Preprint typeset in JHEP style - HYPER VERSION
Nahm sums, quiver A-polynomials and topological
recursion
H ́elder Larragu ́ıvel
1
, Dmitry Noshchenko
1
, Mi losz Panfil
1
and Piotr Su lkowski
1
,
2
1
Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
2
Walter Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA
Abstract:
We consider a large class of
q
-series that have the structure of Nahm sums, or
equivalently motivic generating series for quivers. First, we initiate a systematic analysis
and classification of classical and quantum A-polynomials associated to such
q
-series. These
quantum quiver A-polynomials encode recursion relations satisfied by the above series, while
classical A-polynomials encode asymptotic expansion of those series. Second, we postulate
that those series, as well as their quantum quiver A-polynomials, can be reconstructed by
means of the topological recursion. There is a large class of interesting quiver A-polynomials
of genus zero, and for a number of them we confirm the above conjecture by explicit calcula-
tions. In view of recently found dualities, for an appropriate choice of quivers, these results
have a direct interpretation in topological string theory, knot theory, counting of lattice
paths, and related topics. In particular it follows, that various quantities characterizing
those systems, such as motivic Donaldson-Thomas invariants, various knot invariants, etc.,
have the structure compatible with the topological recursion and can be reconstructed by its
means.
CALT-2020-014
arXiv:2005.01776v1 [hep-th] 4 May 2020
Contents
1
. Introduction
2
2
. Classical and quantum quiver A-polynomials
6
2.1
Classical quiver A-polynomials
7
2.2
Quantum quiver A-polynomials
8
2.3
Framing and reciprocity
11
3
. Classification and examples of quiver A-polynomials
12
3.1
One-vertex quiver,
m
= 1
13
3.2
Two-vertex quivers,
m
= 2
14
3.3
Uniform quivers
24
3.4
Yet another family of quivers
27
4
. Topological recursion and quantum curves
28
5
. Nahm sums and quantum quiver A-polynomials from topological recursion
30
5.1
One-vertex quiver,
m
= 1
33
5.2
Uniform two-vertex quivers,
m
= 2
37
5.3
Uniform quivers of arbitrary size
m
39
5.4
Quiver
[
2 1
1 1
]
42
5.5
Quiver
[
3 1
1 1
]
45
5.6
Quiver
[
2 0
0
−
1
]
45
5.7
Quiver
[
2 2
2 1
]
47
5.8
Quiver
[
3 2
2 1
]
48
5.9
Quiver
[
2 0
0 2
]
49
– 1 –
1. Introduction
There is an interesting class of
q
-series that arise in various context in mathematics and
physics, whose structure is encoded in a matrix
C
with integer entries
C
i,j
, and which in
addition depend on
m
generating parameters
x
1
,...,x
m
P
C
(
x
1
,...,x
m
) =
∑
d
1
,...,d
m
≥
0
(
−
q
1
/
2
)
∑
m
i,j
=1
C
i,j
d
i
d
j
(
q
;
q
)
d
1
···
(
q
;
q
)
d
m
x
d
1
1
···
x
d
m
m
.
(1.1)
The sums in the above expression are made over nonnegative integers
d
1
,...,d
m
.
First, such series were found in certain statistical models and identified as characters of
coset conformal field theories [1, 2].
Second, the series (
1.1
) can be interpreted as Nahm sums [3, 4]. For special choices of
C
and generating parameters
x
i
such Nahm sums characterize integrable field theories, however
it is also of interest to consider them for more general
C
and
x
i
. Among others, Nahm sums
have interesting modularity properties and are closely related to knot invariants [5, 6].
Third, the matrix
C
can be interpreted as defining a symmetric quiver, with
C
i,j
=
C
j,i
encoding the number of arrows from vertex
i
to vertex
j
. In this case the above series is
identified as the motivic generating series for the quiver defined by the matrix
C
. A certain
product decomposition of this generating series into quantum dilogarithms encodes motivic
Donaldson-Thomas invariants associated to this quiver, which characterize a moduli space of
representations of this quiver [7–11].
The series (
1.1
) plays also a prominent role in the knots-quivers correspondence formu-
lated in [12, 13], and analyzed and generalized further in [14–21]. According to this cor-
respondence, the generating series of knot polynomials, as well as certain open topological
string partition functions, can be written in the form (
1.1
), for an appropriate choice of
C
.
Among others, this enables to express certain open BPS invariants, referred to as the LMOV
invariants, in terms of motivic Donaldson-Thomas invariants, thereby providing a long sought
after proof of integrality of these former invariants [22–27]. The knots-quivers correspondence
relates to each other also various other objects from the realms of knot theory, quiver repre-
sentation theory, and topological string theory, and thus it provides one of the motivations
for this work. Furthermore, as an important byproduct of the knots-quivers correspondence,
it turns out that various series of the form (
1.1
) capture counts of certain classes of lattice
paths [17].
The series (
1.1
) has an interesting asymptotic expansion, which can be determined via the
saddle point (WKB) approximation, and which is encoded in the following Nahm equations
(
−
1)
C
i,i
x
i
m
∏
j
=1
z
C
i,j
j
+
z
i
−
1 = 0
, i
= 1
,...,m.
(1.2)
It turns out that an interesting information is also encoded in the so called mixed resultant
(with respect to
z
i
) of the above equations and an additional equation
y
=
∏
m
i
=1
z
i
. The
– 2 –
resultant, denoted
A
(
x
1
,...,x
m
,y
), is an irreducible polynomial that vanishes only if Nahm
equations and
y
=
∏
i
z
i
have a common root. Furthermore, introducing non-commutative
operators
̂
z
l
that satisfy the Weyl algebra
̂
z
l
̂
x
k
=
q
δ
kl
̂
x
k
̂
z
l
, it turns out that a quantum
version Nahm equation holds, which is a statement that the following operators annihilate
the generating function (
1.1
)
(
(
−
q
1
/
2
)
C
k,k
̂
x
k
m
∏
j
̂
z
C
kj
j
+
̂
z
k
−
1
)
P
C
(
x
1
,...,x
m
) = 0
.
(1.3)
Furthermore, eliminating
̂
z
i
from (
1.3
) and introducing
̂
y
=
∏
m
i
=1
̂
z
i
, leads to the following
operator relation
̂
A
(
̂
x
1
,...,
̂
x
m
,
̂
y
)
P
C
(
x
1
,...,x
m
) = 0
,
(1.4)
where
̂
A
(
̂
x
1
,...,
̂
x
m
,
̂
y
) is a polynomial in
̂
x
i
and
̂
y
whose coefficients depend on
q
, and which
can be interpreted as the quantum version of the resultant, i.e. it reduces to the resultant
A
(
x
1
,...,x
m
,y
) in the limit
q
= 1.
In certain situations it is of interest to consider a single generating parameter
x
, by
imposing that each
x
i
variable – or, more generally, a subset of
x
i
’s – is proportional to a single
x
(and treating proportionality factors, or remaining generating parameters, as constants or
moduli). The asymptotics of (
1.1
) with such identifications can be encoded in an algebraic
curve
A
(
x,y
) = 0
,
(1.5)
which arises from the solution of the Nahm equations, or equivalently from an appropriate
specialization of
x
i
’s in the resultant
A
(
x
1
,...,x
m
,y
). In what follows we refer to this curve as
the (classical) quiver A-polynomial. Quiver A-polynomials have been already introduced and
discussed in [17,18]. For quivers corresponding to knots, quiver A-polynomials are identified as
various generalizations of A-polynomials for knots (hence the name “quiver A-polynomials”).
