The theory of virtual fundamental class defines important invariants such as the Gromov–Witten and the Donaldson– Thomas invariants. It has been generalized to the cosection localized virtual cycle which has applications in Seiberg– Witten, Fan–Jarvis–Ruan–Witten and other invariants. In this paper, we prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles.

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This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014. We describe and characterize all random subsets $K$ of simply connected domain which satisfy the ``conformal restriction" property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\HH$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\HH$ fixing 0 and $\infty$. In the radial case, the random set $K$ in the upper half-plane $\HH$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\HH$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\HH$ fixing 0 and $i$. It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.

We compute the almost-sure Hausdorff dimension of the double points of chordal SLE$_\kappa$ for $\kappa > 4$, confirming a prediction of Duplantier--Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE$_\kappa$ for $\kappa > 4$ as well as analogous dimensions for the radial and whole-plane SLE$_\kappa(\rho)$ processes for $\kappa > 0$. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field $e^{ih/\chi}$, where $h$ is a Gaussian free field and $\chi > 0$, of different angles with each other and with the domain boundary.

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