Operators from mirror curves and the quantum dilogarithm

0.32 MB·English

by  Rinat Kashaev

OPERATORS FROM MIRROR CURVES AND THE QUANTUM
DILOGARITHM
RINAT KASHAEV AND MARCOS MARIN˜O
5
1 Abstract. MirrormanifoldstotoricCalabi–Yauthreefoldsareencodedinalgebraiccurves.
0 The quantization of these curves leads naturally to quantum-mechanical operators on the
2 real line. We show that, for a large number of local del Pezzo Calabi–Yau threefolds, these
2
c operators are of trace class. In some simple geometries, like local , we calculate the
e integral kernel of the corresponding operators in terms of Faddeev’s qPuantum dilogarithm.
D
Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the

4 state-integralsappearinginthree-manifoldtopology,andweshowthattheycanbeevaluated
1 explicitlyinsomecases. Ourresultsprovidefurtherverificationsofarecentconjecturewhich
gives an explicit expression for the Fredholm determinant of these operators, in terms of
]
h enumerative invariants of the underlying Calabi–Yau threefolds.
t
-
p
e
h
[ Contents

3
1. Introduction 2
v
4 Acknowledgments 3
1
2. Trace class operators and mirror curves 3
0
1 2.1. Mirror curves 3
0 2.2. Quantization 5
.
1 2.3. Generic three-term operator 8
0 2.4. Local F 10
5 0
1 2.5. Perturbed operators 11
: 2.6. Symmetries and equivalences 12
v
i 3. Calculation of spectral traces 14
X
3.1. General considerations 14
r
a 3.2. Some explicit results 17
3.3. Refined traces 18
4. Comparison with the conjecture of [GHM] 19
5. Conclusions and outlook 24
Appendix A. Some useful properties of the quantum dilogarithm 25
References 26
Date: December 16, 2015.
R.K. and M.M. are supported in part by the Swiss National Science Foundation, subsidies 200020-
141329, 200020-149226, 200021-156995, and by the NCCR 51NF40-141869 “The Mathematics of Physics”
(SwissMAP).
2010 Mathematics Subject Classification: Primary 34K08. Secondary 14J33,14J81,81T30.
Key words and phrases: mirror symmetry, quantum dilogarithm.
1
2 RINATKASHAEVANDMARCOS MARIN˜O
1. Introduction
The problem of “quantizing” in a suitable way an algebraic curve appears in many dif-
ferent contexts in mathematical physics. For example, the quantization of spectral curves
associated to integrable systems leads to the Baxter equation. Solving the spectral problem
for the Baxter equation, after imposing appropriate boundary conditions, gives the solution
of the original eigenvalue problem. In other cases, the classical limit of a quantum theory is
encoded in an algebraic curve, and a suitable quantization of the curve makes it possible to
go beyond the classical limit and to obtain the quantum corrections.
One important example of such a situation is local mirror symmetry [KKV, CKYZ]. The
mirror manifold to a toric Calabi–Yau manifold X reduces to an algebraic curve in the
exponentiated variables x, y, of the form
(1) W(ex,ey) = 0.
The genus zero Gromov–Witten invariants of X can be obtained by computing periods of
this curve. In [ADKMV] it was pointed out that the higher genus invariants (which can be
regarded as “quantum corrections”) might be obtained by quantizing in an appropriate way
the mirror curve. Building on work on supersymmetric gauge theories and quantum inte-
grable systems [NS], it was shown in [ACDKV] that mirror curves can be formally quantized
by using the WKB approximation (this method was previously used in [MM], in the context
of Seiberg–Witten curves). The quantum corrections obtained in this way do not correspond
to the conventional higher genus Gromov–Witten invariants, but to a peculiar combination
of the “refined” BPS invariants of the toric Calabi–Yau manifold [IKV, CKK, NO].
The quantization prescription of [ACDKV] leads to “quantum” periods which are formal
WKB series in powers of ~. However, in [KM] it was pointed out that one can associate
operators with a well-defined, discrete spectrum, to the mirror curves of toric Calabi–Yau
threefolds. The WKB series considered in [ACDKV] turns out to be insufficient to determine
this spectrum, as it misses non-perturbative corrections of the instanton type. It was sug-
gested in [KM] that these corrections would involve the conventional higher genus Gromov–
Witten invariants of X, and this suggestion was verified in some examples in [KM, HW].
