Invertible Topological Field Theories and Symmetry Protected Topological Phases

Transcription

1 Invertible Topological Field Theories and Symmetry Protected Topological Phases Peter Teichner Max Planck Institute for Mathematics, Bonn Following Dan Freed and Michael Hopkins [FH, 2017] Last part joint with Matthias Kreck and Stephan Stolz 3 hours of Physical Mathematics at the Bad Honnef Physics School, September 20-21, 2018 Part 1

2 Outline of my two talks 1. A mathematical 10-fold way: Super division algebras 2. Magic agreement of SPT phase-tables with PTFT-tables 3. Topological field theory via Atiyah-Segal axioms 4. Main Result: Invertible TFT = SKK manifold invariant 5. Derive the entire table from a certain spectral sequence

3 In this approach to SPT phases, we invoke two physical principles to pass from quantum systems, in practice defined on a lattice, to continuum systems: (i) The deformation class of a quantum system is determined by its low energy behavior, and (ii) The low energy physics of a gapped system is well approximated by an invertible topological field theory. We will use the Atiyah-Segal axioms for a TFT to propose a moduli space M n,a of certain TFTs and compute its deformation classes π0(m n,a ) := Symmetry protected topological (SPT) phases. =: TPn,A Here n is the space dimension, and A is a real super division algebra (one of 10 possibilities) which codifies the internal structure of the material in the Freed-Hopkins preprint [FH] on Reflection positivity and invertible topological phases.

4 Real super division algebras A even K even super division algebra A internal group K := S(A) Cartan label particle hole P R {±1} R {±1} D P 2 = 1 time reversal T R {±1} Cliff+1 Pin+1 BDI P 2 = 1 T 2 = 1 R {±1} Cliff-1 Pin-1 DIII P 2 = 1 T 2 = -1 C U1 C U1 A C U1 Cliff+2 Pin+2 AI T 2 = 1 C U1 Cliff-2 Pin-2 AII T 2 = -1 C U1 Cliff1 Pin c 1 AIII T 2 arbitrary H SU2 H SU2 C P 2 = -1 H SU2 Cliff+3 Pin+3 CI P 2 = -1 T 2 = 1 H SU2 Cliff-3 Pin-3 CII P 2 = -1 T 2 = -1

5 Fermionic structure groups Definition: A fermionic group is a Lie group K, together with a homomorphism - : K Z/2 and a central involution 1 c K. Further examples are Pin±n in Cliff±n and Lorentz forms in Cliffn-1,1. Two Fermionic groups can be multiplied, inspired by the graded tensor product of super algebras: K K := (K x K )/ c=c where (k1, k1 ) (k2, k2 ) := c k 1 k 2 (k1 k2, k1 k2 ) and (k, k ) := k + k Definition: Hn,A := Pinn KA is the structure group of the material. The internal group KA is the group of (homogenous) elements of norm 1 in a real super division algebra A and n Z+ is the space dimension. Our describes exactly the groups given in [FH].

6 The groups Hn,A := Pinn KA above fit into two exact sequences: 1 KA Hn,A On 1 and 1 Pinn Hn,A KA / c 1 There are Wick rotated versions of these groups, obtained by simply changing Pinn to Pinn-1,1, hence changing On to On-1,1 but keeping KA. These (non-compact) groups are relativistic symmetry groups, where KA is the internal symmetry and On-1,1 rotates Minkowski space. This continues to hold for Hn,A in flat Euclidean space but not in curved cases. The following computations are in magic agreement with recent physical arguments by various groups, e.g.: Gu, Furusaki, Hughes, Kapustin, Kitaev, Lu, Ludwig, Potter, Qi, Ryu, Schnyder, Seiberg, Senthil, Thorngren, Turzillo, Wang, Wen, Witten, Vishvanat, Zirnbauer, Zhang,

7 [FH]-Computations of topological phases TPn,A super division algebra A FF1,A TP1,A FF2,A TP2,A FF3,A TP3,A R Z/2 Z/2 Z Z 0 Cliff+1 Z Z/8 0 0 Cliff-1 Z/2 Z/2 Z/2 Z/2 Z Z/16 C 0 Z Z 2 0 Cliff+2 0 Z/2 0 0 Z/2 Cliff-2 0 Z/2 Z/2 Z/2 (Z/2) 3 Cliff1 Z Z/4 0 Z Z/8 x Z/2 H 0 Z Z 2 0 Cliff+3 0 Z/2 0 Z Z/4 x Z/2 Cliff-3 Z (Z/2) 2 0 Z/2 (Z/2) 3

8 Invertible susy topological field theories Definition: For G On+1 a homomorphism of Lie groups, define TFTn+1(G) X := Fun (Bordn+1(G), sline X ). If H is fermionic, for example H=Hn+1,A, we will use G:=H ev.and the entire group H is only used for an involution T on Bordn+1(G). The bordism category Bordn+1(G) of Riemannian n-manifolds with tangential G-structure will be explained below. sline is simply the category of complex super lines. Both are considered with their symmetric monoidal structures respectively. In sline X, we only use invertible morphisms and hence get the -invertible part of svect. By definition, we can consider isomorphism classes of invertible TFTs and in this approach, topological phases TPn,A will be defined as deformation classes of such, after introducing reflection positive TFTs.

