The weakly interacting tenfold way

THE WEAKL Y INTERA CTING TENFOLD W A Y LUCAS C.P .A.M. MÜSSNICH 1 , REN A TO V . VIEIRA 2 A bstra ct . W e present implementations of the topological K-theory spectra K U and K O in terms of time ev olution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. W e also introduce a geometric definition of weakly interacting time e volution operators, and show ho w associated spec- tra K U wi and K O wi deformation retract to K U and K O . W e thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions. C ontents 1. Introduction 2 1.1. Structure of the article 3 1.2. Notation and terminology 4 1.3. Ackno wledgements 4 2. Classification of free fermion systems with symmetries 4 2.1. Nambu space model for fermions 4 2.2. Nambu space symmetries 6 2.3. Reduction to grotesque fermion systems 7 2.4. Con venient choice of basis 9 2.5. Spaces of equi variant free Hamiltonian 11 2.6. Classification by symmetric spaces of compact type 13 3. K-theoretical classification 16 3.1. Cohomology theories and representing spectra 16 3.2. Construction of K U and K O in terms of time-ev olution operators 18 4. W eakly interacting systems 24 5. Concluding remarks 27 5.1. T wisted equiv ariant K-theory 27 5.2. Classification of interacting fermion systems by cobordism 28 References 28 1 I nstituto de C i ˆ encias M a tem ´ a ticas e de C omput a ¸ c ˜ a o , U niversidade de S ˜ a o P a ulo (ICMC-USP), S ˜ a o C arlos , SP , B rasil 2 I nstituto de M a tem ´ a tica , E st a t ´ istica e C omput a ¸ c ˜ a o C ient ´ ifica , U niversidade E st adual de C ampinas (IMECC-UNICAMP), C ampin as , SP , B rasil . 1 1. I ntr oduction The tenfold way is a classification scheme for the building blocks of non-interacting fermion systems. More precisely , it classifies the isomorphism classes of spaces of equi- v ariant free Hamiltonians in irreducible fermion systems with symmetries. This classifi- cation scheme naturally leads to the K-theoretical classification of topological phases of matter , known as the periodic table of topological insulators and superconductors. In this article we show how the topological K-theory spectra K U and K O hav e imple- mentations in terms of time e volution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. W e further giv e a geometric definition of weakly interacting time ev olution operators. W e sho w ho w asso- ciated spectra K U wi and K O wi of weakly interacting time e volution operators deformation retract to K U and K O , which means the y represent the same cohomology theories. Thus, we pro vide a stable homotopy theoretical proof that the tenfold way is stable to weak interactions. One perspectiv e on the tenfold way is to consider that fermion state spaces can be mod- eled by representations of Cli ff ord algebras, of which there are exactly 10 Morita equi v- alence classes. Another perspectiv e is that, due to Schur’ s lemma, the automorphisms of irreducible super group representations form associati ve super di vision algebras, of which there are exactly ten. These points of view are connected by the fact that e very Cli ff ord algebra is Morita equi valent to a super di vision algebra [5, 12]. Another perspecti ve on the tenfold way is in terms of the 10 infinite families of compact symmetric spaces classified by Cartan [7, 8]. In [14] Heinzner , Huckleberry and Zirnbauer extended Dyson’ s threefold way [10], by classifying fermion systems with equiv ariant Hamiltonians that are quadratic on creation and annihilation operators. There they showed that the systems’ spaces of time ev olution operators are compact symmetric spaces, and that all 10 classes classified by Cartan can be obtained this way . In [1] Agarwala, Haldar and Sheno y sho wed that all classes can be obtained by the subspace of free Hamiltonians. They further explicitly lay out the structure of equiv ariant Hamiltonians that preserve particle number , both for free systems and interacting ones. The role of topological K-theory in the tenfold way was made explicit by Kitaev in [20]. Here he sho ws that by deforming gapped Hamiltonians by a process of spectral flat- tening you can obtain classifying spaces for free-fermion phases, which are all compact symmetric spaces. He also points out that if you ha ve a family of irreducible systems parametrized by a space Λ , then its topological phases are classified by homotopy classes of maps from Λ to one of the classifying spaces, which are in bijection with di ff erence classes of equiv ariant vector bundles over Λ that model free fermion ground states. These homotopy classes form groups isomorphic to some topological K-group of Λ , which one depending on the internal symmetries of the system and on the dimensionality of Λ . As Freed and Moore explain in [12], in a crystalline system where we consider Λ as the Bril- louin zone, we must further consider the action of the magnetic point group. This leads to a classification of topological phases by twisted equi variant K-theory . 2 In [20] Kitae v points out that the tenfold way should be stable to weak interactions, and further giv es an e xample of how two topological phases distinguished by K-theoretical in- v ariants can be connected by a continuous path of interacting phases. Thus, though su ffi - ciently strong interactions may break the classification scheme of the tenfold way , weakly interacting fermion systems should still be classified by topological K-theory . In this ar- ticle we giv e a stable homotopy theoretical proof of this stability to weak interactions. In the full interacting re gime topological phases are classified by cobordism, represented by Thom’ s bordism spectra [11, 17]. From the point of view of stable homotopy theory , the connection between the classi- fication of free fermion systems and topological K-theory arrises from the fact that the symmetric spaces of time e volution operators form the underlying spaces of spectra K U and K O . These spectra represent, in the sense of Bro wn’ s representability theorem, com- plex and real topological K-theory . This means the K-groups of Λ are isomorphic to stable homotopy groups of mapping spectra K U Λ and K O Λ . Stable homotopy theory provides an e ffi cient computational tool for topological classification problems, for instance in [9] the formalism of equiv ariant ring spectra is used to obtain the complete classification of topologically distinct quantum states of 3D crystalline topological insulators and super- conductors for ke y space-groups, by considering fermion systems parametrized by the Brillouin zone torus. In this article, in order to pro ve the stability of the tenfold way to weak interactions, we start by revie wing the structure of the spaces of free time e volution operators underlying K U and K O , as shown in [1, 14]. Using this construction we provide explicit formulas for the suspension maps, which are determined by Cartan embeddings, and by the ho- motopy equiv alences Bott used in his periodicity theorem [6]. These suspension maps are closely related to the diagonal map Kennedy and Zirnbauer used in their homotopy- theoretic proof of the periodic table [18, 19]. W e further give a geometric definition of weakly interacting time ev olution operators in terms of the complement of the cut locus of the submanifold of free operators within the full interacting space. This allows us to define spectra K U wi and K O wi composed of weakly interacting operators, with our weak interaction condition guaranteeing that these spectra deformation retract to K U and K O . Since the cut locus is closed our definition of weak interaction is stable to small perturba- tions. 1.1. Structure of the article. In section 2 we revie w the classification of irreducible free fermion systems with symmetries by compact symmetric spaces. Follo wing [14] we consider the Nambu space model for fermions, and as in [1] we present con venient choice of basis to describe the structure of the symmetries and free equiv ariant Hamiltonians. W e further show how the symmetries determine the symmetric structure of the spaces of time ev olution operators, and giv e their structure in the chosen con venient basis for each symmetry class. In section 3 we revie w the construction of the topological K-theory functors, and the definition of the spectra that represent cohomology theories. W e then construct spectra K U and K O that represent complex and real topological K-theories in terms of free time e volution operators, including e xplicit formulas for the structural suspension maps. 3 In section 4 we gi ve our geometric definition of weakly interacting time ev olution op- erators, and show how these form spectra K U wi and K O wi that deformation retract onto the subspectra of free operators. In section 5 we make some remarks concerning possible extensions of our w ork, in par - ticular regarding the classification of crystalline fermion systems by twisted equi variant K-theory , and the classification of interacting systems by cobordism. 