从高维到四维：费曼积分和残基配对的新应用

103 浏览量更新于2024-07-16 收藏 645KB PDF 举报

"从无穷大到四个维度：更高的残基配对和费曼积分" 在本文中，我们研究了一个令人惊讶的现象，即在D = 4 − 2ε时空维度中ε→0的费曼积分可以通过其在相反极限ε→∞中的行为来充分表征。这种现象可以通过对费曼积分的矢量束的研究来解释，其中连接对ε具有多项式依赖性，并且已知受扭曲形式的相交数控制。费曼积分是一种数学工具，用于计算量子场论中的散射幅度。在本文中，我们使用费曼积分来研究高维空间中的残基配对。残基配对是一种数学结构，用于描述高维空间中的几何形状。在本文中，我们使用Saito的较高残基配对来研究费曼积分。 Saito的较高残基配对是一种数学工具，用于描述高维空间中的几何形状。这种方法可以用于计算高维空间中的散射幅度，并且可以用于研究高维空间中的几何结构。在本文中，我们使用Saito的较高残基配对来研究费曼积分，并得到了精确的计算结果。在本文中，我们还讨论了费曼积分在D = 4 − 2ε时空维度中的性质，并得到了精确的计算结果。我们发现，费曼积分在ε→0和ε→∞极限中的行为可以通过相互作用来描述。这种相互作用可以用于研究高维空间中的几何结构，并且可以用于计算高维空间中的散射幅度。本文的结果对无质量量子场理论的散射幅度一致，表明了我们的方法可以用于研究高维空间中的物理现象。此外，本文的结果还可以用于研究高维空间中的几何结构，并且可以用于计算高维空间中的散射幅度。本文讨论了费曼积分在高维空间中的应用，并得到了精确的计算结果。本文的结果可以用于研究高维空间中的物理现象，并且可以用于计算高维空间中的散射幅度。知识点： 1. 费曼积分是一种数学工具，用于计算量子场论中的散射幅度。 2. 残基配对是一种数学结构，用于描述高维空间中的几何形状。 3. Saito的较高残基配对是一种数学工具，用于描述高维空间中的几何形状。 4. 费曼积分可以用于研究高维空间中的几何结构。 5. 费曼积分在D = 4 − 2ε时空维度中的性质可以通过相互作用来描述。 6. 费曼积分可以用于计算高维空间中的散射幅度。 7. 本文的结果对无质量量子场理论的散射幅度一致。 8. 本文的结果可以用于研究高维空间中的物理现象。 9. 本文的结果可以用于计算高维空间中的散射幅度。

JHEP02(2020)159

It is important to note that the pairing (2.8) is not unique. As a matter of fact, we

can embed it into the following formalism. Consider the exact sequence

0 O

Ω

··· Ω

∇

(2.11)

called a twisted de Rham complex, where we introduced a diﬀerential

∇

:= d + τdW ∧, (2.12)

which deﬁnes an integrable connection, or equivalently a ﬂat line bundle, since ∇

= 0.

As was the case before, we can deﬁne cohomology groups based on this complex

(M, ∇

) :=

{ϕ

∈ Ω

|∇

= 0}

{∇

k−1

∈ Ω

|ϕ

k−1

∈ Ω

k−1

}

, (2.13)

which are spaces of ∇

-closed modulo ∇

-exact forms. We will call H

:=H

(M,∇

)

for short from now on. As in the case of (2.4), only k = m gives a non-trivial cohomol-

ogy [40] and hence we have

dim H

= (−1)

χ(M). (2.14)

Let us introduce a dual twisted cohomology H

−dW

deﬁned in the same way as H

but

with W → −W . We will often refer to cohomology classes [ϕ

] ∈ H

±dW

as twisted cocycles

and speciﬁc representatives ϕ

as twisted forms. Duality of the two cohomologies is induced

by the intersection pairing

−dW

× H

→ C (2.15)

deﬁned by

hϕ

−

|ϕ



2π

√

−1



−

∧ ϕ

(2.16)

and called an intersection number. The overall normalization is chosen for later conve-

nience. Since M is non-compact, for this deﬁnition to make sense one needs to use a

compactly-supported form ϕ

(i.e., one which vanishes in inﬁnitesimal neighbourhoods of

the boundary divisor ∂M) in the same cohomology class as ϕ

for direct computations.

This not only makes the result depend on the potential W , but also means the above in-

tegral localizes on ∂M, as in the bulk of M we have ϕ

−

∧ ϕ

= ϕ

−

∧ ϕ

= 0 for two top

holomorphic forms, see, e.g., [5].

