Higher gauge fields and fermions in lattice models

Introduction and summary
Basic notions
	Geometric setup and field configurations
	Degrees of freedom and holonomies
	Gauge and electric transformations
	Interesting field configurations — examples
Hamiltonian models
	Construction
	An explicit example
	Symmetries
	Vacuum states
	A peek at dynamics
Kernel and cokernel of partial
Twisted cohomology
Classifying spaces
	Classifying spaces of groups
	Classifying spaces of crossed modules
	Postnikov class
	Homomorphisms and weak equivalences
	Construction of classifying spaces
Introduction
Description of the model
	Degrees of freedom, action and gauge freedom
	Nonlocal order parameters and symmetries
	Reduction of dynamics to simpler models
	Phase diagram proposal for D = 4
Monte Carlo study
	Simulation method
	Numerical results for local observables
	Numerical results for non-local observables
Summary and conclusions
Non-spherical Wilson surfaces
Comparison with continuous theories
Introduction
Geometric setup
Fermions — generators and relations
Gamma model
	Definition of the model
	Choice of a representation
	Modified constraints and Z(2) gauge fields
	Example: toroidal geometries
	Example: quadratic fermionic hamiltonians
Deformed Z(2) gauge theories
	Gauge invariant operators
	Classification of Gauss\' operators
	Local formulations
Duality with higher gauge theory
Summary and outlook
Canonical transformations for Ising degrees of freedom
Graphs with vertices of odd degree
Bosonization of Majorana modes and edge states
	Abstract
	I Introduction
	II The bosonization method
	III Examples
		A Honeycomb lattice and Kitaev\'s model
		B Decagonal lattice
		C Hubbard model
	IV Boundary effects
		A Rectangular lattice with a boundary
		B Comment about topological phases
	V Euclidean representation of unconstrained ``Majorana spins\"
		A Basic idea and the (1+1)-dimensional example
			1 YY terms - the phases.
		B (2+1)-dimensional system
		C The sign problem
			1 (1+1)-dimensions
			2 (2+1)-dimensions
			3 The sign problem - summary
	VI Conclusions and outlook
	Acknowledgments
	References