1.2. Notation and terminology. For A = h a i j i ∈ M N C we denote its conjugate by A ∗ = h a ∗ i j i , its transpose by A t = h a ji i , and its conjugate transpose by A † = A ∗ t = A t ∗ = h a ∗ ji i . W e will denote by K the anti-linear complex conjugation operator K : C N → C N , K v = h v ∗ i i . For any A ∈ M N C we ha ve K AK = A ∗ . For p , q ∈ N we set the matrix 1 p , q = h 1 p 0 0 − 1 q i . For N even we set J N = h 0 1 N / 2 − 1 N / 2 0 i and F N = h 0 1 N / 2 1 N / 2 0 i . W e note that the compact symplectic group S p ( N / 2 ) is composed of the unitary matrices U ∈ U ( N ) such that J N U J † N = U ∗ . The smash product of pointed spaces X , Y ∈ Top ∗ is X ∧ Y B X × Y X ×{ y 0 }∪{ x 0 }× Y . For I = [0 , 1] the standard interval we set S 1 B I / 0 ∼ 1 ∈ Top ∗ , with the identified extremities as base point. W e also set I + B I ⊔ { x 0 } ∈ Top ∗ the space obtained by adding a disjoint base point. A pointed map h : X ∧ I + → Y is a base point preserving homotopy , and a pointed map σ : X ∧ S 1 → Y is a base point preserving homotopy from the constant map to itself. Let G be a Lie group, and g be its associated Lie algebra. W e will denote the group le vel commutator by [ g , h ] = ghg − 1 h − 1 for g , h ∈ G , and the algebra le vel commutator by { A , B } = A B − BA for A , B ∈ g . 1.3. Acknowledgements. Renato V . V ieira was financed by Conselho Nacional de De- sen volvimento Científico e T ecnológico – Brasil (CNPq), grant no. 150669 / 2024-0. Lucas C.P .A.M. Müssnich was financed by Coordenação de Aperfeiçoamento de Pes- soal de Nível Superior - Brasil (CAPES). The authors would like to thank N. Javier Buitrago Aza for fruitful discussions during the initial stages of this work. 2. C lassifica tion of free fermion systems with symmetries 2.1. Nambu space model for fermions. The construction we now describe is com- pletely standard in the mathematical-physics literature. See, for instance, [2, 15, 24]. W e present it here for completeness. The choice here made for the Cli ff ord-algebraic frame work relates to presenting results in the same setting as the main references here used [14, 20]. For a complete discussion on the mathematical aspects of quantization, see, for example, [16]. The state space of a single fermion is modeled by a Hilbert space. W e shall only consider the finite-dimensional case, which already presents complexities regarding the classification of symmetries. Giv en N ∈ N , let then V N be a Hilbert space with dimension N . The Fock space of man y-fermion states is the exterior algebra V V N B L N n = 0 V ∧ n N , 4 where each V ∧ n N is the space of n -particle anti-symmetrized states with hermitian structure D ∧ i u i    ∧ j v j E V ∧ n N B det h D u i    v j E V N i . Recall that V ∧ 1 N  V N , and that the direct sum goes up to N due to anti-symmetry and finite dimension. The subspace V ∧ 0 N = C is associated with the vacuum state : we set | 0 ⟩ B (1 , 0 , 0 , . . . ) ∈ V ∧ 0 N , and interpret it as the zero-particle state. Then, for each v ∈ V N , we associate cr eation and annihilation operators, that allow the construction of n -particle states from the vacuum. These are, respectiv ely , ε ( v ) ∈ B ( V V N ) (degree 1) and ι ( v † ) ∈ B ( V V N ) (degree − 1), gi ven by ε ( v )( u 1 ∧ · · · ∧ u n ) B v ∧ u 1 ∧ · · · ∧ u n , ι ( v † )( u 1 ∧ · · · ∧ u n ) B P n i = 1 ( − 1) i + 1 v † ( u j ) u 1 ∧ · · · ∧ u j − 1 ∧ u j + 1 ∧ · · · u n , where v † ∈ V ∗ N is the dual of v via the Fréchet-Riesz representation theorem. This being gi ven, the following notation is usual: denote by B = { | i ⟩ } 1 ≤ i ≤ N an orthonormal basis for V N , and and by B ∗ = { ⟨ i | } 1 ≤ i ≤ N its dual base. W e set a † i B ε ( | i ⟩ ) , a i B ι ( ⟨ i | ) , so that a † i | 0 ⟩ = | i ⟩ , a i | j ⟩ = δ i j | 0 ⟩ . These operators satisfy the so called canonical anti-commutation r elations (CAR), char- acteristic of fermionic systems: a † i a † j + a † j a † i = 0 , a i a j + a j a i = 0 , a † i a j + a j a † i = δ i j . Definition 2.1. The Namb u space associated with V N is the Hilbert space W N B V N ⊕ V ∗ N equipped with the symmetric bilinear form b : W N × W N → C , b ( v 1 + f 1 , v 2 + f 2 ) B f 1 v 2 + f 2 v 1 . The Nambu space embeds into B ( V V N ) via the creation and annihilation operators. Elements of W N are called fermionic field operators. The bilinear structure encodes the CAR, in the sense that for all Ψ 1 , Ψ 2 ∈ W we ha ve Ψ 1 Ψ 2 + Ψ 2 Ψ 1 = b ( Ψ 1 , Ψ 2 ) ˆ 1 as operators on V V N . The associati ve algebra generated by W N in B ( V V N ) is isomorphic to the Cli ff ord algebra C l ( W N , b ) B L 2 N n = 0 W ⊗ n N / Ψ 1 ⊗ Ψ 2 +Ψ 2 ⊗ Ψ 1 − b ( Ψ 1 , Ψ 2 ) . W e hav e a natural filtered algebra structure C l ( W N , b ) = L 2 N n = 0 C l n ( W N , b ) with C l n ( W N , b )  W ∧ n N . Definition 2.2. Let ( W N , b ) be a Namb u space. The space of fr ee Hamiltonians is H N B { ˆ H = P i j H i j a † i a j | H ∈ M N C , H † = H } ⊂ C l 2 ( W N , b ) . The adjoint action of H N on C l ( W N , b ) is defined as ad ˆ H Ψ B { ˆ H , Ψ } . 5 The Hilbert space V N ⊂ C l ( W N , b ) is closed under this action, with ad ˆ H v = P i j H i j ⟨ j | v ⟩ | i ⟩ , [ ad ˆ H ] B = H . Thus the adjoint action of free Hamiltonians is faithfully represented in V N . In the next section we will describe the symmetries of fermion systems at the Nambu space lev el, so we will have to consider the Hamiltonian adjoint actions on W N , repre- sented in the basis e B = B ⊔ B ∗ by [ ad ˆ H ] e B = " H 0 0 − H ∗ # . The exponential map associates to each free Hamiltonian ˆ H ∈ H N a unitary operator exp  − i ˆ H  in C l ( W N , b ), with associated adjoint action Ad exp ( − i ˆ H ) Ψ = exp  − i ˆ H  Ψ exp  i ˆ H  . Definition 2.3. The space of time evolution oper ators on the Namb u space ( W N , b ) is M N B { Ad exp ( − i ˆ H ) ∈ Aut ( C l ( W N , b )) | ˆ H ∈ H N } . The Hilbert space V N is closed under the action of time e volution operators, with [ Ad exp ( − i ˆ H ) ] B = exp ( − i H ) . At the Nambu space le vel we ha ve [ Ad exp ( − i ˆ H ) ] e B = " exp ( − i H ) 0 0 exp ( i H ∗ ) # . Since i H N  u ( N ) we hav e M N  U ( N ). 2.2. Nambu space symmetries. Let ˆ U : W N → W N be a bijection. W e say ˆ U is a linear Nambu space symmetry if it is a unitary operator such that b ( ˆ U Ψ 1 , ˆ U Ψ 2 ) = b ( Ψ 1 , Ψ 2 ), and that it is an anti-linear Nambu space symmetry if it is an anti-unitary operator such that b ( ˆ U Ψ 1 , ˆ U Ψ 2 ) = b ( Ψ 1 , Ψ 2 ). A Nambu space symmetry ˆ U , unitary or anti-unitary , is usual if it preserves the Nambu space decomposition, ie if ˆ U ( V N ) = V N and ˆ U ( V ∗ N ) = V ∗ N . A symmetry is transposing if ˆ U ( V N ) = V ∗ N and ˆ U ( V ∗ N ) = V N . The set of all Namb u space symmetries forms a group U , which decomposes as a disjoint union U = U U L ⊔ U T A ⊔ U T L ⊔ U U A . The set U U L of r e gular symmetries is composed of the usual linear symmetries, which forms a normal subgroup of U . For all ˆ U ∈ U U L compatibility with the bilinear form b means that ˆ U f ( v ) = b ( ˆ U f , v ) = b ( f , ˆ U † v ) = f ( ˆ U † v ) , so ˆ U | V ∗ N = ( ˆ U | † V N ) t . In the basis e B a regular symmetry is then represented by a matrix [ ˆ U ] e B = " U 0 0 U ∗ # , 6 for some U ∈ U ( N ). The set U U A of time-r ever sal symmetries is composed of the usual anti-linear sym- metries. Since anti-unitary operators are unitary operators composed with the complex conjugation, the time-re versal symmetries are represented by [ ˆ T ] e B = " T 0 0 T ∗ # K , for some T ∈ U ( N ). The set U T L of char ge conjugation symmetries is composed of the transposing linear symmetries. For all ˆ C ∈ U U A compatibility with the bilinear form b no w means that g ( ˆ C f ) = b ( ˆ C f , g ) = b ( f , ˆ C † g ) = f ( ˆ C † g ) , so again we ha ve ˆ C | V ∗ N = ( ˆ C | † V N ) t . Thus a charge conjugation symmetry is represented by [ ˆ C ] e B = " 0 C C ∗ 0 # , for some C ∈ U ( N ). The set U T A of sublattice , or chiral , symmetries is composed of the transposing anti- linear symmetries. A sublattice symmetry ˆ S ∈ U T A is represented by [ ˆ S ] e B = " 0 S S ∗ 0 # K , for some S ∈ U ( N ). Any non-re gular symmetry squares to a regular one, ie if ˆ U ∈ U T A ⊔ U T L ⊔ U U A then ˆ U 2 ∈ U U L . Also, the product of tw o distinct type of non-regular symmetries is of the third type. This means the quotient U / U U L is isomorphic to the Klein 4-group Z 2 2 . Definition 2.4. A Nambu space with symmetries is a representation ρ : G → U of a compact Lie group G . For an y Nambu space with symmetries we ha ve an induced decomposition G = G U L ⊔ G T A ⊔ G T L ⊔ G U A . 