Super p-brane Theory. Super Homotopy Theory

Transcription

1 Super p-brane Theory emerging from Super Homotopy Theory Urs Schreiber (CAS Prague and HCM Bonn) talk at String Math 2017 Based on arxiv: arxiv: with D. Fiorenza and H. Sati with J. Huerta these slides are kept at ncatlab.org/schreiber/print/stringmath2017

2 Notorious Open Problem of String Theory: What is the full non-perturbative Theory?

3 Notorious Open Problem of String Theory: What is the full non-perturbative Theory? We still have no fundamental formulation of M-theory - Work on formulating the fundamental principles underlying M-theory has noticeably waned. [... ]. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue. G. Moore, Physical Mathematics and the Future, at Strings 2014

4 Notorious Open Problem of String Theory: What is the full non-perturbative Theory? What is even its Principle?

5 Principles physics mathematics gauge principle & Pauli exclusion homotopy theory super-geometry = super-homotopy theory

6 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s)

7 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s) Quillen 69, Sullivan 77 : infinitesimal methods in homotopy theory

8 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s) Nieuwenhuizen 82, D Auria-Fré 82: FDAs efficiently construct SuGra-s

9 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s) Schwede-Shipley 03 : stable homotopy theory subsumes homological algebra

10 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s) Schlegel

11 Homotopy Theory stable parameterized stable plain plain spectra parameterized over the point parameterized spectra underlying parameter space spaces rational cochain complexes dg-modules dgc-algebras super rational super cochain complexes super dg-modules super dgc-algebras ( FDA s)

12 Cohomology extension homotopy fiber space twist twisted cocycle parameter space parameterized spectrum

13 Cohomology extension homotopy fiber space twist twisted cocycle parameter space parameterized spectrum Nikolaus-Schreiber-Stevenson 12, Ando-Blumberg-Gepner-Hopkins-Rezk 14

14 We now work out in rational super-homotopy theory a tower of extensions, each invariant wrt automorphisms modulo R-symmetries.

15 We now work out in rational super-homotopy theory a tower of extensions, each invariant wrt automorphisms modulo R-symmetries. Beware: Everything in the following holds in (super-) rational homotopy theory.

16 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 In the beginning the atom of space: the superpoint T 2,1 2+2 T 2,1 2 T T 0 1

17 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 Its maximal torus extension is the super-line T 3,1 4+4 T 3,1 4 T 1 1 T T 0 1

18 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 its type II version: the N = 2 superpoint T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

19 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 maximal torus extension: d = 3, N = 1 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

20 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 type II version: d = 3, N = 2 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

21 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 maximal invariant torus extension: d = 4, N = 1 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

22 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 5,1 8 T 5,1 8+8 T 3,1 4 type II version: d = 4, N = 2 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

23 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 maximal invariant torus extension: d = 6, N = 1 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

24 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 type IIB version: d = 6, N = (2, 0) super-minkowski spacetime. T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

25 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 type IIA version: d = 6, N = (1, 1) super-minkowski spacetime. T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

26 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 maximal invariant torus extension: d = 10, N = 1 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

27 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 type IIB version: d = 10, N = (2, 0) super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

28 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 and its type IIA version: d = 10, N = (1, 1) super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

29 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 maximal invariant torus extension: d = 11, N = 1 super-minkowski spacetime T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

30 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 5,1 8+8 T 3,1 4+4 T 2,1 2+2 T 9,1 16 T 5,1 8 T 3,1 4 T 2,1 2 T 9, T 5,1 8+8 T 1 1 In summary: Theorem (Huerta-Schreiber 17 ): There exists a diagram as shown of maximal torus extensions at each stage invariant with respect to the semi-simple part of automorphisms modulo R-symmetry which happens to be the Lorentzian Spin-groups. T T 0 1

31 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 5,1 8+8 T 9,1 16 T 5,1 8 T 9, T 5,1 8+8 T 3,1 4+4 T 2,1 2+2 T T 3,1 4 T 2,1 2 T 0 1 T 1 1

32 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 5,1 8 T 5,1 8+8 the higher invariant extensions: superstrings condense T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

33 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 hofib(µ F 1 ) T 9, µ F 1 T 5,1 8+8 T 3,1 4+4 T 2,1 2+2 T 5,1 8 T 3,1 4 T 2,1 2 T 5,1 8+8 T 1 1 B 3 Q these extensions are classified by WZW-term for the GS-Superstring µ F 1 = iψ Γ a ψ e a (Green-Schwarz 81, Henneaux-Mezincescu 85) T T 0 1