Similarly, for quivers arising in the reformulation of open topological string amplitudes, their
A-polynomial are identified as mirror curves [19]. However, apart from these special cases, it
is of interest – and the primary aim of this work – to analyze A-polynomials associated to an
arbitrary matrix
C
. In particular, interpreting the matrix
C
as encoding a symmetric quiver,
the corresponding A-polynomial captures certain classical (i.e. numerical) Donaldson-Thomas
invariants or some combinations thereof, as also discussed in [17, 28].
Furthermore, there exists a quantum counterpart of (classical) quiver A-polynomial, i.e.
an operator
̂
A
(
̂
x,
̂
y
), which arises from the appropriate specialization of
x
i
’s in
̂
A
(
̂
x
1
,...,
̂
x
m
,
̂
y
),
and which encodes recursion relations for the series (
1.1
) (with the same specialization)
̂
A
(
̂
x,
̂
y
)
P
C
(
x
) = 0
.
(1.6)
In this case
̂
y
̂
x
=
q
̂
x
̂
y
. Similarly as in other contexts, we call this operator the quantum quiver
A-polynomial. Its existence follows also from the fact that (
1.1
) is a generating series of
q
-
holonomic functions (i.e. the summand in (
1.1
) is expressed in terms of
q
-Pochhammers and
– 3 –
at most quadratic powers of
q
); it then follows from the theory of
q
-holonomic functions that
it satisfies a
q
-difference equation, which can be written in terms of the operator
̂
A
(
̂
x,
̂
y
) [29].
In the limit
q
= 1, quantum quiver A-polynomial reduces to the classical quiver A-polynomial.
From quantum (quiver) A-polynomials one can also extract motivic Donaldson-Thomas in-
variants, or combinations thereof, or related BPS invariants arising upon an appropriate
choice of the quiver, as discussed in [14]. Nonetheless, explicit identification of a quantum
A-polynomial for a given
C
is usually a difficult task.
There are two main goals of this work. The first one is to initiate a systematic analysis and
classification of quiver A-polynomials. Their various properties have been reported in [17],
however mainly in the context of other systems which were the main focus of that work.
In this paper we are primarily interested in quiver A-polynomials in general, irrespective of
their relation to knot theory or topological string theory. Among others, we discuss how
to determine classical and quantum A-polynomials, and we identify various classes of A-
polynomials of some specific genus (in particular genus 0).
Furthermore, the second important goal of this work is to show that the generating
series (
1.1
) and corresponding quantum A-polynomials are encoded in the classical quiver
A-polynomial, and this information can be extracted by means of the topological recursion,
or equivalently the B-model formalism.
The topological recursion is a framework which assigns an infinite family of multi-
differential forms to a given algebraic curve, which is usually called a spectral curve in this
context [30]. Topological recursion was discovered in the context of matrix models, as a refor-
mulation of loop equations [31–33]. Subsequently it was shown that it has vast applications
and enables to extract various quantities and invariants in problems, in which a certain al-
gebraic curve plays a prominent role. For example, in the form of the remodeling conjecture
it can be interpreted as the B-model formalism that enables to compute topological string
amplitudes from the mirror curve [34], it computes various enumerative invariants (e.g. vari-
ous generalizations of Hurwitz numbers) [35], expansions of colored knot polynomials [36, 37],
etc. In [38, 39] it was shown in general how to determine quantum curves perturbatively by
means of the topological recursion. In recent years various generalization of the topological
recursion have been formulated, which include its refined version [40], curves with higher ram-
ification points [41], formulation in terms of Airy structures [42, 43] and their supersymmetric
generalization [44], etc.
More precisely, in this work we conjecture that the motivic generating series (
1.1
) can be
interpreted as a wave function, which is annihilated by the quantum A-polynomial operator
that arises as a quantization of a quiver A-polynomial. By means of the topological recursion
we determine the expansion of (
1.1
) in
~
= log
q
(so that
q
=
e
~
) and confirm this statement
in various examples. The quantization of classical A-polynomials that we consider follows
general lines discussed in [38, 39]. However, note that in order to consider algebraic curves,
we must consider specialization of various generating parameters
x
i
to a single
x
, as explained
above, so in general the topological recursion does not provide information about (
1.1
) with
– 4 –
the full dependence on
x
i
, but about its specialized form that depends on a single
x
. In
consequence, for a fixed specialization, the topological recursion encodes e.g. not all motivic
Donaldson-Thomas invariants associated to the series (
1.1
), but some combinations thereof.
Nonetheless, such specializations are also of interest in various applications (e.g. in the knots-
quivers correspondence or the relation to topological string theory, as already mentioned
above), so it is of importance to consider them. Furthermore, for a given
C
one could conduct
topological recursion calculations for different specializations of
x
i
, and then reconstruct more
general information about (
1.1
).
Note that, even though the topological recursion is defined abstractly for spectral curves
of arbitrary genus, in practice it can be computed for curves of genus 0 and 1. For this
reason, it is difficult, or just impossible, to test it explicitly in many situations of interest,
for example for knots whose A-polynomials are of genus larger than 1, or for toric Calabi-
Yau manifolds whose mirror curves have genus larger than 1; in those cases the number of
interesting, directly computable examples is limited. Happily, for
q
-series of the form (
1.1
)
we have much larger testing ground – there are many matrices
C
, of various size, for which
corresponding A-polynomials are of genus 0 or 1, and thus our conjecture can be explicitly
verified. In this work we conduct such tests and confirm the above conjecture for a number
of representative quivers whose A-polynomials have genus 0 (apart form one subtlety for a
class of “diagonal” quivers).
Furthermore, also note that in some special cases our conjecture follows from combining
various earlier results. For example, it is shown in [19] that open topological string am-
plitudes for toric “strip” geometries take form of motivic generating functions for certain
quivers. On the other hand, according to the remodeling conjecture [34] these amplitudes are
reproduced by the topological recursion for mirror curves, and it follows from [19] that in this
case mirror curves can be identified with quiver A-polynomials. A prototype example in this
case are topological string amplitudes and the quantum curve for (framed)
C
3
, for which the
topological recursion formalism is discussed e.g. in [38, 45], and for which the corresponding
quiver consists of one vertex with a number of loops. Combining these facts we conclude,
that the topological recursion for mirror curves for strip geometries reproduces motivic gen-
erating functions for quivers corresponding to such geometries. The results of this paper can
be regarded as a generalization of this observation to much larger family of
q
-series (
1.1
),
whose structure is encoded in arbitrary matrices
C
. This also means that various quantities
associated to those
q
-series – such as Donaldson-Thomas invariants, various knot invariants
that arise via the knots-quivers correspondence upon an appropriate choice of
C
, certain
amplitudes in integrable theories, etc. – have the structure compatible with the topological
recursion, and can be computed using this recursion. We believe this is an important con-
ceptual statement, which should be of advantage in further analysis of the above mentioned
quantities, and which shows yet another powerful application of the topological recursion.