These observations were deepened and put on a firmer ground in [GHM], where it was con-
jectured that, given a mirror curve to a toric Calabi–Yau threefold X, one can associate to
it a trace class operator. Furthermore, [GHM] proposed an exact formula for the Fredholm
determinant of this trace class operator in terms of the enumerative geometry of X. Both
conjectures were tested in detail in a number of examples, both analytically and numerically.
The proposal of [GHM] focused on the case in which the mirror curve is of genus one, i.e.
in the case in which X is a local del Pezzo Calabi–Yau threefold, but it can be suitably
generalized to the case of higher genus curves.
In this paper we will perform a detailed study of the operators associated to mirror curves
by the procedure explained in [GHM]. We will prove that, indeed, for a large number of
local del Pezzo threefolds, they lead to positive-definite, trace class operators on L2( ).
R
Therefore, their spectrum is discrete and positive (as it was shown numerically in some
examples in [KM, HW, GHM]), and their Fredholm determinants are well-defined. One of
themostinteresting aspectsofourstudy isthattheoperatorsobtainedinthiswayareclosely
related to those appearing in the quantization of Teichmu¨ller theory [K1, K2, K3, K4, FC].
OPERATORS FROM MIRROR CURVES 3
In particular, in the simple case of three-term operators, we compute explicitly their integral
kernels, which involve in an essential way Faddeev’s quantum dilogarithm function [Fad95,
FK94]. The spectral traces of these operators lead then to multi-dimensional integrals that
are formally very similar to the state-integral invariants of three-manifolds studied recently
in [Hik01, DGLZ09, AKa, KLV12, DG13, AKb, Dim]. We use recent techniques for the
evaluation of state integrals [GK] to calculate these spectral traces in various cases, and we
verify that the results fully agree with the predictions of [GHM].
This paper is organized as follows. In section 2 we give a brief review of the construction
of the operators from mirror curves, and after introducing some of the necessary ingredients,
we prove that the relevant operators are of trace class. In addition, we calculate the explicit
expression of their integral kernels in the simple case of three-term operators. In section 3
we write down formulae for the spectral traces of the three-term operators, and we evaluate
themexplicitly insome cases. Insection 4 we compareour results totheconjecture of [GHM]
for the Fredholm determinant of these operators. Finally, in section 5 we conclude and list
some interesting open problems.
Acknowledgments. We would like to thank Andrea Brini, Alba Grassi, Jie Gu, Yasuyuki
Hatsuda, Albrecht Klemm, Jonas Reuter, Shamil Shakirov and Leon Takhtajan for useful
conversations and correspondence.
2. Trace class operators and mirror curves
2.1. Mirror curves. The operators which we will study in this paper arise by quantizing
mirror curves (in an appropriate sense). We will focus for simplicity on toric (almost) del
Pezzo Calabi–Yauthreefolds, which aredefinedasthetotalspaceofthecanonical linebundle
on a toric (almost) del Pezzo surface S,
(2) X = (K ) S.
S
O →
They aresometimes called“localS,”soforexample ifS = 2, thetotalspaceofitscanonical
P
line bundle will be called local 2. Examples of toric (almost) del Pezzos include, besides
P
2, the Hirzebruch surfaces , n = 0,1,2, and the blowups of 2 at n points, denoted by
n
P F P
, for n = 1,2,3 (note that = , and that = 1 1).
n 1 1 0
B F B F P ×P
By standard results in toric geometry (see for example [HKP, CR]), toric, almost del
Pezzo surfaces can be classified by reflexive polyhedra in two dimensions. The polyhedron
∆ associated to a surface S is the convex hull of a set of two-dimensional vectors
S
(3) ν(i) = ν(i),ν(i) , i = 1, ,k +2,
1 2 ···
(cid:16) (cid:17)
together with the origin. In order to construct the total space of the canonical line bundle
over S, we have to consider the extended vectors
ν(0) = (1,0,0),
(4)
ν(i) = 1,ν(i),ν(i) , i = 1, ,k +2.
