String Theory, the Witten Index and the Seiberg-Lebesgue Problem

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:

    \[S_p^D = - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{{D_{\mu \nu }}L}}{{{\partial _{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

for contextualization, note that a necessary condition for the world-sheet Dirac propagator {\delta ^{\left( 2 \right)}}\left( {{\sigma _i} - {\sigma _j}} \right):

    \[S = i\int {{d^2}} {\sigma _1}{d^2}{\sigma _2}\sum\limits_{i,j = + , - } {{\psi _i}\left( {{\sigma _1}} \right)} {A_{ij}}\left( {{\sigma _1},{\sigma _2}} \right)\psi \left( {{\sigma _2}} \right)\]

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

    \[{S_\eta } = \frac{1}{\beta }\sum\limits_{\frac{{i2\pi }}{\beta }} {{{\left( {i\frac{{2n + 1}}{\beta }} \right)}^\pi }} W + \alpha '{R_{\left( 2 \right)}}\Phi \]

with

    \[W \equiv {h^{mn}}{\partial _m}{X^a}{\partial _n}{X^b}{g_{ab}}\left( X \right)\]

and \beta the bosonic frequency, be analytically summable. The string world-sheet is given by:

    \[{S_{ws}} = \frac{1}{{4\pi \alpha '}}\int\limits_{c + o} {d\tilde \sigma } d\tau '\sqrt h \left( {W + \alpha '{R_{\left( 2 \right)}}\Phi } \right)\]

A major problem is that by the Heisenberg’s uncertainty principle:

    \[\left( {\Delta A/} \right)\left( {\left| {\frac{{d\left\langle A \right\rangle }}{{dt}}} \right|} \right)\left( {\Delta H} \right) \ge \hbar /2\]

the string time-parameter on the world sheet {\sigma _t} with Feynman propagator in Euclidean signature being:

    \[\begin{array}{c}G\left( {x,y} \right) = \int_0^\infty {d{\sigma _t}} G\left( {x,y;{\sigma _t}} \right)\\ = \int {\frac{{{d^D}p}}{{{{\left( {2\pi } \right)}^D}}}} \exp \left[ {ip \cdot \left( {y - x} \right)} \right]\frac{2}{{{p^2} + {m^2}}}\end{array}\]

violates the integrability condition for the action:

    \[S_p^D = - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{{D_{\mu \nu }}L}}{{{\partial _{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

and hence, in light of the principle of superposition{S_\eta }(n) as a function of n runs into the Riemann-Lebesgue Lemma problem, given that the Fourier transform of {S_\eta }(n):

    \[\mathop {{S_\eta }}\limits^ \sim (n) = \int_{ - \infty }^\infty {{e^{i\varphi t}}} {S_\eta }(n)\,dt\]

is non-convergent, with \varphi real, since the quantization of spacetime is an anti-smoothing dynamical breaking of the Ricci scalar {R_{\mu \nu }}. Hence we get,

    \[\int_{ - \infty }^\infty {\left| {{S_\eta }(n)} \right|} \,dt > \infty \]

which is incoherent. To see this, note that the anti-smoothing of spacetime implies that {S_\eta }(n) cannot be recovered from \mathop {{S_\eta }}\limits^ \sim (n) via

    \[{S_\eta }(n) = \left( {1/2\pi } \right)\int_{ - \infty }^\infty {{e^{i\varphi t}}} \mathop {{S_\eta }(n)\,d\varphi }\limits^ \sim \]

and that implies that the gravitonic wave-propagation travels in spacetime at infinite speed, given

    \[\int_{ - \infty }^\infty {\left| {{S_\eta }(n)} \right|} \,dt > \infty \]

