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 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 determinants. 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 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 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 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 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 generate 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).

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

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