We study a toy model of the Kerr/CFT correspondence using string theory on AdS_3 × S^3. We propose a single trace irrelevant deformation of the dual CFT generated by a vertex operator with spacetime dimensions (2,1). This operator shares the same quantum numbers as the integrable T\bar{J} deformation of two-dimensional CFTs where \bar{J} is a chiral U(1) current. We show that the deformation is marginal on the worldsheet and that the target spacetime is deformed to null warped AdS_3 upon dimensional reduction. We also calculate the spectrum of the deformed theory on the cylinder and compare it to the field theory analysis of T\bar{J}-deformed CFTs.

We use modular invariance to derive constraints on the spectrum of warped conformal field theories (WCFTs) --- nonrelativistic quantum field theories described by a chiral Virasoro and U(1) Kac-Moody algebra. We focus on holographic WCFTs and interpret our results in the simplest holographic set up: three dimensional gravity with Compere-Song-Strominger boundary conditions. Holographic WCFTs feature a negative U(1) level that is responsible for negative norm descendant states. Despite the violation of unitarity we show that the modular bootstrap is still viable provided the (Virasoro-Kac-Moody) primaries carry positive norm. In particular, we show that holographic WCFTs must feature either primary states with negative norm or states with imaginary U(1) charge, the latter of which have a natural holographic interpretation. For large central charge and arbitrary level, we show that the first excited primary state in any WCFT satisfies the Hellerman bound. Moreover, when the level is positive we point out that known bounds for CFTs with internal U(1) symmetries readily apply to unitary WCFTs.

To understand the new physics and richness of quantum many-body system phenomena is one of the stimuli driving the condensed matter community forward. Importantly, the new insights and solutions for condensed matter theory sometimes come from the developed and developing knowledge of high energy theory, mathematical and particle physics, which is also true the other way around: Condensed matter physics has been providing crucial hints and playgrounds for the fundamental laws of high energy physics. In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states (SPTs) and topologically ordered states (TOs). We first explore the ground state structures of SPTs and TOs: the former can be symmetry twisted and the latter has robust degeneracy. The Berry phases generated by transporting and overlapping ground state sectors potentially provide universal topological invariants that fully characterize the SPTs and TOs. This framework provides us the aspects of symmetry and topology. We establish a field theory representation of SPT invariants in any dimension to uncover group cohomology classification and beyond — the former for SPTs with gapless boundary gauge anomalies, the latter for SPTs with mixed gauge-gravity anomalies. We study topological orders in 3+1 dimensions such as Dijkgraaf-Witten models, which support multi-string braiding statistics; the resulting patterns may be analyzed by the mathematical theory of knots and links. We explore the aspects of surface anomalies of bulk gapped states from the bulk-edge correspondence: The gauge anomalies of SPTs shed light on the construction of bosonic anomalies including Goldstone-Wilczek type, and also guide us to design a non-perturbative lattice model regularizing the low-energy chiral fermion/gauge theory towards the Standard Model while overcoming the Nielsen-Ninomiya fermion-doubling problem without relying on Ginsparg-Wilson fermions. We conclude by utilizing aspects of both quantum mechanical topology and spacetime topology to derive new formulas analogous to Verlinde’s via geometric-topology surgery. This provides new insights for higher dimensional topological states of matter.

We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic proof. First, we propose a set of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models at $\theta=\pi$, with N $=2,3,4$ and others, by enlisting all possible 3d cobordism invariants and selecting the matched terms. Second, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric SU(N)-Yang-Mills (YM) gauge theory at $\theta=\pi$, via 5d cobordism invariants (higher symmetry-protected topological states) such that compactifying YM theory on a 2-torus matches the constrained 3d cobordism invariants from sigma models. Based on algebraic/geometric topology, QFT analysis, manifold generator dimensional reduction, condensed matter inputs and additional physics criteria, we derive a correspondence between 5d and 3d new invariants, thus broadly prove a more complete anomaly-matching between 4d YM and 2d $\mathbb{CP}^{\mathrm{N}-1}$ models via a twisted 2-torus reduction, done by taking the Poincar\'e dual of specific cohomology class with $\mathbb{Z}_2$ coefficients. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis theorem" to constrain the low-energy dynamics.

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new `quantum surgery' formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies. This article is meant to be an extended and further detailed elaboration of our previous work [arXiv:1602.05951] and Chapter 6 of [arXiv:1602.05569]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.