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We construct balanced metrics on the family of non-Khler Calabi-Yau threefolds that are obtained by smoothing after contracting ( 1, 1) -rational curves on a Khler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of ( 1, 1) copies of ( 1, 1) .

We establish a compact analogue of the P=W conjecture. For a projective irreducible holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result: one on the cohomology of the base and fibers of a Lagrangian fibration, and the other on the refined Gopakumar–Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.

We propose a renormalization group flow with emergent supersymmetry in two dimensions from a non-Lagrangian theory. The ultraviolet theory does not have supersymmetry while the infrared theory does. The flow is constrained analytically by topological defect lines including a new spin constraint, and further supported by numerics from the truncated conformal space approach.

We investigate the angular distribution of Ly$\alpha$ photons transferring in or emergent from an optically thick medium. Since the evolutions of specific intensity $I$ in the frequency space and the angular space are coupled with each other, we first develop the WENO numerical solver in order to find the time-dependent solutions of the integro-differential equation of $I$ in frequency and angular space simultaneously. We first show that the solutions with the Eddington approximation, which assume $I$ to be linearly dependent on the angular variable $\mu$, yield similar frequency profiles of the photon flux as that without the Eddington approximation. However, the solutions of the $\mu$ distribution evolution are significantly different from that given by Eddington approximation. First, the angular distribution of $I$ are found to be substantially dependent on the frequency of photons. For photons with the resonant frequency $\nu_0$, $I$ contains only a linear term of $\mu$. For photons with frequency at the double peaks of the flux, the $\mu$-distribution is highly anisotropic, in which most photons are in the direction of radial forward. Moreover, either at $\nu_0$ or at the double peaks, the $\mu$ distributions actually are independent of the initial $\mu$ distribution of photons of the source. This is because the photons with frequency either of $\nu_0$ or of the double peaks undergo the process of forgetting their initial conditions due to the resonant scattering. We also show that the optically thick medium is a collimator of photons at the double peaks. Photons from the double peaks form a forward beam with very small spread angle.