9 Examples of categories & functors A group G is the same as a groupoid with one element pt, also written as pt/g. A space X forms the objects of the path category PX, with morphisms = pairs (t 0,p) where p:[0,t] X is a continuous path. Example 1: Fun(pt/G, Vect) = Rep(G), the category of representations: F(pt) is a vector-space on which F(g): F(pt) F(pt) acts for every g G. Vect has a symmetric monoidal structure given by and pt/g is symmetric monoidal if and only if G is abelian. Example 2: Fun(PX, Vect) = Vect c (X), Fun (Bord1(X), Vect) = Vect c (X)fd, the category of vector-bundles (resp. finite dimensional) with connection on X. For every x X, F(x) is a vector-space, the fibre over x. For a path p from x0 to x1, F(p): F(x0) F(x1) is the parallel transport along p. As a sym. monoidal category with duals, Bord1(X) is freely generated by PX.

10 The bordism category, n=2: Objects: Closed Riemannian n-manifolds with a tangential lift to G On+1 on x R. Morphisms := {Compact (n+1)-dim. G-manifolds}/G-diffeo.

11 Invertible PTFTs and SPT Phases, Part 2 Peter Teichner Max Planck Institute for Mathematics, Bonn This part is joint with Matthias Kreck and Stephan Stolz Freed-Hopkins [FH 2017] proposed a moduli space M n,a of certain TFTs using higher categories and equivariant stable homotopy theory. We are giving a simplified approach using just the original Atiyah-Segal axioms, still arriving at the same deformation classes: π0(m n,a ) := Symmetry protected topological phases =: TPn,A

12 [FH]-Computations of topological phases TPn,A super division algebra A FF1,A TP1,A FF2,A TP2,A FF3,A TP3,A R Z/2 Z/2 Z Z 0 Cliff+1 Z Z/8 0 0 Cliff-1 Z/2 Z/2 Z/2 Z/2 Z Z/16 C 0 Z Z 2 0 Cliff+2 0 Z/2 0 0 Z/2 Cliff-2 0 Z/2 Z/2 Z/2 (Z/2) 3 Cliff1 Z Z/4 0 Z Z/8 x Z/2 H 0 Z Z 2 0 Cliff+3 0 Z/2 0 Z Z/4 x Z/2 Cliff-3 Z (Z/2) 2 0 Z/2 (Z/2) 3

13 Recall: Atiyah-Segal axioms for a TFT A TFT is a symmetric monoidal functor F: Bordn+1(G) Vect. It associates a vector-space V = F( ) to any closed n-dimensional G-manifold and a linear map F(Y): V1 V2 for any G-bordism Y, a compact (n+1)-mfd. with boundary 1 2. Important property: F(Y) only depends on the diffeomorphism class (rel. boundary) of Y.

14 Euler (characteristic) TFTs Here are the easiest invertible TFTs that work for any tangential structure, i.e. G=On+1 : Fix a non-zero complex number z. For any closed n-manifold we assign the line C as the (ground) state vector-space. For a compact (n+1)-dimensional bordism Y we define F(Y) := z E(Y,in), where E(Y,in) Z is the Euler characteristic of Y, relative to its incoming boundary. The usual gluing formulas for the Euler characteristic show that F is a functor, i.e. multiplicative under gluing of bordisms. These Euler TFTs will not contribute to our topological phases TP since they can be deformed to the trivial TFT z=1.

15 Classification of Piccard groupoids Invertibility reduces TFTs from functors between symmetric monoidal categories to those between Picard groupoids B. Such B are classified by the triple (π0(b), π1(b), k) where the k-invariant k: π0(b) π1(b) is given by k(x) π1(b) {isomorphism classes of objects in B} This classification can be understood in terms of symmetric cocycles as follows: Let G:= π0(b) and M:= π1(b) then the symmetric structure on B, fx,y : X Y Y X, gives as above a map f: G x G M which satisfies associativity and communitivity axioms. It turns out that they are equivalent to being a cocycle in the following cochain complex Cs(G;M) which is a variation on the usual complex computing group cohomology:

16 The differential of f: G x G M has two components, the standard one takes care of the associativity constraint and the non-standard one in the G G-component is responsible for the symmetry contraint: It is given by df(g g ):=f(g,g ) - f(g,g) and so a cocycle is simply a symmetric f. It represents a class in H 2 (Cs(G;M)) which computes spectrum cohomology. Thm.: There is an isomorphism H 2 (Cs(G;M)) Hom(G/2,M) which sends the cocycle f to the k-invariant of B.