34 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 5,1 8+8 T 9,1 16 T 5,1 8 T 9, T 5,1 8+8 T 3,1 4+4 T 2,1 2+2 T T 3,1 4 T 2,1 2 T 0 1 T 1 1

35 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 5,1 8 T 5,1 8+8 the next higher invariant extensions: D-branes condense T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

36 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 5,1 8 T 5,1 8+8 the next higher invariant extensions: D-branes condense T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

37 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 these extensions are classified by WZW-terms for the super D-branes exp(f 2 ) c p ψ Γ a1 a p ψ e a1 e ap T 2,1 2+2 T 2,1 2 T 1 1 Azcárraga et al. 99, Sakaguchi 00 Fiorenza-Sati-Schreiber 13 T T 0 1

38 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het (pb) string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 3,1 4+4 T 5,1 8 T 3,1 4 T 5,1 8+8 D0-brane condensate is 11-th dimension via homotopy pullback T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

39 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het (pb) string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 5,1 8 T 5,1 8+8 its own higher invariant extension: M2-branes condense T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

40 d5brane d3brane d1brane d0brane d2brane d4brane d7brane T 10,1 32 d6brane d9brane string IIB string het (pb) string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8+8 T 5,1 8 T 5,1 8+8 its next higher extension: M5-branes condense T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 1 1 T T 0 1

41 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 2,1 2+2 T 5,1 8 T 3,1 4 T 2,1 2 T 5,1 8+8 T 1 1 spacetime and M-branes have emerged from the superpoint as iterated higher invariant extensions T T 0 1

42 d5brane d3brane d1brane d7brane T 10,1 32 Perhaps we need to understand the nature of time itself better. [...] understand in what sense time itself is an emergent concept, [...] how pseudo-riemannian geometry can emerge from more fundamental and abstract notions such as categories of branes. (G. Moore, Physical Mathematics and the Future, at Strings 2014) d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 2,1 2+2 T 5,1 8 T 3,1 4 T 2,1 2 T 5,1 8+8 T 1 1 spacetime and M-branes have emerged from the superpoint as iterated higher invariant extensions T T 0 1

43 d5brane d3brane d1brane d7brane T 10,1 32 d6brane d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 2,1 2+2 T 5,1 8 T 3,1 4 T 2,1 2 T 5,1 8+8 T 1 1 spacetime and M-branes have emerged from the superpoint as iterated higher invariant extensions T T 0 1

44 d5brane d3brane d1brane d7brane T 10,1 32 d9brane string IIB string het string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 consider the M-brane sector T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T T 0 1

45 B 7 Q hofib(µ M2 ) d7brane T 10,1 32 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 the M2-extension is classified by a 4-cocycle: the GS-WZW-term of the M2-brane µ M2 = i 2 ψ Γ a 1 a 2 ψ e a 1 e a 2 D Auria-Fré 82, Bergshoeff-Sezgin-Townsend 87, Fiorenza-Sati-Schreiber 13 T T 0 1

46 hofib(µ M5 ) µ M5 B 7 Q hofib(µ M2 ) d7brane T 10,1 32 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 the M5-extension is classified by a 7-cocycle: the GS-WZW-terms of the M5-brane µ M5 = 1 5! ψ Γ a 1 a 5 ψ e a 1 e a c ψ Γ a 1 a 2 ψ e a 1 e a 2 D Auria-Fré 82, Pasti-Sorokin-Tonin 97, Fiorenza-Sati-Schreiber 13 T T 0 1

47 hofib(µ M5 ) µ M5 B 7 Q hofib(µ M2 ) d7brane T 10,1 32 hofib(ρ) (B 7 Q)/B 3 Q µ M2 d9brane string IIB string het B 4 Q string IIA d8brane ρ T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 to descend this means to ask for analogous fiber sequence on the coefficients Nikolaus-Schreiber-Stevenson 12 T 2,1 2+2 T 2,1 2 T T 0 1

48 hofib(µ M5 ) µ M5 S 7 hofib(µ M2 ) d7brane T 10,1 32 S 4 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 5,1 8 T 5,1 8+8 this comes out to be: quaternionic Hopf fibration (rationally) Fiorenza-Sati-Schreiber 15 T T 0 1

