\Spin-statistics" is a categorification of \Hermitian", Theo Johnson-Freyd, 22 February 2016. p 1/2. Talk based on arXiv:1507.06297. Prop [Li-Bland]: When X is 1-topos, SPANSd (X) is an (1; d)-category internal to X, i.e. k-morphisms can vary 0. Talk punchline in X-parameterized moduli. C Orientations Hermiticity 2 Defn: Treat BORDd as X-internal category via unique ge- SUPERVECT Spin structures Spin-Statistics ometric morphism SPACES X. Let Xfptg= = fpointed X-objectsg.(fptg 2 X is initial object.) The G-structured 1. Local structures on manifolds bordism category is the pullback of X-internal categories Let MANd = fd-manifolds and local diffeomorphismsg. G BORD SPANSd (Xfptg=) (Topologized (1; 1)-category.) Let X an 1-topos. d p Forget the pointing Defn: Topological; local structure = sheaf G :MAN ! X. d G BORD SPANS (X) Non-e.g.: M 7! fmetrics on Mg is a sheaf on the strict d d strict category MANd but not on the (1; 1)-category. (Internal local structure can vary in moduli.) Lemma (\Cobordism Hypothesis"): Topological local Defn: Let;V be X-internal sym mon (1; d)-category. A structures valued in are classified by O(d) = -objects X X fX V-valued G-structured field theory is functor of X-internal equipped with O(d)-actiong. G sym mon (1; d)-categories BORDd ! V. d d Pf: Use O(d) ' homMANd (R ; R ) together with exis- Lemma: Let SPANSd (X; V) = (1; d)-category whose k- tence of good open covers. morphisms are k-fold spans in X equipped with bundles 2O(d) E.g.: Given O(d) y X in X , corresponding sheaf is of k-morphisms in V [Haugseng]. V-valued G-structured M 7! mapsO(d)(Fr(M);X). field theory = lift of (non-internal!) functors: • Trivial action topological sigma-model. SPANSd (X; V) • fG-tangential; structuresg ! X = O(d)=G. • forientationsg ! X = π0(O(1)) = Z=2. Forget the V-bundle • fspin strsg ! X = π≤1(O(1)) = Z=2×B(Z=2). G BORDd SPANSd (X) Why allow other topoi X? So that local structures can Pf: Unpack some adjunctions. vary in moduli. If G is algebraic group, then LocG(−) is valued in algebraic stacks. If G is super Lie group, then Question: Explain \qfts fibered2 over G" as \G-V-twisted" fG-bundles with connectiong is valued in topos of sheaves field theories a la Stolz{Teichner. on site of supermanifolds (and smooth maps). 3. Hermitian field theory 2. Locally structured bordism category In SPACES, sheaf Or = forientationsg is classified by unique smooth Let BORDd = BORDd [Calaque{Scheimbauer, Lurie]. Z=2-torsor. Let X = STACKSR. Then Spec(C) is another I.e. k-morphisms are k-dim smooth cobordisms between Z=2-torsor, and is the unique nontrivial one. (k − 1)-morphisms, with no extra local data. Defn: Sheaf Her : MANd ! STACKSR of Hermitian struc- Let SPANSd (X) be sym mon (1; d)-cat with k-morphisms O(d) tures is classified by Spec(C) 2 (STACKSR) . = k-fold spans in X, ◦ = fiber product [Haugseng]. Or(−) × Spec( ) Lemma: Her(−) = C . Given G :MANd ! X, can assign to each bordism in Z=2 BORDd a span in X: 2 Recall stack of categories (i.e. STACKSR-internal) cate- fG-structures on Mg gory QCOH : A 7! MOD . Normal to demand that for G A 7! restrict restrict quantum field theory, d-manifolds 7! numbers, (d − 1)- M fG-strs on N1g fG-strs on N2g manifolds 7! vector spaces. In our case, this becomes: N1 N2 Ωd−1V = QCOH. This defines a sym mon functor G :BORDd ! SPANSd (X). Sheaf condition ) composition. Topological (i.e. use (Sym mon (1; k)-cat C sym mon (1; k −1)-cat ΩC = strict EndC(1) called its looping. E.g.: ΩQCOH = O.) (1; 1)-cat MANd , not just MANd ) ) units. ; \Spin-statistics" is a categorification of \Hermitian", Theo Johnson-Freyd, 22 February 2016. p 2/2. Her E.g.: Let Z :BORDd ! V. 5. Spin{statistics field theories • Suppose Md is connected orientable. Then Her(M) =∼ Take X = CATSTACKS . There is a canonical nontrivial Spec( ), but not canonically. Z(M) 2 O(Her(M)). Each R C π O(1) = =2 × B( =2) torsor, namely Galois action orientation