2.3. Reduction to grotesque fermion systems. In a Namb u space with symmetries, the subgroup G U L acts unitarily on the Hilbert space V N . Let b G U L be the set of isomorphism classes of irreducible unitary G U L -representations. For each class λ ∈ b G U L fix a repre- sentati ve ρ λ : G U L → U ( R λ ), and set d λ B dim R λ . Setting E λ B Hom G U L ( R λ , V N ) and E ∗ λ B Hom G U L ( R ∗ λ , V ∗ N ) on each ( E λ ⊗ R λ ) ⊕ ( E ∗ λ ⊗ R ∗ λ ) we hav e the natural G U L -action g ( h ⊗ r + f ⊗ t ) B h ⊗ ρ λ ( g ) r + f ⊗ ρ λ ( g ) t . The map Φ : L λ ∈ b G U L ( E λ ⊗ R λ ) ⊕ ( E ∗ λ ⊗ R ∗ λ ) → W N , Φ ( L λ h λ ⊗ r λ + f λ ⊗ t λ ) B P λ h λ r λ + f λ t λ 7 is a G U L -equi variant isomorphism. W e can define an inner product on E λ by ⟨ h 1 | h 2 ⟩ E λ = ⟨ h 1 r 1 | h 2 r 2 ⟩ V N ⟨ r 1 | r 2 ⟩ R λ for arbitrary r 1 , r 2 ∈ R λ . The map Φ is an isometry , which induces the symmetric bilinear form b ( h 1 ⊗ r 1 + f 1 ⊗ t 1 , h 2 ⊗ r 2 + f 2 ⊗ t 2 ) B 1 / d λ (( f 2 h 1 )( t 2 r 1 ) + ( f 1 h 2 )( t 1 r 2 )) . W e now want to consider the action of the full symmetry group G . If g ∈ G U L ⊔ G U A then for each λ ∈ b G U L there is some λ g ∈ b G U L such that ρ λ ( g )( E λ ⊗ R λ ) = E λ g ⊗ R λ g . If g ∈ G T L ⊔ G T A then for each λ ∈ b G U L there is some λ g ∈ b G U L such that ρ λ ( g )( E λ ⊗ R λ ) = E ∗ λ g ⊗ R ∗ λ g . Defining the equi v alence relation ∼ on b G U L by λ 1 ∼ λ 2 i ff ∃ g ∈ G : λ 2 = λ g 1 , we can define for each Λ ∈ b G U L / ∼ the blocks A Λ B L λ ∈ Λ E λ ⊗ R λ , B Λ B A Λ ⊕ A ∗ Λ . Then L Λ ∈ b G U L / ∼ B Λ is a sum of G in v ariant blocks, which is isomorphic to W N by Φ . For ˆ H ∈ H N the adjoint action on V N induces an action on each E Λ by ( ad ˆ H h ) r B ad ˆ H ( hr ) = { ˆ H , hr } . The action on each E λ ⊗ R λ is then ad ˆ H ( h ⊗ r ) B ( ad ˆ H h ) ⊗ r . Thus, on each A Λ , G U L acts trivially on the E λ terms, and the Hamiltonians act trivially on the R λ terms. Since Hom G U L ( E λ 1 ⊗ R λ 1 , E λ 2 ⊗ R λ 2 )  Hom( E λ 1 , E λ 2 ) ⊗ Hom G U L ( R λ 1 , R λ 2 ), we can reduce the classification problem to the cases where G U L acts tri vially on the underlying one-particle Hilbert space. W e can then consider from now on fermionic systems with trivial G U L -actions. Free Hamiltonians always commute with the particle number operator ˆ N B P j a † j a j . W e can then consider “grothesque” fermionic system, which are Nambu spaces with symmetries such that G U L = n exp  − i 2 π t ˆ N  | t ∈ S 1 o  U (1) . If g ∈ G U A ⊔ G T A then g 2 ∈ G U L , so g 2 = z ˆ 1 for some z ∈ U (1). By anti-linearity zg = g 2 g = gg 2 = gz = z ∗ g , which implies z = ± 1. This means that if ˆ T ∈ G U A then T T ∗ = ± 1 N , and if ˆ S ∈ G T A then S 2 = ± 1 N . If ˆ C ∈ G T L , then we again hav e ˆ C 2 = z ˆ 1, and since [ ˆ C ] e B =  0 C C ∗ 0  we get C C ∗ = z 1 N . Since ˆ C 2 ˆ C = ˆ C ˆ C 2 implies z C = z ∗ C , we again conclude z = ± 1. Let ˆ S ∈ G T A . If ˆ S 2 = − ˆ 1 then (exp  i π / 2 ˆ N  ˆ S ) 2 = ˆ 1. Thus, in the presence of an idempotent chiral symmetry we can always assume a basis such that ˆ S 2 = ˆ 1. Thus, in a grotesque fermion system, the symmetry group is assumed to be generated by G U L  U (1) and at most two non-ordinary symmetries that square to ± ˆ 1. If one of the generators is a chiral symmetry , we may assume it to square to the identity . If all 3 types of non-ordinary symmetries are present, we choose the generators such that ˆ S = ˆ T ˆ C . In the presence of time-rev ersal symmetry , we set  T ∈ { 1 , − 1 } such that ˆ T 2 =  T ˆ 1. Similarly , in the presence of charge conjugation we set  C ∈ { 1 , − 1 } such that ˆ C 2 =  C ˆ 1. W e also set  S = 1 whene ver the system has chiral symmetry . W e then get 10 classes of grotesque Nambu spaces with symmetries. W e can index them by a signature (  S  C  T ) ∈ { 0 , 1 } × {− 1 , 0 , 1 } 2 , where 0 indicates the absence of the associated symmetry type. 8 CAZ (  S  C  T ) S C T [ ad ˆ H ] B A I I I (100) 1 m , N − m − − H = " 0 b b † 0 # A (000) − − − − BD I (111) 1 m , N − m 1 m , N − m 1 N H = " 0 b b t 0 # , b = b ∗ A I (001) − − 1 N H = H t C I (1-11) 1 N / 2 , N / 2 − J N F N H = " 0 b b ∗ 0 # , b t = b C (0-10) − J N − H = " a b b ∗ − a ∗ # , a † = a , b t = b C I I (1-1-1) 1 m , N − m " − J m 0 0 J N − m # " J m 0 0 J N − m # H = " 0 B B † 0 # , B = " b 0 b 1 − b ∗ 1 b ∗ 0 # A I I (00-1) − − J N H = " a b − b ∗ a ∗ # , a † = a , b t = − b t D I I I (11-1) 1 N / 2 , N / 2 F N J N H = " 0 b − b ∗ 0 # , b t = − b D (010) − 1 N − H = − H t T able 1. The first column lists the Cartan-Atland-Zirnbauer (CAZ) sym- metry class label, follo wed by their corresponding symmetry signature; Columns three through fiv e specify the matrices representing the chiral, charge conjugation and time-in version symmetries; The final column de- scribes the structure of the adjoint representation of the equiv ariant free Hamiltonians. 2.4. Con venient choice of basis. W e no w show how to choose con venient basis for the 10 symmetry classes so the unitary matrices representing the symmetries have a conv e- nient form, which are listed in T able 1. Class A (000) : This class of systems have no non-ordinary symmetries, so any or - thonormal basis may be chosen. Let’ s consider now the classes whose non-ordinary symmetries are generated by a sin- gle operator . Class AIII (100) : Let ˆ S ∈ G T A be a chiral symmetry such that ˆ S 2 = ˆ 1, so its associated matrix satisfies S 2 = 1 N . Since S is unitary that means S † = S , and so the eigen v alues of S are all ± 1. This means that we hav e a decomposition V N = V + N ⊕ V − N into the positiv e and negati ve eigenspaces of ˆ S . Let m = dim V + N . If B ± are basis for V ± N and B = B + ⊔ B − , 9 then S = 1 m , N − m . Consider no w systems with only a charge conjugation symmetry ˆ C satisfying ˆ C 2 =  C ˆ 1, or equi valently C C ∗ =  C 1 N . This means C † =  C C ∗ , or equi valently C =  C C t . Class D (010) : Let ˆ C ∈ G T L be a charge conjugation symmetry such that ˆ C 2 = ˆ 1, so that it is represent by a symmetric matrix C . By T akagi decomposition, there is some unitary matrix A such that C = AA t . If R is some unitary matrix representing a change of basis in V N , it induces a change of basis in W N represented by  R 0 0 R ∗  . This change of basis transforms the representation of ˆ C into h 0 RCR t R ∗ C ∗ R † 0 i . W e may then conclude, using R = A , that in some basis we hav e C = 1 N . Class C (0 - 10) : Suppose now that ˆ C ∈ G T L satisfies ˆ C 2 = − ˆ 1, and so that C is ske w- symmetric. Since C C ∗ = − 1 N we ha ve that | det C | 2 = ( − 1) N , which can only be the case if N is even. T akagi decomposition tells us there is some matrix A such that C = A J N A t . Thus there is some basis in which C = J N . Class AI (001) : For a time-re versal symmetry ˆ T ∈ G U A such that ˆ T 2 = ˆ 1 we have T T ∗ = 1 N , so by the same argument as for systems in the class (010) there is some basis such that T = 1 N . Class AII (00 - 1) : For ˆ T ∈ G U A such that ˆ T 2 = − ˆ 1 we have T T ∗ = − 1 N , which forces N to be ev en. As in class (0-10) there is some basis such that T = J N . Let’ s now consider systems in which all 3 types of non-ordinary symmetries are present. The assumption ˆ S = ˆ T ˆ C is equi valent to S = T C ∗ . W e may start by choosing a basis where S = 1 m , N − m . Since T =  T T t we can write T = h a b  T b t d i , with a ∈ M m C satisfying a =  T a t , and d ∈ M N − m C satisfying d =  T d t . W e then ha ve C = T t S ∗ = h  T a −  T b b t −  T d i . Suppose now  T =  C . The conditions  T 1 N = T T ∗ = C C ∗ gi ve us further restraints on a , b and d . The second equation simplifies to h bb † ab ∗ db † b t b ∗ i =  0 0 0 0  , which implies the first equation is equi valent to h aa ∗ bd ∗ b t a ∗ dd ∗ i = h  T 1 m 0 0  T 1 N − m i . Class BDI (111) : Suppose  T =  C = 1. From the abov e equations we can conclude both a and d are symmetric unitary matrices, and that b = 0. T akagi decomposition then tells us that there is a unitary matrix of the form R =  r 0 0 s  such that RR t = T . Since R 1 m , N − m R † = 1 m , N − m , in the basis determined by R we hav e S = 1 m , N − m , C = 1 m , N − m , T = 1 N . Class CII (1 - 1 - 1) : If  T =  C = − 1 then a and d are ske w-symmetric, and b = 0. The fact that aa ∗ = − 1 m and d d ∗ = − 1 N − m implies both m and N − m are ev en. Again, T akagi 10 decomposition gi ves us a basis change such that a = J m and d = J N − m . Thus, we hav e a basis in which S = 1 m , N − m , C = " − J m 0 0 J N − m # , T = " J m 0 0 J N − m # . Suppose now  T = −  C . Note that either  T or  C must be − 1, which implies N is e ven. In this case we hav e T T ∗ = − CC ∗ , which is equiv alent to h aa ∗ bd ∗ b t a ∗ dd ∗ i =  0 0 0 0  . From T T ∗ =  T 1 N we no w get h bb † ab ∗ db † b t b ∗ i = h  T 1 m 0 0  T 1 N − m i . This last equation forces m = N / 2 , a = 0 and also c = 0. Class CI (1 - 11) : Suppose no w  T = 1 and  C = − 1. The matrix R = h b † 0 0 1 N / 2 i gi ves us a change of basis, in which we no w hav e S = 1 N / 2 , N / 2 , C = − J N , T = F N . Class DIII (11 - 1) : If  T = − 1 and  C = 1 then the same matrix R = h b † 0 0 1 N / 2 i provides a change of basis where S = 1 N / 2 , N / 2 , C = F N , T = J N . 2.5. Spaces of equivariant free Hamiltonian. W e are interested in classifying the spaces of equi v ariant free Hamiltonians. For ˆ U ∈ G and ˆ H ∈ H N we ha ve ˆ U ad ˆ H ˆ U † Ψ = ˆ U ˆ H ˆ U † Ψ − Ψ ˆ H , thus ˆ U ˆ H ˆ U † = ˆ H ⇐ ⇒ ˆ U ad ˆ H ˆ U † = ad ˆ H . Definition 2.5. Let ( W N , b , G ) be a Namb u space with symmetries of signature (  S  C  T ). Its space of equivariant fr ee Hamiltonians is H  S  C  T N = { ˆ H ∈ H N | ∀ ˆ U ∈ G : ˆ U ˆ H ˆ U † = ˆ H } . For a usual symmetry , conjugation of the adjoint representation corresponds, in a gi ven orthonormal basis, to conjugation of the representati ve matrices. Explicitly for an ordi- nary symmetry ˆ U ∈ G U L and free Hamiltonian ˆ H , since [ ˆ U ad ˆ H ˆ U † ] e B = " U H U † 0 0 − U ∗ H ∗ U t # we hav e that ˆ U ˆ H ˆ U † = ˆ H ⇐ ⇒ U H U † = H . For ˆ S ∈ G T A , ˆ C ∈ G T L and ˆ T ∈ G U A we hav e [ ˆ S ad ˆ H ˆ S † ] e B = " 0 S S ∗ 0 # K " H 0 0 − H ∗ # K " 0 S t S † 0 # = " − S H S † 0 0 S ∗ H ∗ S t # , [ ˆ C ad ˆ H ˆ C † ] e B = " 0 C C ∗ 0 #" H 0 0 − H ∗ #" 0 C t C † 0 # = " − C H ∗ C † 0 0 C ∗ HC t # , [ ˆ T ad ˆ H ˆ T † ] e B = " T 0 0 T ∗ # K " H 0 0 − H ∗ # K " T † 0 0 T t # = " T H ∗ T † 0 0 − T ∗ H T t # . 