Bruhat-Tits树上的自旋场理论边界理论研究
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Bruhat-Tits树上的自旋场理论的边界理论
Bruhat-Tits树是一种特殊的p-adic数空间，它在数论和表示论中有着重要的应用。在物理学中，Bruhat-Tits树也被用于研究场论，特别是自旋场理论。本文研究的主题是关于Bruhat-Tits树上的自旋场理论的边界理论，我们通过整合Bruhat-Tits树的内部来计算其边界论的作用和分配函数。
自旋场理论是场论的一种，它描述了自旋粒子的行为。在高能物理学中，自旋场理论是研究粒子相互作用的重要工具。然而，在研究自旋场理论时，我们经常会遇到计算复杂度高的问题。为了解决这个问题，我们可以使用边界理论来研究自旋场理论的行为。
边界理论是一种研究场论的方法，它通过研究场论在边界上的行为来了解场论的性质。在Bruhat-Tits树上的自旋场理论中，我们可以使用边界理论来研究其行为。通过整合Bruhat-Tits树的内部，我们可以计算出自旋场理论的边界论的作用和分配函数。
我们的研究结果表明，Bruhat-Tits树上的自旋场理论的边界理论与p-adic数上的标量场理论非常相似。这意味着，我们可以使用p-adic数上的标量场理论来研究Bruhat-Tits树上的自旋场理论。这将有助于我们更好地理解自旋场理论的行为。
在我们的研究中，我们还发现了Bruhat-Tits树上的自旋场理论的其他一些性质。例如，我们发现了自旋场理论的边界论的作用和分配函数之间的关系。这将有助于我们更好地理解自旋场理论的行为。
本文研究了Bruhat-Tits树上的自旋场理论的边界理论，我们发现了自旋场理论的边界论与p-adic数上的标量场理论之间的相似性。这将有助于我们更好地理解自旋场理论的行为，并为进一步研究提供了新的方向。
关键词：Bruhat-Tits树、自旋场理论、边界理论、p-adic数、标量场理论。
在这篇文章中，我们还讨论了Bruhat-Tits树上的自旋场理论的其他一些性质，例如其边界论的作用和分配函数之间的关系。这将有助于我们更好地理解自旋场理论的行为。
本文的研究结果将有助于我们更好地理解Bruhat-Tits树上的自旋场理论的行为，并为进一步研究提供了新的方向。
Physics Letters B 803 (2020) 135331
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The boundary theory of a spinor field theory on the Bruhat-Tits tree
Feng Qu
a,b,∗
, Yi-hong Gao
a,b
a
School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, China
b
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received 26 October 2019
Received in revised form 21 February 2020
Accepted 21 February 2020
Available online 26 February 2020
Editor: M. Cveti
ˇ
c
Keywords:
Anti-de Sitter/conformal field theory
correspondence
Gauge/gravity correspondence
p-adic number
Spinor field
Bruhat-Tits tree
For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of
its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary
theory is very similar to a scalar field theory over p-adic numbers.
© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The applications of p-adic analysis to string theory have provided numerous insights in the study of the AdS/CFT correspondence [1–3].
In the early time, Freund and Olson [4] considered a kind of string world sheets over p-adic numbers (Q
p
), and gave some expressions
for the p-adic string amplitudes. Later on, Zabrodin [5]found a specific realization of such kind of world sheets in term of the Bruhat-Tits
tree (T
p
). According to Zabrodin’s paper, a boundary theory, which is different from the “CFT” in the AdS/CFT correspondence, can be
obtained by integrating out the interior of T
p
. The AdS/CFT correspondence on T
p
is proposed in [6,7]. Some further developments are
given in [8–21].
Zabrodin only considered a massless scalar field. Recently, the spinor field theory on T
p
has been proposed by Gubser, Jepsen and
Trundy [15]. They succeed in taking the square root of the Laplacian “”. Let φ
a
denote a field on the vertices (a vertex-field) of T
p
. φ
a
’s
on all vertices can be organized into a column vector φ ≡ (φ
a
, φ
b
, φ
c
, ···)
T
, where “T” represents the transposition. The action of on φ
can be written as the matrix multiplication:
(φ)
a
:=
b∈∂a
(φ
a
− φ
b
), ()
a,a
= p + 1and()
a,b∈∂a
=−1 . (1)
(·)
a
gives the entry in row a, and (·)
a,b
gives that in row a and column b. b ∈ ∂a means that b is one of the nearest neighboring vertices
of the given vertex a, in other words, b belongs to the boundary of a. Imposing a directed structure on T
p
, the Laplacian has a square
root d, which is a matrix whose row index takes value in edges and column index takes value in vertices. The result of d’s action on a
vertex-field is a field on edges (an edge-field), and the result of d
T
’s action on an edge-field is a vertex-field. Let s(e) and t(e) denote the
starting point and the terminal point of edge e. The matrix d satisfies
(dφ)
e
:= φ
t(e)
− φ
s(e)
,(d
T
χ)
a
=
t(e)=a
χ
e
−
s(e)=a
χ
e
, = d
T
d . (2)
*
Corresponding author at: CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China.
E-mail addresses: qufeng@itp.ac.cn (F. Qu),
gaoyh@itp.ac.cn (Y.-h. Gao).
https://doi.org/10.1016/j.physletb.2020.135331
0370-2693/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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