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 {A_3} \equiv 1/3!{A_{3\mu \nu }}\,d{x^\mu }d{x^\nu }d{x^\rho }, with the field strength {F_4} \equiv d{A_3}, it is surprisingly:

    \[1.\quad {S_{11}} = \frac{1}{{2\kappa _{11}^2}}\int {\left[ {\sqrt G \left( {{R_G} - \frac{1}{2}{{\left| {{F_4}} \right|}^2}} \right) - \frac{1}{6}{A_3} \wedge {F_4} \wedge {F_4}} \right] + {\rm{fermions}}} \]

with \kappa 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 N = 11 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 A_4^ + with a self-dual field strength. Nonetheless, one may still derive an action that involves both dualities of {A_4}. Then, by imposing the self-duality as a supplementary equation, we get:

    \[\begin{array}{l}2.\quad {S_{IIB}} = + \frac{1}{{4\kappa _B^2}}\int {\sqrt G } {e^{ - 2\Phi }}\left( {2{R_G} + 8{\partial _\mu }\Phi - {{\left| {{H_3}} \right|}^2}} \right)\\ - \frac{1}{{4\kappa _B^2}}\int {\left[ {\sqrt G \left( {{{\left| {{F_1}} \right|}^2} + {{\left| {\mathop {{F_3}}\limits^ \sim } \right|}^2} + A_4^ + \wedge {H_3} \wedge {F_3}} \right)} \right]} + {\rm{ fermions}}\end{array}\]

with field strengths: {F_1} = dC, {H_3} = dB, {F_3} = d{A_2}, {F_5} = dA_4^ +, \mathop {{F_3}}\limits^ \sim = {F_3} - C{H_3}, \mathop {{F_5}}\limits^ \sim = {F_5} - \frac{1}{2}{A_2} \wedge {H_3} + \frac{1}{2}B \wedge {F_3} with the self-duality condition * \mathop {{F_5}}\limits^ \sim = \mathop {{F_5}}\limits^ \sim.

Note that the above action arises from the string low-energy limit and:

    \[{S_{IIB}} = + \frac{1}{{4\kappa _B^2}}\int {\sqrt G } {e^{ - 2\Phi }}\left( {2{R_G} + 8{\partial _\mu }\Phi {\partial ^\mu }\Phi - {{\left| {{H_3}} \right|}^2}} \right)\]

naturally yields the NS-NS sector of the theory, while:

    \[ - \frac{1}{{4\pi _B^2}}\int {\left[ {\sqrt G \left( {{{\left| {{F_1}} \right|}^2} + {{\left| {\mathop {{F_3}}\limits^ \sim } \right|}^2}} \right) + A_4^ + \wedge {H_3} \wedge {F_3}} \right]} + {\rm{ fermions}}\]

is derivable from the RR sector of the theory. Now, Type IIB supergravity theory is invariant under the non-compact symmetry group SU(1,1) \sim SL(2,\mathbb{R}) and the key is that this symmetry is not manifest in:

    \[\begin{array}{l}2.\quad {S_{IIB}} = + \frac{1}{{4\kappa _B^2}}\int {\sqrt G } {e^{ - 2\Phi }}\left( {2{R_G} + 8{\partial _\mu }\Phi - {{\left| {{H_3}} \right|}^2}} \right)\\ - \frac{1}{{4\kappa _B^2}}\int {\left[ {\sqrt G \left( {{{\left| {{F_1}} \right|}^2} + {{\left| {\mathop {{F_3}}\limits^ \sim } \right|}^2} + A_4^ + \wedge {H_3} \wedge {F_3}} \right)} \right]} + {\rm{ fermions}}\end{array}\]

To make it so, one must redefine fields, from the string metric {G_{\mu \nu }} in (2) to the Einstein metric {G_{E\mu \nu }}, along with a complexification of the tensor fields:

    \[3.\quad {G_{E\mu \nu }} \equiv {e^{ - \Phi /2}}{G_{\mu \nu }},{\rm{ }}\tau \equiv C + i{e^{ - \Phi }}{\rm{, }}G \equiv \left( {{F_3} - \tau {H_3}} \right)/\sqrt {{\mathop{\rm Im}\nolimits} \tau } \]

