AdS5×S5中无张力IIB弦的单环自由能：精确计算与N=4 SYM对齐
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本文主要探讨了AdS 5×S 5背景下的无张力IIB型弦论的单环自由能问题。在这个高维空间中，N=4超 Yang-Mills 理论在零't Hooft耦合极限下，其单迹算子的精确谱可以通过so(2,4)的基本乘方字符来获取。这个理论的进展得益于最近发展起来的zeta函数字符积分表示法（CIRZ），这是一种用于计算量子场论中物理量的有效工具。
文章的核心部分聚焦于计算第(p-1)个Regge轨迹中弦态的单环自由能。作者们首先通过CIRZ方法得到了这些特定轨迹上的自由能，发现它们与p成比例，即单个N=4超Yang-Mills理论中的麦克斯韦多重峰的p倍。这一计算结果导致了一个发散级数，形式为∑p=2∞p，这意味着在常规正则化过程中出现了未被完全处理的部分。
为了得到完整的单环自由能的有限结果，作者们对所有Regge轨迹进行了状态重组，并在求和之后提取了自由能的有限部分。这一过程揭示了单环自由能对于N=4多重态的贡献是负值，这与边界理论中与N²-1因子相关的-1项相吻合。
最终得出的结论是，通过这种方式计算的无张力IIB型弦在AdS 5×S 5背景下的单环自由能，不仅提供了对弦理论的一个重要数值结果，还验证了理论预测与实验观察之间的关联，加深了我们对强相互作用和弦理论本质的理解。这篇论文发表于《Journal of High Energy Physics》(JHEP) 2017年6月，具有开放获取性质，展示了理论物理学家们在探索高能物理领域的最新进展。
JHEP06(2017)155
2 Single-particle states/single-trace operators
In the tensionless limit (α
0
→ ∞ while L
AdS
finite) of type IIB string theory in the
bulk [33–47], the ’t Hooft coupling of the boundary N = 4 SYM goes to zero,
λ = N g
2
YM
=
L
2
AdS
α
0
2
→ 0 , (2.1)
hence all the operators become free from any anomalous dimensions. In this case, the
spectrum of single-trace operators can be exactly identified by group theoretical and com-
binatorial methods [48–51]. Via the AdS/CFT correspondence, we identify this with the
spectrum of single-particle states in the bulk and consider the corresponding bulk theory,
following [52], as our working definition for tensionless string theory. By exploiting this
definition, one can collect and sum over the one-loop free energies of all single-particle
states in the string spectrum [31, 32].
For the group theoretical identification of the operators made by p insertions of field
in a single trace,
Tr(Φ
1
· · · Φ
p
) , Φ
i
: any field (or its derivative) of N = 4 SYM , (2.2)
we need to consider the p-th tensor product of the N = 4 Maxwell multiplet and project
it to the singlet under cyclic permutations. Each of the conformal primaries lying in this
tensor product space is identified via the AdS/CFT correspondence to a field in AdS
5
as
mentioned above. Henceforth, for notational convenience, we define the (p − 1)-th Regge
trajectory to be the set of all such fields.
The issue of tensor products and decompositions can be conveniently analyzed in terms
of the so(2, 4) character. The cyclicity is required by the property of trace operation, and
handling the cyclic projection is equivalent to the combinatorial problem of counting the
number of different necklaces with p beads where each bead corresponds to a state — either
primary or descendant — of N = 4 Maxwell multiplet. Polya’s enumeration theorem solves
this problem by making use of the cyclic index.
Partition function of N = 4 Maxwell multiplet. The cyclic index can be used to
account the tensor products of so(2, 4) representation H, if the index variables are replaced
by Tr
H
(g
k
) with g ∈ SO(2, 4) . In this way, the cyclic index of a bosonic system gives
nothing but the so(2, 4) character. For more general systems with fermionic degrees of
freedom, we should consider the weighted partition function [13, 53],
Z
H
(q, x
1
, x
2
) = Tr
H
(−1)
F
q
M
05
x
M
12
1
x
M
34
2
, (2.3)
where M
ab
are the generators of so(2, 4) and F is the fermion number. Even though we
refer to this as partition function following previous references, it should be distinguished
from the exponential of the one-loop free energy that we shall calculate in the next section.
It can be equally viewed as a generalized Witten index. For an irreducible representation
of so(2, 4) , it is proportional to the character up to a sign factor:
Z
D(∆,[j
+
,j
−
])
= (−1)
2(j
+
+j
−
)
χ
D(∆,[j
+
,j
−
])
, (2.4)
– 3 –
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