BB-correspondence in solid state theory

A review of topological methods applied in solid-state theory is given. First, we recall the basic provisions of Bloch theory describing the properties of solids with a crystal lattice. Then we construct an algebra of observables of a topological dielectric and the resulting classes of symmetries and pseudosymmetries. Next, a description of the algebras of observables is given in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory of graded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras and the accompanying topological invariants of a solid. The algebra of boundary observables is defined in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory proposed by Kasparov. In conclusion, we describe the correspondence between the topological invariants of the body and its boundary (BB-correspondence). In the particular case of a periodic unitary model, this correspondence can be described explicitly.