This problem exemplifies the difference between topological anomaly and geometric anomaly: isomorphism class of this line bundle is determined by third Stiefel-Whitney W3(i/), may vanish even if holonomy nontrivial.

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group deformation classes using stable homotopy theory.We apply these theory considerations to lattice systems, assuming existence validity low energy effective approximations, thereby produce a general formula Symmetry Protected Topological (SPT) phases in terms Thom's bordism spectra; only input is dimension symmetry type.We provide...

We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected group has mixed anomaly between its one-form global symmetry (associated center) periodicity $\theta$-parameter. This is at root many recently discovered properties these theories, including their phase transitions interfaces. These new can be used this understanding systems without discrete symmetries (such...

It is customary to couple a quantum system external classical fields. One application the global symmetries of (including Poincaré symmetry) background gauge fields (and metric for symmetry). Failure invariance partition function under transformations these reflects ’t Hooft anomalies. also common view ordinary (scalar) coupling constants as fields, i.e. study theory when they are spacetime dependent. We will show that notion anomalies can be extended naturally include scalar Just allow us...

We introduce a definition and framework for internal topological symmetries in quantum field theory, including “noninvertible symmetries” “categorical symmetries”. outline calculus of defects which takes advantage well-developed theorems techniques theory. Our discussion focuses on finite symmetries, we give indications generalization to other symmetries. treat quotients quotient (often called “gauging” “condensation defects”), electromagnetic duality, duality defects, among topics. include...

This is the first in a series of papers investigating relationship between twisted equivariant K-theory compact Lie group G and "Verlinde ring" its loop group. In this paper we set up foundations K-groups, more generally groupoids. We establish enough basic properties to make effective computations. Using Mayer-Vietoris spectral sequence compute K-groups connected with torsion free fundamental relate computation representation theory at level related twisting.

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models 2 dimensions, with electromagnetic for finite gauge theories 3 dimensions.The relation is mediated by notion boundary are pure theory one dimension higher.Thus order/disorder operators endpoints Wilson/'t Hooft defects theory.Symmetry breaking on low-energy states reflects multiplicity topological states.In process we describe as (extended) boundaries...

We study eta-invariants on odd dimensional manifolds with boundary. The dependence boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of (inverse) determinant line prove a gluing law and variation formula for this invariant. This yields new, simpler proof holonomy bundle family Dirac operators, also known ``global anomaly'' formula. paper written using AMSTeX 2.1, which can be obtained via ftp from American Mathematical Society (instructions...

The space ΩG of based loops on a compact Lie group admits Kahler metric.Its curvature is expressed in terms Toeplitz operators, and we define Chern classes by analogy with Chern-Weil theory finite dimensions.In infinite dimensions extra geometric structure-a Fredholm structure-must be imposed before characteristic are defined.There natural structure induced from the family operators.We use index theorem for families Fredholms parametrized (proved [20]) to show that agree defined...

Quantum field theories with an energy gap can be approximated at long-range by topological quantum theories. The same should true for suitable condensed matter systems. For those short range entanglement (SRE) the effective theory is invertible, and so amenable to study via stable homotopy theory. This leads concrete invariants of gapped SRE phases which are finer than existing invariants. Computations in examples demonstrate their effectiveness.