Z[0,∞)N Yi<j i − j Ni=1(ce−m+zi)
∞ πN−1(x;q)e−21g(lnx)2 Q
(3.1) = γ dx =: γ h (λ),
N−1 λ+x N−1 N−1
Z0
where π (x) is the monic Stieltjes-Wigert polynomial [16], a polynomial of order N 1, given
N−1
−
explicitly in next Section, λ = ce−m and γ = 1/c2 and is a numerical prefactor that will be
N − N
appearing often below, in the determinantal expressions.
We emphasize that the polynomial γ h (λ)computes the integral in (2.6), normalized
N−1 N−1
by a matrix integral which is the same ensemble but without the characteristic polynomial
insertion. That is, normalized by a Stieltjes-Wigert ensemble [16]
(3.2) ZNU(fN=)0 = ZNSW = (2πg1)N/2 Z[0,∞)N dNz i<j(zi −zj)2e−21gPNi=1(lnzi)2.
Y
We will need its explicit expression below, so let us collect it here [16, Eq. (2.13)]
N−1
(3.3) ZNSW = N!q−N(2N−16)(2N+1) (1−qj)N−j.
j=1
Y
Thus, the polynomial h (λ), which is the Stieltjes (or Cauchy) transform of the Stieltjes-
N−1
Wigert polynomial π (x;q)
N−1
∞ πN−1(x;q)e−21g(lnx)2dx
(3.4) h (λ):= ,
N−1
λ+x
Z0
itself provides a solution, for all N, of the matrix integral for N = 1, whereas with the Mordell
f
integral method [10, 12] a N N determinant has to be computed. To obtain (3.4) is non-
×
trivial and will be the subject of the next Section. Therefore, using characteristic polynomials is
essentially equivalent to having the orthogonal polynomials for the Mordell ensemble [10] (2.6).
3.1. Determinantal formulas. Suppose 1 N N and let γ = 1/c2, where c is the
≤ f ≤ n − n n
normalization constant defined above (2.8). Then the determinantal expression holds [13, 14]
(3.5) Nf DN−1[ǫj,H] = (−1)Nf∆(N(2fǫ−,1)...,ǫjN=−N1)−Nf γj (cid:12) hN−N...f(ǫ1) ... hN−1(ǫ1) (cid:12),
*jY=1 + 1 QNf (cid:12)(cid:12)(cid:12) hN−Nf(ǫNf) ... hN−1(ǫNf) (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where D [ǫ ,H] is given by (2.4), the average is taken over a random matrix ensemble (2.1)
N j (cid:12) (cid:12)
and h (ǫ) are the Stieltjes transform (2.10) of the polynomials orthogonal with regards to the
n
measure dα(x) in (2.1). In the case of the Stieltjes-Wigert ensemble, these are given by (3.1).
Normally, to avoid divergences, the spectral parameters ǫ are taken with a (small) imaginary
j
part [13], as in the definition of the corresponding Stieltjes transform. The range of integration
in (2.10) is R+ in our case, since it is the domain of our random matrix ensemble. In general,
6 MIGUELTIERZ
ourparameter is either λ > 0 orλ C,4 and, sincex R+, thecorrespondingCauchy transform
∈ ∈
is well-defined.
Notice that this formalism is extensive enough to solve the more general setting where all the
masses of the multiplets are different. Namely, to
(3.6) ZU(N) = 1 dNz (z z )2 e−21gPi(lnzi)2 ,
Nf ZNSW Z[0,∞)N i<j i − j Nj=f1 Ni=1(ce−mj +zi)
Y
b
for which, using (3.5), we find Q Q
(3.7) ZNUf(N) = (−1)N∆f(N(2λf−1,1)..Q.,λjN=N−Nf1)−Nf γj (cid:12)(cid:12)(cid:12) hhN−N...f((λλ1)) ...... hhN−1((λλ1)) (cid:12)(cid:12)(cid:12),
(cid:12) N−Nf Nf N−1 Nf (cid:12)
b
(cid:12) (cid:12)
where λ = ce−mj and the polynomials are exac(cid:12)tly (3.1). Notice that this correla(cid:12)tor admits an
j (cid:12) (cid:12)
interpretation asN non-compactbranesatthedifferentpositionsgivenbytheλ ontheCalabi-
f j
Yau threefold P1 [18]. For N = 1 it can be interpreted as the amplitude of a non-compact
f
brane.