49 hofib(µ M5 ) µ M5 S 7 hofib(µ M2 ) d7brane T 10,1 32 µ M2/M5 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane S 4 T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 M5-cocycle descends: unified M2/M5-cocycle T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 Fiorenza-Sati-Schreiber 15 T T 0 1

50 hofib(µ M5 ) µ M5 S 7 hofib(µ M2 ) d7brane T 10,1 32 µ M2/M5 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane S 4 T 9, T 9,1 16 T 9, dgc-model for S 4 : T 5,1 8 T 5,1 8+8 dω 4 = 0 dω 7 = 1 2 ω 4 ω 4 T 3,1 4+4 T 3,1 4 11d SuGra C-field equation of motion: dg G 4 G 4 = 0 T 2,1 2+2 T 2,1 2 T T 0 1

51 d5brane d3brane d1brane d7brane T 10,1 32 µ M2/M5 µ M2 d9brane string IIB string het B 4 Q string IIA d8brane S 4 T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 consider this unified M-brane cocycle T 2,1 2+2 T 2,1 2 T T 0 1

52 d5brane d3brane d1brane d7brane CycExt(T 9, ) T 10,1 32 µ M2/M5 S 4 CycExt(S 4 /S 1 ) µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 remember that 11d spacetime is (maximal invariant) extension of type IIA spacetime T 2,1 2+2 T 2,1 2 T T 0 1

53 d5brane d3brane d1brane d7brane CycExt(T 9, ) T 10,1 32 µ M2/M5 S 4 CycExt(S 4 /S 1 ) µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 5,1 8 T 5,1 8+8 similarly S 4 is homotopy extension of its S 1 homotopy quotient via canonical SU(2)-action on S 4 S(R H) T T 0 1

54 d5brane d3brane d1brane d7brane CycExt(T 9, ) T 10,1 32 µ M2/M5 S 4 CycExt(S 4 /S 1 ) µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 5,1 8 T 5,1 8+8 This orbifold S 4 /C n S 4 /S 1 happens to be the same as in the near-horizon geometry of the black M5-brane at an A-type singularity Medeiros, Figueroa-O Farrill 10 T T 0 1

55 d5brane d3brane d1brane d7brane CycExt(T 9, ) µ M2/M5 T 10,1 32 S 4 CycExt(S 4 /S 1 ) µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, T 3,1 4+4 T 3,1 4 T 5,1 8 T 5,1 8+8 hence the unified M2/M5-cocycle is really of this form T 2,1 2+2 T 2,1 2 T T 0 1

56 d5brane d3brane d1brane d7brane CycExt(T 9, ) µ M2/M5 T 10,1 32 S 4 CycExt(S 4 /S 1 ) µ M2 d9brane string IIB string het B 4 Q string IIA d8brane T 9, T 9,1 16 T 9, Theorem (Fiorenza-Sati-Schreiber 17): Ext has a derived right adjoint SuperHomotopyTypes T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T 5,1 8 T 5,1 8+8 Extension SuperHomotopyTypes /BS 1 Cyclification given by passing to twisted loop spaces / cyclic cohomology T T 0 1

57 d5brane d3brane d1brane d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) T 10,1 32 S 4 CycExt(S 4 /S 1 ) µ F 1 d9brane string IIB string het S 3 string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 apply the right adjoint T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T T 0 1

58 T 9, S 4 /S 1 η T 9, η S 4 /S 1 d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) T 10,1 32 S 4 CycExt(S 4 /S 1 ) d9brane string IIB string het S 3 string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 and compose with the adjunction unit T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T T 0 1

59 T 9, S 4 /S 1 η T 9, µ M2/M5 η S 4 /S 1 d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) T 10,1 32 S 4 CycExt(S 4 /S 1 ) d9brane string IIB string het S 3 string IIA d8brane T 9, T 9,1 16 T 9, T 5,1 8 T 5,1 8+8 to obtain the Ext Cyc-adjunct of the unified M-brane cocycle T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T T 0 1

