环 O(n) 模型中的离散应力-能量张量
Discrete stress-energy tensor in the loop O(n) model
AI总结：
本文在蜂窝晶格环 O(n) 模型中通过局部非平面变形构造离散应力-能量张量，猜想其离散全纯并收敛到连续应力-能量张量，并在 n=1（Ising 模型）情形下证明了该猜想及相关关联函数收敛。
摘要：
我们研究蜂窝晶格上的环 O(n) 模型。通过对晶格进行局部非平面变形，我们构造了一个离散应力-能量张量。对于 n∈[0,2]，它给出了一个满足部分柯西-黎曼方程的新可观测量。我们猜想它近似离散全纯，并在连续极限下收敛到应力-能量张量，已知后者是具有 Schwarzian 共形协变性的全纯函数。为支持这一猜想，我们在 n=1 的情形下证明了它，该情形对应于 Ising 模型。此外，在这一情形下，我们证明了离散应力-能量张量与主场的关联函数收敛到其连续对应物，后者满足中心荷 c=1/2 的共形场论给出的算子乘积展开。证明该猜想对其他 n 值成立仍然是一个挑战。特别地，这将为在标度极限下建立界面到相应 SLE_κ 的收敛开辟一条道路。
英文摘要：
We study the loop $O(n)$ model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [0,2]$, it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of $n=1$ which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge $c=1/2$. Proving the conjecture for other values of $n$ remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding $\mathrm{SLE}_κ$ in the scaling limit.
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