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

By George Shiber Monday, September 14, 2015Friday, October 4, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Physics, Super String Theory Atiyah-Bott, chern

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 generator of the Clifford group action, such a representation will have a character valued index with respect to $H$ and $P$ eigenvectors despite the non-compactness of the $L({R_{ws}})$ induced flow generated by $H$ . Note that the index can only involve information about the loop space $L({R_{ws}})$ . The character valued index is only a function of ${\tau _1}$ ; however,

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

involves all of $\tau$ , so to deal with the infinitary degeneracy problem and to calculate the path integral in the small ${\tau _z}$ limit, we need the quadratic approximation of the super-Lagrangian:

${L_2} = \overline {\not \partial } {X^\mu }\not \partial {X^\mu } + \overline {\not \partial } \,{\psi ^\mu } \cdot {\psi ^\nu } + {\not R_{\mu \nu }}\overline {\not \partial } {X^\mu }{X^\nu }$

with $X$ and $\psi$ interpreted as fluctuations about the classical solutions $X_0^\mu$ and $X_0^\nu$ , with:

${\not R_{\mu \nu }} \equiv \frac{1}{2}{\not R_{\mu \nu ,{Q_\sigma }}}\psi _0^\sigma \overline \psi _0^\sigma$

Hence, one gets an index that is given by:

${I_P} = \frac{1}{{{{\left( {2\pi } \right)}^{d/2}}}}\int {{d^d}} {X_0}\,{d^d}{\psi _0}{\left[ {{\rm{de}}{{\rm{t}}^\dagger }\left( {\, - \not \partial \overline {\not \partial } + \not R\overline {\not \partial } } \right)} \right]^{ - 1/2}}{\left[ {{\rm{de}}{{\rm{t}}^\dagger }\left( {\overline {\not \partial } } \right)} \right]^{1/2}}$

where $\dagger$ denotes the omission of the zero modes. So we get:

$\begin{array}{c}{I_P} = \frac{1}{{{{\left( {2\pi } \right)}^{d/2}}}}\int {{d^d}} {X_0}\,{d^d}{\psi _0}\exp \left[ {\sum\limits_j {\sum\limits_\Lambda ^ \to {\left( {\frac{{2{\tau _2}i{x_j}}}{{m + n\overline \tau }}} \right) + \frac{1}{2}\sum\limits_j {\sum\limits_\Lambda ^ \leftarrow {{{\left( {\frac{{2{\tau _2}i{x_j}}}{{m + n\overline \tau }}} \right)}^2}} } } } } \right]\\ \times \prod\limits_j {\frac{{2{\tau _2}i{x_j}}}{{\sigma \left( {2{\tau _2}i{x_j}\overline \tau } \right)}}} \cdot \frac{1}{{{{\left( {2{\tau _2}} \right)}^{d/2}}}} \cdot \frac{1}{{{{\left[ {1 - \left( {\overline q } \right)} \right]}^d}}}\end{array}$

with $\Lambda$ the periodic lattice determined by a parallelogram with sides 1 and $\tau$ , and ${x_j}$ are the supersymmetric entries in the block diagonalization of the ${\not R_{\mu \nu }}/2\pi$ quasi-matrix, and ${\eta _ \to }\left( {\overline q } \right)$ is the Dedekind Eta-function. Let me define:

${\eta _ \pm }\left( {\overline q } \right) \equiv {\left( {\overline q } \right)^{1/24}}\prod\limits_1^\infty {\left( {1 \pm {{\left( {\overline q } \right)}^n}} \right)}$

Now, the Weierstrass

$\sigma$

function can be derived as:

$\sigma \left( {{z_j},\tau } \right) = z\prod\nolimits_\Lambda ^\dagger {\left( {1 - \,\frac{z}{{m + n\tau }}} \right)} \exp \left[ {\sum\nolimits_\Lambda ^\dagger {\frac{z}{{m + n\tau }} + \frac{1}{2}\sum\nolimits_\Lambda ^\dagger {{{\left( {\frac{z}{{m + n\tau }}} \right)}^2}} } } \right]$

Given that integration of fermionic zero modes $\psi _0^\alpha$ are all that contribute to the string harmonics, we can see then that the Weierstrass $\sigma$ function is equivalent to:

${I_P} = \int\limits_{{R_{ws}}} {\exp \left[ {\sum\limits_j {\sum\nolimits_\Lambda ^\dagger {\frac{{i{x_j}/2\pi }}{{m + n\bar \tau }} + \frac{1}{2}\sum\limits_j {\sum\nolimits_\Lambda ^\dagger {{{\left( {\frac{{i{x_j}/2\pi }}{{m + n\bar \tau }}} \right)}^2}} } } } } \right]} \times \prod\limits_j {\frac{{i{x_j}/2\pi }}{{\sigma \left( {i{x_j}/2\pi \widetilde \tau } \right)}}} \cdot \frac{1}{{\left[ {{\eta _ \to }{{\left( {\overline q } \right)}^d}} \right]}}$

Since the first Chern class is given by:

${p_1}\left( {{R_{ws}}} \right) = \sum\limits_j {x_j^2}$

we must hence note that unless the left-hand-side vanishes, we will have divergences. The vanishing of ${p_1}\left( {{R_{ws}}} \right)$ is equivalent to having a pure string-like phenomenon that is not present in any finite array-truncation of the infinite spectrum of the string theory, and that is because the sum is logarithmically divergent, hence all energy scales have equal contributions. Thus, any truncation of:

does not necessarily entail the vanishing of the string Pontrjagin class. To make progress on this front, let us discard terms of the form:

$\sum\limits_j {{x_j}} \sum\nolimits_\Lambda ^\dagger {\frac{1}{{m + n\tau '}}}$

we can, since string regulators respect symmetries of the lattice-period. Hence, the string world-sheet will satisfy both:

$\left[ {\begin{array}{*{20}{c}}{\sum\limits_j {{x_j}{c_1}\left( {{T^{\left( {0,1} \right)}}{R_{ws}}} \right) = 0} }\\{{p_1}\left( {{R_{ws}}} \right) = 0}\end{array}} \right.$

for ${c_1}$ the first Chern class of the string holomorphic tangent bundle. Noting that such bundles are what embed the compactification spaces in string theory. So now we can see that the vanishing of the Pontrjagin class is substitutable by the Green–Schwarz relation:

${\rm{Tr}}{\not R^2} - {\rm{Tr}}\,{{\rm{F}}^2} = 0$

Observe now that one loop renormalization of sigma-models have the form:

${\rm{log}}\,\Lambda \cdot {\not R_{\mu \nu }}\overleftarrow {\not \partial } X_0^\mu \wedge \overrightarrow {\not \partial } X_0^\nu$

with $\Lambda$ the string cutoff. Now one can tell that the background ${X_0}$ is constant and no term appears: this is central since the index is topological. Moreover, some anomalies are hard to observe since the compactification torus has a flat metric. We are now in a position to show the equivalence of:

with the expression obtained by applying the character index theorem to $L({R_{ws}})$ on the metaplectic manifold M, which gets us:

${I_\theta } = \int\limits_{{N_b}} {\prod\limits_j {\frac{{i{x_j}/2\pi }}{{\sin i\,{x_j}/2\pi }}} } \prod\limits_r {\frac{1}{{2i\sin \left( {{{\overline \theta }_r}/2\pi } \right)}}}$

with ${N_b}$ being the fixed point set of the renormalization group action and:

${\overline \theta _r} \equiv \alpha \,{\theta _r} + {x_n}$

with ${\theta _r}$ the eigenvalues of the group action on the tangent bundle of M, ${T^b}(M)$ , and ${X_r}$ the eigenvalues of the curvature 2-form at the group fixed points and $\alpha$ the parameter of the infinitesimal transformation. We have thus:

${N_b} = L{({R_{ws}})_b} = M$

and ${T^b}(L{(M)_b})$ is the infinite dimensional vectorial space generated by the string Fourier modes. Now, ${\theta _r} \equiv \theta _r^\alpha = 2\pi n$ if $\alpha = {\tau _1} - i{\tau _2} = \overline \tau$ corresponds to the simultaneous $H$ and $P$ actions. Applying the character function theorem gives us:

$\prod\limits_r {\frac{1}{{2i\sin \left( {{{\bar \theta }_r}/2\pi } \right)}}} = \prod\limits_j {\prod\limits_{n = 1}^\infty {\left( {\frac{{\exp \left( {{\mkern 1mu} - {\mkern 1mu} \pi i\bar \tau n + {x_j}/2\pi } \right)}}{{1 - \exp \left( { - 2\pi i\bar \tau n + {x_j}} \right)}}} \right)} } \cdot \left( {\frac{{\exp \left( { - \pi i\overline \tau n - {x_j}/2\pi } \right)}}{{1 - \exp \left( { - 2\pi i\overline \tau n - {x_j}} \right)}}} \right) = \prod\limits_j {\prod\limits_{n = 1}^\infty {\frac{{{{\left( {\overline q } \right)}^n}}}{{1 - {{\left( {\overline q } \right)}^n}{e^{ - {x_j}}}}}} }$

