K-theory of the torus equivariant under the 2-dimensional - PowerPoint PPT Presentation

Main theorem Gapped system and K -theory Equivariant twist K-theory of the torus equivariant under the 2-dimensional crystallographic point groups Kiyonori Gomi Oct 21, 2015

Main theorem Gapped system and K -theory Equivariant twist The theme of my talk The equivariant K -theory of the torus acted by the point group of each 2 -dimensional space groups, or equivalently the finite subgroups of the mapping class group GL (2 , Z ) . My talk is based on joint works with Ken Shiozaki and Masatoshi Sato. Our computational result will be the main theorem. The computation is motivated by the classification of 3 -dimensional topological crystalline insulators.

Main theorem Gapped system and K -theory Equivariant twist 1 Main theorem 2 Gapped system and K -theory 3 Equivariant twist

Main theorem Gapped system and K -theory Equivariant twist The space group As is well-known, the group of isometries of R d is the semi-direct product of O ( d ) and R d : 1 → R d → O ( d ) ⋉ R d → O ( d ) → 1 . A d -dimensional space group (crystallographic group) is a subgroup S , → R d − → O ( d ) ⋉ R d − 1 − → O ( d ) − → 1 ∪ ∪ ∪ 1 − → Π − → S − → P − → 1 , such that = Z d of translations, S contains a rank d lattice Π ∼ the point group P = S/ Π is a finite subgroup of O ( d ) .

Main theorem Gapped system and K -theory Equivariant twist The space group → R d − → O ( d ) ⋉ R d − 1 − → O ( d ) − → 1 ∪ ∪ ∪ 1 − → Π − → S − → P − → 1 S is not necessarily a semi-direct product of P and Π . S is called symmorphic if it is a semi-direct product. S is called nonsymmorphic if not. In the nonsymmorphic case, S contains for example a glide, which is a translation along a line ℓ followed by a mirror reflection with respect to ℓ .

Main theorem Gapped system and K -theory Equivariant twist The space group The space groups are identified if they are conjugate under the affine group R d ⋉ GL + ( d, R ) . d = 2 ⇒ 17 classes. d = 3 ⇒ 230 classes. · · · In the case of d = 2 , the space group is also called the plane symmetry group, the wallpaper group, etc. To denote the 17 classes of space groups, I will follow: D. Schattschneider, The plane symmetry groups: their recognition and their notation. American Mathematical Monthly 85 (1978), no.6 439–450.

Main theorem Gapped system and K -theory Equivariant twist The 2 -dimensional space groups (1/2) label P symmorphic? P ⊂ SO (2)? p1 1 yes yes p2 Z 2 yes yes p3 Z 3 yes yes p4 Z 4 yes yes p6 Z 6 yes yes These point groups are generated by rotations of R 2 . The other points groups are the dihedral group D n of degree n and order 2 n : D n = � C, σ | C n , σ 2 , CσCσ � . (For example, D 1 ∼ = Z 2 , D 2 ∼ = Z 2 × Z 2 , D 3 = S 3 .)

Main theorem Gapped system and K -theory Equivariant twist The 2 -dimensional space groups (2/2) label P symmorphic? P ⊂ SO (2)? pm D 1 yes no pg D 1 no no cm D 1 yes no pmm D 2 yes no pmg D 2 no no pgg D 2 no no cmm D 2 yes no p3m1 D 3 yes no p31m D 3 yes no p4m D 4 yes no p4g D 4 no no p6m D 6 yes no

Main theorem Gapped system and K -theory Equivariant twist The point group acts on T 2 Naturally, the point group P = S/ Π acts on the 2 -dimensional torus T 2 = R 2 / Π . By this construction, we get all the 13 classes of finite subgroups in the mapping class group GL (2 , Z ) of T 2 . In the case of p1–p6, the cyclic group Z n = � C n | C n n � ( n = 1 , 2 , 3 , 4 , 6 ) is embedded into SL (2 , Z ) through: C 1 = C 6 6 = C 4 4 = 1 , ( ) 0 − 1 C 2 = C 3 6 = C 2 4 = − 1 , C 4 = , 1 0 ( ) 0 − 1 C 3 = C 2 6 , C 6 = . 1 1