11 Thus ˆ S ˆ H ˆ S † = ˆ H ⇐ ⇒ S H S † = − H , ˆ C ˆ H ˆ C † = ˆ H ⇐ ⇒ C H C † = − H ∗ , ˆ T ˆ H ˆ T † = ˆ H ⇐ ⇒ T H T † = H ∗ . W e can now characterize the matrices that represent the equi variant free Hamiltonians for each symmetry class in the con venient basis of the last section. W e summarize the results from this section in the last column of T able 1. Class AIII (100) : Let ˆ S ∈ G T A be the chiral symmetry represented by the matrix S = 1 m , N − m . For all a ∈ M m C , b ∈ M m , N − m C , c ∈ M N − m , m C and d ∈ M N − m C we hav e 1 m , N − m  a b c d  1 m , N − m =  a − b − c d  . By self-adjointness, for a Hamiltonian ˆ H its representing matrix is of the form H = h a b b † d i . Since ˆ S ˆ H ˆ S † = ˆ H i ff S H S = − H , we conclude the equi variant Hamiltonians in this class are represented by matrices of the form H = h 0 b b † 0 i . Class D (010) : If ˆ C ∈ G T L is represented by C = 1 N then a Hamiltonian ˆ H commutes with ˆ C i ff H = − H ∗ . Class C (0 - 10) : If ˆ C ∈ G T L is represented by C = J N then a Hamiltonian ˆ H commutes with ˆ C i ff J N H J † N = − H ∗ . If H = h a b b † d i for a , b , d ∈ M N / 2 C , with a and d self-adjoint, then J N H J † N = h d − b † − b a i . So a Hamiltonian ˆ H commutes with ˆ C i ff H =  a b b ∗ − a ∗  , a = a † and b = b t . Class AI (001) : If ˆ T ∈ G U A is represented by T = 1 N then a Hamiltonian ˆ H commutes with ˆ T i ff H = H ∗ . Class AII (00 - 1) : If ˆ T ∈ G U A is represented by T = J N then ˆ H commutes with ˆ T i ff J N H J † N = H ∗ . Thus ˆ H commutes with ˆ T i ff H =  a b − b ∗ a ∗  , a = a † and b = − b t . Class BDI (111) : From the restrictions in classes AI I I and A I , ˆ H is equi v ariant in this class i ff H = h 0 b b t 0 i and b = b ∗ . Class DIII (11 - 1) : From the restrictions in classes A I I I and A I I a Hamiltonian in this class is equi variant i ff H =  0 b − b ∗ 0  , with b = − b t . Class CI (1 - 11) : From the restriction in A I I I and equiv ariance with regards to time- reflection symmetry we again ha ve that H = h 0 b b † 0 i , and now F N H F N = H ∗ is equi v alent to b = b t . Class CII (1 - 1 - 1) : From the restriction in A I I I we hav e H = h 0 B B † 0 i , where B =  a b c d  with a , b , c , d ∈ M m / 2 , N − m / 2 C . Equiv ariance with re gards to time-reflection symmetry implies J m B J † N − m = B ∗ , thus d = a ∗ and c = − b ∗ . 12 2.6. Classification by symmetric spaces of compact type. The central observ ation of the classification of free fermion system with symmetries is that H  S  C  T N is the tangent space of an irreducible compact symmetric space, and that all irreducible compact sym- metric space classes are realized this way . Definition 2.6. Let ( W N , b , G ) be a Namb u space with symmetries of signature (  S  C  T ). The spaces of equi variant time-e volution operators is M  S  C  T N B { Ad exp ( − i ˆ H ) ∈ M N | ˆ H ∈ H  S  C  T N } . Let M be a connected d -dimensional Riemannian manifold and p ∈ M . A di ff eomor- phism f on a neighborhood U of p is a geodesic symmetry around p if it fixes p and re verses geodesics around p , ie if γ is a geodesic with γ (0) = p then f γ ( θ ) = γ ( − θ ). In particular this implies d f p = − I d T p M . W e say M is locally symmetric if its geodesic symmetries are isometries, and that a locally symmetric manifold is (globally) symmetric if all its geodesic symmetries can be extended to all of M . The space M  S  C  T N is a compact symmetric space. The symmetric structure comes from the following construction: Let M be a connected compact Lie group equipped with a Cartan in volution, ie an automorphism τ : M → M with the in voluti ve property τ 2 = id M . Define the subgroup of τ -fixed elements as Fix τ B { u ∈ M | u = τ ( u ) } . Then the quotient space M Fix τ is a symmetric space of compact type. T o define its Riemannian structure let m = T e M be the Lie algebra of M , m = k ⊕ p be its de- composition into the positiv e and negati ve eigenspaces of the in volution d τ ( e ), and let ⟨ · | · ⟩ p be an inner product which is in v ariant under the adjoint action of Fix τ on p , ie ⟨ v | w ⟩ p = ⟨ Ad k v | Ad k w ⟩ p for all v , w ∈ p and k ∈ Fix τ . Then the Riemannian metric at T u Fix τ M Fix τ is g u Fix τ ( v , w ) B D d L − 1 u ( v )    d L − 1 u ( w ) E p , where L u ( v Fix τ ) = uv Fix τ . The submani- fold In v τ B { u ∈ M | τ ( u ) = u − 1 } of τ -in verted elements is isometric to M Fix τ by the Cartan embedding c τ : M Fix τ → In v τ ⊂ M , c τ ( u Fix τ ) B u τ ( u − 1 ) . The geodesic in version with respect to y ∈ In v τ is s y ( x ) = y x − 1 y . If v ∈ In v ν then c τ ( v Fix τ ) = v τ ( v − 1 ) = v 2 . Thus if v is a square root of u in In v τ , then c − 1 τ ( u ) = v Fix τ . W e no w deriv e the symmetric spaces associated with each class of grotesque fermion systems. In all cases we assume the con venient basis of the previous sections, and their structure is listed in T able 2. Class A (000) : In the absence of non-ordinary symmetries M 000 N = M N  U ( N ) . In the presence of a charge conjugation symmetry we will ha ve to consider the Lie subgroup of elements fixed by its action. Let ˆ C ∈ G T L be a charge conjugation symmetry 13 CAZ (  S  C  T ) [ Ad ˆ U ] B Symmetric space A I I I (100) U = " a b − b † d # , a † = a d † = d U ( N ) U ( m ) × U ( N − m ) A (000) − U ( N ) BD I (111) U = " a b b t d # , a ∗ = a t = a d ∗ = d t = d b ∗ = − b O ( N ) O ( m ) × O ( N − m ) A I (001) U = U t U ( N ) O ( N ) C I (1-11) U = " a b − b ∗ a ∗ # , a † = a b t = b S p ( N / 2 ) U ( N / 2 ) C (0-10) U = " a b − b ∗ a ∗ # S p ( N / 2 ) C I I (1-1-1) U =                  a 0 a 1 b 0 b 1 − a ∗ 1 a ∗ 0 b ∗ 1 − b ∗ 0 − b † 0 − b t 1 d 0 d 1 − b † 1 b t 0 − d ∗ 1 d ∗ 0                  , a † 0 = a 0 a t 1 = − a 1 d † 0 = d 0 d t 1 = − d 1 S p ( N / 2 ) S p ( m / 2 ) × S p ( N − m / 2 ) A I I (00-1) U = " a b c a t # , b t = − b c t = − c U ( N ) S p ( N / 2 ) D I I I (11-1) U = " a b b ∗ a ∗ # , a † = a b t = − b O ( N ) U ( N / 2 ) D (010) U = U ∗ O ( N ) T able 2. The third column describes the structure of equi v ariant time e vo- lution unitary operators within the manifold M  S  C  T N . The fourth column identifies the corresponding symmetric space classified by Cartan. such that ˆ C 2 = ± ˆ 1. Then Ξ : M 000 N → M 000 N , Ξ ( Ad ˆ U ) B ˆ C Ad ˆ U ˆ C † , Ξ " U 0 0 U ∗ #! = " C U ∗ C † 0 0 C ∗ U C t # is a Cartan in v olution. A Hamiltonian ˆ H commutes with ˆ C i ff Ad e − i ˆ H is Ξ -fixed. At the matrix le vel the Ξ -fix ed elements satisfy C U C † = U ∗ . Class D (010) : Ad ˆ U ∈ M 010 N = Fix Ξ i ff U = U ∗ . Thus M 010 N  O ( N ). Class C (0 - 10) : Ad ˆ U ∈ M 0 − 10 N = Fix Ξ i ff U =  a b − b ∗ a ∗  . Thus M 0 − 10 N  S p ( N / 2 ). Let ˆ S ∈ G T A be a chiral symmetry such that ˆ S 2 = ˆ 1. Then Σ : M 000 N → M 000 N , Σ ( Ad ˆ U ) B ˆ S Ad ˆ U ˆ S † , Σ " U 0 0 U ∗ #! = " S U S † 0 0 S ∗ U ∗ S t # 14 is a Cartan in volution. Due to ˆ S being anti-linear we no w hav e that ˆ H commutes with ˆ S i ff Ad exp ( − i ˆ H ) ∈ In v Σ . The Σ -fixed elements are those such that S U S † = U , and the Σ -in verted elements satisfy S U S † = U † . The associated Cartan embedding is c Σ ( Ad ˆ U Fix Σ ) = [ Ad ˆ U , ˆ S ] . Class AIII (100) : In our choice of basis we hav e S = 1 m , N − m for some m ≤ N . Let Ad ˆ U ∈ M 000 N , a ∈ M m C , b ∈ M m , N − m C , c ∈ M N − m , m C and d ∈ M N − m C be such that U =  a b c d  . Since Ad ˆ U ∈ Fix Σ i ff b = 0 and c = 0, so that Fix Σ  U ( m ) × U ( N − m ), then M 100 N = In v Σ  M 000 N Fix Σ  U ( N ) U ( m ) × U ( N − m ) . Thus Ad ˆ U ∈ M 100 N i ff U = h a b − b † d i , a = a † and d = d † . Note we get di ff erent spaces depending on m ≤ N , so we will denote the spaces of time-e volution operators in class A I I I as M 100 N , m to di ff erentiate between them. Let ˆ T ∈ G U A be a chiral symmetry such that ˆ T 2 = ± ˆ 1. Then Θ : M 000 N → M 000 N , Θ ( Ad ˆ U ) B ˆ T Ad ˆ U ˆ T † , Θ " U 0 0 U ∗ #! = " T U ∗ T † 0 0 T ∗ U T t # is a Cartan in volution. Due to ˆ T being anti-linear we now ha ve that ˆ H commutes with ˆ T i ff Ad exp ( − i ˆ H ) ∈ In v Θ . The Θ -fixed elements are those such that T U T † = U ∗ , and the Θ -in verted elements satisfy T U T † = U t . The associated Cartan embedding is c Θ ( Ad ˆ U Fix Θ ) = [ Ad ˆ U , ˆ T ] . Class AI (001) : Since Fix Θ  O ( N ) we hav e M 001 N = In v Θ  U ( N ) O ( N ) . Thus Ad ˆ U ∈ M 001 N i ff U t = U . Class AII (00 - 1) : Since Fix Θ  S p ( N / 2 ) we ha ve M 00 − 1 N = In v Θ  U ( N ) S p ( N / 2 ) . Thus Ad ˆ U ∈ M 00 − 1 N i ff U = h a b c a t i , b t = − b and c t = − c . When all 3 types of non-ordinary symmetries are present, we consider the subgroup of Ξ -fixed elements. In all cases, this group is closed under the in volution Σ , and the space of time-e volution operators is the space of Ξ -fix ed elements that are Σ -in verted. The same results would be obtained if we considered the in volution Θ instead of Σ . Class BDI (111) : The Ξ -fixed elements are represented by unitary matrices U =  a b c d  such that C U C = U ∗ for C = 1 m , N − m , which means  a − b − c d  = h a ∗ b ∗ c ∗ d ∗ i , ie those where a ∈ M m R , d ∈ M N − m R , b ∈ M m , N − m i R and c ∈ M N − m , m i R . W e then ha ve an isomorphism Fix Ξ  − → O ( N ) , U 7→ " 1 m 0 0 − i 1 N − m #" a b c d #" 1 m 0 0 i 1 N − m # = " a ib − ic d # . The Σ -fixed elements are those such that b = 0 and c = 0, so M 111 N , m = Fix Ξ ∩ In v Σ  O ( N ) O ( m ) × O ( N − m ) . Thus Ad ˆ U ∈ M 111 N , m i ff U = h a b b t d i , a = a t = a ∗ , b ∗ = − b and d = d t = d ∗ . 