The authors construct here a lattice path integral for a (3+1)D topological phase with U(1) symmetry, realizing (2+1)D topological order on its boundary. The U(1) background gauge field can possibly have a nontrivial Chern number. This allows the derivation of various formulas for anomalies of topological order involving U(1) symmetry, and leads to Walker-Wang construction of a topological insulator and superconductor. Mathematically, they correspond to new combinatorial state sums for invariants of topologically nontrivial, curved U(1) bundles over 4-manifolds, including spin${}^{c}$, pin${}^{c}$ structures.
The authors construct here a lattice path integral for a (3+1)D topological phase with U(1) symmetry, realizing (2+1)D topological order on its boundary. The U(1) background gauge field can possibly have a nontrivial Chern number. This allows the derivation of various formulas for anomalies of topological order involving U(1) symmetry, and leads to Walker-Wang construction of a topological insulator and superconductor. Mathematically, they correspond to new combinatorial state sums for invariants of topologically nontrivial, curved U(1) bundles over 4-manifolds, including spinc, pinc structures.
Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group G=U(1)⋊H, we construct a (3+1)D topologically invariant path integral in the presence of a curved background G gauge field, as an exact combinatorial state sum. Specifically, the U(1) component of the G gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat G bundles. Our construction expresses the U(1) gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form R gauge field and 2-form Z gauge field. We develop a new graphical calculus for anyons interacting with "Villain symmetry defects", associated with the 1-form and 2-form background gauge fields. This graphical calculus is used to define the (3+1)D path integral, which can describe either a bosonic or fermionic symmetry-protected topological (SPT) phase. For example, we can construct the topological path integral on curved U(1) bundles for the (3+1)D fermionic topological insulator in class AII and topological superconductor in class AIII given appropriate (2+1)D fermionic symmetry fractionalization data; these then give invariants of 4-manifolds with Spinc or Pinc structures and their generalizations. The (3+1)D path integrals define anomaly indicators for the (2+1)D topological orders; in the case of Abelian (2+1)D topological orders, we derive by explicit computation all of the mixed U(1) anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spinc generalization of the Gauss-Milgram sum, valid for super-modular categories.
We study anomalies in time-reversal-(Z2T) and U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)-dimensional system but can be realized on the surface of a (3+1)-dimensional [(3+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of (3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk (3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1) symmetry.
The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a D-dimensional system under modular transformation, or modular anomaly, signals the presence of a (D+1)-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a (2+1)-D and a (3+1)-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in (2+1)-D and (3+1)-D are modular invariant. However, we show in (3+1)-D that when the edge theory is coupled to a background U(1) gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a (4+1)-D bulk. Using the modular anomaly, we find that the edge theory of a (4+1)-D insulator with space-time inversion symmetry (PT) and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
By definition, the physics of the d-dimensional (dim) boundary of a (d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a d-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a d-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d boundary of a 5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N topological order with topological degeneracy, but this Z2N topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d space.
Quasiparticle excitations in 3+1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in (3+1)-dimensional (3+1d) topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics $\ensuremath{\mu}=\ifmmode\pm\else\textpm\fi{}1$, that distinguishes two bosonic topological orders that both superficially resemble 3+1dZ2 gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic Z2 gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a nontrivial (4+1)-dimensional (4+1d) invertible phase, stable without any symmetries, i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable 4+1d model using a method that bootstraps a boundary theory into a bulk Hamiltonian, analogous to that of Walker and Wang, and computing the loop self-statistics in the fermionic Z2 gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a nontrivial Z2-classified 4+1d invertible phase with action $S=\frac{1}{2}\ensuremath{\int}{w}_{2}{w}_{3}$.