17 As a consequence, Picard groupoids B are classified by the fundamental triple (π0(b), π1(b), k). One first reduces to a skeleton and then uses that f - f is a coboundary if and only if the corresponding Picard groupoids B, B are isomorphic. By similar cocycle techniques, one can understand functors between Picard groupoids up to natural isomorphisms: Thm. [Sinh, 1970, student of Grothendieck]: For Picard groupoids B and C, isomorphism classes of functor Fun (B, C)/ are in bijection to the set { (f0, f1) f0 Hom(π0(B), π0(c)), f1 Hom(π1(B), π1(c)) and kc f0 = f1 kb } The bijection is given by evaluation a functor F on π0 and π1.

18 Partition functions classify super line bundles Corollary: For any symmetric monoidal category B, sending F to F1 : Fun (B, sline X )/ Hom(π1(B), C X ) gives a bijection. Here the k-invariant is useful: Otherwise, the other invariant of functors is needed, namely the induced homomorphism F0: π0(b) {even, odd}. However, it is determined because k(sline) : π0 π1 = C X is injective: Susy is essential: Without it, the map above is not onto since the composition π0(b) π1(b) C X is trivial for the target Line X. For B = Bordn+1(G), the above isomorphisms sends a field theory to its partition function, i.e. its value on closed (n+1)-dim. G-manifolds.

19 Thm. [Kreck-Stolz-T]: π1(bordd(g)) =: SKKd(G) = -Grothendieck group on closed G-manifolds of dimension d, modulo the 4-term SKK relations: SKK stands for Schneiden-und -Kleben, -Kontrolliert, best translated by controlled cut- and -paste, [Karras-Kreck-Neumann-Ossa, book 1970]. Example: We have [S k x S d-k ] = (1+(-1) k ) [S d ] in SKKd. For k=1:

20 Classification of invertible field theories Corollary [KST, 2018]: TFTn+1(G) X / = Hom(SKKn+1(G), C X ) is the set of closed (n+1)-dim. G-manifold invariants that are multiplicative under and satisfy the SKK-relations. These two properties characterize partition functions of invertible TFTs among all G-manifold invariants. They are the holonomies (along closed (n+1)-dim. G-bordisms) of the line bundle on the moduli space of our n-dim. closed G-manifolds. To understand field theories, it thus suffices to understand the groups SKKd(G) for d=n+1. The following result uses handle decompositions of compact G-bordisms and is a generalization of [KKNO, 1970]: Thm: If G Od extends well to Gd+1 Od+1, e.g. if G is the even part of Hd,A, then there is an isomorphism SKKd(G) / S d Ωd(G). Here Ωd(G) is the group of closed d-dimensional G-manifolds up to (d+1)-dimensional Gd+1-bordisms.

21 Uncontrolled cut- and -paste invariants Question: What are multiplicative invariants of closed d-manifolds that are fixed by all (uncontrolled) cut- and -paste operations along closed codimension 1 submanifolds? Answer depends strongly on the structure group G, for example [KKNO]: If G = Od, i.e. no tangential structure, then the Euler number is the only invariant. For G = SOd, i.e. for oriented manifolds, we have exactly the Euler number and the signature. So the Euler number always gives a homomorphism E: SKKd(G) Z which exponentiates to the partition function of the Euler TFTs discussed above and is non-trivial exactly for even dimensions n, e.g. E: SKK2 Z.. The image of E is unknown for general G. Note that SKK2(SO2) 2 Z..

22 Proof of our central SKK computation By definition, there is an epimorphism SKKd(G) Ωd(G) and we are claiming that the kernel is generated by the d-sphere S d. Consider a closed d-manifold M in the kernel, i.e. assume that M is the boundary of a compact (d+1)-manifold W. Pick a handle structure on W to see that M is obtained from S d by a finite sequence of surgeries: Def.: M is obtained from N by one surgery (of index k) if M = N \ (S k x D d-k ) (D k+1 x S d-k-1 ), both glued along (S k x S d-k-1 ). Now apply the SKK-relation to see that:

23 Unitarity: Reflection positive TFTs For the fermionic groups H = Hn,A, the stability condition is satisfied thanks to the exact sequences involving On. So the last step in explaining the (torsion part of) the Freed-Hopkins table is to control the value of S n+1 from a field theoretic point of view. Surprisingly, this can be achieved by introducing unitarity into the notion of a TFT. Following Osterwalder-Schrader, this can be achieved in Euclidean signature via reflection positive TFTs. Compared to [FH], we ll give a non-extended and simplified version of this notion. There are two steps: 1. Change the target to super hermitian lines =: sherm X. This category come with a notion of adjoint operator which gives a *-structure. 2. Notice that for the groups G = even part of Hn+1,A, there is a canonical *-structure on Bordn+1(G) by changing the G-structure only. Definition: A reflection positive invertible TFT is a symmetric monoidal *-functor F: Bordn+1(G) sherm X such that all F( ) are positive definite.