1 2 ···
(cid:16) (cid:17)
4 RINATKASHAEVANDMARCOS MARIN˜O
They satisfy the relations
k+2
(5) Qαν(i) = 0,
i
i=0
X
where Qα is a matrix of integers (called the charge matrix) which characterizes the geometry.
i
The construction of the mirror geometry to (2) goes back to Batyrev, and it has been
recently reviewed in [CR], to which we refer for further details. In order to write down the
equation for the mirror curve to the Calabi–Yau (2), we note that it depends on k complex
moduli z , α = 1, ,k, which can be parametrized in many possible ways. The most useful
α
···
parametrization involves a modulus u˜ and a set of “mass” parameters ζ , i = 1, ,r, where
i
···
r depends on the geometry under consideration [HKP, HKRS]. In terms of these variables,
the mirror curve for a local del Pezzo Calabi–Yau threefold can be written as,
(6) W(ex,ey) = (x,y)+u˜ = 0,
S
O
where
k+2
(i) (i)
(7) (x,y) = exp ν x+ν y +f (ζ ) ,
OS 1 2 i j
Xi=1 (cid:16) (cid:17)
and f (ζ ) are suitable functions of the parameters ζ .
i j j
Example 2.1. The simplest case of a local del Pezzo is local 2. In this case, we have k = 1.
P
The vectors (3) are given by
(8) ν(1) = (1,0), ν(2) = (0,1), ν(3) = ( 1, 1).
− −
In this geometry there is one complex deformation parameter u˜, and the function is
2
given by OP
(9) (x,y) = ex +ey +e−x−y.
2
OP
Example 2.2. The previous example can be generalized by considering the canonical line
bundle over the weighted projective space (1,m,n), where m,n . This is not a
>0
P ∈ Z
smooth manifold, but it can be analyzed by using extensions of Gromov–Witten theory, see
for example [BC] for a study of the case n = 1. The vectors are in this case
(10) ν(1) = (1,0), ν(2) = (0,1), ν(3) = ( m, n),
− −
and the function appearing in the mirror curve (6) is given by
O
(11) (x,y) = ex +ey +e−mx−ny.
m,n
O
Some of these geometries can arise as degeneration limits of toric del Pezzos. For example,
the mirror curve to local is characterized by the function
2
F
(12) (x,y) = ex +ey +e−2x−y +ζe−x,
OF2
and when ζ = 0 we recover the geometry (11) with m = 2 and n = 1.
Some examples of functions obtained from mirror curves of local del Pezzos can be found
in table 1. Details on the corresponding geometries can be found in for example [HKP].
OPERATORS FROM MIRROR CURVES 5
S (x,y)
S
O
2 ex +ey +e−x−y
P
ex +ζe−x +ey +e−y
0
F
ex +ζe−x +ey +e−x−y
1
F
ex +ζe−x +ey +e−2x−y
2
F
ex +ey +e−x−y +ζ e−y +ζ e−x
2 1 2
B
ex +ey +e−x−y +ζ e−x +ζ e−y +ζ ex+y
3 1 2 3
B
Table 1. The functions (x,y) associated to some local del Pezzo Calabi–Yaus.
S
O
2.2. Quantization. The “quantization” of the mirror curve (6), in the case of local del Pez-
zos, is based on the promotion of the function (x,y) to an operator, which will be denoted
S
O
by O . This is achieved by simply promoting x, y to self-adjoint Heisenberg operators x, y
S
satisfying the commutation relation
(13) [x,y] = i~.
Possible ordering ambiguities are resolved by using Weyl’s prescription. As noted in [GHM],
instead of studying O (which is not of trace class), one should rather consider its inverse
S
(14) ρ = O−1.
S S
One of our goals in this paper is to show that, for a large number of choices of S, this
operator exists and is of trace class.
It will be useful to introduce normalized Heisenberg operators, p and q, satisfying the
commutation relation
(15) [p,q] = (2πi)−1.
The “coordinate representation” is given by a realisation in the Hilbert space L2(R) by the
formulae
1 ∂
(16) x q = x x , x p = x , x y = δ(x y), 1 = x dx x ,
h | h | h | 2πi∂xh | h | i − | i h |
R
Z
while the “momentum representation” is given by
i ∂
(17) (x p = x(x , (x q = (x , (x y) = δ(x y), 1 = x)dx(x ,
| | | 2π∂x | | − | |
R
Z
and the transition between these two representations is given by the Fourier kernel
(18) x y) = e2πixy.
h |
The following result is elementary (see also [Si]).
Lemma 2.3. For any f,g L2(R), the operator
∈
(19) G f(q)g(p)
≡
is a Hilbert–Schmidt operator.
