29 Sep 2015

## String Theory, Dirac-Ramond Analysis Of The Euler And Hirzebruch Indices, SuperSymmetry, And The Atiyah-Bott Theorem

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. Note in my last post, the AdS/CFT duality is expressed as:

$$(a+b)^2 = a^2 + b^2 + 2ab$$

$$\cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+\dotsb }}}$$

$$X(e^{j\omega}) = \displaystyle \sum_{n=-\infty}^\infty x[n].e^{-j\omega}$$

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

$$e^{\i \pi} + 1 = 0$$
and ϕ(0) coupled to ϑ(x). Note 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 L(Rws),with Rws being the Riemannian string world-sheet manifold, has S1 action given via sending the loop-parameter σ to σ+Δ, with Δ a constant. Let representations of S1 be denoted by integers n corresponding to 2-dimensional momentum of the string state. The Dirac-Ramond index can be best analyzed by a N=1/2 supersymmetric quantum algebra in parallel with the Atiyah-Singer index (theorem) context. The Lagrangian for a N=1/2 2-dimensional field theory with a gravitational field background is given by: