超对称与瓜图：Melonic超张量模型的探索
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更新于2024-07-16
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"这篇研究论文探讨了在Melonic超张量模型上的理论，这是一种具有两个超荷的超对称量子力学理论，其中的自由度是以张量形式存在的。该研究在大N极限下进行，主要关注瓜图（melon diagrams）如何控制这类系统的性质。文章的目的是探索超对称性与黑旋律优势（melonic dominance）之间的相互作用，以及这种相互作用可能对全息原理的玩具模型构建产生的影响。
研究发现，在这样的理论体系中，超对称性（包含动态玻色子）与黑旋律优势之间存在明显的张力。具体来说，这些理论在低能量状态下会达到一个非超对称的共形固定点。超对称性的破坏源于需要分别对玻色子和费米子的自由度进行独立的规范化处理。
作者Chi-Ming Chang、Sean Colin-Ellerin和Mukund Rangamani来自加州大学戴维斯分校的量子数学和物理中心，他们在论文中深入研究了这个现象。他们通过分析不同类型的图，尤其是瓜图，揭示了模型中的动态行为。瓜图是一种特定的图论构造，在某些统计力学和量子场论模型中特别重要，它们在大N极限下主导着系统的演化。
此外，由于这类模型可以作为全息原理的简化模型，因此研究它们对于理解量子引力和黑洞物理学有着重要的意义。全息原理提出信息的存储和计算能力可以在一个系统的边界上完全表达，而无需考虑其内部的复杂性。超对称性和黑旋律优势的相互作用为构建更有效的全息模型提供了新的视角。
在论文中，作者还讨论了如何通过不同的规范方法来处理玻色子和费米子的度规，以及这些方法如何影响超对称性的保持或破坏。他们还探讨了这些发现可能对理解和解决超对称性破缺问题的潜在贡献，特别是在高能物理和弦理论的背景下。
这项工作提供了对超对称量子力学理论的新见解，尤其是在大N极限下，它揭示了超对称性与黑旋律优势之间微妙的平衡，以及这一平衡如何在特定条件下被打破。这将促进对全息原理更深层次的理解，并可能启发新的理论构造和计算技术。"
JHEP10(2018)157
βJ = 50
βJ = 200
0.5 1.0 1.5 2.0 2.5 3.0
2 π τ
β
0.002
0.004
0.006
0.008
0.010
β
-1
G(τ)
Figure 2. Comparison of numerical (red) and analytic (blue) solutions of the regularized bosonic
Schwinger-Dyson equations (2.17) for two different values of βJ as indicated. The numerical sim-
ulation is carried out with the imaginary time circle discretized by a lattice with 200 points (see
footnote 9).
Note that the problem is unique to bosonic degrees of freedom. Fermionic tensor models
are much better behaved; indeed, the self-energy integral suffers from no UV divergence
issues either in the conformal limit or along the flow. The reason can be traced to the Fermi
statistics which in the IR limit give rise to a conformal propagator G
c
(τ) =
b
|τ |
2∆
sgn(τ) at
zero temperature. The sign function ends up ensuring the self-energy is free of divergences.
We will take inspiration from this analysis for the case of the supersymmetric tensor model
we introduce shortly.
3 The N = 2 supersymmetric tensor model
We now turn to the main model we wish to analyze, a quantum mechanical supertensor
model with N = 2 supersymmetry. This amount of supersymmetry turns out to provide
an interesting interaction term. Similar attempts to construct a theory with one super-
charge lead to an interaction involving an odd number of fermions, while higher number of
supercharges lead to derivative couplings between the component fields (see appendix A).
We will start by introducing the model. It will be convenient to start out in superspace
R
1|2
with coordinates t, θ,
¯
θ (t is the real time coordinate). The basic superfield Φ
A
q
will be tensor-valued as in the bosonic model, so much of the structure is actually quite
straightforward to intuit if we stick to superspace.
3.1 The model
We consider an N = 2 supersymmetric model in (0 + 1)-dimensions with superfields Φ
A
q
transforming in the (q − 1)-fundamental representation of O(N )
q−1
for q ≥ 4 even. These
superfields can be written in terms of component fields on superspace as
Φ
A
q
(t, θ,
¯
θ) = φ
A
q
(t) + i θ
¯
ψ
A
q
(t) + i
¯
θ ψ
A
q
(t) + θ
¯
θ F
A
q
(t), (3.1)
where φ
A
q
, F
A
q
are bosonic and ψ
A
q
,
¯
ψ
A
q
are fermionic.
The action will be given as a superspace integral with canonical kinetic terms along
with a superpotential W (Φ
A
q
). Taking inspiration from the non-supersymmetric tensor
– 9 –
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