The matrix model (2.6) is a particular case of (3.6) which follows by taking λ = ce−m
j
for j = 1,...,N . For this, as usual with these determinants, we only need to consider the
f
correspondinglimitin thedeterminantal expression(3.7), which can beeasily foundby applying
l’Hˆopital’s rule 5. In this way, we have that, for N > 1 the determinant is the Wronskian of
f
the polynomials
h (λ) ... h (λ)
N−1 N−Nf
N−1 h′ (λ) h′ (λ)
U(N) Nf(Nf−1) (cid:12) N−1 N−Nf (cid:12)
(3.8) ZNf = (−1) 2 γj(cid:12)(cid:12) ... (cid:12)(cid:12).
j=NY−Nf (cid:12)(cid:12)(cid:12) h(Nf−1)(λ) ... h(Nf−1)(λ) (cid:12)(cid:12)(cid:12)
(cid:12) N−1 N−Nf (cid:12)
(cid:12) (cid:12)
Therefore, the characteristic polynomials metho(cid:12)d gives results for arbitrary N(cid:12)and, for N > 1
(cid:12) (cid:12) f
arbitrary, a N N determinant is then required. We emphasize that the difference with [10]
f f
×
is that here the full N dependence is encapsulated in the polynomial itself, rather than in a
determinant.
We willcharacterize below theγ prefactors fortheStieltjes-Wigert polynomialsbut,given their
j
definition as γ = 1/c2 and the inclusion of the partition function Z = N! N−1c2 , as a
j − j N k=0 k
normalization factor in the average of the characteristic polynomial, the simpler route will be
Q
to directly cancel common factors.
3.2. The theory with 2N Chiral Hypermultiplets. This theory was analyzed in detail in
f
[10] for N =1. The partition function, expressed as a matrix integral, is
f
(3.9) ZU(N) = 1 dNµ i<j4sinh2(12(µi−µj)) e−21gPiµ2i+iηPiµi ,
2Nf (2π)N N!Z Qi 4cosh(21(µi+m))cosh(12(µi−m)) Nf
although in [10] we did not includethe FQI t(cid:0)erm (η = 0). Thesimple change of(cid:1)variables is [9, 10]
(3.10) zi = ceµi , c eg(N+Nf−iη) ,
≡
4There will be, for a given N and N at least one value of k that will imply λ∈R and λ<0, in which case
f
one has to becareful understandingtheCauchy transform as a principal value.
5ThisisidenticaltothediagonallimitoftheChristoffel-Darbouxkernelofarandommatrixensemble[4],and
hencestandard in random matrix theory.
7
which recasts the integral in the form
(3.11)
ZU(N) = e−g2N(N+Nf−iη)(N−5Nf+iη) dNz (z z )2 e−21gPi(lnzi)2 .
Nf (2π)N N! Z[0,∞)N i<j i − j i(ce−m+zi)Nf (cem +zi)Nf
Y
As shown in [10], this matrix integral can be performed with tQhe help of Mordell integrals [11].
This method is also used in [12] for computations with N > 1. Notice also that we have 2N
f f
and half of the masses are m and the other half m (and hence, the prefactors in (3.11) and
−
(2.6)are completely consistent). Therefore, the solution in terms of orthogonal polynomials also
follows from (3.7) by considering two, instead of one, parameters λ = ce−m and µ = cem. We
thus have, for N = 1
f
(3.12) ZU(N) = 1 dNz (z z )2 e−21gPi(lnzi)2
Nf=1 ZNSW Z[0,∞)N Yi<j i − j Ni=1 1+ziecm 1+zie−cm
e = −γN−2γN−1 (h (λ)h (µ)Q h (cid:0) (λ)h (cid:1)(cid:16)(µ)), (cid:17)
N−2 N−1 N−1 N−2
2csinhm −
U(N)
where again, as before, Z refers to the integral in (3.11) normalized by the SW partition
Nf=1
function (3.2). In the massless limit
e
ZU(N)(m = 0) = γ γ c−1 h (c)h′ (c) h (c)h′ (c) .
