Dirac-Ramond Analysis, The Euler-Hirzebruch Indices, SuSy, And The Atiyah-Bott Theorem

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:

    \[{\left\langle {{e^{\int {{d^4}} x{\phi _{(0)}}(x)\vartheta (x)}}} \right\rangle _{CFT}} = {\not Z_{{\rm{String/AdS}}}}\left\langle {{r^{\Delta - d}}\phi {{(x,r)}_{\left| {_{r = 0}} \right.}} = {\phi _{(0)}}(x)} \right\rangle \]

where

    \[\phi (x,r)\]

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

    \[{\phi _{(0)}}(x) \equiv {r^{\Delta - d}}\phi {(x,r)_{\left| {_{r = 0}} \right.}}\]

and {\phi _{(0)}} coupled to \vartheta (x). 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 L({R_{ws}}),with {R_{ws}} being the Riemannian string world-sheet manifold, has {S^1} action given via sending the loop-parameter \sigma to \sigma + \Delta, with \Delta a constant. Let representations of {S^1} 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:

    \[L = {g_{\mu \nu }}(X)\,\overline {\not \partial } {X^\mu }\not \partial {X^\nu } + {g_{\mu \nu }}(X)\left[ {\overline {\not \partial \,} {\psi ^\mu } + \overline {\not \partial } {X^k}\Gamma _{k\lambda }^\mu (X){\psi ^\lambda }} \right]{\psi ^\nu }\]

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

    \[{g_{\mu \nu }}d\,{X^\mu }/d\sigma \]

corresponding to {S^1} translations:

    \[Q = \int {d\sigma {\psi ^\mu }} (\sigma )\left( { - \frac{{\not D}}{{\not D{X^\mu }(\sigma )}} + {g_{\mu \nu }}(X)\frac{{d\,{X^\nu }}}{{d\sigma }}} \right)\]

Now, Q is well-defined, since the Pontrjagin class {P_1}({R_{sw}}) of {R_{sw}} 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 \tau = {\tau _1} + i{\tau _2}, {\tau _2} > 0. Now, the path integral computes as:

    \[{I_P} = {\rm{Tr}}{\left( { - 1} \right)^F}\exp \left( { - 2\pi {\tau _2}H + 2\pi i{\tau _1}P} \right)\]

with H the Hamiltonian:

    \[H = \left( {{L_0} + \varepsilon } \right) + \left( {{{\overline L }_0} + \overline \varepsilon } \right)\]

and P the momentum:

    \[P = \left( {{L_0} + \varepsilon } \right) - \left( {{{\overline L }_0} + \overline \varepsilon } \right)\]

with \varepsilon = \, - c/24, \overline \varepsilon = \overline c /24, and \overline c is the Virasoro central charges for the right and left RR sectors. Calculating {I_P} explicitly gives us a representation for the character valued index of the Dirac-Ramond operator. One must appreciate here the deep and crucial relation:

    \[{L_0} + \varepsilon = {Q^2}\]

since it allows us to rewrite:

    \[{I_P} = {\rm{Tr}}{\left( { - 1} \right)^F}\exp \left( { - 2\pi {\tau _2}H + 2\pi i{\tau _1}P} \right)\]

as:

    \[{I_P} = {\rm{Tr}}{\left( { - 1} \right)^{{F_q}\,{L_0} + \,\varepsilon }}{\left( q \right)^{\overline {{L_0}} + \overline {\,\varepsilon } }}\quad \quad ,\quad \quad q = {e^{2\pi i\tau }}\]

Now, since Q anti-commutes with {\left( { - 1} \right)^F}, one gets a fermionic number pairing of {Q^2} eigenstates whenever the {Q^2} eigenvalue is strictly greater than zero. Hence, the only terms that do not vanish in the trace are the supersymmetric ones! Hence, from:

    \[{I_P} = {\rm{Tr}}{\left( { - 1} \right)^{{F_q}\,{L_0} + \,\varepsilon }}{\left( q \right)^{\overline {{L_0}} + \overline {\,\varepsilon } }}\quad \quad ,\quad \quad q = {e^{2\pi i\tau }}\]

one can derive:

    \[{I_P} = \underbrace {{\rm{Tr}}}_{{\rm{Supersymmetry}}}{\left( { - 1} \right)^F}{\left( {\overline q } \right)^{ - P}}\not D{\psi ^\mu }\left( X \right)\]

In the string context, for the Riemannian world-sheet manifold {R_{ws}}, 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 P are \lambda, then:

    \[{I_P} = \underbrace {{\rm{Tr}}}_{{\rm{Supersymmetry}}}{\left( { - 1} \right)^F}{\left( {\overline q } \right)^{ - P}}\not D{\psi ^\mu }\left( X \right)\]

becomes:

    \[{I_P} = \sum\limits_\lambda {{I_\lambda }} {\left( {\overline q } \right)^{ - \lambda }}\not D\,{\psi ^\mu }{\left( {\overline X } \right)^\lambda }\]

where {I_\lambda } is the Witten index on the finite subspace with momentum \lambda. Note that the states in the kernel of Q can have any allowed {\overline L _0} + \overline \varepsilon eigenvalues, and in our case, string theory, \lambda are of the form - \left( {\overline \varepsilon + n} \right), n > 0. In conformal field theory, \lambda will be of the form - \left( {\overline \varepsilon + \overline h + n} \right), with \overline h the conformal weight of the highest supersymmetric weight vectors that induce Verma modules in the L({R_{ws}}) spectrum. So we get:

    \[{I_P} = {\left( {\overline q } \right)^{\overline \varepsilon + \,{{\overline h }_1}}}\sum\limits_{n = 0}^\infty {{a_n}} {\left( {\overline q } \right)^n} + {\left( {\overline q } \right)^{\overline \varepsilon + {{\overline {\,h} }_2}}}\sum\limits_{n = 0}^\infty {{b_n}} {\left( {\overline q } \right)^n}\]

with H = \, - P holding. So:

    \[{I_P} = \underbrace {{\rm{Tr}}}_{{\rm{Supersymmetry}}}{\left( { - 1} \right)^F}{\left( {\overline q } \right)^{ - P}}\not D{\psi ^\mu }\left( X \right)\]

can be derived as:

{I_P} = {\rm{Tr}}{\left( { - 1} \right)^F}{e^{2\pi i{\tau _1}\,P}}{e^{ - \,2\pi i{\tau _2}\,H}}

Given that H is the …