JHEP03(2019)185
The vacuum is gauge invariant (a singlet of U(N)). Any state in the dynamics can
be accessed by raising operators acting on the vacuum. However, we need to impose the
Gauss’ law constraint. A fermion ψ
a
b
has an upper and lower U(N) index. These need to
be fully contracted to form a singlet. These contractions are in the form of traces. For
example, a single trace state acting on the vacuum is given by
|ki = tr(ψ
k
) |0i (2.2)
Using the cyclic property of the trace and the fermionic character of ψ it can be easily
shown that
tr(ψ
k
) = (−1)
k−1
tr(ψ
k
) (2.3)
so that only traces with an odd number of fields ψ are allowed. Each of these traces has
fermi statistics and has energy k = 2s + 1, where s = 0, 1, . . . . At infinite N each trace is
supposed to correspond to a different ‘particle’.
If we normalize the energies in units of 1/2, we get a single particle state for each half
integer (s + 1/2). This is the same spectrum of states as a free chiral fermion in 1 + 1
dimensions (a c = 1/2 CFT) on a circle with NS boundary conditions [13] (antiperiodic
boundary conditions χ(θ + 2π) = −χ(θ)).
This suggests that this is a matrix model for a c = 1/2 CFT in a similar way that
a bosonic gauged matrix model gives rise to a c = 1 chiral boson in 1 + 1 dimensions.
The main insight of this map to the chiral boson is that in the bosonic matrix model one
can reduce the dynamics to the eigenvalues of the matrix X. When the Gauss’ law is
implemented the eigenvalues act as fermions. This dynamics for bosons is usually best
described in terms of a first order dynamics
S =
Z
dt tr
¯
ZiD
t
Z −m
¯
ZZ
(2.4)
where Z = X + iP and
¯
Z are complex and the Gauss’ law constraint requires them to
commute. The corresponding fermions are in the phase space of X and the ground state
can be described by a droplet in phase space. This is a familiar story of the quantum hall
effect. The eigenvalue dynamics is the effective field theory of free electrons in the lowest
Landau level in 2 + 1 dimensions, which has been slightly deformed by a binding potential
proportional to x
2
+ y
2
. The traces tr(Z
k
) are collective excitations of the droplet with
angular momentum k on the edge.
Unlike the bosonic matrix model, in the fermionic setup we can not choose a gauge
where we diagonalize the fermionic degrees of freedom. The interpretation of an edge
dynamics is harder to do and will not be pursued here.
Because the bosonic system can also be interpreted in terms of first quantized fermions
in a bulk 2+1 system, we can also write the wave functions in terms of Slater determinants
of single particle states. These wave functions are governed by Schur polynomials of the
matrices. These are obtained by traces in irreducible representations of U(N), which are
labeled by Young diagrams. The map from multi-traces to Schur functions in the bosonic
model is non-trivial [14]. It is also obtained from character expansions of the symmetric
– 3 –
剩余17页未读，继续阅读
weixin_38719635
- 粉丝: 3
- 资源: 972
我的内容管理
展开
最新资源
- 贵州煤矿矿井水分类与处理策略：悬浮物、酸性与非酸性
- 醛固酮增多症肾上腺静脉采样对比：ACTH后LR-CAV的最优评估
- 开源云连接传感器监控平台：农业土壤湿度远程监测
- 母婴用品企业年度生产计划线性规划优化模型：实证与应用
- 井下智能变电站：Rogowski线圈电流检测系统的研发与性能验证
- 霍州矿区煤巷稳定性分析及支护策略
- ARM嵌入式系统远程软件更新方案：基于TFTP协议
- 煤炭选煤中汞分布规律与洗选脱汞效果
- 提升码垛机器人性能：拉格朗日动力学模型与滑模模糊控制的应用
- 增强现实技术提升学前手写教学：设计与开发案例
- 不规则工作面沉陷三角剖分算法提升与应用
- 卡尔曼滤波在瞬变电磁干扰压制中的应用研究
- 煤矿安全能力研究：理论与系统构建
- LonWorks总线技术在斜巷运输车辆定位与跑车防护中的应用
- 神东煤炭集团高效煤粉锅炉系统：节能环保新实践
- Ti/SnO2+Sb2Ox/PbO2电极分形维数与电催化性能研究
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