Nf=1 − N−2 N−1 N−2 N−1 − N−1 N−2
Wehavethuesestablishedthesolutionofthem(cid:0)atrixintegralsintermsofthepolyno(cid:1)mialshn−1(λ)
(3.1). Therefore, in the next Section, we study and fully characterize analytically these polyno-
mials, which are Stieltjes transforms of the Stieltjes-Wigert polynomials.
4. The Stieltjes/Cauchy transform of the Stieltjes-Wigert polynomials
The Stieltjes–Wigert polynomials [19]-[22]
( 1)nqn/2+1/4 n n
(4.1) P (x;q) = − ( 1)jqj2+j/2xj, n = 0,1,...
n
(q;q) j −
n j=0(cid:20) (cid:21)q
X
are orthogonal with regards topthe weight function
(4.2) ω (x) = 1 ke−k2log2x,
SW
√π
with q = e−1/2k2. We used the standard notation
n
(q;q) = 1, (q;q) = (1 qj), n = 1,2,...
0 n
−
j=1
Y
and
n (q;q)
n
(4.3) = , 0 j n.
j (q;q) (q;q) ≤ ≤
(cid:20) (cid:21)q j n−j
There are other slightly different definitions of the polynomials, but (4.1) is the original by
Wigert [22] and the most appropriate in our context, since the corresponding weight function
is (4.2). Let us first construct the monic version of the polynomials because, as we have seen
above, those are the ones that are identified with the average of characteristic polynomials. To
obtain the monic version of (4.1) we need to divide it by γ = qn2+n+1/4/ (q;q) obtaining
n n
n
P (x) n p
(4.4) πn(x) = n = ( 1)nq−n2−n/2 e ( 1)jqj2+j/2xj.
γ − j −
n j=0(cid:20) (cid:21)q
X
e
8 MIGUELTIERZ
Notice that we will need the polynomials for q roots of unity. While they are better understood
for q < 1 (or q > 1), they also hold for q root of unity [23]. Notice for example, that the
q-binomial coefficient is a polynomial in q and hence also well-defined [23]. In general, the whole
four-parameter Askey-Wilson polynomials are studied also for q root of unity [24] (the Stieltjes-
Wigert are at the bottom of the Askey classification scheme). In terms of basic hypergeometric
functions they can be written
πn(x) = (−1)nq−n2−n21φ1(q−n,0;q,−qn+3/2x).
Because the polynomials (4.1) are actually orthonormal, it follows that the c coefficients in
k
the orthogonality relationship (2.8) for the monic polynomials (4.4) are given such that γ =
n
1/c2 = γ2. However, as pointed out before, we will use the fact that these γ prefactors will
− n − n
cancel with factors in the normalizing partition function and directly will only need to use, in
the final exepression a lower-rank Stieltjes-Wigert partition function, as we shall see below.
We need to compute now the Stieltjes transform of these Stieltjes-Wigert polynomials. Even
for classical polynomials, such as Hermite, Laguerre or Jacobi polynomials, these polynomials
are not straightforward to obtain [25, 26, 13]. In addition, in the Stieltjes-Wigert case, there are
some subtleties associated with giving a fully complete solution for complex variable z6, but for
our matrix model applications here, this will not play a role.
ToobtaintheStieltjestransformofthepolynomialsweshalluseresultsonassociatedStieltjes-
Wigert polynomials. These are defined by
P (x) P (y)
n n
(4.5) Q (x) := − ω (y)dy n 0.
n SW
x y ≥
Z −
These polynomials have been computed explicitly in [21] for a slightly different definition of the
Stieltjes-Wigert polynomials. Adapting the result there immediately gives
(4.6) Qn(x) = (−1)nqn/2+1/4 n−1q−p22 n n ( 1)jqj22+(p+21)j xp.
