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首页N=2超弦阈值校正与Eichler积分的量子一元性研究
本文档探讨了N=2超弦理论中的一个重要课题,即阈值校正、广义势函数和Eichler积分在N=2杂合真空中BPS饱和振幅的环积分分析。作者们主要关注的是这些积分在大体积极限下的行为,通过将它们转换成复体积中的傅立叶级数形式,进而用Niebur-Poincaré级数来表示傅立叶系数。Niebur-Poincaré级数的模导数封闭性质揭示了这些积分与全纯正电势fn有直接联系,这是一种推广了N=2超重理论中熟知的电势概念。 Eichler积分的独特特性在这个背景下发挥了关键作用,它们在T对偶下展现出异常的变换行为。利用这一特性,研究者能够计算量子一元性的性质,这是一种在N=2理论中描述物理系统的量子效应的重要量。作者的分析进一步扩展到了与Hecke同余子组相关的模块积分,这些积分在不可分解的托里和自由作用圆管的紧凑化过程中自然出现,从而产生了新的闭式表达式。 具体来说,文中提到的积分涉及到Γ0(N)主模的积分,这些结果不仅包括了量子一元性的全纯势能的全面刻画,而且还提供了全纯Yukawa耦合的精确公式。这些计算的结果对于理解N=2超弦理论的非平凡行为和结构具有重要意义,尤其是在复杂结构模量空间中的动态演变。 这篇文章是理论物理学中一个深入且技术性的研究,它结合了高级数学工具如Niebur-Poincaré级数和Eichler积分,来探索N=2超弦理论的精细结构和对称性,这对于进一步理解和预测高能物理现象具有潜在的理论价值。
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C. Angelantonj et al. / Nuclear Physics B 897 (2015) 781–820 787
and of the Hecke operator
4
F(s,κ,w)=H
(z)
κ
F(s, 1,w). (2.10)
For the special values s =
w
2
and s = 1 −
w
2
, within the domain of analyticity, F(s, κ, w)
becomes a harmonic Maaß form. For negative weight, the case of interest for most applica-
tions to string theory [6], the Niebur–Poincaré series is absolutely convergent at the latter value
s = 1 −
w
2
. Moreover, since the first cusp form for SL(2; Z) appears at weight twelve, for
w ∈{0, −2, −4, −6, −8, −12} the shadow of F(1 −
w
2
, κ, w) vanishes and the Niebur–Poincaré
series is actually holomorphic, hence valued in the polynomial ring generated by E
4
, E
6
and
Δ
−1
, where E
2n
are the holomorphic Eisenstein series and Δ is the modular discriminant. Other
Niebur–Poincaré series of interest are F(1 −
w
2
+n, κ, w), with n a positive integer, which can
be reached from the harmonic Maaß form F (1 −
w
2
+ n, κ, w − 2n) by acting with n modular
derivatives. In fact, one can prove [6–8] that any weak almost holomorphic modular form of
negative weight in the polynomial ring generated by
ˆ
E
2
, E
4
, E
6
and Δ
−1
, with
ˆ
E
2
being the
almost holomorphic Eisenstein series of weight two, can be uniquely decomposed into a linear
combination of Niebur–Poincaré series,
Φ(z) =
0<m≤κ
n
=0
d
(m) F
1 −
w
2
+, m, w
. (2.11)
The coefficients d
(m) are determined by the principal part of the q-Laurent expansion of Φ
itself [7]. The integer n counts the maximal power of
ˆ
E
2
in the ring decomposition of Φ, and is
known as the depth. An important fact is that for each value of , the shadows cancel in the linear
combination (2.11).