Now defining ${\prod\limits_{n = 1}^\infty {\left( {\overline q } \right)} ^n}$ by a zeta functional regularization gives us:

${I_g} = \int\limits_M {\prod\limits_{} j } \frac{{i{x_j}/2\pi }}{{\sin \left( {i{x_j}/2\pi } \right)}}\frac{{{{\left( {\overline q } \right)}^{ - 1/2}}}}{{\prod\limits_{n = 1}^\infty {\left( {1 - {{\left( {\overline q } \right)}^n}{e^{{x_j}}}} \right)\prod\limits_{n = 1}^\infty {\left( {1 - {{\left( {\overline q } \right)}^n}{e^{ - {x_j}}}} \right)} } }}$

Crucially note that if ${p_1}{(M)_b} = 0$ , then the sigma-model representation reduces to:

$\sigma \left( {z;\overline \tau } \right) = {e^{{\eta _1}\,{z^2}}}\frac{{\sin \left( {\pi z} \right)}}{\pi }\prod\limits_1^\infty {\left( {1 - {{\left( {\overline q } \right)}^n}{e^{2\pi iz}}} \right)} {\prod\limits_1^\infty {\left( {1 - {{\left( {\overline q } \right)}^n}} \right)} ^{ - 2}}$

Now, and finally, deriving the Euler and Hirzebruch indices is obtained via the isomorphism between the exterior algebra and the tensor product of the spinor algebra. Let $\phi$ be a spinor and $\widetilde \phi$ a dual spinor, then $\phi \widetilde \phi$ is a bispinor that is decomposable into complete series of gamma matrices and the anti-symmetrization:

$\phi \widetilde \phi \propto {\rm{Tr}}\left( {\widetilde \phi \,\phi } \right) + \sum {{\gamma ^\alpha }} {\rm{Tr}}\left( {\widetilde \phi \,{\gamma ^\mu }\phi } \right) + \sum\limits_{\mu \, < \nu } {{\gamma ^\mu }{\gamma ^\nu }} {\rm{Tr}}\left( {\widetilde {\phi \,}{\gamma ^\mu }{\gamma ^\nu }\phi } \right) + ...$

This is the classic duality between spinors and exterior algebras obtained by the association of:

$d{x^1} \wedge d{x^2} \wedge ... \wedge d{x^k}$

with the anti-symmetric products of the gamma matrices. Thus, we can now introduce a right-moving fermionic Ramond superfield that transforms as a co-vector:

${N_\mu } = {\eta _\mu } + {\theta _{{\varphi _\mu }}}$

and so the above-mentioned Euler characteristic compares forms of even degrees with odd ones and hence can be gotten as:

$\chi = {\rm{Tr}}{\left( { - 1} \right)^{{F_L} + {F_R}}}$

while the Hirzebruch signature self-dual forms with anti-self-dual ones (in Dirac-language). The duality transformation is implemented via a left-multiplication by ${\gamma _5}$ . The associated index is $\tau = {\rm{Tr}}{\left( { - 1} \right)^{{F_R}}}$ . So, we have to derive the path integral with periodic temporal boundary conditions for $\psi$ and $\eta$ in the Euler context, and periodic $\psi '$ and anti-symmetric $\eta$ in the Hirzebruch signature context. Therefore, one gets:

$\left\{ {\begin{array}{*{20}{c}}{{I_\chi } = {\rm{Tr}}{{\left( { - 1} \right)}^{{F_L} + \,{F_R}}}{q^{{L_0} + \varepsilon }}{{\left( {\overline q } \right)}^{{{\overline L }_0} + \,\overline \varepsilon }}}\\{{I_\tau } = {\rm{Tr}}{{\left( { - 1} \right)}^{{F_R}}}{q^{{L_0} + \,\varepsilon }}{{\left( {\overline q } \right)}^{{{\overline L }_0} + \,\overline \varepsilon }}}\end{array}} \right.$