24 Explaining the TPn,A -entries in [FH]-table Thm: [FH and later KST]: For tangential structures G=Gn+1,A as above, the isomorphism classes of reflection positive invertible TFTs are Hom(Ωn+1(G), U1) for n even, and PTFTn+1(G) X / = Hom(Ωn+1(G), U1) x R+ for n odd (positive Euler TFTs). In the usual topologies on U1 and R, the deformation classes TPn,A of such PTFTs are isomorphic to the torsion subgroup of Ωn+1(G). So we can really work modulo (n+2)-dim. G-bordisms which gives the coefficients of a generalized homology theory: Computations possible! Exact sequence of groups 1 Spind Gd,A KA/ c 1 gives twisted Leray-Serre spectral sequence for generalized homology with E 2 p,q-term equal to Hp(BK/ c ; Ωq(Spin)) and converging to Ωp+q(G).

25 Let s see how to get the Z/16 for A = Cliff-1 and n = 3. Our exact sequence is 1 Spind Pind {±1} 1 where {±1} acts nontrivially on Spind and the extension splits. So we just need to know the homology of BZ/2 with coefficients in Spin-bordisms to get E 2 p,q = = Hp(BZ/2; Ωq(Spin)) = q For p+q=4, there are no differerentials and the E - term gives a filtration F1 F2 F3 Ω4(Pin) with all successive quotients equal to Z/2. Checking on RP 4 shows that the Dirac η-invariant Ω4(Pin) Z/16 is an isomorphism. Z/2 0 0 Z/2 Z/2 Z/2 Z/2 Z/2 Z/2 Z/2 Z/2 0 Z/2 0 Z/2 p

26 Turning the pages in our spectral sequence In our range of dimensions, we only need to move from E 2 p,q to E 3 p,q = q This shows that Ωd(Pin) = Z/2 for d = 0, 2, 3 and Ω1(Pin) = 0. This shows why there can t be any other topological invariants than the ones that we already know. Z/ Z/2 Z/2 0 Z/2 0 Z/2 Thanks for your attention! Z/ Z/2 p

27 Appendix 1: Particle hole symmetry Let V be a complex vector-space, a fermionic one-particle system. Then a fixed state in Λ top (V) induces a complex linear isomophism Λ mid-r (V) Λ mid+r (V)* by the wedging operation. We might think that r holes are identified with r anti-particles. Let k:= mid+r. For a particle-hole symmetry P, we need a second isomorphism, P: Λ k (V)* Λ k (V) of complex vector-spaces (on which a symmetry group acts). There are 3 possible cases for the representation Λ k (V): C-type: It is not isomorphic to its dual (so P does not exist), R-type: P exists with P 2 = 1, i.e. P is a real structure on Λ k (V), H-type: P exists with P 2 = -1, i.e. a quaternion structure on Λ k (V). This explains the P-column in our table in dimension n=0, identifying the 3 cases R, C and H for A even with the 3 possibilities for P.

28 App. 2: Time reversal symmetry in [FH] Recall Hn,A := Pinn KA On and consider a reflection r0 On which fixes a hyperplane of dimension n-1 with a lift e0 Pinn. We need an odd element 0 k KA so that T:= [e0,k] is an element in structure group G := even part of Hn,A that projects to the reflection r0. Then T 2 = - k 2 by the definition of our tensor product. Wick rotation changes the time coordinate e0 to have negative square in Pinn-1,1. So relativistically, we have T 2 = k 2 and there are the following cases: 3 cases A = R, C or H is even: Then k and hence T don t exist, 6 cases A = Cliff±n, i.e. even part of center is R: k exists with k 2 = ± 1, 1 case A = Cliff1, i.e. even part of center is C: k exists and k 2 is arbitrary: k = z e1, z S(C)=U1 and hence k 2 = z 2 U1 is arbitrary. For n=3, we assume that k 2 lies in the center of A (physical reason?)

29 App.3: Wick rotation gives many Hilbert spaces Recall that in the easiest case, a TFT is a symmetric monoidal functor F: Bordn+1 Vect, that associates a vector-space V=F( ) to any closed n-dimensional manifold. The axioms allow one to interpret the following bordism as an inner product on V, so that we get many Hilbert spaces V from F: In a relativistic quantum theory we are used to having only one Hilbert space. However, in order to perform a Wick rotation, we need to chose a codimension one subspace of, say, Minkowski space. Each such subspace leads to a different Riemannian n-manifold, each equipped with the given Hilbert space.