In particular, we stress that some of our results explicitly illustrate the conjecture that
colored knot invariants are reconstructed by the topological recursion. For example, the
generating function (
1.1
) and A-polynomials for the quiver which has one vertex and
f
loops
– 5 –
(the same one as mentioned above, encoding topological string amplitudes for
C
3
), capture
extremal invariants of the unknot in framing
f
. On the other hand, the quiver
[
2 1
1 1
]
encodes
full colored HOMFLY polynomials for the unknot in framing
f
= 1 [13]. More complicated
examples involve quivers
[
2 0
0 0
]
and
[
2 1
1 0
]
(and their framed relatives), whose A-polynomials also
have genus 0, and which capture colored extremal invariants of left-handed and right-handed
trefoil knot, and (in some specific framing) also encode counting of certain Duchon paths [17].
In what follows, among others, we discuss properties of these particular quivers, and show
how they are captured by the topological recursion. This supports the conjecture relating
knot invariants and topological recursion [36–38], and also illustrates new links between path
counting problems and topological recursion. We conjecture that analogous relations to the
topological recursion hold for other other types of Duchon paths, at least those which are
related (as discussed in [17]) to other quivers.
The plan of this paper is as follows. In section
2
we introduce classical and quantum
A-polynomials for quivers and summarize their general properties. In section
3
we initiate
systematic classification of quiver A-polynomials; among others we identify various families of
quivers of some specific genus (in particular genus 0) and derive their classical and quantum
A-polynomials. In section
4
we concisely review the formalism of the topological recursion and
its relations to quantum curves. In section
5
we conduct explicit tests of our conjecture, and
confirm (apart from one subtlety for “diagonal” quivers mentioned above) that (specializations
of) the series (
1.1
) and corresponding quantum curves, associated to various representative
quivers identified in section
3
, are determined by the topological recursion.
2. Classical and quantum quiver A-polynomials
One of the main characters in this work is the following series
P
C
(
x
1
,...,x
m
) =
∑
d
1
,...,d
m
≥
0
(
−
q
1
/
2
)
∑
m
i,j
=1
C
i,j
d
i
d
j
(
q
;
q
)
d
1
···
(
q
;
q
)
d
m
x
d
1
1
···
x
d
m
m
,
(2.1)
where
C
i,j
are entries of a fixed symmetric matrix
C
of size
m
, and
x
1
,...,x
m
are interpreted
as generating parameters, and the
q
-Pochhammer symbol is (
x
;
q
)
d
=
∏
d
−
1
i
=0
(1
−
xq
i
). We also
introduce the following notation for coefficients in this series
P
C
(
x
1
,...,x
m
) =
∑
d
1
,...,d
m
≥
0
P
d
1
,...,d
m
x
d
1
1
···
x
d
m
m
,
P
d
1
,...,d
m
=
(
−
q
1
/
2
)
∑
m
i,j
=1
C
i,j
d
i
d
j
(
q
;
q
)
d
1
···
(
q
;
q
)
d
m
.
(2.2)
Furthermore, for
q
=
e
~
, such series has the following asymptotic expansion
P
C
(
x
1
,...,x
m
) = exp
(
1
~
S
0
(
x
) +
S
1
(
x
) +
~
S
2
(
x
) +
...
)
.
(2.3)
As discussed in the introduction, such series play an important role in physics and math-
ematics. On one hand such series are referred to as Nahm sums, and for specific values of
– 6 –
C
and
x
i
arise as characters of integrable theories. On the other hand, they arise as mo-
tivic generating series associated to a symmetric quiver determined by the matrix
C
(so that
C
i,j
=
C
j,i
is the number of arrows from vertex
i
to vertex
j
); a product decomposition of
such a series into quantum dilogarithms encodes motivic Donaldson-Thomas invariants for a
quiver in question. For specific choices of
C
and identification of
x
i
with a single parameter
x
, such series encode knot invariants and topological string amplitudes. In this section we
discuss how to derive classical and quantum quiver A-polynomials associated to series (
2.1
)
and summarize some of their properties.
2.1 Classical quiver A-polynomials
An interesting information is encoded in the asymptotics of the series (
2.1
), in the limit
~
→
0.
It can be obtained by replacing the sums over
d
i
by integrals over
z
i
=
e
~
d
i
P
C
(
x
1
,...,x
n
)
'
∫
dz
1
···
dz
m
z
1
···
z
m
exp
(
1
~
S
0
(
x,z
) +
...
)
,
(2.4)
with the leading term
S
0
(
x,z
) =
1
2
m
∑
i,j
=1
C
i,j
log
z
i
log
z
j
+
m
∑
i
=1
(
log
z
i
log
x
i
+ Li
2
(
z
i
)
−
Li
2
(1) +
iπC
i,i
log
z
i
)
,
(2.5)
whose form follows from the relation (
x
;
q
)
d
'
e
1
~
(Li
2
(
x
)
−
Li
2
(
q
d
x
))+
...
, and where the dilogarithm
function is Li
2
(
x
) =
∑
∞
i
=1
x
i
i
2
. Exponentiating the equation for stationary points
∂
z
i
S
0
(
x,z
) =
0 leads to a set of Nahm equations
H
i
≡
(
−
1)
C
i,i
x
i
m
∏
j
=1
z
C
i,j
j
+
z
i
−
1 = 0
, i
= 1
,...,m.
(2.6)
We denote the solution of these equations by
z
= (
z
i
)
i
=1
,...,m
, and we also introduce
y
i
=
y
i
(
x
1
,...,x
m
) =
e
x
i
∂
x
i
S
0
(
x,
z
)
=
z
i
(
x
1
,...,x
m
)
,
(2.7)
as well as
y
(
x
1
,...,x
m
) =
e
∑
m
i
=1
∂
x
i
S
0
(
x,
z
)
=
m
∏
i
=1
y
i
(
x
1
,...,x
m
) =
m
∏
i
=1
z
i
.
(2.8)
The crucial role in this work is played by the quiver A-polynomial, whose form follows
from the above equations. One way to introduce it is to consider first the following mixed
resultant
A
(
x
1
,...,x
m
,y
) = res
z
1
,...,z
m
{
y
−
m
∏
j
=1
z
j
= 0
, H
1
= 0
,..., H
m
= 0
}
.
(2.9)
For a proper definition of the mixed resultant see [46]; essentially, given a collection of Laurent
polynomials
f
0
,...,f
n
, its mixed resultant is the unique (up to a sign) irreducible polynomial
– 7 –
in coefficients of
f
i
, which vanishes if and only if
f
0
,...,f
n
have a common root. The mixed
resultant, as a polynomial in
y
and
x
i
, has integer coefficients, it is irreducible over integers,
and it is tempered. In various situations it is of interest to identify all (or some)
x
i
in the quiver
generating function
P
C
(
x
1
,...,x
m
) and in
A
(
x
1
,...,x
m
,y
) with a single variable
x
(possibly
up to some proportionality factors). For example, in the knots-quivers correspondence, all
x
i
’s
are proportional to such a single
x
[12, 13], while in the relation to topological strings on strip
geometries only
x
1
is identified with
x
(and other
x
i
’s are identified with K ̈ahler moduli) [19].