It will also be useful to introduce the non-holomorphic weight-w Eisenstein series
E(s,w;z) =
1
2
γ ∈Γ
∞
\Γ
y
s−
w
2
|
w
γ
=
1
2
(c,d)=1
y
s−
w
2
(c z + d)
w
|cz+d|
2s−w
(2.12)
which is absolutely convergent for Re (s) > 1, and admits the Fourier series expansion
E(s,w;z) = y
s−
w
2
+
4
1−s
i
−w
πΓ(2s −1)ζ (2s −1)
Γ(s+
w
2
)Γ(s −
w
2
)ζ(2s)
y
1−s−
w
2
+
m=0
i
−w
(4π|m|)
w
2
π
s
Γ(s+
w
2
sgn(m)) ζ(2s)
|m|
−s
σ
2s−1
(|m|) W
s,w
(my) e
2iπmx
. (2.13)
The Eisenstein series E(s, w) can be obtained formally as the limit κ → 0of |4πκ|
w
2
−s
F(s,
κ, w), and thus shares several of the properties of the Niebur–Poincaré series. Namely, it is an
eigenmode of the weight-w hyperbolic Laplacian
4
The Hecke operator H
(z)
n
acts on the Fourier modes of a weight-w modular form Φ =
m
Φ(m, y) e
2iπmx
as
H
(z)
n
Φ(m,y) = n
1−w
d|(m,n)
d
w−1
Φ(nm/d
2
,d
2
y/n).
It satisfies the commutative algebra H
(z)
κ
H
(z)
λ
=
d|(κ,λ)
d
1−w
H
(z)
κλ/d
2
.
![](https://csdnimg.cn/release/download_crawler_static/12318570/bg8.jpg)
788 C. Angelantonj et al. / Nuclear Physics B 897 (2015) 781–820
Δ
w
+
1
2
s(1 −s) +
1
8
w(w + 2)
E(s,w;z) = 0 , (2.14)
is closed under the action of the modular derivatives
D
z
E(s,w;z) =
1
2π
(s +
w
2
)E(s,w+ 2;z) ,
¯
D
z
E(s,w;z) =
π
2
(s −
w
2
)E(s,w− 2;z) , (2.15)
and becomes harmonic in z when s =
w
2
or s = 1 −
w
2
. For w>2, E(
w
2
, w) coincides with the
holomorphic Eisenstein series E
w
.
2.2. Evaluation of the modular integral
We can now proceed to the study of the integral (1.1). Since any weak almost holomorphic
modular form Φ can be decomposed as in (2.11) it is sufficient to focus our attention on the basic
regularised modular integral
I(s, κ;T,U)=R.N.
F
dμΓ
2,2
(T , U ;τ)F(s, κ, 0;τ), (2.16)
where Γ
2,2
(T , U ; τ) is the partition function of the even self-dual lattice of signature (2, 2), and
R.N. stands for the modular invariant renormalisation prescription of [5,6,28] for treating the
infra-red divergences arising at the cusp. For s = 1, the integral (2.16) develops a simple pole,
which must be subtracted [6,8]. By ab
use of notation we denote by I(1, κ; T, U) the result of
this subtraction. The partition function
Γ
2,2
(T , U ;τ)=τ
2
m
1
,m
2
,n
1
,n
2
∈Z
q
1
4
|p
L
|
2
¯q
1
4
|p
R
|
2
, (2.17)
which we refer to as the Narain lattice, is a sum over integers m
i
and n
i
identified with the
Kaluza–Klein momentum and winding numbers along a two-dimensional torus. The left-moving
and right-moving momenta on the two-dimensional lattice
p
R
=
m
2
−Um
1
+Tn
1
+TUn
2
√
T
2
U
2
,
p
L
=
m
2
−Um
1
+
¯
Tn
1
+
¯
TUn
2
√
T
2
U
2
, (2.18)
depend on the Kähler modulus T =T
1
+iT
2
and the complex structure modulus U =U
1
+iU
2
of the two-torus. Γ
2,2
(T , U ; τ) is modular invariant under SL(2; Z), which requires the modular
weight of the Niebur–Poincaré series in (2.16) to vanish.
Eq. (2.17) corresponds to the so-called Hamiltonian representation of the Narain lattice, and
e
xhibits manifest invariance under the duality group O(2, 2; Z) ∼ SL(2; Z)
T
× SL(2; Z)
U
σ
T,U
, where the Z
2
generator σ
T,U
exchanges the T and U moduli. A Poisson summation over
the momenta m
i
yields the so-called Lagrangian representation
Γ
2,2
(T , U ;τ)=T
2
A∈Mat
2×2
(Z)
e
−2πiT det(A)
exp
−
πT
2
τ
2
U
2
( 1 U)
A
−τ
1
2
, (2.19)
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