6 RINATKASHAEVANDMARCOS MARIN˜O
Proof. By using the integral kernel in the mixed basis
(20) x G y) = f(x)g(y) x y),
h | | h |
we have
(21) Tr(G∗G) = (y G∗ x x G y)dxdy = x G y) 2dxdy
| | ih | | |h | | |
R2 R2
Z Z
= f(x)g(y) 2dxdy = f 2 g 2 <
| | k k k k ∞
R2
Z
where we use the L2-norm
(22) f 2 f(x) 2dx.
k k ≡ | |
R
Z
(cid:3)
In order to study the properties of the operators associated to local del Pezzo geometries,
we will proceed in two steps. First, we will consider the three-term operators O , obtained
m,n
by quantization of the function (11), and we will establish that their inverse operators
(23) ρ O−1
m,n ≡ m,n
exist and are trace class operators on L2( ). The operators associated to more general del
R
Pezzos can be regarded as perturbations of three-term operators, and this makes it possible
to show that they are as well of trace class. The operator associated to local is somewhat
0
F
special, but it can be analyzed with similar techniques, as we will see.
A key ingredient to study the operators ρ is an explicit determination of their integral
m,n
kernels, which involves Faddeev’s quantum dilogarithm Φ (x) [Fad95, FK94]. A summary of
b
the properties of this function can be found in the Appendix. Here, we list some preliminary
results for the analysis of the three-term operators. We fix a positive real number b and
define a set
b+b−1
(24) ∆ (a,c) R2 a+c < h .
b ≡ ∈ >0 b ≡ 2
(cid:26) (cid:12) (cid:27)
(cid:12)
We will also denote (cid:12)
(cid:12)
(25) c ih .
b b
≡
Define a function
e2πax
(26) Ψ (x) , x R, (a,c) ∆ ,
a,c ≡ Φ (x i(a+c)) ∈ ∈ b
b
−
which is a nowhere vanishing Schwartz function in variable x, i.e. a smooth and rapidly
decreasing function at infinity. Indeed, the conditions on parameters a and c are such that
Ψ (x) is a restriction of a meromorphic function in the complex plane whose poles and
a,c
zeros do not belong to the real axis R C, and the formula
⊂
(27) lim Φ (x+iy) = 1, y R,
b
x→−∞ ∀ ∈
OPERATORS FROM MIRROR CURVES 7
implies that
(28) lim (Ψ (x)e−2πax 1) = 0.
a,c
x→−∞ −
By using the equalities
(29) Ψ (x) = Φ ( x+i(a+c))e2πax−πi(x−i(a+c))2 = Φ ( x+i(a+c)) e−2πcx,
a,c b b
| | − | − |
(cid:12) (cid:12)
we also have
(cid:12) (cid:12)
(cid:12) (cid:12)
(30) lim ( Ψ (x) e2πcx 1) = 0.
a,c
x→+∞ | | −
Lemma 2.4. For any (a ,a ) ∆ , one has
1 2 b
∈
3
(31) Ψ 2 = Φ (c 2ia ) , a h a a .
k a1,a2k | b b − j | 3 ≡ b − 1 − 2
j=1
Y
Proof. The formula follows from the integral Ramanujan formula
Φ t+ z
(32) ϕ(z,w) b 2 e2πitwdt = φ (z c )φ (w +c )φ ( z w +c ),
≡ Φ t z b − b b b b − − b
ZR b(cid:0) − 2(cid:1)
where the normalized quantum dilogarithm function
(cid:0) (cid:1)
Φ (z)
(33) φ (z) b e−πiz2/2
b
≡ Φ (0)
b
has the properties
φ (z ib/2)
b
(34) φ (z)φ ( z) = 1, φ (z) = φ ( z¯), − = 2cosh(πbz).
b b b b
− − φ (z +ib/2)
b
We have
Φ (t+i(a +a ))
(35) Ψ 2 = b 1 2 e4πa1tdt = ϕ(2i(a +a ), 2ia )
k a1,a2k Φ (t i(a +a )) 1 2 − 1
R b 1 2
Z −
= ϕ(2i(h a ), 2ia ) = φ (c 2ia )φ (c 2ia )φ (c 2ia ),
b 3 1 b b 3 b b 1 b b 2
− − − − −
which is exactly (31), as we have the equality
(36) Φ (z) = φ (z) if (z) (z) = 0.