(q;q)  j − 
n p=0 j=p+1(cid:20) (cid:21)q
X X
The difference between thepse polynomials Q (x) and the ones we need to obtain is, manifestly,
n
the Stieltjes transform of the log-normal itself (4.2), because, it immediately follows from (4.5)
that
ω (y)dy P (y)ω (y)
SW n SW
(4.7) Q (x)= P (x) dy.
n n
x y − x y
Z − Z −
Thus, we need to compute the first Stieltjes transform in (4.7). We proceed by using Mordell’s
integral [11, 10], after a change of variables. Specifically, we need to compute
ωSW(x)dx 1 em ∞ dx −log2x
(4.8) I(λ) = = e 2g ,
λ+x √2πg c 1+xem
Z Z0 c
with λ R+ or λ C. By changing variables x/c = eµ, as(cid:0)in [10] an(cid:1)d where c is again given by
∈ ∈
(2.5), we obtain
eme−lo2gg2c ∞ eµ(1−N+Nf/2+iη) −µ2
I(λ(c,m)) = dµ e 2g,
√2πg 1+eµ+m
−∞
Z
which is the Mordell integral [10]. We write it in terms of I(ℓ,m) (1.4)
e−g2(N+Nf/2−iη)2
(4.9) I(λ(c,m)) = I(ℓ = N N /2+iη,m).
f
√2πg − −
6This is related to the undetermined nature of the associated moment problem and to the fact that, in the
Nevanlinnaparametrization of theStieltjes transform of ameasure with log-normal moments, thecorresponding
Pick function for thelog-normal weight is not explicitly known. See [21] for details.
9
Therefore, gathering all the previous results, and taking into account that we need the Stieltjes
transform of the monic orthogonal polynomials π (x) (4.4), we have that
n
(4.10) h (λ) = Q∗(λ)+π (λ)I(λ),
n − n n
where Q∗(λ) is now
n
π (x)+π (y)
(4.11) Q∗(x) := n n ω (y)dy n 0,
n x+y SW ≥
Z
which using the result (4.6) explicitly gives
n−1 n
Q∗n(λ) = (−1)nq−n2−n/2 (−1)pq−p22  nj qj22+(p+12)jλp,
p=0 j=p+1(cid:20) (cid:21)q
X X
 
and hence we have an explicit expression for (4.10). Besides, as shown in detail in [10], there
are two settings where I(ℓ,m) admits a very explicit form in terms of a finite sum expression:
(1) When the q-parameter q = exp( g) = exp( 2πi/k) is a root of unity, regardless of λ,
− −
which is the case in the physical setting of Chern-Simons-matter theory. This is one
of the main results by Mordell [11]. In this case, and for the finite order n case, the
Stieltjes-Wigert polynomials are still well defined. See below for a discussion of the large
N limit and q roots of unity.
(2) For arbitrary q and the mass being m = pg with p N [10]7. In this case, the lambda
∈
parameter, which is λ = cexp( m)= exp(g(N+N /2 p)), is an integer or half-integer
f
− −
power of the q-parameter.
We focus on the first case above, having the following explicit expression for the Mordell
integral for k > 0 [10, 12]8
(4.12) I(ℓ,m) = 2πe−iπ(ℓ+k4) e−m(ℓ+k2)+ik4mπ2G+ k,1, ℓ 1+ikm k ,
− − 2π − 2
(cid:18) (cid:19)
with
k
km k 1 i
(4.13) G+ k,1,−ℓ−1+i2π − 2 = e−2πiℓ−km 1 − k eikπ(r−ℓ−1−k2+ik2mπ)2 +i .
(cid:18) (cid:19) − r r=1 !
X
We will present the corresponding explicit expression of the polynomial (4.10) below. The
appearance of the Mordell integral is natural and will also provide consistency checks of the
generic formulas obtained here, by comparison with some of the results in [10]. Before that,
we show in what follows how to alternatively obtain the polynomials using the Mordell integral
result (4.8) and the three-term recurrence satisfied by the polynomials.
4.1. Three-term recurrence and explicit expressions. There is an alternative way to gen-
erate the polynomials h (λ) from the Mordell integral I(λ). For this, we can use a fundamental
n
result in the theory of orthogonal polynomials [25, 26]: the polynomials h (λ) satisfy the same
n
three-term recurrence relation as the Stieltjes-Wigert polynomials
(4.14) h (λ) = (λ b )h (λ) a h (λ).
n+1 n n n n−1
− −
The difference lies on the initial values, which for the h (λ) are
n
ω (x)dx
SW
(4.15) h (λ) = 1 and h (λ) = = I(λ),
−1 0
− λ+x
Z
7Weworked in [10] with η=0 so we refer hereto this setting.