Upon such an identification, we denote by
P
C
(
x
) the quiver generating function with identified
parameters, while the above resultant reduces to a polynomial in
x
and
y
(possibly depending
on some extra parameters), which we refer to as the quiver A-polynomial, and whose zeros
define a complex curve in
C
∗
×
C
∗
A
(
x,y
) = 0
.
(2.10)
This equation can be also obtained as a direct solution of the set of equations (
2.6
) and (
2.8
),
with appropriate identification of
x
i
’s. Moreover, the leading term
S
0
in the expansion (
2.3
),
when various
x
i
are identified with
x
, is given by
S
0
=
∫
log
y
dx
x
,
(2.11)
where
y
=
y
(
x
) is the solution of the equation (
2.10
). This solution can be equivalently
obtained as the limit
y
=
y
(
x
) = lim
q
→
1
P
C
(
qx
)
P
C
(
x
)
.
(2.12)
Higher order terms
S
k
(
x
) in (
2.3
) can be determined by WKB method.
One of the main aims of this paper is the analysis of such quiver A-polynomials.
2.2 Quantum quiver A-polynomials
Information about the generating series (
2.1
) is equivalently encoded in recursion relations
it satisfies. At the same time, these recursion relations can be thought of as quantum gen-
eralizations of classical equations from the previous section. Such recursion relations can be
determined in a number of ways, as we summarize in this section. In the last section of this
paper we show, how such a quantization arises from the formalism of topological recursion.
One way to determine recursion relations satisfied by (
2.1
) is to consider a shift of one of
the subscripts in (
2.2
). Such a shift can be written as
P
d
1
,...,d
k
+1
,...,d
m
=
(
−
q
1
/
2
)
∑
m
i,j
=1
C
ij
(
d
i
+
δ
ik
)(
d
j
+
δ
jk
)
(
q
;
q
)
d
1
···
(
q
;
q
)
d
k
+1
···
(
q
;
q
)
d
m
=
(
−
q
1
/
2
)
C
kk
+2
∑
m
i
C
ki
d
i
1
−
q
d
k
+1
P
d
1
,...,d
m
.
(2.13)
Introducing operators
̂
z
l
such that
̂
z
l
̂
x
k
=
q
δ
kl
̂
x
k
̂
z
l
,
(2.14)
– 8 –
we can write difference equations for the generating series of the above coefficients as follows
∑
(
d
1
,...,d
m
)
≥
0
P
d
1
,...,d
k
+1
,...,d
m
m
∏
i
x
d
i
+
δ
ki
i
=
(
(1
−
̂
z
k
)
−
1
(
−
q
1
/
2
)
C
kk
̂
x
k
m
∏
j
̂
z
C
kj
j
)
·
P
(
x
1
,...,x
m
)
.
(2.15)
Multiplying both sides by (1
−
̂
z
k
) leads to the following set of equations
̂
H
k
P
(
x
1
,...,x
m
) = 0
,
for
̂
H
k
= (
−
q
1
/
2
)
C
kk
̂
x
k
m
∏
j
̂
z
C
kj
j
+
̂
z
k
−
1
.
(2.16)
These are quantum counterparts of the classical equations (
2.6
).
Furthermore, analogously as in the classical case, one can identify various
x
i
with a single
variable
x
. It is then natural to introduce the corresponding shift operator; in particular, for
all
x
i
=
x
, the shift operator takes form
̂
y
=
m
∏
k
=1
̂
z
k
,
(2.17)
and it satisfies
̂
y
̂
x
=
q
̂
x
̂
y
. One can then eliminate appropriate
̂
z
k
’s from (
2.16
), which leads
to a single difference equation for the generating function (
2.1
)
̂
A
(
̂
x
1
,...,
̂
x
m
,
̂
y
)
P
C
(
x
1
,...,x
m
) = 0
,
(2.18)
which is a quantum counterpart of the resultant (
2.9
). Furthermore, appropriately identifying
x
k
’s (e.g. as
x
i
=
c
i
x
, or simply
x
i
=
x
), so that a single variable
x
remains, we obtain a
difference equation of the form
̂
A
(
̂
x,
̂
y
)
P
C
(
x
) = 0
.
(2.19)
We refer to
̂
A
(
̂
x,
̂
y
) as the quantum quiver A-polynomial. The above equation can be also
expanded in
~
, leading to a set of differential equations for coefficients
S
k
(
x
) introduced in
(
2.3
). On the other hand, in the classical
q
→
1 limit,
̂
A
(
̂
x,
̂
y
) reduces to the classical quiver
A-polynomial (
2.10
), which is determined just by
S
0
(
x
).
Another strategy to determine quantum quiver A-polynomial is to take advantage of
computer programs for symbolic computations, such as qZeil and related ones [47]. For
brevity, let us consider a quiver with 2 vertices, and suppose that we identify the generating
parameters as
x
1
=
x
2
=
x
. The generating function (
2.1
) in this case can be written as
P
C
(
x
) =
∞
∑
r
=0
P
r
x
r
,
(2.20)
where
P
r
=
∑
d
1
+
d
2
=
r
(
−
q
1
/
2
)
∑
2
i,j
=1
C
i,j
d
i
d
j
(
q
;
q
)
d
1
(
q
;
q
)
d
2
=
=
r
∑
d
1
=0
(
−
1)
C
1
,
1
d
1
+
C
2
,
2
d
2
q
1
2
(
C
1
,
1
d
2
1
+2
C
1
,
2
d
1
(
r
−
d
1
)+
C
2
,
2
(
r
−
d
1
)
2
)
(
q
;
q
)
d
1
(
q
;
q
)
r
−
d
1
.
(2.21)
– 9 –
Importantly, a recursion relation satisfied by
P
C
(
x
) is equivalent to a recursion relation sat-
isfied by
P
r
. Moreover,
P
r
given above are
q
-holonomic functions, so it is guaranteed that
they satisfy recursion relations of finite order in operators
̂
M
and
̂
L
, which act as
̂
MP
r
=
q
r
P
r
,
̂
LP
r
=
P
r
+1
,
(2.22)
and which satisfy the relation
̂
L
̂
M
=
q
̂
M
̂
L
. Such recursion relations for
P
r
can be found
e.g. by the above metioned qZeil environment [47]. In what follows we will determine such
recursions for various matrices
C
using this tool. Furthermore, the operators
̂
M
and
̂
L
are
dual to
̂
x
and
̂
y
, i.e.
̂
MP
C
(
x
) =
∞
∑
r
=0
P
r
(
qx
)
r
=
P
C
(
qx
)
,
̂
LP
C
(
x
) =
∞
∑
r
=0
P
r
+1
x
r
=
1
x
(
P
C
(
x
)
−
P
0
)
.
(2.23)
Therefore, if
̃
A
(
̂
M,
̂
L
) is an operator that encodes recursion relations for
P
r
, i.e.