Intersection numbers are rational functions of α

’s and τ . Diﬀerent ways of evaluating

them in practice were given in [4, 5, 27, 41–49]. For spaces M admitting a ﬁber bundle

decomposition (or, more precisely, such that the connection decomposes generically on the

ﬁbers), the most eﬃcient computation method is currently given by recursion relations [5].

When M = M

0,n

is the moduli space of genus-zero curves with n marked points, intersec-

tion numbers have an intrinsic interpretation as computing tree-level scattering amplitudes

Since M is ﬁnite-dimensional, one can equivalently think of our setup in terms of Batalin-Vilkovisky

(BV) formalism, see, e.g., [39]. More precisely, for a space MV

•

(M) := Γ(∧

•

T M ) of antisymmetric mul-

tivector ﬁelds on M (a counterpart of Ω

•

(M) = Γ(∧

•

∗

M)) and a BV diﬀerential ∂

:= div + ι

, the

corresponding BV complex (MV

•

(M), ∂

) is isomorphic to (Ω

m−•

(M), ∇

– 6 –

JHEP02(2020)159

of quantum ﬁeld theories [4, 5]. Worldsheet models that might underlie these computa-

tions were studied in [50–53]. In those cases the corresponding potential W can be given

an interpretation as an electrostatic potential for a system of particles [54].

In the context of Landau-Ginzburg models, intersection numbers between basis ele-

ments compute entries of a metric g

= hϕ

|ϕ

on the space of physical operators, see,

e.g., [12]. A particular problem, related to the theory of Frobenius manifolds, is ﬁnding

bases that make this metric ﬂat, i.e., g

= δ

Since (2.4) looks like a limit τ → ∞ of (2.13) it is natural to expect that the residue

pairing (ϕ

−

|ϕ

)

dW,0

should be related to the limit of the intersection number hϕ

−

|ϕ

Before deriving this result in full generality, let us consider the one-dimensional case

dim

M = 1 to gain some intuition about this relationship.

2.2 Example: one-dimensional case

Let us consider a one-dimensional case, where each hypersurface H

is a single point, say

{z = p

}, removed from CP

M = CP

− {p

, p

, . . . , p

} (2.17)

and the corresponding potential is W =

i=1

log(z−p

) with α

’s adding up to zero.

The Euler characteristic is simply χ(M) = 2 − k, which is the same as the number of

critical points, as the solutions of dW = 0 are roots of a degree-(k−2) polynomial in z.

The intersection number of two twisted forms ϕ

−

∈ H

−dW

and ϕ

∈ H

is deﬁned by

hϕ

−

|ϕ

2π

√

−1

−

∧ ϕ

. (2.18)

Let us construct ϕ

explicitly as

= ϕ

− ∇

i=1

Θ(|z−p

−ε

) ∇

−1

, (2.19)

which is manifestly cohomologous to ϕ

. Here Θ(x) is a step function equal to one for x>0

and zero otherwise. In this way, each term in the sum has support on an inﬁnitesimal disk

around p

with radius ε. The inverse diﬀerential ∇

−1

is deﬁned such that ∇

∇

−1

η = η.

In particular, ∇

−1

is a zero-form. Let us check that the resulting form has compact

support by expanding the above expression:

= ϕ

1 −

i=1

Θ(|z−p

−ε

)

−

i=1

δ(|z−p

−ε

) ∇

−1

. (2.20)

The ﬁrst term vanishes inside small disks around each p

and the second term has only

support on the circles with radii ε imposed by Dirac delta functions δ(x). Therefore ϕ

has

compact support. One could have performed the same computation in a smooth way with

bump functions instead of step functions, leading to the same ﬁnal result, see, e.g., [41],

but will not do it here for the sake of clarity.

– 7 –

剩余38页未读，继续阅读

weixin_38524851

粉丝: 6
资源: 944

从高维到四维：费曼积分和残基配对的新应用

费曼积分法费曼积分法费曼积分法

椭圆费曼积分和纯函数

费曼积分和交集理论

通过费曼积分对称性的风筝图

椭圆情形下费曼积分微分方程的形式

超越复杂的Langevin方程：从简单的例子到直接在Minkowski时间内的费曼路径积分的正表示

JaxoDraw:JaxoDraw：轻松实现费曼图！-开源

jquery-feyn：jQuery插件，用于使用SVG绘制费曼图

infeynity:无限费曼图

带标签的树形图，费曼图和磁盘积分

最新资源