60 T 9, S 4 /S 1 η T 9, µ F IIA 1 /D0/D2/D4/NS5 η S 4 /S 1 d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) T 10,1 32 S 4 CycExt(S 4 /S 1 ) d9brane string IIB string het S 3 string IIA d8brane T 9, T 9,1 16 T 9, Theorem (Fiorenza-Sati-Schreiber 17) : This is the Green-Schwarz WZW term of the double dimensional T 5,1 8 reduction T 5,1 8+8 of M2/M5 to F1 IIA /D0/D2/D4/NS5 : dgc-algebra for CycExt(S 4 /S 1 ): T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 dh 3 = 0, dh 7 = F 2 F F 4 F 4 df 2 = 0, df 4 = H 3 F 2, df 6 = H 3 F 4 T T 0 1

61 ?? T 9, S 4 /S 1 η T 9, µ F IIA 1 /D0/D2/D4/NS5 η S 4 /S 1 d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) T 10,1 32 S 4 CycExt(S 4 /S 1 ) d9brane string IIB string het S 3 string IIA d8brane T 9, T 9,1 16 T 9, This gives rise to two questions: 1) Where are the D(p 6)-branes (gauge enhancement)? T 3,1 4+4 T 3,1 4 T 5,1 8 T 5, ) Is there a dashed lift as above? T 2,1 2+2 T 2,1 2 T T 0 1

62 T 9, S 4 /S 1 η T 9, µ F 1/D2/D4/D6/NS5 η S 4 /S 1 d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) CycExt(S 4 /S 1 ) d9brane string IIB S 3 string IIA d8brane T 9, T 9,1 16 T 9, let us first make some room... T 5,1 8 T 5,1 8+8 T 3,1 4+4 T 3,1 4 T 2,1 2+2 T 2,1 2 T T 0 1

63 T 9, S 4 /S 1 Ω S 3 Σ S 3 (S 4 /S 1 ) η T 9, µ F 1/D2/D4/D6/NS5 η S 4 /S 1 Ω S 3 Σ S 3 (η S 4 /S 1) d7brane CycExt(T 9, Cyc(µ M2/M5 ) ) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 ) d9brane string IIB S 3 string IIA d8brane consider the Goodwillie-linearized lifting problem: form the fiberwise suspension spectrum over S 3 to obtain an S 3 parameterized spectrum

64 T 9, Ω S 3 Σ S 3 (S 4 /S 1 ) η T 9, d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) Ω S 3 Σ S 3 (η S 4 /S 1) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 ) d9brane string IIB S 3 string IIA d8brane

65 T 9, ku/b 2 Z Ω S 3 Σ S 3 (S 4 /S 1 ) η T 9, Ω S 3 Σ S 3 (η S 4 /S 1) d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 ) d9brane string IIB S 3 string IIA d8brane Theorem (Roig-Saralegi 00) : rationally, a direct summand of Ω S 3 Σ S 3 (S 4 /S 1 ) is twisted connective K-theory ku/b 2 Z

66 T 9, µ IIA F 1/Dp ku/b 2 Z Ω S 3 Σ S 3 (S 4 /S 1 ) η T 9, Ω S 3 Σ S 3 (η S 4 /S 1) d7brane CycExt(T 9, ) Cyc(µ M2/M5 ) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 ) d9brane string IIB S 3 string IIA d8brane and now there is a lift: the unified cocyle of all the type IIA D-branes dgc-algebra for B 3 Z Q S 3 : dh 3 = 0 dg-module for ku/b 2 Z: df 2p+2 = H 3 F 2p p N

67 T 9, µ IIA F 1/Dp ku/b 2 Z d5brane d3brane Ext IIA (T 8, ) d7brane CycExt(T 9, ) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 ) Conclusion: Double dimensional reduction of unified M-brane cocycle via cyclification is unified IIA-brane cocycle

68 T 9, µ IIA F 1/Dp ku/b 2 Z KU/B 2 Z d5brane d3brane Ext IIA (T 8, ) d7brane CycExt(T 9, ) CycExt(S 4 /S 1 ) Ω S 3 Σ S 3 CycExt(S 4 /S 1 Conclusion: Double dimensional reduction of unified M-brane cocycle via cyclification is unified IIA-brane cocycle

69 Cyc(T 9, Cyc(µ IIA F 1/Dp ) Cyc(KU/B 2 Z rane d3brane CycExt IIA (T 8, ) rane T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, ) Cyc(ΣKU/B 2 Z) we repeat the process: and consider the double dimensional reduction of the IIA-cocycle to 9d super-spacetime T 8,

70 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) rane T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, ) Cyc(ΣKU/B 2 Z) hence apply cyclification