Since ${\overline L _0} + \overline \varepsilon$ can be expressed as the square of a SuSy signature, only states satisfying ${\overline L _0} + \overline \varepsilon = 0$ can contribute to the trace and hence, the zero modes must be factored in ${I_\chi }$ . Now, by functional integration, one gets:

$\begin{array}{c}{I_\chi } = \frac{1}{{{{\left( {2\pi } \right)}^{d/2}}}}\int {{d^d}} {X_0}\,{d^d}{\psi _0}{\left[ {{\mathop{\rm de}\nolimits} t'\left( { - \,\overleftarrow {\not \partial } \,\overrightarrow {\not \partial } + \not R\overline {\not \partial } } \right)} \right]^{ - 1/2}}\\ \times {\left[ {{\mathop{\rm de}\nolimits} t'\left( {\overline {\not \partial } } \right)} \right]^{1/2}} \cdot {\left[ {{\mathop{\rm de}\nolimits} t'\left( { - \not \partial - \not R} \right)} \right]^{1/2}}\end{array}$

Now, by Witten indexicalization, we can derive:

${I_\chi } = \frac{1}{{2{\pi ^{d/2}}}}\int {{d^d}} {X_0}\,{d^d}{\psi _0}{\left[ {{\mathop{\rm de}\nolimits} t'\left( { - \not R} \right)} \right]^{1/2}}$

which, as promised contains only contributions from zero modes. Integrating over the fermionic zero modes ${\psi _0}$ gives us the Euler characteristic of $L({R_{ws}})$ , which, by supersymmetry, is identical to the Euler number of ${R_{ws}}$ itself:

${I_\chi } = \chi \left( {{R_{ws}}} \right) = \int\limits_{{R_{ws}}} {\prod\limits_j {\left( { - {x_j}} \right)} }$

Now, via elliptical functional analysis, and defining ${z_j} \equiv {z_i}{\tau _2}{x_j}$ , we get:

$\begin{array}{c}\left[ {\det {{\left[ {\overline {\not \partial } + \not R} \right]}^{1/2}}} \right] = \prod\limits_j {\prod\limits_\Lambda {\left( {n\overline {\,\tau } + m\frac{1}{2} - 2i\,{{\overline \tau }_2}{x_j}} \right)} } \\ = \left[ {\sqrt {2{\eta _ + }} {{\left( {\overline q } \right)}^d}} \right]\prod\limits_j {\exp \left( {\frac{1}{2}{e^{{z_j}}}} \right)} \,{\sigma _1}({z_j})\end{array}$

By multiplicatively adding the term:

$\exp \left\{ { - \sum\limits_j {\sum\limits_\Lambda {\left[ {\frac{{{z_j}}}{{n\overline {\,\tau } + m - 1/2}}} \right] + \frac{1}{2}{{\left( {\frac{{{z_j}}}{{n\,\overline \tau + m - 1/2}}} \right)}^2}} } } \right\}$

with ${\sigma _1}$ being the elliptical generalization of the cosine function defined via:

${\sigma _1}(z) = \exp \left( { - {\eta _1}{z_j}} \right) \cdot \sigma {\left( {z + \frac{1}{2}/\sigma \left( {\frac{1}{2}} \right)} \right)^2}$

And finally getting to the Atiyah-Bott fixed point theorem: using it allows us to derive the Hirzebruch signature of $L({R_{ws}})$ :

${I_\tau } = \int\limits_{{R_{ws}}} {\left( {\prod\limits_j {2\frac{{i{x_j}}}{{2\pi }}\frac{{{\sigma _1}\left( {i{x_j}/2{\pi _j}\overline \tau } \right)}}{{\sigma \left( {i{x_j}/2{\pi _j}\overline \tau } \right)}}} } \right)} \cdot {\left( {\frac{{\eta + \left( {\overline q } \right)}}{{{\eta _\_}\left( {\overline q } \right)}}} \right)^d}$

and for the sigma-model, we get:

${\tau _g} = \int\limits_{{N_b}} {\prod\limits_j {\frac{{{x_j}}}{{\tan \left( {i{x_j}/2\pi } \right)}}} } \prod\limits_r {\cot \left( {{{\overline \theta }_r}/2} \right)}$

Hence, we have used 2-D $N = 1/2$ supersymmetry to get the topology of $L({R_{ws}})$ , which, via the Dirac-Ramond index, yields a finite theory of quantum gravity.