Main theorem Gapped system and K -theory Equivariant twist The point group acts on T 2 The groups D n = � C, σ | C n , σ 2 , CσCσ � ( n = 1 , 2 , 4 ) are generated by the following matrices in GL (2 , Z ) : label P C σ pm/pg D 1 = Z 2 C 1 = 1 σ x cm D 1 = Z 2 C 1 = 1 σ d pmm/pmg/pgg D 2 = Z 2 × Z 2 C 2 = − 1 σ x cmm D 2 = Z 2 × Z 2 C 2 = − 1 σ d p4m/p4g D 4 C 4 σ x ( ) ( ) ( ) 0 − 1 − 1 0 0 1 C 4 = , σ x = , σ d = . 1 0 0 1 1 0

Main theorem Gapped system and K -theory Equivariant twist The point group acts on T 2 The groups D n = � C, σ | C n , σ 2 , CσCσ � ( n = 3 , 6 ) are generated by the following matrices in GL (2 , Z ) : label P C σ C 3 = C 2 p3m1 D 3 σ x 6 C 3 = C 2 p31m D 3 σ y 6 p6m D 6 C 6 σ y ( ) ( ) ( ) 0 − 1 − 1 − 1 1 1 C 6 = , σ x = , σ y = . 1 1 0 1 0 − 1

Main theorem Gapped system and K -theory Equivariant twist Nonsymmorphic space group and twist It happens that a symmorphic group and a nonsymmorphic group share the same point group P . If a space group is nonsymmorphic, then there is an associated group 2 -cocycle with values in the group C ( T 2 , U (1)) of U (1) -valued functions, which is regarded as a right module over the point group P . Such group cocycles provide equivariant twists, namely the data playing roles of local systems for equivariant K -theory, classified by F 2 H 3 P ( T 2 ; Z ) ⊂ H 3 P ( T 2 ; Z ) . (This classification of twists will be reviewed later.)

Main theorem Gapped system and K -theory Equivariant twist F 2 H 3 P ( T 2 ) label P ori basis p1 1 + 0 p2 + 0 Z 2 p3 + 0 Z 3 p4 + 0 Z 4 p6 + 0 Z 6 pm/pg D 1 − τ pg Z 2 cm D 1 − 0 Z ⊕ 3 pmm/pmg/pgg D 2 − τ pmg , τ pgg , c 2 cmm D 2 − c Z 2 p3m1 D 3 − 0 p31m D 3 − 0 Z ⊕ 2 p4m/p4g D 4 − τ p4g , c 2 p6m D 6 − c Z 2

Main theorem Gapped system and K -theory Equivariant twist Main result Associated to an action of a finite group P on T 2 and an equivariant twist τ on T 2 , we have the P -equivariant τ -twisted K -theory K τ + n ( T 2 ) ∼ = K τ + n +2 ( T 2 ) . P P The equivariant twisted K -theory is a module over the representation ring R ( P ) = K 0 P (pt) . Our main result is the determination of the R ( P ) -module structure of K τ + n ( T 2 ) , where P n = 0 , 1 , P ranges the point groups of the 2d space groups, τ ranges twists classified by F 2 H 3 P ( T 2 ; Z ) . In the following, the module structure will be omitted.

