We study invariants of bosonic and fermionic (Grassmann-valued) matrices under the adjoint action of $U(N)$, weighted by the fermion number. Such models naturally appear as the supersymmetric indices of supersymmetric gauge theories and are captured by $U(N)$ matrix models. We discuss two features of the fermionic models that are qualitatively different from bosonic models. Firstly, the $2N^\text{th}$ power of a Grassmann matrix vanishes, which gives rise to many new trace relations. Secondly, trace relations in models involving fermions could cause an increase in the supersymmetric index as $N$ decreases, in contrast with purely bosonic models. We focus on a simple model involving one fermion and one derivative that corresponds to a $\frac14$-BPS supersymmetric index in $\mathcal{N}=4$ SYM theory, in which we find that the index is independent of $N$. We prove this rank-independence analytically, and experimentally study the cancellations between bosonic and fermionic trace relations that lead to it. Based on these observations, we make some conjectures on resulting algebraic structures, including the analogue of the polarized Cayley-Hamilton identities and the Second Fundamental Theorem of invariants in the presence of Grassmann matrices. Finally, we present various (smooth and singular) limits of the most general supersymmetric index in $\mathcal{N}=4$ SYM theory, and study some patterns in their behavior as a function of $N$.
Mirror symmetry of 3d $\mathcal{N}=4$ gauge theories and supersymmetric indices
我们计算超对称指数来检验三维$\mathcal{N}=4$规范理论的镜像对称性和四维$\mathcal{N}=4$ Super Yang-Mills理论中半BPS富集边界条件和界面的对偶性。在IIB型弦论的S-对偶作用下，我们发现指数的匹配是Gaiotto和Witten提出的三维界面的各种对偶性的有力证据.
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；School of Physics and Shing-Tung Yau Center, Southeast University（东南大学物理学院及陈省身数学中心）
We compute the contribution to the anomalous dimension of the twist-2 operators in N=4 SYM theory, which is proportional to the number of fermion loops inside Feynman diagrams or, formally, to the number of fermions. The result was obtained by the method based on the calculation of critical indices at the critical point by analogy with previous similar computations in scalar theories and in QCD. The obtained result is much simpler with compare to analogous results in QCD and almost satisfies the maximal transcedentality principle. A possible relation between the obtained result and integrabili…
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）；School of Physics and Shing-Tung Yau Center, Southeast University（东南大学物理学院及陈省身数学中心）
We propose a supersymmetric quantum field theory with exotic symmetry related to fracton phases. We use superfield formalism and write down the action of a supersymmetric version of the $φ$ theory in 3+1 dimensions. It contains a large number of ground states due to the fermionic higher pole subsystem symmetry. Its residual entropy is proportional to the area instead of the volume. This theory has a self-duality similar to that of the $φ$ theory. We also write down the action of a supersymmetric version of a tensor gauge theory, and discuss BPS fractons.
Fermionic T-duality in massive type IIA supergravity on AdS_{10-k} x M_k
研究了质量为AdS_{10-k} x M_k（k=3和5）的质量IIA型超引力N=1超对称解的费米T对偶变换.我们推导出这些背景的杀旋量，并使用它们作为输入的费米T-对偶变换。由此产生的双重解决方案形成一个大家庭的超对称变形的原始解决方案的复值RR通量。我们观察到Romans质量参数在费米T对偶下不发生变化，并证明了它在k=3的情况下是不变性的.
We construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the "Area-law" entropy due to fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and the ground-state degeneracy shows the "Volume-law" entropy. Gauging the fermionic subsystem symmetry has an t'Hooft anomaly by itself, but the would-be gauged theory may possess a fermionic defect that is immobile in ce…
It is known previously that the operator product expansion (OPE) between the first ${\cal N}=3 $ multiplet and itself contains the second ${\cal N}=3$ multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$ where the group of the coset contains $SU(N+M)$. By supersymmetrizing the above OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities between the currents, we determine the ${\cal N}=3$ supersymmetric…
We study three dimensional $\mathcal{N}=2$ supersymmetric theories on $I \times M_2$ with 2d $\mathcal{N}=(0,2)$ boundary conditions at the boundaries $\partial (I \times M_2)=M_2 \sqcup M_2$, where $M_2=\mathbb{C}$ or $ T^2$. We introduce supersymmetric indices of three dimensional $\mathcal{N}=2$ theories on $I \times T^2$ that couple to elliptic genera of 2d $\mathcal{N}=(0,2)$ theories at the two boundaries. We evaluate the $I \times T^2$ indices in terms of supersymmetric localization and study dualities on the $I \times M_2$. We consider the dimensional reduction of $I \times T^2$ to $I …
This thesis studies the algebro-geometric aspects of supersymmetric abelian gauge theories in three dimensions. The supersymmetric vacua are demonstrated to exhibit a window phenomenon in Chern-Simons levels, which is analogous to the window phenomenon in quantum K-theory with level structures. This correspondence between three-dimensional gauge theories and quantum K-theory is investigated from the perspectives of semi-classical vacua, twisted chiral rings, and twisted indices. In particular, the twisted index admits an algebro-geometric interpretation as the supersymmetric index of an effect…
Proposals are made to describe 1D, N = 4 supersymmetrical systems that extend SYK models by compactifying from 4D, N = 1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via integrals over the whole superspace, while 2(q-1)-point fermionic vertices are generated via superpotentials. The coupling constants in the superfield Lagrangians are arbitrary, and can be chosen to be Gaussian random. In that case, these 1D, N = 4 supersymmetric SYK models would exhibit Wishart-Laguerre randomness, which share the same feature among …