71 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, ) Cyc(ΣKU/B 2 Z) and compose with the adjunction unit

72 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, ) Cyc(ΣKU/B 2 Z) to obtain the double dimensional reduction

73 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, Cyc(ΣKU/B 2 Z) but there was also the type IIB extension

74 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane Cyc(T 9, Cyc(µ IIB F 1/Dp) Cyc(ΣKU/B 2 Z) whatever cocycle it carries

75 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / η IIB rane string IIB CycExt IIB (T 8, ) S 3 string IIA d8brane µ Cyc(T 9, ) Cyc(µ IIB F 1/Dp) Cyc(ΣKU/B 2 Z) has itself a double dimensional reduction

76 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/BT Ω S 3 Σ S 3 CycExt(S 4 / η IIB rane string IIB CycExt IIB (T 8, ) µ Cyc(T 9, ) Cyc(µ IIB F 1/Dp) Cyc(ΣKU/B 2 Z) by adjunction this defines µ in terms of µ IIA

77 Cyc(T 9, ) Cyc(µ IIA Cyc(KU/B 2 Z) F 1/Dp ) rane d3brane CycExt IIA (T 8, ) η IIA rane T 8, µiia F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/T Ω S 3 Σ S 3 CycExt(S 4 /S 1 η IIB rane string IIB CycExt IIB (T 8, ) µiib F 1/Dp Cyc(T 9, ) Cyc(µ IIB F 1/Dp ) Cyc(ΣKU/B 2 Z) Theorem A: (Fiorenza-Sati-Schreiber 17): This is the cocycle in twisted K 1 for the F1/Dp-branes in type IIB

78 Cyc(T 9, ) Cyc(KU/B 2 Z) rane d3brane CycExt IIA (T 8, ) µ IIA F 1/Dp rane T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/T Ω S 3 Σ S 3 CycExt(S 4 /S 1 rane string IIB CycExt IIB (T 8, ) µ IIB F 1/Dp Cyc(T 9, ) Cyc(ΣKU/B 2 Z) Theorem B: (Fiorenza-Sati-Schreiber 17): The commutativity of this diagram is equivalently the Buscher rules for the RR-fields (Hori 99)

79 Cyc(T 9, ) Cyc(KU/B 2 Z) rane d3brane CycExt IIA (T 8, ) µ IIA F 1/Dp rane T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/T BT rane string IIB CycExt IIB (T 8, ) µ IIB F 1/Dp Cyc(T 9, ) Cyc(ΣKU/B 2 Z) Theorem C: (Fiorenza-Sati-Schreiber 17): The commutativity of this diagram is equivalently the rules of topological T-duality ( Bouwknegt-Evslin-Mathai 04, Bunke-Rumpf-Schick 08 ) rationally

80 Cyc(T 9, ) Cyc(KU/B 2 Z) ane d3brane CycExt IIA (T 8, ) µ IIA F 1/Dp T 8+(1,1),1 32 hofib T 8, CycExt(S 4 /S 1 ) (KU ΣKU)/T BT ane string IIB CycExt IIB (T 8, ) µ IIB F 1/Dp Cyc(T 9, ) Cyc(ΣKU/B 2 Z) Theorem D: (Fiorenza-Sati-Schreiber 17): The homotopy fiber is the doubled generalized geometry 10d super-spacetime

81 T 10+(1,1),1 32 ne T 9+(1,1),1 32 (pb) T 10,1 32 T 9, T 8+(1,1),1 32 hofib T 8, Cyc(KU/B 2 Z) µ IIA F 1/Dp CycExt(S 4 /S 1 ) (KU ΣKU)/T BT ne string IIB CycExt IIB (T 8, ) µ IIB F 1/Dp Cyc(T 9, ) Cyc(ΣKU/B 2 Z) Theorem E: (Fiorenza-Sati-Schreiber 17): The homotopy pullback of type II doubled super-spacetime back to 11d super-spacetime is the local model for an F-theory fibration

82 Conclusion: A fair bit of the expected structure of M-theory emerges out of the superpoint in rational super-homotopy theory. Evident Conjecture: The full theory emerges once passing beyond the rational approximation in full super-geometric homotopy theory. (arxiv: ).

83 Epilogue In full super-geometric homotopy theory the superpoint R 0 1 itself emerges from id id R 0 1 R D Et R (Schreiber 16, FOMUS proceedings)