of M Z(M) 2 ; orientation reversal = ≤1 Z Z C on Spec(SUPERVECT ). Just as in Hermitian case, cor- complex conjugation. R ; responding local structure SpinStat : MAN ! X is (cate- • Suppose Nd−1 is connected orientable. Each ori- d gorified) ´etale-locally equivalent to Spin = fspin structuresg. entation gives iso Her(N) =∼ Spec( ), hence Z(N) 2 C Moreover, just as in Hermitian case: QCOH(Spec(C)) = VECTC. Z(N × ) is a nondegenerate Spin(−) × Spec(SUPERVECT ) symmetric sesquilinear form on Z(M). (Not necessarily SpinStat(−) = C positive definite.) Hence name \Hermitian." Z=2 × B(Z=2) Punchline: \Oriented" and \Hermitian" are the two ver- How does Z=2 × B(Z=2) act on Spin(−)? The Z=2 part acts by orientation reversal. The B( =2) part acts by the sions of \´etale-locally-over-R oriented." Z nontrivial Spin automorphism of a spin manifold, called 4. Categorified torsors \ " or \rotate by 360◦". Let N be Spin; then the map- ping cylinder of N is the Spin manifold N ×[0; 1] with Spin Why does the nontrivial =2-torsor over exist? Z R structure N × . N.b.: = (belt trick). 0. C is non-zero finite dim com R-algebra. 1. (Field) Any non-zero map C ! A of finite dim com Suppose dim N = d − 1 and Z is a SpinStat field theory. R-algebras is injective. Then Z(N) 2 QCOH(SpinStat(N)) = disjoint union of 2. (Algebraically closed) Any non-zero finite-dim com Spec(SUPERVECTC)s, one for each (Z=2 × B(Z=2))-orbit R-algebra A admits A ! C. in Spin(N). So for each spin structure, Z(N) gives a com- 3. (Galois) MOD = MOD . plex supervector space. Compatibility with =2-action ) R CoGal(C=R) Z ∼ Hermitian supervector space. Then G-torsors (i.e. G y T ! Spec(R) s.t. T=G ! ∼ Defn: In a field theory valued in supervector spaces, a Spec(R) and T × G ! T ×Spec( ) T ) are classified by R fermion is a (−1)-eigenstate of (−1)f . In a spin field maps(B Gal(C=R); BG). theory, a spinor is a (−1)-eigenstate of . Defn: A categorified com R-algebra is a (nice!) R-linear In a SpinStat field theory, = (−1)f , i.e. they are spin- sym mon cat. E.g.: fcom R-algsg ,! fcat com R-algsg statistics in the sense that fermions=spinors. via A 7! (MODA; ⊗A). Defn: CATSTACKSR = stacks on site of cat com R-algs. E.g.: QCOH : C 7! C. 6. A topological spin-statistics theorem Defn: Cat com R-alg (C; ⊗;::: ) is finite dim if (a) C' There are spin theories that are not spin-statistics, and MOD for finite dim associative alg A (so that underlying A super field theories that are not spin. However: cat is finite-dim), and (b) projective ) dualizable (so that \internal" and \external" notions of “finite” agree). Thm [TJF]: If Z is ´etale-locally-spinTFT over R such that the unoriented theory R Z is positive, then E.g.: SUPERVECT , as non-sym monoidal category, is spin structures C Z is Hermitian and spin-statistics. REPC(Z=2). Its braiding is determined by σ = −1 : Π ⊗ Π ! Π ⊗ Π. (Π = sign rep of Z=2.) (Hermitian + positive = unitary.) Thm: I. The categorified algebraic closure of ≡ VECT R R 7. Conjectures about categorified Galois groups is SUPERVECTC. II. The extension is Galois with Galois Conj [TJF]: Under further categorification, the infinitely- group Gal(SUPERVECTC=R) = Z=2 × B(Z=2). categorified absolute Galois group of is O(1). Pf: I. Deligne's \existence of super fiber functors" plus R ¯ small modifications. II. The Z=2 acts by complex conju- Conj [Ostrick]: If p ≥ 3, the cat alg closure of Fp is the f gation. The B(Z=2) acts by \(−1) " = natural auto of Verlinde category VERp, a mod-p version of SU(2)-at- f f identity s.t. (−1) j1 = +1 and (−1) jΠ = −1. level-(p − 2). Conj [Etingof]: Weirder p = 2 case. 2 Rmk: Extension VECT ! VER is not separable. Max- Exercise: VECTR has five field extensions: VECTR,VECTC, p p imal separable subextension is VECT ! SUPERVECT . SUPERVECTR,SUPERVECTC, and \SUPERVECTH", which p p So SUPERVECT is still the universal torsor. is VECTR MODH with interesting sym mon str..