Main theorem Gapped system and K -theory Equivariant twist Theorem [Shiozaki–Sato–G] (1/3) K τ +0 K τ +1 ( T 2 ) ( T 2 ) label P τ P P Z ⊕ 2 Z ⊕ 2 p1 1 0 Z ⊕ 6 p2 Z 2 0 0 Z ⊕ 8 p3 Z 3 0 0 Z ⊕ 9 p4 Z 4 0 0 Z ⊕ 10 p6 Z 6 0 0 Z ⊕ 3 Z ⊕ 3 pm D 1 0 pg D 1 τ pg Z ⊕ Z 2 Z Z ⊕ 2 Z ⊕ 2 cm D 1 0

Main theorem Gapped system and K -theory Equivariant twist Theorem [Shiozaki–Sato–G] (2/3) K τ +0 K τ +1 ( T 2 ) ( T 2 ) label P τ P P Z ⊕ 9 pmm D 2 0 0 Z ⊕ 4 pmm D 2 c Z  τ pmg , τ pmg + c   Z ⊕ 4 pmg D 2 τ pmg + τ pgg , Z  τ pmg + τ pgg + c  Z ⊕ 3 pgg D 2 τ pgg , τ pgg + c Z 2 Z ⊕ 6 cmm D 2 0 0 Z ⊕ 2 Z ⊕ 2 cmm D 2 c

Main theorem Gapped system and K -theory Equivariant twist Theorem [Shiozaki–Sato–G] (3/3) K τ +0 K τ +1 ( T 2 ) ( T 2 ) label P τ P P Z ⊕ 5 p3m1 D 3 0 Z Z ⊕ 5 p31m D 3 0 Z Z ⊕ 9 p4m D 4 0 0 Z ⊕ 3 Z ⊕ 3 p4m D 4 c Z ⊕ 6 p4g D 4 τ p4g 0 Z ⊕ 4 p4g D 4 τ p4g + c Z Z ⊕ 8 p6m D 6 0 0 Z ⊕ 4 Z ⊕ 2 p6m D 6 c

Main theorem Gapped system and K -theory Equivariant twist Examples of module structures I only show the module structures of K τ + n ( T 2 ) in the P case of p3m1 and p31m. The point group is D 3 ∼ = S 3 , and R ( D 3 ) = Z [ A, E ] / (1 − A 2 , E − AE, 1 + A + E − E 2 ) , where A is the sign representation, E is the unique 2 -dimensional irreducible representation. K 0 D 3 ( T 2 ) K 1 D 3 ( T 2 ) label R ( D 3 ) ⊕ (1 + A − E ) ⊕ 2 = Z ⊕ 5 p3m1 (1 − A ) = Z R ( D 3 ) ⊕ ( R ( D 3 ) / ( E )) ⊕ 2 = Z ⊕ 5 p31m (1 − A ) = Z

Main theorem Gapped system and K -theory Equivariant twist Some comments: proceeding works In the papers: W. L¨ uck and R. Stamm, 1 . . Computations of K - and L -theory of cocomapct planar groups. K -theory 21 249–292, 2000, M. Yang, 2 Crossed products by finite groups acting on low dimensional complexes and applications. PhD Thesis, University of Saskatchewan, Saskatoon, 1997. the K -theory K n ( C ∗ r ( S λ )) is determined as an abelian group for each 2 d space group S λ , which agrees with our result about K τ λ + n ( T 2 ) . P λ

Main theorem Gapped system and K -theory Equivariant twist Some comments: twists There are twists which cannot be realized as group cocycles. The equivariant K -theories twisted by such twists are not completely computed yet. The role of such a twist in condensed matter seems to be open.

Main theorem Gapped system and K -theory Equivariant twist Some comments : Z 2 Recall that there appeared Z 2 -summands: K τ +0 K τ +1 ( T 2 ) ( T 2 ) label P τ P P pg D 1 τ pg Z ⊕ Z 2 Z Z ⊕ 3 pgg D 2 τ pgg , τ pgg + c Z 2 A consequence is that these Z 2 -summands imply a ‘new’ class of topological insulators which are: classified by Z 2 , { time-reversal realized without the symmetries. particle-hole (The well-known topological insulators classified by Z 2 correspond to KR − 1 (pt) = Z 2 or KR − 2 (pt) = Z 2 , and are realized with TRS or PHS.) This is detailed in PRB B91, 155120 (2015).

Main theorem Gapped system and K -theory Equivariant twist 1 Main theorem 2 Gapped system and K -theory · · · How twisted K -theory arises? 3 Equivariant twist · · · a review of twists and their classification