SLE环测度的共形场论与Kontsevich–Suhov猜想
The CFT of SLE loop measures and the Kontsevich--Suhov conjecture
AI总结：
本文研究SLE_κ环测度的共形场论，构造Virasoro代数的可交换表示，证明Kac表奇异向量存在性等，首次证明Kontsevich–Suhov猜想的限制测度唯一性，为后续研究奠定基础。
摘要：
本文启动了对κ∈(0,4]时SLE_κ环测度ν的共形场论研究，该区间内的环几乎必然是简单的。首先，我们在L²(ν)上构造了Virasoro代数的两个可交换表示(L_n, L̄_n)_{n∈ℤ}，其中心电荷c_M=1-6(2/√κ - √κ/2)²≤1，为（无界）一阶微分算子。其次，我们引入最高权表示并刻画其结构：特别地，证明了Kac表上任意阶消失奇异向量的存在性。第三，我们证明了SLE环测度的分部积分公式，并用其定义该表示的Shapovalov型，这是L²(ν)上非退化（但非正定）的Hermitian型Q，具有极为简洁的几何表达式。Q与L²(ν)内积的差异是非幺正性的体现。最后，我们利用L₀与L̄₀的联合对角化写出Q的谱分解。作为这些结果的应用，我们首次证明了Kontsevich和Suhov猜想的限制测度唯一性。我们的结果为深入研究SLE的CFT奠定了基础：后续工作中，我们将定义黎曼曲面上的关联函数，并证明共形Ward恒等式、BPZ方程和共形自举公式。
英文摘要：
This paper initiates the study of the conformal field theory of the SLE$_κ$ loop measure $ν$ for $κ\in(0,4]$, the range where the loop is almost surely simple. First, we construct two commuting representations $(\mathbf{L}_n,\bar{\mathbf{L}}_n)_{n\in\mathbb{Z}}$ of the Virasoro algebra with central charge $c_\mathrm{M}=1-6(\frac{2}{\sqrtκ}-\frac{\sqrtκ}{2})^2\leq1$ as (unbounded) first order differential operators on $L^2(ν)$. Second, we introduce highest-weight representations and characterise their structure: in particular, we prove the existence of vanishing singular vectors at arbitrary levels on the Kac table. Third, we prove an integration by parts formula for the SLE loop measure, and use it to define the Shapovalov form of the representation, a non degenerate (but \emph{not} positive definite) Hermitian form $\mathcal{Q}$ on $L^2(ν)$ with a remarkably simple geometric expression. The fact that $\mathcal{Q}$ differs from the $L^2(ν)$-inner product is a manifestation of non-unitarity. Finally, we write down a spectral resolution of $\mathcal{Q}$ using the joint diagonalisation of $\mathbf{L}_0$ and $\bar{\mathbf{L}}_0$. As an application of these results, we provide the first proof of the uniqueness of restriction measures, as conjectured by Kontsevich and Suhov. Our results lay the groundwork for an in-depth study of the CFT of SLE: in forthcoming works, we will define correlation functions on Riemann surfaces, and prove conformal Ward identities, BPZ equations, and conformal bootstrap formulas.
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