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Why should things be easy to understand?
~ Thomas Pynchon!

In this post, I will relate the Dirac-Ramond operator to the Euler and Hirzebruch indices in the context of string theory and connect some dots that lead us to supersymmetry and use the Atiyah-Bott theorem in a narrow context to probe finiteness for quantum gravity. The AdS/CFT duality is expressed as:

where

is the bulk-field, the radial coordinate that is dual to the renormalization group in the boundary theory, with:

and coupled to . Note also that the integral term on the left-hand-side of the AdS/CFT duality has integral-measure over string world-sheets that must not go to infinity as the string world-sheet dynamically evolves with respect to time, as time goes to infinity. Moreover, since the string world-sheet dynamics ‘Heisenberg-Hilbert’ creates the graviton, then given that gravity is universally sensitive, the string world-sheet becomes an infinite dimensional Riemannian manifold. Both infinity-problems can only be avoided via a Dirac-Ramond operator analysis. To see this, one needs to split such a gravitonic infinitary degeneracy and do analysis on the finite dimensional subspaces of the kernels of the Dirac-Ramond (DR) operator. Note that the equivariant DR operator effectively does the splitting, and so the loop space ,with being the Riemannian string world-sheet manifold, has action given via sending the loop-parameter to , with a constant. Let representations of be denoted by integers corresponding to 2-dimensional momentum of the string state. The Dirac-Ramond index can be best analyzed by a supersymmetric quantum algebra in parallel with the Atiyah-Singer index (theorem) context. The Lagrangian for a 2-dimensional field theory with a gravitational field background is given by:

Note that the supersymmetry charge here is the equivalent Dirac-Ramond operator with generators:

corresponding to translations:

Now, is well-defined, since the Pontrjagin class of vanishes completely in integral cohomology. That implies that the index admits a symplectic path integral representation. Let me analyse this in the case of the Lagrangian given above with periodic boundary conditions on a torus with a skew structure and sides given by 1 and , . Now, the path integral computes as:

with the Hamiltonian:

and the momentum:

with , , and is the Virasoro central charges for the right and left RR sectors. Calculating explicitly gives us a representation for the character valued index of the Dirac-Ramond operator. One must appreciate here the deep and crucial relation:

since it allows us to rewrite:

as:

Now, since anti-commutes with , one gets a fermionic number pairing of eigenstates whenever the eigenvalue is strictly greater than zero. Hence, the only terms that do not vanish in the trace are the supersymmetric ones! Hence, from:

one can derive:

In the string context, for the Riemannian world-sheet manifold , the spectrum of the Beltrami-Laplace operator is discrete, thus allowing Heisenberg-Hilbert ‘creation’ of the supersymmetric partner of the graviton. If the eigenvalues of are , then:

becomes:

where is the Witten index on the finite subspace with momentum . Note that the states in the kernel of can have any allowed eigenvalues, and in our case, string theory, are of the form , . In conformal field theory, will be of the form , with the conformal weight of the highest supersymmetric weight vectors that induce Verma modules in the spectrum. So we get:

with holding. So:

can be derived as:

Given that is the …