AI 中文总结
研究双标度 SYK 模型圆盘配分函数\(Z(\beta)\)低温展开，利用准模艾森斯坦级数及其微分恒等式和模\(S\)对偶性质，推导半经典低温展开，确定非微扰修正，证明其与双局部 - 刘维尔鞍点壳上作用指数相关。
AI 中文摘要
我们研究了双标度 SYK 模型（DSSYK）在固定耦合\(\lambda = 2p^{2}/N\)下圆盘配分函数\(Z(\beta)\)的低温展开，其中\(N\)是马约拉纳费米子数，\(p\)是每个相互作用项中的费米子数，二者均趋于无穷。我们表明，\(Z(\beta)\)在大自变量（对应低温）下展开的精确贝塞尔函数表示，可以根据准模艾森斯坦级数\(E_{2}\)、\(E_{4}\)、\(E_{6}\)的经典环及其微分恒等式来组织。利用该环的模\(S\)对偶性质，我们推导了\(Z(\beta)\)的半经典（小\(\lambda\)）低温展开，将其分为微扰塔和由\(\widetilde q = e^{-4\pi^{2}/\lambda}\)控制的非微扰部分。在\(\widetilde q\)的每一阶，我们确定了\(\lambda\)二阶以内的非微扰修正的封闭形式；所得级数可重整为同一艾森斯坦级数中的紧凑表达式，将先前的半经典结果扩展到严格的\(\beta\to\infty\)极限之外。我们进一步表明，整个结构源于一个将模导数与温度导数耦合的单一精确微分方程。最后，我们证明\(Z(\beta)\)的非微扰部分在\(\lambda\)的所有阶上，精确地由与 DSSYK 施瓦茨极限的已知双局部 - 刘维尔鞍点的壳上作用相同的指数所支持，这表明这些非微扰修正有明确的体起源。
英文摘要
We study the low-temperature expansion of the disk partition function $Z(β)$ of the double-scaled SYK model (DSSYK) at fixed coupling $λ=2p^{2}/N$, where $N$ is the number of Majorana fermions and $p$ is the number of fermions in each interaction term, both taken to infinity. We show that the exact Bessel-function representation of $Z(β)$, expanded at large argument (corresponding to low temperature), can be organized in terms of the classical ring of quasi-modular Eisenstein series $E_{2},E_{4},E_{6}$ and their differential identities. Exploiting the modular $S$-duality properties of this ring, we derive the semiclassical (small $λ$) low-temperature expansion of $Z(β)$, splitting it into a perturbative tower and a non-perturbative sector controlled by $\widetilde q=e^{-4π^{2}/λ}$. At each order in $\widetilde q$, we determine the non-perturbative correction in closed form up to second order in $λ$; the resulting series resums into a compact expression in the same Eisenstein series, extending previous semiclassical results beyond their strict $β\to\infty$ limit. We further show that this entire structure follows from a single, exact differential equation coupling a modular derivative to derivatives with respect to temperature. Finally, we prove that the non-perturbative sector of $Z(β)$ is exactly supported, to all orders in $λ$, on the same exponents as the on-shell actions of known bilocal-Liouville saddles of the DSSYK Schwarzian limit, pointing to a well-defined bulk origin for these non-perturbative corrections.
Comments23 pages. v2: more references
