用于DSSYK的ETH矩阵模型：非微扰修正与相交理论
The ETH matrix model for DSSYK: non-perturbative corrections and intersection theory
AI总结：
本文研究用于DSSYK的ETH矩阵模型，揭示其非微扰修正由\widetilde q控制，将其与稳定曲线模空间的κ类相交数关联，扩展了相关计算至(3,1)等未得闭式的情形。
AI中文摘要：
在亏格展开的领头阶，用于DSSYK的ETH矩阵模型可按构造重现其关联函数，而其更高亏格修正被猜想为捕捉对偶 sine-dilaton 引力中的更高拓扑贡献——迄今为止，这种对应仅在圆盘和虫洞情形下得到验证。在固定亏格下，关联函数由离散体积N_{g,n}构建，该体积是q-变形ζ值ζ_q(2k)的多项式，其中q=e^{-λ}，λ为DSSYK耦合常数。这些离散体积属于由艾森斯坦级数E_2、E_4、E_6生成的拟模形式环，其S-对偶性给出λ→0时领头阶非微扰修正的精确闭式，由\widetilde q=e^{-4π²/λ}控制。该尺度此前仅在圆盘水平已知，本文证明其同样支配固定亏格的更高边界振幅。我们表明，λ领头阶下与\widetilde q线性相关的项完全由q-变形的Weil-Petersson体积捕捉，且可简化为稳定曲线模空间\overline{\mathcal M}_{g,n}上κ类相交数的有限和，无需重复生成N_{g,n}的拓扑递归即可计算。我们对所有已得q-变形体积闭式的(g,n)情形编制了该相交数的表格，并将其扩展至(3,1)、(3,2)、(4,1)，这些情形尚无对应闭式。该构造并不限于领头阶：我们还针对相同情形计算了O(\widetilde q²)阶的项。
英文摘要：
At leading order in the genus expansion the ETH matrix model for DSSYK reproduces its correlators by construction, while its higher-genus corrections are conjectured to capture higher-topology contributions in the dual sine-dilaton gravity -- a correspondence established so far only for the disk and the wormhole. At fixed genus the correlators are built from discrete volumes $N_{g,n}$, polynomial in $q$-deformed zeta values $ζ_q(2k)$ with $q=e^{-λ}$, $λ$ being the DSSYK coupling. These lie in the ring of quasimodular forms generated by the Eisenstein series $E_2,E_4,E_6$, whose $S$-duality yields an exact closed form for the leading non-perturbative correction as $λ\to0$, controlled by $\widetilde q=e^{-4π^2/λ}$. Known at disk level, this scale is shown here to govern the fixed-genus, higher-boundary amplitudes as well. We show that the term linear in $\widetilde q$, at leading order in $λ$, is captured entirely by the $q$-deformed Weil--Petersson volumes, and reduces to a finite sum of intersection numbers of $κ$-classes on the moduli space $\overline{\mathcal M}_{g,n}$ of stable curves, computable without repeating the topological recursion that produced the $N_{g,n}$. We tabulate it for every $(g,n)$ whose $q$-deformed volume is known in closed form, and extend it to $(3,1),(3,2),(4,1)$, where none is available. The construction is not restricted to leading order: we work out $O(\widetilde q^2)$ for the same cases.