15 Class DIII (11 - 1) : Since in this class C = F N the Ξ -fixed elements are represented by matrices U =  a b b ∗ a ∗  . W e then ha ve an isomorphism Fix Ξ  − → O ( N ) , U 7→ 1 2 " 1 N / 2 1 N / 2 − i 1 N / 2 i 1 N / 2 #" a b b ∗ a ∗ #" 1 N / 2 i 1 N / 2 1 N / 2 − i 1 N / 2 # = " Re( a + b ) − Im( a − b ) Im( a + b ) Re( a − b ) # . The Σ -fix ed elements are those such that b = 0. The abov e isomorphism maps Fix Σ to the subgroup of O ( N ) composed of matrices of the form h Re( a ) − Im( a ) Im( a ) Re( a ) i , which is isomorphic to U ( N / 2 ), so M 11 − 1 N = Fix Ξ ∩ In v Σ  O ( N ) U ( N / 2 ) . Thus Ad ˆ U ∈ M 11 − 1 N , m i ff U =  a b b ∗ a ∗  , a = a † and b = − b t . Class CI (1 - 11) : In this class we have Fix Ξ  S p ( N / 2 ), as in class C. The subgroup of Σ -fixed elements are represented by matrices of the form  a 0 0 a ∗  , which is isomorphic to U ( N / 2 ), so M 1 − 11 N = Fix Ξ ∩ In v Σ  S p ( N / 2 ) U ( N / 2 ) . Thus Ad ˆ U ∈ M 1 − 11 N i ff U =  a b − b ∗ a ∗  , a = a † and b = b t . Class CII (1 - 1 - 1) : In this class we hav e that the Ξ -fixed elements are of the form U =        a 0 a 1 b 0 b 1 − a ∗ 1 a ∗ 0 b ∗ 1 − b ∗ 0 c 0 c 1 d 0 d 1 c ∗ 1 − c ∗ 0 − d ∗ 1 d ∗ 0        , a i ∈ M m / 2 C , b i ∈ M m / 2 , N − m / 2 C , c i ∈ M N − m / 2 , m / 2 C , d i ∈ M N − m / 2 C . Setting V =                  1 m / 2 0 0 0 0 0 − i 1 m / 2 0 0 1 N − m / 2 0 0 0 0 0 − i 1 N − m / 2                  (1) we then hav e an isomorphism Fix Ξ  − → S p ( N ) , U 7→ V U V † =                  a 0 ib 0 a 1 ib 1 − ic 0 d 0 − ic 1 d 1 − a ∗ 1 ib ∗ 1 a ∗ 0 − ib ∗ 0 − ic ∗ 1 − d ∗ 1 ic ∗ 0 d ∗ 0                  . The Σ -fixed elements are those such that b i = 0 and c i = 0, so M 1 − 1 − 1 N , m = Fix Ξ ∩ In v Σ  S p ( N / 2 ) S p ( m / 2 ) × S p ( N − m / 2 ) . Thus Ad ˆ U ∈ M 1 − 1 − 1 N , m i ff U =          a 0 a 1 b 0 b 1 − a ∗ 1 a ∗ 0 b ∗ 1 − b ∗ 0 − b † 0 − b t 1 d 0 d 1 − b † 1 b t 0 − d ∗ 1 d ∗ 0          , a 0 = a † 0 , a t 1 = − a 1 , d † 0 = d 0 and d t 1 = − d 1 . 3. K- theoretical classifica tion 3.1. Cohomology theories and repr esenting spectra. A cohomology theory is a con- trav ariant functor E n : Top o p ∗ → AbGrp Z equipped with degree -1 natural isomorphisms σ n : E n + 1 ( X ∧ S 1 ) → E n X satisfying the Eilenber g-Steenrod homotopy in variance, exact- ness and additi ve axioms. 16 Complex and real topological K-theory are cohomology theories induced by the alge- braic structure of isomorphism classes of vector bundles over spaces [4]. T opological K- theory can be defined from the Murray-von Neumann category functor [28]. Let F = C , R and X ∈ Top . The Murray-von Neumann category pr F X has as objects orthogonal pro- jection matrices ov er C ( X , F ), and morphisms are matrices witnessing the Murray-von Neumann relation: Ob pr F X B ` p ∈ N { P ∈ M p C ( X , F ) | P = P † = P 2 } , pr F X ( P , Q ) B { U ∈ M q , p C ( X , F ) | P = U † U , Q = U U † } . Composition is defined by matrix multiplication, and id P = P . The category pr F X is equi valent to the category of vector bundles over X and bundle isomorphisms. For ev ery P ∈ pr F X we hav e that ` x ∈ X Im P ( x ) ⊂ X × F p is a vector b undle ov er X , and if U ∈ pr F X ( P , Q ) then ˜ U : ` x ∈ X Im P ( x ) → ` x ∈ X Im Q ( x ), with ˜ U ( x , v ) B ( x , U ( x ) v ), is a bundle isomorphism. Every vector bundle ov er X is isomorphic to one induced by an object of pr F X . The fiberwise direct sum gives us a permutativ e structure on the category of vector bundles over X . This permutativ e structure is encoded in pr F X by the direct sum of matrices U ⊕ V B  U 0 0 V  . The neutral element is the empty matrix 0 B h i , and the symmetry braiding is τ P , Q B h 0 Q P 0 i : P ⊕ Q → Q ⊕ P . This permutati ve structure induces a commutativ e monoid structure on the decategorification π 0 | pr F X | , ie the set of isomorphism classes of the objects in pr F X . Applying Grothendieck’ s construction we functorialy get the abelian group completion K 00 F X B G r π 0 | pr F X | . The functor K 00 F induces a cohomology theory K • F defined as follows: F or X ∈ Top ∗ the 0-th K-group is K 0 F X B ker( i ∗ x 0 : K 00 F X → K 00 { x 0 }  F ). This group is isomorphic to the quotient of K 00 F X by the subgroup of trivial b undles ov er the connected component of x 0 . For n < 0 we define K n F X B K 0 F ( X ∧ S | n | ). Bott’ s periodicity theorem [6, 3] tells us complex topological K-theory has periodicity 2, meaning we ha ve a natural isomorphism K n + 2 C X  K n C X , and that real topological K-theory has periodicity 8, ie K n + 8 R X  K n R X . This periodicity allo ws us to define the positiv e degree K-groups. For all n < 0 we ha ve natural isomorphisms S | n | − 1 ∧ S 1  S | n | . These induce natural suspensions σ n : K n + 1 F ( X ∧ S 1 ) → K n F X . The functors K n F equipped with the suspensions σ n form the complex and real topological K-theories. Bro wn’ s representability theorem tells us that cohomology theories can be represented by objects called spectra. There are many versions of the cate gory of spectra, but for our purposes the follo wing simple definition will su ffi ce. Definition 3.1. A (sequential pre-)spectrum Y is a sequence of pointed topological spaces { Y • } •∈ N ∈ Top N ∗ equipped with structural suspension maps { σ Y • : Y • ∧ S 1 → Y • + 1 } •∈ N . A spectrum map φ : Y → Z is a sequence of pointed maps { φ • : Y • → Z • } •∈ N such that σ Z • ( φ • ( y ) , θ ) = φ • + 1 ( σ Y • ( y , θ )). For n ∈ Z the n -th stable homotopy group of a spectrum Y is π S n Y = colim •→∞ Y S • n + • , 17 with the colimit induced by the duals of the structural suspension maps. For X ∈ Top ∗ the mapping spectrum Y X is defined by ( Y X ) • B Y X • , σ Y X • ( f , θ )( x ) B σ Y • ( f ( x ) , θ ) . Every cohomology theory E • is represented by a spectrum Y , in the sense that for all X ∈ Top ∗ we hav e a natural graded isomorphism E n X = π S n Y X . A homotopy between spectra maps f , g : Y → Z is a map h : Y ∧ I + → Z such that h • ( y , 0) = f • ( y ) and h • ( y , 1) = g • ( y ). A homotop y equiv alence is a spectra map that admits an in verse up to homotopy , and spectra that are homotopy equi valent represent the same cohomology theory . In particular , if a subspectrum Y ⊂ Z is a deformation retract, ie if there is a map r : Z → Y homotopy equiv alent to the identity , then Y and Z represent the same cohomology theory . Complex and real topological K-theory are represented by spectra K U and K O . There are many constructions of these spectra, see for instance [22, 26]. W e no w giv e a def- inition of K U and K O in terms of spaces of free time ev olution operators of grotesque fermion systems, with structural suspension maps induced by the homotopy equi valences in Bott’ s proof of the periodicity theorem [6]. T o the best of our knowledge such ex- plicit expressions have not up to now appeared in the literature. In section 4 we will see that spectra K U wi and K O wi constructed from weakly interacting time e volution operators deformation retract to K U and K O . Thus, the tenfold way is stable to weak interactions. 