标题： 超弦JT引力中的代数和状态 显示英文标题

标题： Algebras and states in super-JT gravity

摘要： 在玻色子JT引力中，与体物质最小耦合，在每个物质QFT的$SL(2,R)$表示中，对于左边界和右边界上的任意正能对$E_L, E_R$，存在一个单个的、δ函数可归一化的态。 在$\mathcal{N} = 2$超JT引力与物质耦合的情况下，我们证明在每个$SU(1,1|1)$物质表示中存在一个单个的可归一化态（给定适当的R电荷），该态在两个边界处的能量恰好为零。 对于非BPS表示，这些态具有奇特的性质，即它们在体中破坏所有超对称性，而在两个边界处保持超对称性。 将边界可观测量的代数投影到这些零能态上，会在每个边界处产生一个类型II$_1$冯诺依曼因子，该因子对每个具有足够小R电荷的超对称物质边界原初算符包含一个单独的算符。 对于中性边界原初算符，类型II$_1$因子在物质QFT希尔伯特空间（没有额外的引力自由度）上有自然的作用，使得QFT真空是唯一的迹态。 此外，中性物质算符的乘积可以非常明确地找到，并且具有令人惊讶的简单形式。 当包含具有非零物质R电荷的初级粒子时，迹可以写成每个允许的边界R电荷$J_R$相关的物质真空期望值的和，求和项的权重为$\mathrm{cos}(\pi J_R)$。以这种方式，基态代数编码了每个R电荷区段内BPS微观状态数量的比值。除了上述关于超JT引力的结果外，我们提供了一种纯洛伦兹推导，对规范量子化（玻色子）JT引力加物质的代数结构进行推导，而无需诉诸于之前工作中使用的欧几里得引力路径积分。 显示英文摘要

摘要： In bosonic JT gravity, minimally coupled to bulk matter, there exists a single, delta-function-normalisable state in each $SL(2,R)$ representation of the matter QFT for any pair of positive energies $E_L, E_R$ at the left and right boundaries. In $\mathcal{N} = 2$ super-JT gravity coupled to matter, we show that there exists a single normalisable state in each $SU(1,1|1)$ matter representation (given appropriate R-charges) that has exactly zero energy at both boundaries. For non-BPS representations, these states have the peculiar property that they break all supersymmetry in the bulk, while preserving supersymmetry at both boundaries. Projecting the algebras of boundary observables onto these zero-energy states leads to a Type II$_1$ von Neumann factor at each boundary that contains a single operator for each supersymmetric matter boundary primary with sufficiently small R-charge. For neutral boundary primaries, the Type II$_1$ factor has a natural action on the matter QFT Hilbert space (with no additional gravitational degrees of freedom) such that the QFT vacuum is the unique tracial state. Moreover, the product of neutral matter operators can be found very explicitly and has a remarkably simple form. When primaries with nonzero matter R-charge are included, the trace can be written as a sum over matter vacuum expectation values associated to each allowed boundary R-charge $J_R$, with the terms in the sum weighted by $\mathrm{cos}(\pi J_R)$. In this way, the ground state algebras encode the ratios of the number of BPS microstates within each R-charge sector. In addition to the results on super-JT gravity described above, we provide a purely Lorentzian derivation of the algebraic structure of canonically quantised (bosonic) JT gravity plus matter, without appeal to the Euclidean gravitational path integrals used in previous work.