̃
A
(
̂
M,
̂
L
)
P
r
=
0, then the recursion relations for the generating series
P
C
(
x
) are captured by the operator
̂
A
(
̂
x,
̂
y
) =
̃
A
(
̂
y,
̂
x
−
1
)
,
(2.24)
see also [28]. However, because of the shift in the action of
̂
L
in (
2.23
), the recursion for
P
(
x
)
takes a nice form
̂
A
(
̂
x,
̂
y
)
P
C
(
x
) = 0
(2.25)
only if boundary conditions related to this shift get compensated. If this is not the case,
the recursion may get modified by some extra terms, which can be detected explicitly by
verifying the result of
A
(
̂
x,
̂
y
)
P
(
x
) for first few terms in
x
-expansion of
P
(
x
). In case of
quiver generating functions, we have verified for a large set of matrices
C
of size 2, that such
boundary terms indeed vanish and the equation (
2.25
) holds without any extra terms.
We also state some basic properties of quantum A-polynomials. First, if
ψ
(
x
) is annihi-
lated by quantum A-polynomial,
̂
A
(
̂
x,
̂
y
)
ψ
(
x
) = 0, then
̂
A
(
q
b
̂
x,q
−
a
̂
y
)
̃
ψ
(
x
) = 0
,
for
̃
ψ
(
x
) =
x
a
ψ
(
q
b
x
)
.
(2.26)
Second, rescaling the argument of the wave-function
ψ
(
x
) = exp(
~
−
1
S
0
(
x
) +
S
1
(
x
) +
...
)
(which is annihilated by
̂
A
(
̂
x,
̂
y
)) by
q
, results in shifting
S
1
(
x
) by log
y
:
ψ
(
qx
) = exp
(
1
~
S
0
(
x
) +
(
S
1
(
x
) + log
y
)
+
...
)
,
(2.27)
and of course also in appropriate shifting of subsequent
S
k
with
k >
1.
– 10 –
2.3 Framing and reciprocity
In what follows we consider various operations on the generating function (
2.1
), which do
not change genus of corresponding A-polynomials. The first such operation is a shift of each
element of the matrix
C
by the same number
C
7→
C
f
=
C
+
f f
···
f f
···
.
.
.
.
.
.
.
.
.
(2.28)
We refer to this operation as (a change of)
framing
, because it indeed corresponds to a change
of framing in case the generating function (
2.1
) for appropriately chosen
C
encodes knot
polynomials or topological string amplitudes [17]. Note that the operation (
2.28
) modifies
the summand in (
2.1
) by
(
−
q
1
/
2
)
f
∑
i,j
d
i
d
j
= (
−
q
1
/
2
)
f
(
d
1
+
...d
m
)
2
= (
−
q
1
/
2
)
fr
2
(2.29)
for
r
=
d
1
+
...
+
d
m
. In particular, if in the series (
2.1
) we identify
x
i
=
±
x
and write this
series as
P
C
(
x
) =
∑
r
P
r
x
r
, then the change of framing (
2.28
) is equivalent to multiplying the
coefficient
P
r
by (
−
q
1
/
2
)
fr
2
. It also follows that if such
P
C
(
x
) satisfies a difference equation
̂
A
(
̂
x,
̂
y
)
P
C
(
x
) = 0, then the framed generating function
P
C
f
(
x
) satisfies a difference equation
with
̂
x
replaced by
̂
x
(
−
̂
y
)
f
, i.e.
̂
A
(
̂
x
(
−
̂
y
)
f
,
̂
y
)
P
C
f
(
x
) = 0
.
(2.30)
Clearly, after the framing operation, the corresponding classical curve takes form
A
(
x
(
−
y
)
f
,y
) =
0, where
A
(
x,y
) = 0 is the curve corresponding to the original generating function. An impor-
tant statement is that such an operation does not change the genus of the algebraic curve. In
particular, for curves of genus 0 that have a rational parametrization (
x
(
t
)
,y
(
t
)), the framed
curve has also a rational parametrization (
x
(
t
)(
−
y
(
t
))
f
,y
(
t
)) and thus its genus is also 0.
Another operation that we also consider is
C
7→
1
m
×
m
−
C
=
1 0
···
0 1
···
.
.
.
.
.
.
.
.
.
−
C,
(2.31)
where 1
m
×
m
≡
1 is the identity matrix of size
m
. We call the quiver characterized by the
resulting matrix 1
−
C
as the reciprocal quiver. In particular, in the knots-quivers correspon-
dence, for a matrix
C
corresponding to some knot, the matrix 1
−
C
corresponds to the mirror
image of this knot. Note that the form of quantum and classical A-polynomial associated to
the matrix (1
−
C
) can be deduced from A-polynomials associated to
C
. To show that, we
first denote by
̂
H
1
−
C
i
≡
̂
H
1
−
C
i
(
x
j
,
̂
z
j
,q
) the operator (
2.16
) associated to the matrix (1
−
C
),
– 11 –
and rewrite it as
̂
H
1
−
C
i
(
x
j
,
̂
z
j
,q
) = (
−
q
1
/
2
)
δ
ii
−
C
ii
̂
x
i
m
∏
j
̂
z
δ
ij
−
C
ij
j
+
̂
z
i
−
1 =
= (
−
̂
z
i
)
(
(
−
q
−
1
/
2
)
C
ii
(
q
−
1
/
2
̂
x
i
)
m
∏
j
̂
z
−
C
ij
j
−
1 +
̂
z
−
1
i
)
=
= (
−
̂
z
i
)
̂
H
C
i
(
q
−
1
/
2
x
j
,
̂
z
−
1
j
,q
−
1
)
,
(2.32)
where
̂
H
C
i
(
q
−
1
/
2
x
j
,
̂
z
−
1
j
,q
−
1
) is the operator (
2.16
) associated to
C
, but with modified ar-
guments. Furthermore, we can rewrite the defining equation of
̂
y
in (
2.17
), acting on the
generating function
P
1
−
C
(
x
1
,...,x
m
) associated to (1
−
C
), as follows
0 = (
̂
y
−
m
∏
j
=1
̂
z
α
j
j
)
·
P
1
−
C
(
x
1
,...,x
m
) =
=
(
−
̂
y
m
∏
j
=1
̂
z
α
j
j
)(
̂
y
−
1
−
m
∏
j
=1
̂
z
−
α
j
j
)
P
1
−
C
(
x
1
,...,x
m
)
.
(2.33)
The operator (
2.32
) and the second line of (
2.33
) (ignoring an irrelevant first factor (
−
̂
y
∏
j
̂
z
α
j
j
))
are analogous to the pair of equations (
2.16
) and (
2.17
), and (via modification of arguments)
relate operators associated to the matrix (1
−
C
) and those associated to
C
. Eliminating
̂
z
−
1
j
from the above equations implies, that if
̂
A
C
(
x
j
,
̂
y,q
) is the operator that annihilates the gen-
erating function (
2.1
) associated to
C
, then the same operator but with modified arguments
is associated to 1
−
C
̂
A
1
−
C
(
x
j
,
̂
y,q
) =
̂
A
C
(
q
−
1
/
2
x
j
,
̂
y
−
1
,q
−
1
)
,
(2.34)
and therefore the above operator annihilates the generating function
P
1
−
C
(
x
1
,...,x
m
)
̂
A
C
(
q
−
1
/
2
x
j
,
̂
y
−
1
,q
−
1
)
P
1
−
C
(
x
1
,...,x
m
) = 0
.