3.2. Construction of K U and K O in terms of time-ev olution operators. W e first sho w ho w the complex topological K-theory spectrum K U can be described in terms of time e volution operators of systems without time rev ersal or charge conjugation symmetries. Bott’ s periodicity theorem for comple x K-theory is reflected in the fact that its represent- ing spectrum K U is composed of a 2-periodic sequence of spaces, alternating between classes with only chiral symmetry and without any non-re gular symmetries. Any choice of basis for each Hilbert space V N gi ves us inclusions ι N : V N  → V N + 1 , with ι N ( | i ⟩ ) = | i ⟩ . These induce the maps ι N : M 000 N  → M 000 N + 1 , ι N ( a ) = " a 0 0 1 # . W e can define M 000 B colim N ∈ N M 000 N  U ( ∞ ) . For each N , m ∈ N with m ≤ N the inclusions ι N induce maps ι N : M 100 N , m  → M 100 N + 1 , m . W e can then define M 100 ∞ , m B colim N ∈ N ≥ m M 100 N , m . W e also have inclusions κ N , m : V N  → V N + 1 with κ N , m ( | i ⟩ ) = | i ⟩ if i ≤ m , and κ N , m ( | i ⟩ ) = | i + 1 ⟩ if i > m . These induce maps κ N , m : M 100 N , m  → M 100 N + 1 , m + 1 , κ N , m " a b − b † d #! =            a 0 b 0 1 0 − b † 0 d            , 18 which gi ves us inclusions κ m : M 100 ∞ , m  → M 100 ∞ , m + 1 . W e can define M 100 B colim m ∈ N M 100 ∞ , m  BU ( ∞ ) . The structural suspension maps of K U can be constructed by adapting the homotopy equi valences used by Bott to prove his periodicity theorem [6], which is a consequence of the follo wing proposition: Proposition 3.2. Let G be a compact Lie gr oup equipped with a Cartan in volution τ . Let s be a geodesic se gment on the symmetric space G F ix τ fr om the coset F ix τ to a coset gF ix τ , with g in the normalizer of F ix τ . Let K s be the centralizer of s, and Ω s G F ix τ be the component of s in the space of paths fr om F ix τ to gF ix τ . Define the map f s : F ix τ K s → Ω s G F ix τ , f s ( x K s ) B ( θ 7→ x s ( θ ) x − 1 F ix τ ) . If s contains no conjugate point of e in its interior , then the induced homomorphism f ∗ s : H • ( Ω s G F ix τ , Z 2 ) → H • ( F ix τ K s , Z 2 ) is surjective. Bott used this proposition to sho w that for all N there is a geodesic s in U (2 N ) such that f s : U (2 N ) U ( N ) × U ( N ) → Ω s (2 N ) induces an isomorphism of homotopy groups up to dimension 2 N . T aking colimits this gi ves us a homotopy equi valence BU ( ∞ ) ≃ Ω s U ( ∞ ). W e can adapt Bott’ s construction to define the e ven structural suspensions of K U . Con- sider for N ∈ N the geodesic Ad ˆ s 0 ( θ ) ∈ M 000 2 N , ˆ s 0 ( θ ) B exp  − i πθ ( P N j = 1 a † j a j − a † N + j a N + j )  , (2) [ Ad ˆ s 0 ( θ ) ] B = exp  − i πθ 1 N , N  = " e − i πθ 1 N 0 0 e i πθ 1 N # . Since the centralizer of ˆ s 0 is Fix Σ the map σ ′ 0 : M 000 2 N Fix Σ ∧ S 1 → M 000 2 N , σ ′ 0 ( Ad exp ( − i ˆ H ) Fix Σ , θ ) B [ Ad exp ( − i ˆ H ) , Ad ˆ s 0 ( θ ) ] = Ad [exp ( − i ˆ H ) , ˆ s 0 ( θ )] is well defined. T aking colimits we can define σ 0 : M 100 ∧ S 1 → M 000 , σ 0 ( Ad exp ( − i ˆ H ) , θ ) B σ ′ 0 ( c − 1 Σ Ad exp ( − i ˆ H ) , θ ) = Ad [exp ( − i / 2 ˆ H ) , ˆ s 0 ( θ )] . (3) Under the con venient basis of the pre vious section we hav e σ 0 exp − i " 0 b b † 0 #! , θ ! = exp − i 2 " 0 b b † 0 #! exp i 2 " 0 e − i 2 πθ b e i 2 πθ b † 0 #! . This structural suspension induces a homotopy equi valence of M 100 with the connected component of the tri vial loop in Ω M 000 . The total spaces E U ( m ) of the uni versal U ( m )-principal bundle are contractible, and these contractions induce a homotopy equiv alence U ( ∞ ) ≃ Ω B U ( ∞ ). This giv es us the odd structural suspensions of K U . 19 Recall that the chiral symmetry ˆ S induces orthogonal decomposition V N = V + N ⊕ V − N . Let ˆ N + B P m j = 1 a † j a j and ˆ N − B P N j = m + 1 a † j a j , which restricted to V N gi ve us the orthogonal projections onto V + N and V − N , respecti vely . Consider the subgroups M + N , m B { Ad ˆ U ∈ M 000 N | ˆ N + Ad ˆ U ˆ N + + ˆ N − = Ad ˆ U } , M − N , m B { Ad ˆ U ∈ M 000 N | ˆ N + + ˆ N − Ad ˆ U ˆ N − = Ad ˆ U } . Note that Fix Σ = M + N , m M − N , m = M − N , m M + N , m . Under the isomorphism M 000 N  U ( N ), we hav e M + N , m  U ( m ) × { 1 N − m } and M − N , m  { 1 m } × U ( N − m ). W e can define E 100 N , m B M 000 N M − N , m . The inclusions ι N induce maps E 100 N , m  → E 100 N + 1 , m , and we define E 100 ∞ , m B colim N ∈ N ≥ m E 100 N , m . By construction this space is di ff eomorphic to E U ( m ). W e then hav e fibrations π m : E 100 ∞ , m ↠ M 100 ∞ , m with fibers M + ∞ , m B colim N ∈ N ≥ m M + N , m  M 000 m  U ( m ). The inclusions κ m , N gi ve us maps κ m : E 100 ∞ , m → E 100 ∞ , m + 1 , and we define E 100 B colim m ∈ N E 100 ∞ , m . By construction E 100  E U ( ∞ ). The fibrations π m then induce a fibration π : E 100 ↠ M 100 , with fibers M + B colim m ∈ N M + ∞ , m  M 000  U ( ∞ ), which is di ff eomorphic to the uni versal U ( ∞ )-principal bundle. W e then set the maps ι 1 : H 000 N → H 000 2 N , ι 1 ( H ) B " H 0 0 0 # , which induce a fiber inclusion M 000  → E 100 . For N , m ∈ N with m ≤ N consider the geodesic Ad ˆ s 1 ( θ ) ∈ M 000 N + m , ˆ s 1 ( θ ) B exp  − i πθ / 2 ( P m j = 1 a † j a j − a † j a N + j − a † N + j a j + a † N + j a N + j )  , [ ˆ s 1 ( θ )] B = exp            − i πθ 2            1 m 0 − 1 m 0 0 N − m 0 − 1 m 0 1 m                       =            1 + e − i πθ 2 1 m 0 1 − e − i πθ 2 1 m 0 1 N − m 0 1 − e − i πθ 2 1 m 0 1 + e − i πθ 2 1 m            W e can then define ˜ σ ′ 1 : E 100 N , m ∧ S 1 → M 000 N + m , m Fix Σ , ˜ σ ′ 1 ( Ad exp ( − i ˆ H ) M − N , m , θ ) B Ad exp ( − i ˆ s 1 ( θ ) ι 1 ( ˆ H ) ˆ s 1 ( θ ) † ) Fix Σ , since elements of M − N , m ⊂ M − N + m , m commute with ˆ s 1 ( θ ). This map induces a homotopy from ιπ : E 100 N , m ↠ M 100 N + m , m to the constant map at the base point. Considering that under our choice of basis M 000 N is identified with the fiber of π N : E 100 ∞ , N ↠ M 100 ∞ , N , we can define σ 1 : M 000 ∧ S 1 → M 100 , σ 1 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad exp ( − i ˆ s 1 ( θ ) ι 1 ( ˆ H ) ˆ s 1 ( θ ) † ) , ˆ S ] (4) σ 1  exp ( − i H ) , θ  =         exp         − i         cos  πθ 2  2 H i 2 sin ( πθ ) H − i 2 sin ( πθ ) H sin  πθ 2  2 H                 , ˆ S         . 20 Definition 3.3. The comple x topological K-theory spectrum K U is K U • B        M 100  BU ( ∞ ) , • = 0 mod 2 M 000  U ( ∞ ) , • = 1 mod 2 , σ • ( Ad exp ( − i ˆ H ) , θ ) B        Ad [exp ( − i / 2 ˆ H ) , ˆ s 0 ( θ )] , • = 0 mod 2 [ Ad exp ( − i ˆ s 1 ( θ ) ι 1 ( ˆ H ) ˆ s 1 ( θ ) † ) , ˆ S ] , • = 1 mod 2 . W e no w describe the real topological K-theory spectrum K O in terms of time ev olution operators of systems with time-in version and / or charge conjug ation symmetries. The maps ι N induce inclusions M 010 N  → M 010 N + 1 and M 001 N  → M 001 N + 1 . W e can then define M 010 B colim N ∈ N M 010 N  O ( ∞ ) and M 001 B colim N ∈ N M 001 N  U ( ∞ ) O ( ∞ ) . The maps ι N and κ m , N induce inclusions M 111 N , m  → M 111 N + 1 , m and M 111 N , m  → M 111 N + 1 , m + 1 . W e then define M 111 B colim m ∈ N , N ∈ N ≥ m M 111 N , m  BO ( ∞ ) . The maps ι N + 1 κ m + N − m / 2 , N = κ m + N − m / 2 , N + 1 ι N and κ m + 1 , N + 1 κ m / 2 , N = κ m / 2 , N + 1 κ m , N induce inclusions M 1 − 1 − 1 N , m  → M 1 − 1 − 1 N + 2 , m and M 1 − 1 − 1 N , m  → M 1 − 1 − 1 N + 2 , m + 2 . W e then define M 1 − 1 − 1 B colim m ∈ 2 N , N ∈ 2 N ≥ m M 1 − 1 − 1 N , m  BS p ( ∞ / 2 ) . The maps ι N + 1 κ N / 2 , N = κ N / 2 , N + 1 ι N induce inclusions M 0 − 10 N  → M 0 − 10 N + 2 , M 00 − 1 N  → M 00 − 1 N + 2 , M 1 − 11 N  → M 1 − 11 N + 2 and M 11 − 1 N  → M 11 − 1 N + 2 . W e then define M 0 − 10 B colim N ∈ 2 N M 0 − 10 N + 2  S p ( ∞ / 2 ) , M 00 − 1 B colim N ∈ 2 N M 00 − 1 N + 2  U ( ∞ ) S p ( ∞ / 2 ) , M 1 − 11 B colim N ∈ 2 N M 1 − 11 N + 2  S p ( ∞ / 2 ) U ( ∞ / 2 ) , M 11 − 1 B colim N ∈ 2 N M 11 − 1 N + 2  O ( ∞ ) U ( ∞ / 2 ) . Setting ˆ s 0 ( θ ) as in (2), the 0th structural suspension map is σ 0 : M 111 ∧ S 1 → M 001 , σ 0 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad [exp ( − i / 2 ˆ H ) , ˆ s 0 ( θ )] , ˆ T ] , σ 0 exp − i " 0 b b t 0 #! , θ ! = " exp − i 2 " 0 b b t 0 #! exp i 2 " 0 e − i 2 πθ b e i 2 πθ b t 0 #! , ˆ T # . In order to define the 1st structural suspension map, note first we hav e an inclusion ι 1 : H 001 N → H 000 2 N , ι 1 ( H ) B " H 0 0 − H # such that ι 1 ( ˆ H ) commutes with ˆ C in the class CI (1-11). Setting Ad ˆ s 1 ( θ ) ∈ M 1 − 11 2 N , ˆ s 1 ( θ ) B exp  − i πθ ( P N j = 1 a † j a N + j + a † N + j a j )  , (5) [ Ad ˆ s 1 ( θ ) ] B = exp ( − i πθ F 2 N ) = " cos ( πθ ) 1 N − i sin ( πθ ) 1 N − i sin ( πθ ) 1 N cos ( πθ ) 1 N # 21 we can define σ 1 : M 001 ∧ S 1 → M 1 − 11 , σ 1 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad [exp ( − i / 2 ι 1 ( ˆ H ) ) , ˆ s 1 ( θ )] , ˆ T ] σ 1  exp ( − i H ) , θ  = " exp − i 2 " H 0 0 − H #! exp i 2 " cos ( 2 πθ ) H i sin ( 2 πθ ) H − i sin ( 2 πθ ) H − cos ( 2 πθ ) H #! , ˆ T # . Setting Ad ˆ s 2 ( θ ) ∈ M 0 − 10 N , ˆ s 2 ( θ ) B exp  − i πθ ( P N / 2 j = 1 a † j a j − a † N / 2 + j a N / 2 + j )  , [ Ad ˆ s 2 ( θ ) ] B = exp  − i πθ 1 N / 2 , N / 2  = " e − i πθ 1 N / 2 0 0 e i πθ 1 N / 2 # the 2nd structural suspension map is σ 2 : M 1 − 11 ∧ S 1 → M 0 − 10 , σ 2 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad exp ( − i / 2 ˆ H ) , ˆ s 2 ( θ )] (6) σ 2 exp − i " 0 b b ∗ 0 #! , θ ! = exp − i 2 " 0 b b ∗ 0 #! exp i 2 " 0 e − i 2 πθ b e i 2 πθ b ∗ 0 #! . T o define the 3rd structural suspension we set Ad ˆ s 3 ( θ ) ∈ M 000 2 N , ˆ s 3 ( θ ) B exp  − i πθ / 2 ( P N j = 1 a † j a j − a † j a N + j − a † N + j a j + a † N + j a N + j )  , [ Ad ˆ s 3 ( θ ) ] B = exp − i πθ 2 " 1 N − 1 N − 1 N 1 N #! =       1 + e − i πθ 2 1 N 1 − e − i πθ 2 1 N 1 − e − i πθ 2 1 N 1 + e − i πθ 2 1 N       as in the 1st suspension map of K U . W e also set the inclusion ι 3 : H 0 − 10 N → H 000 2 N , ι 3 ( H ) B " H 0 0 0 # , such that ι 3 ( ˆ H ) commutes with ˆ C in the class CII (1-1-1). This lets us define σ 3 : M 0 − 10 ∧ S 1 → M 1 − 1 − 1 , σ 3 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad exp ( − i ˆ s 3 ( θ ) ι 3 ( ˆ H ) ˆ s 3 ( θ ) † ) , ˆ S ] , σ 3  exp ( − i H ) , θ  =         exp         − i         cos  πθ 2  2 H i 2 sin ( πθ ) H − i 2 sin ( πθ ) H sin  πθ 2  2 H                 , ˆ S         . Letting V be as in (1) then conjugation by V induces the inclusion ι 4 : H 1 − 1 − 1 2 N , N → H 000 N , ι 4                                   0 0 b 0 b 1 0 0 − b ∗ 1 b ∗ 0 b † 0 − b t 1 0 0 b † 1 b t 0 0 0                                   B                  0 ib 0 0 ib 1 − ib † 0 0 ib t 1 0 0 − ib ∗ 1 0 ib ∗ 0 − ib † 1 0 − ib t 0 0                  . 