Now, the action is easily seen to be:

    \[\begin{array}{c}4.\quad {S_{IIB}} = \frac{1}{{4\kappa _B^2}}\int {\sqrt {{G_E}} } \left( {2{R_{{G_E}}} - \frac{{{\partial _\mu }\mathop \tau \limits^ \sim {\partial ^\mu }\tau }}{{{{\left( {{\rm{Im}}\tau } \right)}^2}}} - \frac{1}{2}{{\left| {{F_1}} \right|}^2} - {{\left| {{G_3}} \right|}^2} - \frac{1}{2}{{\left| {\mathop {{F_5}}\limits^ \sim } \right|}^2}} \right)\\ - \frac{1}{{4i\kappa _B^2}}\int {{A_4}} \wedge \mathop {{G_3}}\limits^ \sim \wedge {G_3}\end{array}\]

the metric and A_4^ + fields are invariant under the SU(1,1) \sim SL(2,\mathbb{R}) symmetry of Type IIB supergravity. The axionic dilaton field \tau varies with a Möbius super-transformation:

    \[5.\quad \tau \to \tau ' = \frac{{a\tau + b}}{{c\tau + d}}{\rm{ }}\quad ad - bc = 1\,{\rm{, }}\,a,\,b,\,c,\,d \in \mathbb{R}\]

and {A_{2\mu \nu }}, {B_{\mu \nu }} self-rotate under the Möbius super-transformation, and can most clearly be visualized as a complex 3-form field {G_3}:

    \[6.\quad {G_3} \to G_3^\dagger = \frac{{c{\tau ^\dagger } + d}}{{\left| {c\tau + d} \right|}}{G_3}\]

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 \lambda and the gravitino {\psi _M}:

    \[\begin{array}{c}7.\quad \delta \lambda = \frac{i}{{{\kappa _B}}}{\Gamma ^\mu }{\eta ^ * }\frac{{{\partial _\mu }\tau }}{{{\rm{Im}}\tau }} - \frac{i}{{24}}{\Gamma ^{\mu \nu \rho }}\eta {G_{3\mu \nu \rho }} + {\left( {{\rm{fermi}}} \right)^2}\\\delta {\psi _\mu } = \frac{1}{{{\kappa _B}}}{D_\mu }\eta + \frac{1}{{480}}{\Gamma ^{\mu 1...\mu 5}}{\Gamma _\mu }\eta {F_{5\mu 1...\mu 5}} + \frac{1}{{96}}\left( {\Gamma _\mu ^{\rho \sigma \tau }{G_{3\rho \sigma \tau }} - 9{\Gamma ^{\nu \rho }}} \right){\eta ^ * }\\ + \,{{\rm{(Fermi)}}^2}\end{array}\]

It is crucial to realize that in the SU(1,1) context, the SUSY transformation parameter \eta essentially has 1/2{\rm{ }}\;U(1) charge, implying that \lambda has necessarily, given unitarity, 3/2 and {\psi _\mu } has 1/2.

The geometry of superstring theory in the Ramond-Neveu-Schwarz setting is given by the bosonic world-sheet fields x_{WS}^\mu and the fermionic world-sheet fields \psi _ \pm ^\mu, with \pm expressing chirality, and x_{WS}^\mu, \psi _ \pm ^\mu must be functions of local world-sheet coordinates {\xi ^1}\,,{\rm{ }}\,{\xi ^2}. Both x_{WS}^\mu and \psi _ \pm ^\mu 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 x_{WS}^\mu naturally gives rise to a non-linear sigma model:

    \[8.\quad {S_{x_{WS}^\mu }} = \frac{1}{{4\pi {\alpha ^ * }}}\int_\Sigma {\sqrt \gamma } \left[ {\left\{ {{\gamma ^{mn}}{G_{\mu \nu }}(x_{WS}^\mu ) + {\varepsilon ^{mn}}{B_{\mu \nu }}(x_{WS}^\mu )} \right\}{\partial _m}{x^{\mu \dagger }}{\partial _n}{x^{\nu \dagger }} + {\alpha ^\dagger }R_\gamma ^{(2)}\Phi (x_{WS}^\mu )} \right]\]