(2.35)
In consequence, analogous relation between quantum A-polynomials holds after identifying
x
j
with a single
x
,
̂
A
1
−
C
(
x,
̂
y,q
) =
̂
A
C
(
q
−
1
/
2
x,
̂
y
−
1
,q
−
1
), while classical resultant and A-
polynomial associated to 1
−
C
are related to those associated to
C
by inverting
y
A
1
−
C
(
x,y
) =
A
C
(
x,y
−
1
)
.
(2.36)
3. Classification and examples of quiver A-polynomials
A basic characteristic of an algebraic curve is its genus. In particular it plays a crucial role
from the perspective of the topological recursion – in practice, explicit topological recursion
calculations can be conducted for spectral curves of genus 0 or 1. It is therefore of interest
to determine the genus a quiver A-polynomial associated to a matrix
C
, and identify quiver
A-polynomials whose genus is 0 or 1. In this section we make the first step towards such a
– 12 –
classification. After a quick summary in section
3.1
of the simplest case of
C
of size 1, in
section
3.2
we consider arbitrary symmetric matrices
C
of size 2, determine genus of associated
A-polynomials, and identify and present in more detail all families of A-polynomials of genus
0. Finally, in sections
3.3
and
3.4
we identify two families of matrices
C
of arbitrary size,
whose A-polynomials have genus 0. Note that when discussing particular examples we focus
on analysis of A-polynomials that we call admissible, i.e. they are irreducible and have a
maximal number of ramification points (which cannot be increased by changing the framing).
In section
5
we will show that the topological recursion reproduces quiver generating series
and corresponding quantum curves for those examples.
3.1 One-vertex quiver,
m
= 1
We start the analysis from quivers that consist of one (i.e.
m
= 1) vertex and
f
loops, so
that the quiver matrix reads
C
= [
f
] with
f
∈
Z
. The quiver generating function (
2.1
) in this
case takes form
P
C
(
x
) =
∑
d
≥
0
(
−
q
1
/
2
)
fd
2
(
q
;
q
)
d
x
d
=
∞
∑
d
=0
(
−
1)
fd
q
f
2
d
2
(
q,q
)
d
x
d
≡
∞
∑
d
=0
p
d
x
d
,
(3.1)
and it can be regarded as the framed version (
2.28
) of the
C
= [0] case. Note that
p
d
p
d
−
1
=
(
−
1)
f
q
f
2
(2
d
−
1)
1
−
q
d
⇒
p
d
−
q
d
p
d
−
(
−
1)
f
q
f
2
(2
d
−
1)
p
d
−
1
= 0
,
(3.2)
so multiplying the last equation by
x
d
and summing over
d
we get the expression which can
be written as
̂
A
(
̂
x,
̂
y
)
P
C
(
x
)
∼
0, where
∼
means equality possibly up to some initial terms,
and where the difference operator has form
̂
A
(
̂
x,
̂
y
) = (
−
1)
f
q
f/
2
̂
x
̂
y
f
+
̂
y
−
1
.
(3.3)
We verify that the quantum A-polynomial is satisfied without any extra terms, so that
̂
A
(
̂
x,
̂
y
)
P
C
(
x
) = 0
.
(3.4)
On the other hand, the saddle point approximation of
P
C
(
x
) yields
S
0
(
x
) =
−
Li
2
(1
−
y
)
−
f
2
(log
y
)
2
,
S
1
(
x
) =
−
1
2
log
(1
−
y
)
y
x
−
1
2
log
(
−
dx
dy
)
,
(3.5)
where
y
(
x
) is a solution to the classical A-polynomial
A
(
x,y
) =
x
(
−
y
)
f
+
y
−
1 = 0
,
(3.6)
– 13 –
which is indeed
q
= 1 limit of (
3.3
). For each
f
this curve has the following rational
parametrization
x
(
t
) = (
−
1)
f
1
−
t
t
f
, y
(
t
) =
t,
(3.7)
which immediately implies that it has genus 0. Moreover, solving the equation (
3.6
) for
y
=
y
(
x
) we find
y
(
x
) = 1
−
(
−
1)
f
x
+
fx
2
−
(
−
1)
f
3
f
−
1
2
x
3
+
8
f
2
−
6
f
+ 1
3
x
4
+
−
(
−
1)
f
f
(125
f
3
−
150
f
2
+ 55
f
−
6)
24
x
5
+
O
(
x
6
)
.
(3.8)
We recall that the generating function (
3.1
) also arises as the generating function of extremal
HOMFLY polynomials for the unknot [13], and as the partition function for a brane in
C
3
[17].
3.2 Two-vertex quivers,
m
= 2
Now we consider symmetric matrices
C
of size 2, with the aim of determining the genus
of their A-polynomials, and in particular identifying all A-polynomials of genus 0. These
matrices depend on 3 parameters; however, recalling that the framing operation (
2.28
) does
not change the genus, we can use it to remove off-diagonal terms. Thus it is sufficient to focus
on diagonal matrices
[
a
0
0
b
]
(3.9)
Moreover, let us first consider the specialization of generating parameters
x
1
=
c
1
x
and
x
2
=
c
2
x
with generic
c
1
,c
2
∈
C
, and look for A-polynomials which do not factorize (in
what follows we are primarily interested in specialization
c
1
=
±
c
2
= 1, and then for some
C
the A-polynomial may be reducible). We determine such A-polynomials along the lines
presented in section
2.1
, or equivalently as
q
= 1 limit of quantum A-polynomials, determined
as discussed in section
2.2
. From this analysis we find, first of all, that A-polynomials for a
fixed genus
g >
0 arise for a finite number of pairs (
a,b
), as summarized below.
On the other hand, we find a few infinite families of A-polynomials of genus 0, which
arise for pairs (
a,b
) of the form (
a,
0)
,
(
a,
1)
,
(
a,a
) and (
a,
1
−
a
), for
a
∈
Z
. In fact, the
pairs (
a,
0) and (
a,
1) are related by the reciprocity operation (
2.31
) that does not change
the genus of A-polynomial; however it is of advanatage to write down explicitly results for all
those cases. For all such pairs (
a,b
), the corresponding A-polynomials factorize only when
x
1
=
x
2
(i.e.
c
1
=
c
2
= 1) for the family (
a,
1
−
a
), and for (
a,a
) for
a
6
= 0
,
1; in all other
cases A-polynomials are irreducible (for all
c
1
and
c
2
). Let us now provide explicit results
for quantum and classical curves, as well as their parametrizations, for all these families that
yield curves of genus 0.
– 14 –
Trivial quiver
[
0 0
0 0
]
To start with, consider the simplest uniform case
C
=
[
0 0
0 0
]
(3.10)
The operators that annihilates the generating function (
2.1
), and its specializations (
x
1
=
c
1
x,x
2
=
c
2
x
) and
x
1
=
x
=
±
x
2
, take form
̂
A
(
x
1
,x
2
,
̂
y
) = (1
−
x
1
)(1
−
x
2
)
−
̂
y,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) = (1
−
xc
1
)(1
−
xc
2
)
−
̂
y,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) = (1
−
x
)
2
−
̂
y,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) = (1
−
x
)(1 +
x
)
−
̂
y.