22 Setting Ad ˆ s 4 ( θ ) ∈ M 00 − 1 2 N , ˆ s 4 ( θ ) B exp  − i πθ ( P N / 2 j = 1 a † j a j − a † N / 2 + j a N / 2 + j + a † N + j a N + j − a † 3 N / 2 + j a 3 N / 2 + j )  , [ Ad ˆ s 4 ( θ ) ] B = exp − i πθ " 1 N / 2 , N / 2 0 0 1 N / 2 , N / 2 #! =                  e − i πθ 1 N / 2 0 0 0 0 e i πθ 1 N / 2 0 0 0 0 e − i πθ 1 N / 2 0 0 0 0 e i πθ 1 N / 2                  the 4th structural suspension map is σ 4 : M 1 − 1 − 1 ∧ S 1 → M 00 − 1 , σ 4 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad [exp ( − i / 2 ι 4 ( ˆ H ) ) , ˆ s 4 ( θ )] , ˆ T ] , σ 4                  exp                  − i                  0 0 b 0 b 1 0 0 − b ∗ 1 b ∗ 0 b † 0 − b t 1 0 0 b † 1 b t 0 0 0                                   , θ                  =           exp          − i 2          0 ib 0 0 ib 1 − ib † 0 0 ib t 1 0 0 − ib ∗ 1 0 ib ∗ 0 − ib † 1 0 − ib t 0 0                   exp           i 2           0 e − i 2 πθ ib 0 0 e − i 2 πθ ib 1 − e i 2 πθ ib † 0 0 e i 2 πθ ib t 1 0 0 − e − i 2 πθ ib ∗ 1 0 e − i 2 πθ ib ∗ 0 − e i 2 πθ ib † 1 0 − e i 2 πθ ib t 0 0                     , ˆ T           . T o define the 5th structural map we hav e to consider the mappings ι 5 : H 00 − 1 N → H 000 2 N , ι 5 " a b − b ∗ a ∗ #! B                  a b 0 0 − b ∗ a ∗ 0 0 0 0 − a ∗ − b ∗ 0 0 b − a                  , such that ι 5 ( ˆ H ) commutes with ˆ C in the class DIII (11-1). Definig Ad ˆ s 5 ( θ ) ∈ M 11 − 1 2 N , ˆ s 5 ( θ ) B exp  − i πθ ( P N / 2 j = 1 a † j a N / 2 + j − a † N / 2 + j a j − a † N + j a 3 N / 2 + j + a † 3 N / 2 + j a N + j )  , [ Ad ˆ s 5 ( θ ) ] B = exp − i πθ " 0 J N − J N 0 #! =        cos ( πθ ) 1 N / 2 0 0 − i sin ( πθ ) 1 N / 2 0 cos ( πθ ) 1 N / 2 i sin ( πθ ) 1 N / 2 0 0 i sin ( πθ ) 1 N / 2 cos ( πθ ) 1 N / 2 0 − i sin ( πθ ) 1 N / 2 0 0 cos ( πθ ) 1 N / 2        we hav e σ 5 : M 00 − 1 ∧ S 1 → M 11 − 1 , σ 5 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad [exp ( − i / 2 ι 5 ( ˆ H ) ) , ˆ s 5 ( θ )] , ˆ T ] σ 5 exp − i " a b − b ∗ a ∗ #! , θ ! =       exp − 1 2 " a b 0 0 − b ∗ a ∗ 0 0 0 0 − a ∗ − b ∗ 0 0 b − a #! exp       1 2       cos ( 2 πθ ) a cos ( 2 πθ ) b − i sin ( 2 πθ ) b i sin ( 2 πθ ) a − cos ( 2 πθ ) b ∗ cos ( 2 πθ ) a ∗ − i sin ( 2 πθ ) a ∗ − i sin ( 2 πθ ) b ∗ − i sin ( 2 πθ ) b ∗ i sin ( 2 πθ ) a ∗ cos ( 2 πθ ) a ∗ cos ( 2 πθ ) b ∗ − i sin ( 2 πθ ) a − i sin ( 2 πθ ) b − cos ( 2 πθ ) b cos ( 2 πθ ) a             , ˆ T       . Defining ι 6 : H 11 − 1 N → H 010 N , ι 6 " 0 b − b ∗ 0 #! B " i Im( b ) − i Re( b ) − i Re( b ) − i Im( b ) # 23 and Ad ˆ s 6 ( θ ) ∈ M 010 , ˆ s 6 ( θ ) B exp  − i πθ ( P N / 2 j = 1 ia † j a N / 2 + j − ia † N / 2 + j a j )  , [ Ad ˆ s 6 ( θ ) ] B = exp ( πθ J N ) = " cos ( πθ ) 1 N / 2 sin ( πθ ) 1 N / 2 − sin ( πθ ) 1 N / 2 cos ( πθ ) 1 N / 2 # the 6th structural suspension map is σ 6 : M 11 − 1 N ∧ S 1 → M 010 N , σ 6 ( Ad exp ( − i ˆ H ) , θ ) B [ Ad exp ( − i / 2 ι 6 ( ˆ H ) ) , ˆ s 6 ( θ )] σ 6 exp − i " 0 b − b ∗ 0 #! , θ ! = exp 1 2 h Im( b ) − Re( b ) − Re( b ) − Im( b ) i ! exp − 1 2 h − sin ( 2 πθ ) Re( b ) + cos ( 2 πθ ) Im( b ) − cos ( 2 πθ ) Re( b ) − sin ( 2 πθ ) Im( b ) − cos ( 2 πθ ) Re( b ) − sin ( 2 πθ ) Im( b ) sin ( 2 πθ ) Re( b ) − cos ( 2 πθ ) Im( b ) i ! Defining ˆ s 7 ( θ ) and ι 7 as as in the 1st suspension map of K U the 7th structural suspension map is σ 7 : M 010 ∧ S 1 → M 111 , σ 7 ( Ad exp ( − i ι 7 ( ˆ H ) ) , θ ) B [ Ad exp ( − i ˆ s 7 ( θ ) ι 7 ( ˆ H ) ˆ s 7 ( θ ) † ) , ˆ S ] , σ 7  exp ( − i H ) , θ  =         exp         − i         cos  πθ 2  2 H i 2 sin ( πθ ) H − i 2 sin ( πθ ) H sin  πθ 2  2 H                 , ˆ S         . Definition 3.4. The real topological K-theory spectrum K O is K O • B                                            M 111  BO ( ∞ ) , • = 0 mod 8 M 001  U ( ∞ ) O ( ∞ ) , • = 1 mod 8 M 1 − 11  S p ( ∞ / 2 ) O ( ∞ / 2 ) , • = 2 mod 8 M 0 − 10  S p ( ∞ / 2 ) , • = 3 mod 8 M 1 − 1 − 1  BS p ( ∞ / 2 ) , • = 4 mod 8 M 00 − 1  U ( ∞ ) S p ( ∞ / 2 ) , • = 5 mod 8 M 11 − 1  O ( ∞ ) U ( ∞ / 2 ) , • = 6 mod 8 M 010  O ( ∞ ) , • = 7 mod 8 , σ • ( Ad exp ( − i ˆ H ) , θ ) B              [ Ad [exp ( − i / 2 ι • ( ˆ H ) ) , ˆ s • ( θ )] , ˆ T ] , • = 0 , 1 , 4 , 5 mod 8 Ad [exp ( − i / 2 ι • ( ˆ H ) ) , ˆ s • ( θ )] , • = 2 , 6 mod 8 [ Ad exp ( − i ˆ s • ( θ ) ι • ( ˆ H ) ˆ s • ( θ ) † ) , ˆ S ] , • = 3 , 7 mod 8 , where we assume ι 0 and ι 2 to be the identities. 4. W eakl y interacting systems As is standard, to model interacting systems we must consider operators in the Clif- ford subalgebra C l + ( W N , b ) = L n ∈ N C l 2 n ( W N , b ) generated by e ven degree monomials of creation and annihilation operators. These are the operators that commute with the parity operator ˆ P B e xp  i π ˆ N  , thus preserve the parity in the number of fermions. See [13, 15, 24, 29] for justifications for this assumption. 24 Definition 4.1. Let ( W N , b ) be a Namb u space. The space of interacting Hamiltonians is H i , N B { ˆ H ∈ C l + ( W N , b ) | ˆ H † = ˆ H } . The space of interacting time e volution operator s is M i , N B { Ad exp ( − i ˆ H ) ∈ Aut ( C l ( W N , b )) | ˆ H ∈ H i , N } . In the context of time ev olution operators we can giv e a geometric definition of weak interactions. W e will consider an operator weakly interacting if there is an unambiguous closest free operator to it, and a unique distance minimizing geodesic between them. This condition is satisfied by elements in the complement of the cut locus of M N in M i , N . W e recall here the definition of the cut locus of a submanifold [25, 27]. Definition 4.2. Let N be a compact submanifold of a complete manifold M . The separat- ing set S e ( N ) ⊂ M of N consists of all points u such that at least tw o distance minimizing geodesics from N to u exist. A point u is in the cut locus C u ( N ) ⊂ M of N if there is some distance minimizing geodesic joining N to u such that any extension of it beyond u is not a distance minimizing geodesic. Since C u ( N ) = Se ( N ) the cut locus is closed. T o express our geometric definition of weak interactions we need to consider the orthogonal complement X N ⊂ H i , N of H N . The map M N × X N → M i , N , ( Ad ˆ U 0 , ˆ X ) 7→ Ad ˆ U 0 exp ( − i ˆ X ) is surjecti ve. Definition 4.3. Let ( W N , b ) be a Nambu space. The space of weakly interacting time evolution oper ators is M wi , N B        Ad ˆ U ∈ M i , N | ∃ ( Ad ˆ U 0 , ˆ X ) ∈ M N × X N , ∀ t ∈ [0 , 1] : Ad ˆ U = Ad ˆ U 0 exp ( − i ˆ X ) , Ad ˆ U 0 exp ( − it ˆ X ) ∈ M i , N \ C u ( M N )        . Example 4.4. Consider ˆ U ( t ) B exp  i π ( a † 1 a 1 + a † 2 a 2 )  exp  − i 2 π t ( a 1 a 2 + a † 2 a † 1 )  . In the basis C = { | 0 ⟩ , | 1 ⟩ , | 2 ⟩ , | 12 ⟩ } of V V 2 we hav e [ Ad ˆ U ( t ) ] C =                  cos ( 2 π t ) 0 0 − i sin ( 2 π t ) 0 − 1 0 0 0 0 − 1 0 − i sin ( 2 π t ) 0 0 cos ( 2 π t )                  . Since Ad ˆ U (0) = Ad ˆ U (1) = Ad ˆ P ∈ M 2 and Ad ˆ U ( t ) ∈ M i , 2 \ M 2 for t ∈ (0 , 1), then Ad ˆ U ( 1 / 2 ) = − ˆ 1 V V 2 ∈ M i , 2 is in the cut locus of M 2 . Thus − ˆ 1 V V 2 is an interacting time ev olution operator that is not weakly interacting. 25 If Ad ˆ U = Ad ˆ U 0 exp ( − i ˆ X ) is weakly interacting then Ad ˆ U 0 is the element of M N that is clos- est to Ad ˆ U in the geodesic metric. The cut locus assumption guarantees Ad ˆ U 0 is unique, and further that M wi , N strongly deformation retracts to M N by the homotopy h : M wi , N ∧ I + → M wi , N , h ( Ad ˆ U , t ) B Ad ˆ U 0 exp ( − it ˆ X ) . (7) W e no w want to consider equiv ariant interacting time ev olution operators ov er grotesque Nambu spaces with symmetries ( W N , b , G ). Assuming equi variance with re gard to the full subgroup G U L  U (1) would restrict terms like a j a k + a † k a † j from appearing in the gener - ating Hamiltonians. This precludes consideration of superconducting models, like in the Bogoliubov-de Gennes formalism. W e will thus consider Hamiltonians equi variant under the subgroup G ′ ⊂ G generated by non-ordinary symmetries that square to ± ˆ 1 and by the parity operator ˆ P , so that G ′ U L  O (1). Definition 4.5. Let ( W N , b , G ) be a grotesque Nambu space with symmetries, with un- derlying Hilbert space of dimension N and signature (  S  C  T ). The space of equiv ariant interacting Hamiltonians is H  S  C  T i , N = { ˆ H ∈ H i , N | ∀ ˆ U ∈ G ′ : ˆ U ˆ H ˆ U † = ˆ H } . The space of equi variant interacting time e volution operators is M  S  C  T i , N = { Ad exp ( − i ˆ H ) ∈ Aut ( C l ( W , b )) | ˆ H ∈ H  S  C  T i , N } . The space of equi variant weakly interacting time e volution operators is M  S  C  T wi , N B M  S  C  T i , N ∩ M wi , N . Gi ven conv enient choices of basis we denote by M  S  C  T wi the colimit as in the definition of M  S  C  T . W e are no w ready to define weakly interacting versions of the topological K-theory spectra. The cut locus assumption allows us to define the structural suspension maps by projecting to the free subspectra K U and K O , and then applying their suspension maps. Definition 4.6. The weakly interacting comple x topological K-theory spectrum K U wi is K U wi • B        M 100 wi , • = 0 mod 2 M 000 wi , • = 1 mod 2 , σ wi • ( Ad ˆ U , θ ) B σ • ( Ad ˆ U 0 , θ ) . The weakly interacting r eal topological K-theory spectrum K O wi is K O wi • B                                            M 111 wi , • = 0 mod 8 M 001 wi , • = 1 mod 8 M 1 − 11 wi , • = 2 mod 8 M 0 − 10 wi , • = 3 mod 8 M 1 − 1 − 1 wi , • = 4 mod 8 M 00 − 1 wi , • = 5 mod 8 M 11 − 1 wi , • = 6 mod 8 M 010 wi , • = 7 mod 8 , σ wi • ( Ad ˆ U , θ ) B σ • ( Ad ˆ U 0 , θ ) . 