b b
| | | | ℑ ℜ
(cid:3)
Remark 2.5. More generally, we have the following Fourier transformation formula
Φ (t+i(a +a ))
(37) Ψ (t) 2e2πixtdt = b 1 2 e2πit(x−2ia1)dt
| a1,a2 | Φ (t i(a +a ))
R R b 1 2
Z Z −
= ϕ(2i(a +a ),x 2ia )
1 2 1
−
= φ (2i(a +a ) c )φ (x 2ia +c )φ ( 2i(a +a ) x+2ia +c )
b 1 2 b b 1 b b 1 2 1 b
− − − −
= φ (c 2ia )φ (x 2ia +c )φ ( x 2ia +c )
b b 3 b 1 b b 2 b
− − − −
φ (x 2ia +c )
= φ (c 2ia ) b − 1 b .
b b − 3 φ (x+2ia c )
b 2 b
−
8 RINATKASHAEVANDMARCOS MARIN˜O
Lemma 2.6. One has the following equalities
(38) Φ (p)e2πbqΦ∗(p) = e2πbq +e2πb(p+q),
b b
(39) Φ∗(q)Φ (p)e2πbqΦ∗(p)Φ (q) = e2πbq +e2πb(p+q) +e2πb(p+2q)
b b b b
Proof. We have
Φ (p ib/2)
(40) Φ (p)e2πbqΦ∗(p) = eπbq b − eπbq = eπbq 1+e2πbp eπbq = e2πbq +e2πb(p+q)
b b Φ (p+ib/2)
b
(cid:0) (cid:1)
and, by using the previous formula,
(41) Φ∗(q)Φ (p)e2πbqΦ∗(p)Φ (q) = Φ∗(q) e2πbq +e2πb(p+q) Φ (q)
b b b b b b
Φ (q ib/2)
= e2πbq +eπb(p+q) b − eπb(p+(cid:0)q) = e2πbq +eπb((cid:1)p+q) 1+e2πbq eπb(p+q)
Φ (q+ib/2)
b
= e2πb(cid:0)q +e2πb(p+(cid:1)q) +e2πb(p+2q).
(cid:3)
2.3. Generic three-term operator. Consider theoperatorassociatedtothefunction(11):
(42) O = ex +ey +e−mx−ny, m,n R .
m,n >0
∈
Note that m,n can be a priori arbitrary positive, real numbers, although in the operators
arising from the mirror curves they are integers. By using Lemma 2.6 and the substitutions
(n+1)p+nq mp+(m+1)q
(43) x 2πb , y 2πb ,
≡ m+n+1 ≡ − m+n+1
so that
2πb2
(44) ~ = ,
m+n+1
we have
(45) e−y/2O e−y/2 = ex−y +1+e−mx−(n+1)y = e2πb(p+q) +1+e2πbq
m,n
Φ (q ib/2)
= 1+Φ (p)e2πbqΦ∗(p) = Φ (p) b − Φ∗(p).
b b b Φ (q+ib/2) b
b
By defining an operator
(46) Am,n ≡ Φ∗b(q−ib/2)Φ∗b(p)eπmb(+mn++11)qemπ+bnm+1p,
we obtain the following formula for the inverse operator (23),
(47) ρ = A∗ A .
m,n m,n m,n
Let us now rewrite A in the form
m,n
(48) Am,n = Φ∗b q−ib2 eπmb(+mn++11)qΦ∗b p−i2(mb(m++n+1)1) emπ+bnm+1p = Ψa+c,cn(q)Ψa,c(p),
(cid:18) (cid:19) (cid:18) (cid:19)
OPERATORS FROM MIRROR CURVES 9
where
bm b
(49) a , c .
≡ 2(m+n+1) ≡ 2(m+n+1)
We see that A is a Hilbert–Schmidt operator due to the inclusions
m,n
b(m+1) bn bm b
(50) , ∆ ,
b
2(m+n+1) 2(m+n+1) ∈ ∋ 2(m+n+1) 2(m+n+1)
(cid:18) (cid:19) (cid:18) (cid:19)
and Lemma 2.3.
