We explore 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space B 2 G of the symmetry group G, and they are classified by cohomology classes of B 2 G. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of B 2 G as provided by the so-called W -construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of G-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.

Canada

Germany

Clement Delcamp

Switzerland

Apoorv Tiwari

On 2-form gauge models of topological phases

We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and ’t Hooft anomalies in the context of these higher group symmetries.

United States

From gauge to higher gauge models of topological phases

We study a single-channel Kondo effect using a recently developed [1–4] holographic large-<math><mrow><mi>N</mi></mrow></math> technique. In order to obtain resistivity of this model, we introduce a probe field. The gravity dual of a localized fermionic impurity in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>-dimensional host matter is constructed by embedding a localized two-dimensional Anti-de Sitter (<math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>)-brane in the bulk of three-dimensional <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math>. This helps us construct an impurity charge density which acts as a source to the bulk equation of motion of the probe gauge field. The functional form of the charge density is obtained independently by solving the equations of motion for the fields confined to the <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>-brane. The asymptotic solution of the probe field is dictated by the impurity charge density, which in turn affects the current-current correlation functions and hence the resistivity. Our choice of parameters tunes the near-boundary impurity current to be marginal, resulting in a <math><mi>log</mi><mi>T</mi></math> behavior in the UV resistivity, as is expected for the Kondo problem. The resistivity at the IR fixed point turns out to be zero, signaling a complete screening of the impurity.

Bikash Padhi

Chandan Setty

Philip W. Phillips

Log-rise of the resistivity in the holographic Kondo model