26 Lemma 4.7. Let τ be a Cartan involution of M i , N such that τ ( M N ) = M N . Then the deformation r etract h in (7) leaves the subspaces F ix τ and In v τ in variant. Pr oof. Suppose Ad ˆ U ∈ Fix τ . Since Cartan in volutions preserv e geodesics, τ ( Ad ˆ U 0 ) is the closest element of M N to Ad ˆ U , thus τ ( Ad ˆ U 0 ) = Ad ˆ U 0 . Similarly if Ad ˆ U ∈ In v τ then τ ( Ad ˆ U 0 ) = Ad ˆ U † 0 . Since h is defined in terms of geodesics this guarantees that Fix τ and In v τ are in variant under h . □ Theorem 4.8. The spectr a K U wi and K O wi deformation r etract to K U and K O. Pr oof. By lemma 4.7 the deformation retract h determines well defined homotopy retracts of each M  S  C  T wi onto M  S  C  T . Since h ( σ wi • ( Ad ˆ U , θ ) , t ) = h ( σ • ( Ad ˆ U 0 , θ ) , t ) = σ • ( Ad ˆ U 0 , θ ) = σ wi • ( h ( Ad ˆ U , t ) , θ ) these homotopies are compatible with the structural suspension maps, thus determine spectra deformation retracts. □ 5. C oncluding remarks 5.1. T wisted equivariant K-theory. As mentioned in the introduction, the topological phases of crystalline fermion systems are classified by twisted equi variant K-theory [12]. For a compact Lie group G the G -equi v ariant cohomology theories are represented by G -spectra [21, 23]. A G -spectrum Y is composed of a collection of pointed G -spaces { Y V } index ed by G -representations, equipped for each subrepresentation V  → W with an equi- v ariant structural suspension map σ V , W : Y V ∧ S W − V → E W . There are G -spectra K U G and K O G that represent complex and real G -equi v ariant K-theory . Their base spaces BU G and BO G classify G -v ector b undles, and their structure maps are induced by equi v ariant Bott periodicity [3]. Freed and Moore state their arguments in [12] can be adapted to general- ize the classifying spaces used by Kitaev in [20] to the equiv ariant context. This suggest our construction of K U and K O in terms of time ev olution operators can be extended to G -spectra. A d -dimensional crystal is a solid material whose constituents (atoms, molecules, or ions) are distributed ov er the translation lattice of a crystallographic group. A crystallo- graphic group is a discrete subgroup C ⊂ Iso( R d ) containing a normal subgroup N of translations that is a lattice, and such that the point group P B C N is contained in O ( d ). The Brillouin zone of the crystal is the momentum space modeled by the Pontjagin dual b N B Grp ( N , S 1 ), which is naturally equipped with a P -space structure. It is common, eg [9], to consider crystallographic groups ov er the lattice Z d , whose Brillouin zone is the torus T d . In a crystalline fermion system we further assume each constituent has an internal symmetry group G generated by chiral, charge conjugation and / or time-rev ersal symmetries. If we assume the internal and crystallographic symmetries commute then the symmetry group of the system is G = C × G , and the topological phases are classified by some stable homotopy group of either the mapping equiv ariant spectrum K U b N P or K O b N P , depending on the signature of G . If the internal and crystallographic symmetries don’ t commute the full symmetry group of the system is a semi-direct product G = C ⋉ α G . In this context topological phases 27 are classified by twisted equiv ariant K-groups, which are composed of sections of K U P - bundles or K O P -bundles o ver b N , with the b undle structure induced by the twisting α . Gi ven a full description of twisted equiv ariant K-theory in terms of spectra of time e volution operators, we expect our arguments extend to a stable homotopy theoretical proof that the classification of crystalline topological insulators and superconductors is also stable to weak interactions. 5.2. Classification of interacting fermion systems by cobordism. The formulas for the structural suspension maps of K U and K O also apply for the spaces of equi variant interacting time e volution operators M  S  C  T i , N , which means they also form spectra M U i and M O i . In [11] Freed and Hopkins produce a general formula for symmetry protected phases in terms of Thom’ s bordism spectra, sho wing that in the interacting regime topo- logical phases are classified by cobordism as conjectured in [17]. This suggests the inter- acting spectra M U i and M O i might be weakly equi valent to Thom’ s spectra. W e suspect the filtered algebra structure of C l + ( W N , b ) induces a spectra filtration in- terpolating between K U and M U i , and an analogous filtration between K O and M O i . In [9] the authors hypothesize this filtration might be related to the chromatic filtration in- terpolating between K-theory and cobordism. Our definitions in terms of spaces of time e volution operators may pro vide a frame work to search for e vidence of such relation. R eferences [1] Adhip Agarw ala, Arijit Haldar , and V ijay Shenoy . The tenfold way redux: Fermionic systems with n-body interactions. Annals of Physics , 385:469–511, 2017. [2] H. Araki. Bogoliubov automorphisms and fock representations of canonical anticommutation rela- tions. Contemporary Mathematics , 62:23–141, 1985. [3] Michael Atiyah. Bott periodicity and the index of elliptic operators. The Quarterly J ournal of Mathe- matics , 19(1):113–140, 1968. [4] Michael Atiyah. K-theory . CRC press, 2018. [5] John Baez. The tenfold way . arXiv pr eprint arXiv:2011.14234 , 2020. [6] Raoul Bott. The stable homotopy of the classical groups. Annals of Mathematics , 70(2):313–337, 1959. [7] Élie Cartan. Sur une classe remarquable d’espaces de riemann. Bulletin de la Société mathématique de F rance , 54:214–264, 1926. [8] Élie Cartan. Sur une classe remarquable d’espaces de riemann. ii. Bulletin de la Société Mathématique de F rance , 55:114–134, 1927. [9] Eyal Cornfeld and Shachar Carmeli. T enfold topology of crystals: Unified classification of crystalline topological insulators and superconductors. Physical Revie w Resear ch , 3(1):013052, 2021. [10] Freeman Dyson. The threefold way . algebraic structure of symmetry groups and ensembles in quantum mechanics. Journal of Mathematical Physics , 3(6):1199–1215, 1962. [11] Daniel Freed and Michael Hopkins. Reflection positivity and in vertible topological phases. Geometry & T opolo gy , 25(3):1165–1330, 2021. [12] Daniel Freed and Gregory Moore. T wisted equi v ariant matter . In Annales Henri P oincaré , v olume 14, pages 1927–2023. Springer , 2013. [13] Nicolai Friis. Reasonable fermionic quantum information theories require relati vity . New Journal of Physics , 18(3):033014, 2016. [14] Peter Heinzner , Alan Huckleberry , and Martin Zirnbauer . Symmetry classes of disordered fermions. Communications in mathematical physics , 257:725–771, 2005. [15] W . A. d. S. Pedra J.-B. Bru. C*-Alg ebras and mathematical foundations of quantum statistical me- chanics . Springer , 2023. 28 [16] C. Gérard J. Derezi ´ nski. Mathematics of quantization and quantum fields . Cambridge, 2022. [17] Anton Kapustin, Ryan Thorngren, Alex Turzillo, and Zitao W ang. Fermionic symmetry protected topological phases and cobordisms. J ournal of High Ener gy Physics , 2015(12):1–21, December 2015. [18] Ricardo Kennedy and Martin Zirnbauer . Bott–kitaev periodic table and the diagonal map. Physica Scripta , 164(1):014010, 2015. [19] Ricardo Kennedy and Martin Zirnbauer . Bott periodicity for Z 2 symmetric ground states of gapped free-fermion systems. Communications in Mathematical Physics , 342(3):909–963, 2016. [20] Alex ei Kitaev . Periodic table for topological insulators and superconductors. In AIP confer ence pr o- ceedings , volume 1134, pages 22–30. American Institute of Physics, 2009. [21] Gaunce Lewis, J Peter May , and Mark Steinberger . Equivariant stable homotopy theory , volume 1213. Springer , 2006. [22] J Peter May . E ∞ ring spaces and E ∞ ring spectra . Springer , 1977. [23] J Peter May and Michael Cole. Equivariant homotopy and cohomology theory: Dedicated to the memory of Robert J. Piacenza . Number 91. American Mathematical Soc., 1996. [24] D. W . Robinson O. Bratteli. Operator algebras and quantum statistical mec hanics 2 . Springer , 2002. [25] Sachchidanand Prasad. Cut Locus of Submanifolds: A Geometric and T opolo gical V iewpoint . PhD thesis, Indian Institute of Science Education and Research K olkata, 2022. [26] Stefan Schwede. Symmetric spectra. [27] René Thom. Sur le cut-locus d’une variété plongée. Journal of Di ff er ential Geometry , 6(4):577–586, 1972. [28] Renato V V iera, LCP AM Müssnich, and NJB Aza. Operator k-theory algebra spectra of c*-algebras. arXiv pr eprint arXiv:2203.03050 , 2022. [29] Gian Carlo W ick, Arthur Strong W ightman, and Eugene Paul W igner . The intrinsic parity of elemen- tary particles. Physical Revie w , 88(1):101, 1952. 29