Riemann-Hilbert correspondence for unit FF-crystals on embeddable algebraic varieties

For a separated scheme $X$ of finite type over a perfect field $k$ of characteristic $p>0$ which admits an immersion into a proper smooth scheme over the truncated Witt ring $W_{n}$, we define the bounded derived category of locally finitely generated unit $F$-crystals with finite Tor-dimension on $X$ over $W_{n}$, independently of the choice of the immersion. Then we prove the anti-equivalence of this category with the bounded derived category of constructible \'etale sheaves of ${\mathbb Z}/{p^{n}{\mathbb Z}}$-modules with finite Tor dimension. We also discuss the relationship of $t$-structures on these derived categories when $n=1$. Our result is a generalization of the Riemann-Hilbert correspondence for unit $F$-crystals due to Emerton-Kisin to the case of (possibly singular) embeddable algebraic varieties in characteristic $p>0$.Comment: This is the final version, to appear in Annales de l'Institut Fourie

On logarithmic nonabelian Hodge theory of higher level in characteristic p

Given a natural number $m$ and a log smooth integral morphism $X\to S$ of fine log schemes of characteristic $p>0$ with a lifting of its Frobenius pull-back $X'\to S$ modulo $p^{2}$, we use indexed algebras ${\cal A}_{X}^{gp}$, ${\cal B}_{X/S}^{(m+1)}$ of Lorenzon-Montagnon and the sheaf ${\cal D}_{X/S}^{(m)}$ of log differential operators of level $m$ of Berthelot-Montagnon to construct an equivalence between the category of certain indexed ${\cal A}_{X}^{gp}$-modules with ${\cal D}_{X/S}^{(m)}$-action and the category of certain indexed ${\cal B}_{X/S}^{(m+1)}$-modules with Higgs field. Our result is regarded as a level $m$ version of some results of Ogus-Vologodsky and Schepler

Resonating-valence-bond liquid in low dimensions

The Hubbard model in $D$ dimensions, with the on-site repulsion $U$ and the transfer integral between nearest neighbors $-t/\sqrt{D}$, is studied on the basis of the Kondo-lattice theory. If $U/|t| \gg 1$, $|n - 1| \lesssim |t|/(DU)$, where $n$ is the number of electrons per unit cell, and $D$ is so small that $|J|/D \gg k_{\rm B}T_c$, where $J = -4t^2/U$ and $T_c$ is $0 {\rm K}$ for $D = 1$ and is the highest critical temperature among possible ones for $D \ge 2$, a low-$T$ phase where $T_c < T \ll |J|/(k_{\rm B}D)$ is a frustrated electron liquid. Since the liquid is stabilized by the Kondo effect in conjunction with the resonating-valence-bond (RVB) mechanism, it is simply the RVB electron liquid; in one dimension, it is also the Tomonaga-Luttinger liquid. The Kondo energy of the RVB liquid is $k_{\rm B}T_{\rm K} = O(|J|/D)$; its effective Fermi energy is $O(k_{\rm B}T_{\rm K})$. A midband appears on the chemical potential between the upper and lower Hubbard bands; the Hubbard gap is a pseudogap. As regards the density of states per unit cell of the midband, its bandwidth is $O(k_{\rm B}T_{\rm K})$ or $O(|J|/D)$, its peak height is $O(1/U)$, and its spectral weight is $O[t^2/(DU^2)]$. Since the midband almost disappears in the Heisenberg limit, the RVB electron liquid in the Heisenberg limit is simply the RVB spin liquid. The RVB electron and spin liquids adiabatically continue to each other. Since local moments form in a high-$T$ phase where $T \gtrsim T_{\rm K}$, the high-$T$ phase is simply the Mott insulator.Comment: arXiv admin note: substantial text overlap with arXiv:1210.821