8There is an analogous expression for k <0. Notice also that the prefactor of G in [10] is slightly different.
+
This is dueto thefact that therewe discussed the model with 2N hypermultiplets.
f
10 MIGUELTIERZ
and a has to be assumed non-zero and equal to a = ω(t)dt [25, 26]. The standard Stieltjes-
0 0
Wigert polynomials satisfy instead, the customary
R
(4.16) S (x) = 0 and S (x) = 1.
−1 0
Hence,theh polynomialisMordell’sintegralh (λ)= I(λ). Theuseofthethree-termrecurrence
0 0
is used for an algorithmic-computation of the Stieltjes transform of the classical orthogonal
polynomials in [25, 26]. Let us use (4.14) to generate the first polynomials from Mordell’s
integral and we shall see that we again obtain (4.10).
Theexplicitexpressionsforthecoefficientsb anda ofthemonicStieltjes-Wigertpolynomials
n n
are
b = q−2n−3/2(1+q qn+1),
n
−
a = q−4n(1 qn).
n
−
We can find a few polynomials using the recurrence, leaving for simplicity the coefficients in
generic form
h (λ)= (λ b )I(λ)+a ,
1 0 0
−
h (λ) = (λ b )[(λ b )I(λ)+a ] a I(λ) = I(λ)[(λ b )(λ b ) a ]+(λ b )a ,
2 1 0 0 1 1 0 1 1 0
− − − − − − −
h (λ) = I(λ)[(λ b )(λ b )(λ b ) a (λ b ) a (λ b )]+a a +(λ b )(λ b )a .
3 2 1 0 1 2 2 0 0 2 2 1 0
− − − − − − − − −
Notice that this construction, based solely on the recurrence, must beequivalent to the previous
one (4.10) and hence the polynomial multiplying the Mordell integral in the examples above
must be the Stieltjes-Wigert polynomial. This is indeed the case since, as noted above, they
satisfy (4.14) but with (4.16). Therefore the Stieltjes-Wigert polynomials follow from these
examplesabovebytakinga = 0andtheMordellintegralto1, whichprovesthatthepolynomial
0
multiplying Mordell’s integral is the Stieltjes-Wigert polynomial. The rest is a lower-order
polynomial,whichcorrespondstoQ∗(λ)aboveandismoredifficulttoobtainfromtherecurrence,
but was easily characterized above using the results in [21].
Let us now compare the expressions to what is known by the other methods in [10], which
analyzed the case N = 1. First we look at the Abelian cases
f
(4.17) ZU(1) = γ h (λ) = q1/2I(λ).
Nf=1 0 0
For the matrix model (3.12) if N = N = 1 then c = 1 and we have
f
γ γ q1/2
(4.18) ZU(1) = − −1 0 (h (λ)h (µ) h (λ)h (µ)) = I(λ) I(µ) ,
Nf=1 2sinhm −1 0 − 0 −1 2sinhm −
where we haveeused that γ−1 = 1 and γ0 = q1/2. The latter numerical pre(cid:0)factor is wha(cid:1)t remains
of the Stieltjes-Wigert partition function9 when N = 1 (3.3) and hence the results above are as
in [10].
Themost relevant explicit result within our setting is for theU(N) theory given by (1.3) with
N = 1, whose characterization in terms of the polynomial, namely (3.1), is now totally explicit
f
ZNUf(N=1) = N−2( 1)pq−p22 N−1 N −1 qj22+(p+21)j q−(N+12−iη+i2kπm)p
A − −  j 
N p=0 j=p+1(cid:20) (cid:21)q
X X
N−1   k
N 1 i
+BN − ( 1)jq(j−N+iη−i2kπm)j eikπ(r+N−iη−21−k2+ik2mπ)2 +i ,
j − − k
j=0 (cid:20) (cid:21)q r r=1 !
X X
9Recallthat thepartition function without thecharacteristic polynomial insertion appearsas anormalization
factor and the Cauchy transform of thepolynomial captures this normalization as well (3.1).