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:

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


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

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:

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


The Witten Index and Integral Measure Analysis

In my last post, I studied some aspects of Witten Index. Here, I shall probe ways of measuring the WI via integral analysis, as it plays a central role in supersymmetric Yang-Mills quantum theories with saturated and maximal SuSy. In the context of quantum mechanics, to get the path integral formulation from the operator formulation, one discretizes the temporal direction then injects a complete set at each of the time slice:

with ‘a’ being the lattice spacing. So for a bosonic dimensionless lattice field {\phi ^{{\rm{lat}}}}, the path integral measure is

    \[{\int {D\,\phi } ^{{\rm{lat}}}}\prod\limits_{k = 0}^{N - 1} {\left[ {\left( {\frac{1}{{2\pi }}} \right)\int_{ - \infty }^\infty {d\phi \,_k^{{\rm{lat}}}} } \right]} \]

and in the fermionic variables case,

    \[\int {D{\psi ^ * }} D\psi \prod\limits_{k = 0}^{N - 1} {\int {d{\psi ^ * }} } d{\psi _k}\]

which will allow us to derive and determine the natural-measure for the path integral