(3.11)
The resultant and corresponding classical A-polynomials take form
A
(
x
1
,x
2
,y
) = (1
−
x
1
)(1
−
x
2
)
−
y,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) = (1
−
xc
1
)(1
−
xc
2
)
−
y,
A
(
x
1
=
x,x
2
=
x,y
) = (1
−
x
)(1
−
x
)
−
y,
A
(
x
1
=
x,x
2
=
−
x,y
) = (1
−
x
)(1 +
x
)
−
y,
(3.12)
and their parametrizations read
x
(
t
) =
t, y
(
t
) = (1
−
c
1
t
)(1
−
c
2
t
)
,
x
(
t
) =
t, y
(
t
) = (1
−
t
)
2
,
for
c
1
=
c
2
= 1
,
x
(
t
) =
t, y
(
t
) = (1
−
t
)(1 +
t
)
,
for
c
1
=
−
c
2
= 1
.
(3.13)
Family
[
a
0
0 0
]
The above uniform quiver is a special example of the more general family of the form
C
=
[
a
0
0 0
]
(3.14)
for
a
∈
Z
. Recall that the data associated to such a quiver with
a
= 1 encodes colored
HOMFLY polynomials for the unknot [13], while for
a
= 2 it captures extremal invariants of
the trefoil knot and certain Duchon paths [17].
In general, the form of quantum A-polynomial in this case depends on whether
a
is
positive or not. For
a
≥
1 we find
̂
A
(
x
1
,x
2
,
̂
y
) = (
−
q
1
/
2
)
a
x
1
̂
y
a
+ (
qx
2
;
q
)
a
−
1
̂
y
−
(
x
2
;
q
)
a
,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) = (
−
q
1
/
2
)
a
xc
1
̂
y
a
+ (
qxc
2
;
q
)
a
−
1
̂
y
−
(
xc
2
;
q
)
a
,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) = (
−
q
1
/
2
)
a
x
̂
y
a
+ (
qx
;
q
)
a
−
1
̂
y
−
(
x
;
q
)
a
,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) = (
−
q
1
/
2
)
a
x
̂
y
a
+ (
−
qx
;
q
)
a
−
1
̂
y
−
(
−
x
;
q
)
a
,
(3.15)
– 15 –
where (
x,q
)
a
=
∏
a
−
1
i
=0
(1
−
xq
i
), and for
a
≤
0
̂
A
(
x
1
,x
2
,
̂
y
) = (
−
q
1
/
2
)
−
a
x
1
(
x
2
;
q
)
1
−
a
+ (
q
−
a
x
2
−
1 +
̂
y
)
̂
y
−
a
,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) = (
−
q
1
/
2
)
−
a
xc
1
(
xc
2
;
q
)
1
−
a
+ (
q
−
a
xc
2
−
1 +
̂
y
)
̂
y
−
a
,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) = (
−
q
1
/
2
)
−
a
x
(
x
;
q
)
1
−
a
+ (
q
−
a
x
−
1 +
̂
y
)
̂
y
−
a
,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) = (
−
q
1
/
2
)
−
a
x
(
x
;
q
)
1
−
a
+ (
−
q
−
a
x
−
1 +
̂
y
)
̂
y
−
a
.
(3.16)
Corresponding classical A-polynomials, for
a
≥
1, are
A
(
x
1
,x
2
,
̂
y
) = (
−
1)
a
x
1
y
a
+ (1
−
x
2
)
a
−
1
y
−
(1
−
x
2
)
a
,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) = (
−
1)
a
xc
1
y
a
+ (1
−
xc
2
)
a
−
1
y
−
(1
−
xc
2
)
a
,
A
(
x
1
=
x,x
2
=
x,y
) = (
−
1)
a
xy
a
+ (1
−
x
)
a
−
1
y
−
(1
−
x
)
a
,
A
(
x
1
=
x,x
2
=
−
x,y
) = (
−
1)
a
xy
a
+ (1 +
x
;
q
)
a
−
1
y
−
(1 +
x
)
a
,
(3.17)
and for
a
≤
0
A
(
x
1
,x
2
,y
) = (
−
1)
−
a
x
1
(1
−
x
2
)
1
−
a
+ (
x
2
−
1 +
y
)
y
−
a
,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) = (
−
1)
−
a
xc
1
(1
−
xc
2
)
1
−
a
+ (
xc
2
−
1 +
y
)
y
−
a
,
A
(
x
1
=
x,x
2
=
x,y
) = (
−
1)
−
a
x
(1
−
x
)
1
−
a
+ (
x
−
1 +
y
)
y
−
a
,
A
(
x
1
=
x,x
2
=
−
x,y
) = (
−
1)
−
a
x
(1 +
x
)
1
−
a
+ (
−
x
−
1 +
y
)
y
−
a
.
(3.18)
The parametrizations of classical curves are however universal for all
a
∈
Z
and take form
x
(
t
) =
c
a
−
2
1
(
t
+ (
−
1)
a
c
1
)
t
a
, y
(
t
) =
t
a
−
c
a
−
2
1
c
2
t
+ (
−
1)
a
+1
c
a
−
1
1
c
2
(
−
1)
a
+1
c
1
t
a
−
1
,
x
(
t
) =
(
t
+ (
−
1)
a
)
t
a
, y
(
t
) =
t
a
−
t
+ (
−
1)
a
+1
(
−
1)
a
+1
t
a
−
1
,
for
c
1
=
c
2
= 1
,
x
(
t
) =
(
t
+ (
−
1)
a
)
t
a
, y
(
t
) =
t
a
+
t
+ (
−
1)
a
(
−
1)
a
+1
t
a
−
1
,
for
c
1
=
−
c
2
= 1
.
(3.19)
Family
[
a
0
0 1
]
Furthermore, we consider the family
C
=
[
a
0
0 1
]
(3.20)
with
a
∈
Z
. This family is related to (
3.14
) by the reciprocity operation (
2.31
), and one
can easily derive the form of quantum and classical A-polynomials from those given above.
Nonetheless, for convenience, and in order to illustrate how the reciprocity operation works,
we provide these A-polynomials explicitly. Quantum A-polynomials, for
a
≥
1, take form
̂
A
(
x
1
,x
2
,
̂
y
) =(
−
q
1
/
2
)
a
x
1
(
q
1
/
2
x
2
;
q
)
a
̂
y
a
+ (1
−
q
1
/
2
x
2
)
̂
y
−
1
,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) =(
−
q
1
/
2
)
a
xc
1
(
q
1
/
2
xc
2
;
q
)
a
̂
y
a
+ (1
−
q
1
/
2
xc
2
)
̂
y
−
1
,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) =(
−
q
1
/
2
)
a
x
(
q
1
/
2
x
;
q
)
a
̂
y
a
+ (1
−
q
1
/
2
x
)
̂
y
−
1
,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) =(
−
q
1
/
2
)
a
x
(
−
q
1
/
2
x
;
q
)
a
̂
y
a
+ (1 +
q
1
/
2
x
)
̂
y
−
1
,
(3.21)
– 16 –
and for
a
≤
0 take form
̂
A
(
x
1
,x
2
,
̂
y
) =(
q
1
/
2
x
2
;
q
)
a
+1
̂
y
a
+1
+ (
−
q
−
1
/
2
)
a
x
1
−
(
q
1
/
2
x
2
;
q
)
a
̂
y
a
,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) =(
q
1
/
2
xc
2
;
q
)
a
+1
̂
y
a
+1
+ (
−
q
−
1
/
2
)
a
xc
1
−
(
q
1
/
2
xc
2
;
q
)
a
̂
y
a
,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) =(
q
1
/
2
x
;
q
)
a
+1
̂
y
a
+1
+ (
−
q
−
1
/
2
)
a
x
−
(
q
1
/
2
x
;
q
)
a
̂
y
a
,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) =(
−
q
1
/
2
x
;
q
)
a
+1
̂
y
a
+1
+ (
−
q
−
1
/
2
)
a
x
−
(
−
q
1
/
2
x
;
q
)
a
̂
y
a
.