with {\alpha ^\dagger } being the square root of the Planck length, {\gamma ^{mn}} being the world-sheet metric, and R_\gamma ^{(2)} being the Gaussian curvature. The world-sheet fermionic field \psi _ \pm ^\mu 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:

    \[9.\quad {S_\psi } = \frac{1}{{4\pi {\alpha ^\dagger }}}\int_\Sigma {{d^2}} \xi \left( {{G_{\mu \nu }}(x)\left( {\psi _ + ^\mu {D_{{\gamma _A}}}\psi _ + ^\nu + \psi _ - ^\mu {D_{{\gamma _A}}}\psi _ - ^\nu } \right) + \frac{1}{2}{R_{\mu \nu \rho \sigma }}\psi _ + ^\mu \psi _ + ^\nu \psi _ - ^\rho \psi _ - ^\sigma } \right)\]

and {R_{\mu \nu \rho \sigma }} being the Riemann tensor for the metric {G_{\mu \nu }}. Now the all too important SUSY-covariant derivatives can be derived:

    \[10.1\quad {}^s{D_{{\gamma _A}}}\psi _ + ^\mu = {\partial _{{\gamma _A}}}\psi _ + ^\mu + \left( {\Gamma _{\rho \sigma }^\mu (x) + \frac{1}{2}H{{_3^\mu }_{\rho \sigma }}(x)} \right){\partial _{{\gamma _A}}}{x^\rho }\psi _ + ^\sigma \]

    \[10.2\quad {D_{{\gamma _A}}}\psi _ - ^\mu = {\partial _{{\gamma _A}}}\psi _ - ^\mu + \left( {\Gamma _{\rho \sigma }^\mu (x) - \frac{1}{2}H{{_3^\mu }_{\rho \sigma }}(x)} \right){\partial _{{\gamma _A}}}{x^\rho }\psi _ - ^\sigma \]

with \Gamma _{\rho \sigma }^\mu the Levi-Civita connections for G. Now one is in a position to solve the Seiberg-Lebesgue problem via the functional integral over all x_{W{S_i}}^\mu and {\psi _ \pm } by integrating over all world-sheet metrics {\gamma ^{mn}} and world-sheet gravitini fields {\chi _m} via the amplitude:

    \[\begin{array}{c}{\rm{11}}.\quad {\rm{amplititude}}{\mkern 1mu} = \\\sum\limits_{{\rm{topologies/orbifolds}}} {\int {D{\gamma ^{mn}}} } D{\chi _m}\int D x_{WS}^\mu D\psi {e^{ - {S_x} + {S_\psi }}}\end{array}\]

This clearly solves the Feynman propagation problem. And the upshot is that the vacuum expectation value of the dilaton field is \phi = \left\langle \Phi \right\rangle, and the string vacuum expectation value of the string amplitude is given by the Euler number {\chi ^{(\Sigma )}} of the world-sheet R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}:

    \[12.\quad \frac{1}{{2\pi }}\int_{R_{{S_{({\sigma _s},{\sigma _t})}}}^{2d}} {\sqrt \gamma } {R^{(2)}}_{{\gamma _A}} = {\chi ^{(\Sigma )}} = 2 - 2h - b\]

with h the genus, b is a number that counts the orbifoidal puntures in spacetime as the string world-sheet propagates under quantum fluctuations. Thus, a genus h world-sheet with no boundary gets a multiplicative contribution: {e^{ - (2 - 2h)\phi }} = g_s^{2h - 2}, thus deriving another truly remarkable identity, {g_s} = {e^\phi }, representing the closed string coupling constant which lives on a Dp-brane, and hence by the supersymmetry action and the Witten Index:

    \[{\rm{Tr}}{\left( - \right)^F}{e^{ - \beta H}} = \int_{PBC} {\left[ {d\phi d\psi } \right]} \,{e^{ - {S_E}\left( {\phi ,\psi } \right)}}\]

where F 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.

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