Theorem 2.7. The operator ρ is positive-definite and of trace class.
m,n
Proof. Due to (47), we have that, for any f L2(R),
∈
(51) f∗(x)(ρ f)(x)dx = A f 2,
m,n m,n
k k
R
Z
and since A is invertible, we conclude that ρ is positive-definite. Since the product of
m,n m,n
two Hilbert–Schmidt operators is trace class, and A is Hilbert–Schmidt, we also conclude
m,n
that ρ is of trace class. (cid:3)
m,n
Proposition 2.8. The integral kernel of ρ in the momentum representation is given by
m,n
the formula
Ψ (x)Ψ (y)
a,c a,c
(52) (x ρ y) = .
m,n
| | 2bcosh πx−y+i(a+c−nc)
b
(cid:16) (cid:17)
Proof. From (47) and (48) we obtain
(53) ρ = Ψ∗ (p)Ψ∗ (q)Ψ (q)Ψ (p) = Ψ∗ (p) Ψ (q) 2Ψ (p).
m,n a,c a+c,cn a+c,cn a,c a,c | a+c,cn | a,c
By using the difference functional equation for the quantum dilogarithm and the Fourier
integral
1 e2πixy
(54) = dy,
cosh(πx) cosh(πy)
R
Z
and denoting
(55) h a+c nc,
≡ −
we have
e2π(a+c)x 2 Φ (x+ib/2)
(56) Ψ (x) 2 = = b e4π(a+c)x
| a+c,cn | Φ (x ib/2) Φ (x ib/2)
(cid:12) b − (cid:12) b −
e4π(a+c)(cid:12)x e2πhx(cid:12) e2πhxe2πibxy e2πibx(y−ih/b)
(cid:12) (cid:12)
= (cid:12) = (cid:12) = dy = dy
1+e2πbx 2cosh(πbx) 2cosh(πy) 2cosh(πy)
R R
Z Z
e2πibxz e2πixt
= dz = dt,
2cosh(π(z +ih/b)) 2bcosh(π(t+ih)/b))
ZR−ih/b ZR
10 RINATKASHAEVANDMARCOS MARIN˜O
where in the last equality we have shifted the line of integration by using analyticity of the
integrand in the strip z h /b. Finally, we remark that if we have a Fourier integral of
|ℑ | ≤ | |
the form
(57) f(x) = f˜(t)e2πixtdt,
R
Z
then we have
(58) (x f(q) y) = f˜(t)(x e2πiqt y)dt = f˜(t)(x t y)dt
| | | | − |
R R
Z Z
˜ ˜
= f(t)δ(x t y)dt = f(x y),
− − −
R
Z
so that
(59) (x ρ y) = (x Ψ∗ (p) Ψ (q) 2Ψ (p) y)
| m,n| | a,c | a+c,cn | a,c |
Ψ (x)Ψ (y)
= Ψ (x) x Ψ (q) 2 y Ψ (y) = a,c a,c .
a,c | a+c,cn | a,c 2bcosh πx−y+ih
b
(cid:0) (cid:12) (cid:12) (cid:1) (cid:3)
(cid:12) (cid:12) (cid:0) (cid:1)
2.4. Local F . The operator associated to the mirror curve of local is obtained by quan-
0 0
F
tization of the function F appearing in table 1, and it reads
O 0
(60) OF = ex +ζe−x +ey +e−y.
0
This operator can not be regarded as a perturbation of a three-term operator of the form
O : whenζ = 0, theresulting operatorisprecisely the oneassociatedto geodesic lenghts in
m,n
quantum Teichmu¨ller theory [K1, K2, FC], and it has a continuous spectrum [K3, K4, FT].
We then assume ζ > 0. We will now show that, when this is the case, the inverse operator
ρF0 = O−F01 exists and is of trace class. Let us set
(61) x = πb(p+2q), y = πbp,
so that
(62) ~ = πb2.
By using Lemma 2.6 we have
(63) ex/2OF ex/2 ζ = e2x +ex+y +ex−y = e2πb(p+2q) +e2πb(p+q) +e2πbq
0 −
= Φ∗(q)Φ (p)e2πbqΦ∗(p)Φ (q).
b b b b
Thus,
(64) Φ∗b(p)Φb(q)ex/2OF0ex/2Φ∗b(q)Φb(p) = ζ +e2πbq = ζ 1+e2πb(q−µ)
Φ (q µ ib/2)
(cid:0) (cid:1) b
= ζ − − ,
Φ (q µ+ib/2)
b
−
where we have introduced a new parameter µ through the equation
(65) ζ = e2πbµ.