The abcabc-problem for Gabor systems

A Gabor system generated by a window function $\phi$ and a rectangular lattice $a \Z\times \Z/b$ is given by $${\mathcal G}(\phi, a \Z\times \Z/b):=\{e^{-2\pi i n t/b} \phi(t- m a):\ (m, n)\in \Z\times \Z\}.$$ One of fundamental problems in Gabor analysis is to identify window functions $\phi$ and time-frequency shift lattices $a \Z\times \Z/b$ such that the corresponding Gabor system ${\mathcal G}(\phi, a \Z\times \Z/b)$ is a Gabor frame for $L^2(\R)$, the space of all square-integrable functions on the real line $\R$. In this paper, we provide a full classification of triples $(a,b,c)$ for which the Gabor system ${\mathcal G}(\chi_I, a \Z\times \Z/b)$ generated by the ideal window function $\chi_I$ on an interval $I$ of length $c$ is a Gabor frame for $L^2(\R)$. For the classification of such triples $(a, b, c)$ (i.e., the $abc$-problem for Gabor systems), we introduce maximal invariant sets of some piecewise linear transformations and establish the equivalence between Gabor frame property and triviality of maximal invariant sets. We then study dynamic system associated with the piecewise linear transformations and explore various properties of their maximal invariant sets. By performing holes-removal surgery for maximal invariant sets to shrink and augmentation operation for a line with marks to expand, we finally parameterize those triples $(a, b, c)$ for which maximal invariant sets are trivial. The novel techniques involving non-ergodicity of dynamical systems associated with some novel non-contractive and non-measure-preserving transformations lead to our arduous answer to the $abc$-problem for Gabor systems

Universality for Shape Dependence of Casimir Effects from Weyl Anomaly

We reveal elegant relations between the shape dependence of the Casimir effects and Weyl anomaly in boundary conformal field theories (BCFT). We show that for any BCFT which has a description in terms of an effective action, the near boundary divergent behavior of the renormalized stress tensor is completely determined by the central charges of the theory. These relations are verified by free BCFTs. We test them with holographic models of BCFT and find exact agreement. We propose that these relations between Casimir coefficients and central charges hold for any BCFT. With the holographic models, we reproduce not only the precise form of the near boundary divergent behavior of the stress tensor, but also the surface counter term that is needed to make the total energy finite. As they are proportional to the central charges, the near boundary divergence of the stress tensor must be physical and cannot be dropped by further artificial renormalization.Our results thus provide affirmative support on the physical nature of the divergent energy density near the boundary, whose reality has been a long-standing controversy in the literature.Comment: 19 pages, 1 figure and 3 tables, references added, accepted for publication in JHE

Universality in the Shape Dependence of Holographic R\'enyi Entropy for General Higher Derivative Gravity

We consider higher derivative gravity and obtain universal relations for the shape coefficients $(f_a, f_b, f_c)$ of the shape dependent universal part of the R\'enyi entropy for four dimensional CFTs in terms of the parameters $(c, t_2, t_4)$ of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients $f_a$ and $f_c$ satisfy the differential relation as derived previously for the R\'enyi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of $f_a$ for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we find a combination of our holographic relations which are satisfied by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we find there are similar universal laws for holographic R\'enyi entropy in general dimensions.Comment: 32 pages,0 figures, references added, appendix added, accepted for publication in JHE

Emergent Dark Matter in Late Time Universe on Holographic Screen

We discuss a scenario that the dark matter in late time universe emerges as part of the holographic stress-energy tensor on the hypersurface in higher dimensional flat spacetime. Firstly we construct a toy model with a de Sitter hypersurface as the holographic screen in the flat bulk. After adding the baryonic matter on the screen, we assume that both of the dark matter and dark energy can be described by the Brown-York stress-energy tensor. From the Hamiltonian constraint equation in the flat bulk, we find an interesting relation between the dark matter and baryonic matter's energy density parameters, by comparing with the Lambda cold dark matter parameterization. We further compare this holographic embedding of emergent dark matter with traditional braneworld scenario and present an alternative interpretation as the holographic universe. It can be reduced to our toy constraint in the late time universe, with the new parameterization of the Friedmann equation. We also comment on the possible connection with Verlinde's emergent gravity, where the dark matter is treated as the elastic response of the baryonic matter on the de Sitter spacetime background. We show that from the holographic de Sitter model with elasticity, the Tully-Fisher relation and the dark matter distribution in the galaxy scale can be derived.Comment: 28 pages, 2 figures; Matches published version and we thank the referees for many insightful comments; v3: typos in the Friedmann equations are fixe

Holographic RG flows with nematic IR phases

We construct zero-temperature geometries that interpolate between a Lifshitz fixed point in the UV and an IR phase that breaks spatial rotations but preserves translations. We work with a simple holographic model describing two massive gauge fields coupled to gravity and a neutral scalar. Our construction can be used to describe RG flows in non-relativistic, strongly coupled quantum systems with nematic order in the IR. In particular, when the dynamical critical exponent of the UV fixed point is z=2 and the IR scaling exponents are chosen appropriately, our model realizes holographically the scaling properties of the bosonic modes of the quadratic band crossing model.Comment: 19 pages, 2 figures. References added. Expanded discussion on nematic orde