Schreiber Super T-Duality inside the Superpoint

A talk that I will have given:

• Urs Schreiber

  Super T-Duality inside the Superpoint,

  talk at

  Conference on invertible objects and duality in derived algebraic geometry and homotopy theory,

  April 3-7, 2017, Regensburg

Abstract. Twisted cohomology is maps in the tangent homotopy theory or parameterized spectra¹. There is a Quillen-Sullivan-type model for its rationalization via unbounded L-∞ algebras². Examples appear by iterated maximal higher central extensions of super L-∞ algebras which are invariant with respect to all automorphisms modulo R-symmetries – a super-equivariant version of the Whitehead tower. Applied to the superpoint, this process discovers all the twisted cohomology seen in string/M-theory³$\,$⁴, rationally, in particular twisted topological K-theory with 5-brane correction and with supersymmetric Chern-forms on 10d-superspace⁵. This comes out of the M-brane coefficients, which turn out to be the 4-sphere⁶, via these rational equivalences $\,$:

$$\mathcal{L}S^4/S^1 \underset{\mathbb{Q}, \leq 6}{\simeq} ku/BU(1) \;\;\; \text{and} \;\;\; \Sigma^\infty_{S^3} (S^4/S^1) \underset{\mathbb{Q}}{\simeq} ku/BU(1) \oplus \cdots \,.$$

Passage to cyclic? L-∞ cohomology reflects double dimensional reduction of super p-branes. Applied to twisted K-theory this yields a super L-∞ isomorphism exhibiting supersymmetric topological T-duality, rationally⁷. The super L-∞ algebra it classifies is the local tangent complex of super T-folds.

These rational phenomena indicate that integrally the super-equivariant Whitehead tower should discover extremely interesting twisted super-differential cohomology theories. Possibly one should promote super L-∞ algebras to spectral super-schemes, namely spectral schemes over an even periodic ring spectrum, and then repeat the process.

Lecture notes with more details are here:

• geometry of physics – fundamental super p-branes?.

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1. Urs Schreiber, Differential Cohomology in a Cohesive ∞-Topos ↩

2. Vincent Schlegel, Urs Schreiber Rational parameterized homotopy theory, in preparationi ↩

3. Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264) ↩

4. John Huerta, Urs Schreiber, M-Theory from the Superpoint (2017) ↩

5. Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory, Journal of Geometry and Physics, Volume 114, April 2017 (arXiv:1606.03206) ↩

6. Domenico Fiorenza, Hisham Sati, Urs Schreiber, The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557) ↩

7. Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes (arXiv:1611.06536) ↩