(3.22)
In consequence, classical A-polynomials, for
a
≥
1, read
A
(
x
1
,x
2
,y
) =(
−
1)
a
x
1
(1
−
x
2
)
a
y
a
+ (1
−
x
2
)
y
−
1
,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) =(
−
1)
a
xc
1
(1
−
xc
2
)
a
y
a
+ (1
−
xc
2
)
y
−
1
,
A
(
x
1
=
x,x
2
=
x,y
) =(
−
1)
a
x
(1
−
x
)
a
y
a
+ (1
−
x
)
y
−
1
,
A
(
x
1
=
x,x
2
=
−
x,y
) =(
−
1)
a
x
(1 +
x
)
a
y
a
+ (1 +
x
)
y
−
1
,
(3.23)
and for
a
≤
0 take form
A
(
x
1
,x
2
,y
) =(1
−
x
2
)
a
+1
y
a
+1
+ (
−
1)
a
x
1
−
(1
−
x
2
)
a
y
a
,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) =(1
−
xc
2
)
a
+1
y
a
+1
+ (
−
1)
a
xc
1
−
(1
−
xc
2
)
a
y
a
,
A
(
x
1
=
x,x
2
=
x,y
) =(1
−
x
)
a
+1
y
a
+1
+ (
−
1)
a
x
−
(1
−
x
)
a
y
a
,
A
(
x
1
=
x,x
2
=
−
x,y
) =(1 +
x
)
a
+1
y
a
+1
+ (
−
1)
a
x
−
(1 +
x
)
a
y
a
.
(3.24)
The parametrizations, for all
a
∈
Z
, take form
x
(
t
) =
c
a
−
2
1
(
t
+ (
−
1)
a
c
1
)
t
a
, y
=
(
−
1)
a
+1
c
−
1
1
t
a
+1
t
a
−
c
a
−
2
1
c
2
t
+ (
−
1)
a
+1
c
a
−
1
1
c
2
,
x
(
t
) =
(
t
+ (
−
1)
a
)
t
a
, y
=
(
−
1)
a
+1
t
a
+1
t
a
−
t
+ (
−
1)
a
+1
,
for
c
1
=
c
2
= 1
,
x
(
t
) =
(
t
+ (
−
1)
a
)
t
a
, y
=
(
−
1)
a
+1
t
a
+1
t
a
+
t
+ (
−
1)
a
,
for
c
1
=
−
c
2
= 1
.
(3.25)
Reciprocal quivers
The next family that yields A-polynomials of genus 0 is characterized by
C
=
[
a
0
0 1
−
a
]
(3.26)
It is sufficient to consider
a >
0; the case
a
≤
0 is equivalent to interchanging
x
1
↔
x
2
. For
– 17 –
a >
0 quantum A-polynomials read
̂
A
(
x
1
,x
2
,
̂
y
) =
a
−
1
∏
n
=0
(
q
a
x
1
̂
y
a
−
q
(
a
−
n
)(
a
−
1)+1
/
2
x
2
)
+
+
q
a/
2
a
−
2
∏
n
=0
(
q
n
+
a
x
1
̂
y
a
−
1
−
q
(
a
−
n
)(
a
−
1)+1
/
2
x
2
)
(1
−
̂
y
)(
−
̂
y
)
a
−
1
,
̂
A
(
x
1
=
c
1
x,x
2
=
c
2
x,
̂
y
) =
x
a
∏
n
=1
(
q
(2
n
−
1)
/
2
c
1
̂
y
a
−
c
2
)
+
+ (1
−
̂
y
)(
−
q
−
1
/
2
̂
y
)
a
−
1
a
−
1
∏
n
=1
(
q
n
−
a
+1
/
2
c
1
̂
y
a
−
1
−
c
2
)
,
̂
A
(
x
1
=
x,x
2
=
x,
̂
y
) =
x
a
∏
n
=1
(
q
(2
n
−
1)
/
2
̂
y
a
−
1
)
+
+ (1
−
̂
y
)(
−
q
−
1
/
2
̂
y
)
a
−
1
a
−
1
∏
n
=1
(
q
n
−
a
+1
/
2
̂
y
a
−
1
−
1
)
,
̂
A
(
x
1
=
x,x
2
=
−
x,
̂
y
) =
x
a
∏
n
=1
(
q
(2
n
−
1)
/
2
̂
y
a
+ 1
)
+
+ (1
−
̂
y
)(
−
q
−
1
/
2
̂
y
)
a
−
1
a
−
1
∏
n
=1
(
q
n
−
a
+1
/
2
̂
y
a
−
1
+ 1
)
,
(3.27)
and corresponding classical A-polynomials
A
(
x
1
,x
2
,y
) = (
x
1
y
a
−
x
2
)
a
+ (1
−
y
)(
−
y
)
a
−
1
(
x
1
y
a
−
1
−
x
2
)
a
−
1
,
A
(
x
1
=
c
1
x,x
2
=
c
2
x,y
) =
x
(
c
1
y
a
−
c
2
)
a
+ (1
−
y
)(
−
y
)
a
−
1
(
c
1
y
a
−
1
−
c
2
)
a
−
1
,
A
(
x
1
=
x,x
2
=
x,y
) =
x
(
y
a
−
1)
a
+ (1
−
y
)(
−
y
)
a
−
1
(
y
a
−
1
−
1)
a
−
1
=
= (
y
−
1)
a
[
xδ
a
(
y
)
a
−
(
−
y
)
a
−
1
δ
a
−
1
(
y
)
a
−
1
]
,
A
(
x
1
=
x,x
2
=
−
x,y
) =
x
(
y
a
+ 1)
a
+ (1
−
y
)(
−
y
)
a
−
1
(
y
a
−
1
+ 1)
a
−
1
,
(3.28)
where
δ
a
(
y
) =
a
−
1
∑
i
=0
y
i
=
1
−
y
a
1
−
y
.
(3.29)
We find the following parametrizations
x
(
t
) = (
t
−
1)(
−
t
)
a
−
1
(
c
1
t
a
−
1
−
c
2
)
a
−
1
(
c
1
t
a
−
c
2
)
a
, y
(
t
) =
t,
x
(
t
) = (
−
t
)
a
−
1
δ
a
−
1
(
t
)
a
−
1
δ
a
(
t
)
a
, y
(
t
) =
t,
for
c
1
=
c
2
= 1
,
x
(
t
) = (
t
−
1)(
−
t
)
a
−
1
(
t
a
−
1
+ 1)
a
−
1
(
t
a
+ 1)
a
, y
(
t
) =
t,
for
c
1
=
−
c
2
= 1
.
(3.30)
– 18 –