String Theory 3.0 : Use the algebraic vacuum in eight octonionic dimensions, instead of the 10D Minkowski vacuum.

General Relativity is curvature of 4D Minkowski spacetime.

Octonionic Theory is the unified theory = `curvature' of 10D Minkowski spacetime = 8D octonionic spacetime. Except that the concept of curvature is more sophisticated now.

The new action is simply that for a free particle in octonionic 8D: equivalence principle holds, unified interaction as geometry of spacetime holds.

The octonionic theory could also be called String Theory 3.0 where the vacuum shifts from being 10D Minkowski vacuum to octonionic 8D algebraic vacuum. Also, we did not arrive at String Theory 3.0 by postulating ad hoc existence of extended objects such as strings. Rather, we arrive at strings by solving a foundational problem of quantum theory: find a reformulation of quantum field theory which does not depend on classical time. This leads us to strings in an inevitable unavoidable way, and this time as a successful theory of unification.

All this is made very clear and convincing by Clifford algebras and Jordan algebras, and by extension of the standard model to the Left-Right symmetric model. The Right sector is (would be) gravity.

Consider this connection between Jordan algebra, Lie algebra, and Minkowski spacetime symmetry in different dimensions:

J_2 (R) \sim SL(2,R) \sim SO(1,2) 2+1 Gravity

J_2 (C) \sim SL(2,C) \sim SO(1,3) 3 + 1 GR, Einstein gravity

J_2 (H) \sim SL(2,H) \sim SO(1,5) 5+1 Gravi-Weak theory !

J_2 (O) \sim SL(2,O) \sim SO(1,9) 9+1 Gravi-weak-electro-color theory [L-R symmetric model]

In the language of Clifford algebras: modulo Bott periodicity, one only has to go up to Cl(7) and every Cl(n) then connects to the standard model.

Cl(2) 4D Lorentz algebra made from complex quaternions using two of the three imaginary directions: home for GR, SL(2,C)

Cl(3) 6D Lorentz algebra made from complex quaternions using all three imaginary directions (complex split biquaternions) home for gravi-weak theory SL(2,H)

Cl(4) just the weak interaction

Cl(6) complex octonions : SU(3) X U(1) symmetry, use six out of seven imaginary octonionic directions : electro-color symmetry

Cl(7) Lorentz algebra SO(1,9) \sim SL(2,O) made using all seven imaginary octonionic directions (complex split bioctonions) : gravi-weak-electro-color unified symmetry.

What do we gain by going to algebras? We bring in gravity in an obvious way linking up to SO(1,9) and E_6. Above all, the characteristic equation of the exceptional Jordan algebra determines the free parameters of the standard model.

In summary

4D is GR

6D. unification of gravity with weak interaction

10D unification of gravi-weak with electro-color

all the time staying with forces as geometry, but beyond 4D inevitably going (pre)quantum.

This could be called the return of string theory if one so wishes, but with the limitations now removed because of important changes: use the algebraic vacuum of octonions, not the Minkowski vacuum.

For a detailed review of the Octonionic Theory please see

Quantum Theory without Classical Time: a route to quantum gravity and unification

https://www.tifr.res.in/~tpsingh/Penrose90singh.pdf