Pis'ma v ZhETF, vol. 100, iss. 4, pp. 297-304 © 2014 August 25
Towards matrix model representation of HOMFLY polynomials
A. Alexandrov+*x1), A. Mironovxo 1), A. Morozovx 1), And. MorozovxVA + Freiburg Institute for Advanced Studies, University of Freiburg, 79104 Freiburg, Germany * Mathematics Institute, University of Freiburg, 79104 Freiburg, Germany x Alikhanov Institute of theoretical and Experimental Physics, 123182 Moscow, Russia °Lebedev Physics Institute, 119991 Moscow, Russia vLomonosov MSU, 119991 Moscow, Russia △Laboratory of Quantum Topology, Chelyabinsk State University, Chelyabinsk, Russia
Submitted 16 July 2014
We investigate possibilities of generalizing the TBEM (Tierz, Brini-Eynard-Marino) eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N = 2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in h = log q. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz r-function approach to knot theory, but this remains a subject for future investigation.
DOI: 10.7868/S0370274X14160115
1. Introduction. Knot polynomials [1] are examples of the Hurwitz t-function [2], a new and intriguing generalization of the free-fermion [3] KP/Toda t-functions, probably related to non-Abelian T-functions of [4]. As such they should possess a number of different realizations: as functional integrals in free field and topological theories [5], as matrix models of the ordinary and Kontsevich types [6], as various W-representations [7] a la [8-10]. While the first of these representations is well known: knot polynomials are Wilson line averages in Chern-Simons theory [11, 12] and/or results of R-matrix (modular group) evolution of conformal blocks [12-14], all the other realizations are more-or-less available only for the very specific class of torus knots and links: this story is mostly around the Rosso-Jones formula [15]. In particular, the matrix model representation is known only for the unknot (Chern-Simons partition function) [16]
and for arbitrary [m,n] torus link/knot [17, 18]:
Z(
CS
N N , 2\
J n sinh2 (ui - uj) n exp -2hj
dui (1)
^e-mail: alexandrovsash@gmail.com; mironov@itep.ru; mironov@lpi.ru; morozov@itep.ru; Andrey.Morozov@itep.ru
q\A)
N
x sinh
i<j
q=eh,A=qN
J XR(eU) x < sinh ^ U——^ sinh ^ U——^ x
N ( u2 \
TT exp--i— dui
y mnh J
(2)
(here and everywhere in this paper knot polynomials are non-normalized).
However, despite being now available only for torus knots, all such realizations should exist for an arbitrary family of knots, what is strongly supported by the overwhelming success of the evolution method [19, 20]. Still, it is a long-standing problem to generalize (2), to begin with, beyond the very special family of torus knots. This is the goal of the present letter to make a step towards this generalization. Though the final answer remains not yet reached, we realize a few essential properties of a possible final answer for the knot matrix model.
2. Summary. We are looking for an answer for the HOMFLY polynomial in the matrix model form
H L
(3)
dui
N N ( u2 \
fXB.(eU) n - uj\t) sinh (ui - uj) n exp -^
i<j i=1 \ ' /
N N ( '
/ n vLsinh (ui - uj\t) n exp - Yn dui
i<j i=1 \ J
where ~ means a factor that depends on q, N and representation R in a controllable way, 7 is a yet unknown constant, and we choose an anzatz for the measure to depend on N only through a function t(N, h). We propose to construct such generalization of the TBEM model to the twist knots in a few steps.
• First, one considers the case of N = 2: then the question is, what is the relevant integral representation of the Jones polynomial. The answer is universal: since the inverse of the integral transform
F (P)=exp x
(-12)
x i^(u) sinh ( P— ) exp (--- ) du (4)
J-^ v h y V 2yh/
(we need it in application to odd functions) is
M—) = exp(- dl) F^)
(5)
the measure in the matrix-model integral is made from the Laplace evolution of the Jones polynomial JL(p\h) for the link/knot L, rewritten in appropriate variables (p, h). After making a substitution p = u/7, it is
HL(u\h) = exp ( - — d2
(-YK «) j c( u I*)-
(6)
This, however, gives the answer without dividing by the normalization integral as in (3), which leads to the normalization factor in (3).
Usually the Jones polynomial Jr (q) is a function of q and of the spin (r — 1)/2 of representation of SU(2). Eq. (6) deals with J(p\h) = Jp/n(en) obtained from Jr (q) by the substitution r ^ p = rh, well familiar from the study of Kashaev limit [21] and Hikami invariants [22], and we denote the function of these new variables (p, h) by the calligraphic letter, J(p\h) which implies some analytic continuation in the discrete index r described below. In fact, since we will be not able to perform an exact integration in (3), we study series in h.
