String/M-[F]-theory remains by far the most promising – only? – theoretical paradigm for both, grand unification and quantization of general relativity. With the Dp-action given by:
for contextualization, note that a necessary condition for the world-sheet Dirac propagator :
to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:
and the bosonic frequency, be analytically summable. The string world-sheet is given by:
A major problem is that by the Heisenberg’s uncertainty principle:
the string time-parameter on the world sheet with Feynman propagator in Euclidean signature being:
violates the integrability condition for the action:
is non-convergent, with real, since the quantization of spacetime is an anti-smoothing dynamical breaking of the Ricci scalar . Hence we get,
which is incoherent. To see this, note that the anti-smoothing of spacetime implies that cannot be recovered from via
and that implies that the gravitonic wave-propagation travels in spacetime at infinite speed, given
and by wave-particle duality and the violation of special relativity, the graviton provably cannot exist. Or, by quantum tunnelling and the fact that gravitons self-gravitate, we have the instantaneous collapse of spacetime to a zero-dimensional singularity. Pick your poison.
A solution is to integrate over orbibolds and derive the Lagrangian of N=1 supergravity by orbifoidal D-11 and D-10 SUGRA-Barbero coupled actions. Let us see how this works. One must begin by giving a description of the field contents and the degrees of freedom, which will turn out to be a crucial number. At first, note that in D=11, SUGRA has a simple action: using exterior algebraic notation for the anti-symmetric tensor fields , with the field strength , it is surprisingly:
with the Newtonian constant in 11 dimensions. By dimensional reduction, the Type IIA can be derived from (1). Note that there are D=10 supergravity theories with only SUSY which couple to D=10 super-Yang-Mills theory. We still do not have a workable Type IIB theory since it involves an antisymmetric field with a self-dual field strength. Nonetheless, one may still derive an action that involves both dualities of . Then, by imposing the self-duality as a supplementary equation, we get:
with field strengths: , , , , , with the self-duality condition .
Note that the above action arises from the string low-energy limit and:
naturally yields the NS-NS sector of the theory, while:
is derivable from the RR sector of the theory. Now, Type IIB supergravity theory is invariant under the non-compact symmetry group and the key is that this symmetry is not manifest in:
To make it so, one must redefine fields, from the string metric in (2) to the Einstein metric , along with a complexification of the tensor fields:
Now, the action is easily seen to be:
the metric and fields are invariant under the symmetry of Type IIB supergravity. The axionic dilaton field varies with a Möbius super-transformation:
and , self-rotate under the Möbius super-transformation, and can most clearly be visualized as a complex 3-form field :
The SUSY transformation for Type IIB supergravity on the fermion fields are of the following form, via Seiberg–Witten analysis, without a need for bosonic transformation laws, with the dilaton and the gravitino :
It is crucial to realize that in the context, the SUSY transformation parameter essentially has charge, implying that has necessarily, given unitarity, and has .
The geometry of superstring theory in the Ramond-Neveu-Schwarz setting is given by the bosonic world-sheet fields and the fermionic world-sheet fields , with expressing chirality, and , must be functions of local world-sheet coordinates . Both and vectorially transform under the irreducible representation of the Lorentz group. By using Gliozzi-Scherk-Olive holographic projections, the spacetime supersymmetric derivative can act on (1) and (2) above. It is more informative to work with orientable strings. Type II and heterotic string theories are perfectly suited in this context. Field interactions in second quantization arise from the orbifoidal splitting and joining of the world-sheets, and causality is maintained. Moreover, the genus for orientable world-sheets equals the number of Witten-handles. The world-sheet bosonic field naturally gives rise to a non-linear sigma model:
with being the square root of the Planck length, being the world-sheet metric, and being the Gaussian curvature. The world-sheet fermionic field axially gives rise to a world-sheet supersymmetric completion of the sigma model. It suffices to give its form on a flat world-sheet metric with a vanishing world-sheet gravitino field:
and being the Riemann tensor for the metric . Now the all too important SUSY-covariant derivatives can be derived:
with the Levi-Civita connections for . Now one is in a position to solve the Seiberg-Lebesgue problem via the functional integral over all and by integrating over all world-sheet metrics and world-sheet gravitini fields via the amplitude:
This clearly solves the Feynman propagation problem. And the upshot is that the vacuum expectation value of the dilaton field is , and the string vacuum expectation value of the string amplitude is given by the Euler number of the world-sheet :
with the genus, is a number that counts the orbifoidal puntures in spacetime as the string world-sheet propagates under quantum fluctuations. Thus, a genus world-sheet with no boundary gets a multiplicative contribution: , thus deriving another truly remarkable identity, , representing the closed string coupling constant which lives on a Dp-brane, and hence by the supersymmetry action and the Witten Index:
where is the fermion number, and the trace is over all bound and continuum states of the supersymmetric-Hamiltonian, and PBC being the periodic boundary conditions on both the fermionic and bosonic fields, we get a finite, causal SUGRA action in D=11/ D=10, thus solving the Seiberg-Lebesgue problem.