The next step is equally universal: the N-fold integral for HOMFLY polynomial is made from this measure by direct analogue of (2):
H L
(XR)1
(7)
where
« -I
G(eU) x
N
x n (ui — uj\t) sinh(uj — uj)
i<j
N
exp i = 1 ^
2 '
n
x 11 exp 1- yK) dui
(8)
and hL(u\t) is an odd function of the integration variable u.
However, it is of course impossible to reconstruct HOMFLY polynomial from Jones in a universal way: something in this reconstruction should depend on the type of the knot. For the two simplest families, of torus and twist knots the difference is basically in the choice of the evolution parameter
for torus links/knots t = h = log q,
1 (9)
for twist knots t = ^Nh = log Al/2
what is in perfect accordance with what we know from the study evolution method in [20].
It is an intriguing question, what happens for other families. But now the way is open to study this kind of problems - which look very promising.
3. The role of 7. What happens in the case of torus links/knots, is that there is an additional great simplification: one can choose auxiliary parameter 7 in such a way, that the result of Laplace evolution in (6) gets h-independent and actually the measure gets nearly trivial - namely, reduces to that in (2). The choice is clear from (2)
(10)
7 K"1 = -mn
(note the sign minus indicating a non-naive choice of integration contour, or analytical continuation of the answer, if one prefers, which implicit in (2)).
It is an open question for us, what is the meaning of this spectacular possibility, and if some counterpart of it exists in general. Even for the twist knots we have not yet resolved this problem.
x
Now we provide some evidence in support of above claims. We discuss the family of twist knots, following the description in [20, s. 5.2], which we assume the reader to be familiar with. In this brief presentation all the torus links/knots will be represented by a single trefoil, which is also a member of the twist family. All the claims, illustrated by this is example, are actually true for entire torus family.
4. Jones polynomials. According to general principles of the link differential calculus [23-26], the HOMFLY polynomial is decomposed into a sum of products of the quantities {Aqa} = Aqa — A-1 q—a. In particular, for the Jones polynomial there is usually a hypergeo-metric type expansion [27], which is especially simple for the twist knots [28, 20]:
^fc-i](q) = ËFk n {qr+j} =
= [r] + [r - 1][r][r + 1]F(k) + + [r - 2][r - 1][r][r + 1][r + 2]F2(k) + ... :
(11)
where the square brackets denote quantum numbers M = {qr}/{q}. Note that we shifted the labeling of representations by one to simplify the formulas below. The coefficient functions Fs(k), polynomials in A = qN = eNh and q = eR for the twist knot number k are found in Sec. 5.2 of [20]:
F(k) = qS( s —1)/2as x
^ ( — )j [sji (Aqj —1)2jk {Aq2j—1}
j=0 № - j]!
ПГ=—W}
(-k)s + O(ft).
The figure eight knot 41 corresponds to k = —1, the trefoil 31 to k = 1, unknot arises at k = 0. The Rolfsen table notation [29] is (2|k| + 2)1 for negative k and (2k + 1)2 for positive k (for 31 the Rolfsen labeling is not smooth: 32 is actually 31, since it is the only knot with plane projection having only three intersections).
Since
]J{qr+j} = E(-j
[2s + 1]!
j=o
[j]![2s + 1 - j]!
{q(2s
-1-2j)r
}
(13)
one gets for the measure (after the substitution s =
= P + j)
^L(u\h) =
1
x sinh
(2p + 1)u
sinh u
p=0 (-)
j=o
^q-(2p+1)2/27 x
[2p + 2 j + 1]! (k) [j]![2p + j + 1]! p+j
(14)
A=q2
This q-hypergeometric function is equivalent to (6) and along with (9) gives the answer for the matrix integral
(3).
5. Comments. 5.1. Jones polynomials in variables (p, h). Formulas similar to (14) are not that simple to deal with. When performing checks for the matrix model, we used the h-expansion instead. Let us see how these checks are done.
One of the possibilities is to use (14) directly. Say, in the leading order
œ(_j (2P + 2j + 1)! (_k)P+j ) j!(2p + j + 1)!( k)
j=o
(-k)p • 2Fi(p + 1,p + 3/2; 2p + 2; -4k) 2(-4k)P
V1 - 4k (1 + л/1 - 4k)
2p+1
(15)
and the sum in (14) is easily calculated:
sinh (u/y)
P(0)(u)
1 + 4k sinh2 (u/y)
(16)
Another, simpler possibility is to use formula (6). Indeed, let us make a substitution r —> p = rh:
[r]=2smhp^ {^}[г - 1] = 2sinh(p - ft), etc. (17)
{q}
(12) Then, from (11)
2 {q}J (k)(p, ft) =
= sinh p 1
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