In this post, I will discuss the Witten Index in the context of 4-D N = 1 gauge theories on ${\Sigma _g} \times {T^2}$, a Riemann surface $\Sigma$ of genus g times a torus ${T^2}$ where in 2-D, the WI is

and by Jeffrey-Kirwan path-integration, we obtain

with ${Z_{cl,1l}}$ the classical and 1-loop contribution derived from the index of a topological twist on ${\Sigma _g} \times {T^2}$ via the partition function

and crucially noting that the effective action

is topological on ${\Sigma _g}$

Credit-address for the header photo. In this post, I will discuss and use non-linear multigravity theory to model quantum foam and probe solutions to the cosmological constant cosmic/Planck-scales 'discrepancy paradox', related to the hierarchy problem: namely, the 10-47GeV 4/EZP E ≈ 1071GeV cut-off one.. Note first that spacetime/quantum randomness-foamy-chaos can be interpreted as a large N composition of Schwarzschild wormholes with a scalar curvature $R$ in n-dimensions being

with $\Sigma$ the Regge-Wheeler hypersurface and let me begin with the action involving N massless gravitons without matter fields

with ${\Lambda _i}$ and ${G_i}$ being the cosmological constant and the Newton constant corresponding to the i-th universe, respectively, and the total action takes the following form

In this way, the action ${S_0}$ describes a Bose-Einstein

In this post, continuing from part one on Gromov-Witten invariants, I shall derive two deep properties, the second being central about the genus 1 Gromov-Witten potential, and along the way, discuss some propositions regarding the total ancestor potential. Let me introduce the total ancestor potential

where the genus g ancestor potential $\not \tilde F_t^g$ is defined by

with

referring to the pull-backs of the classes ${\psi _i}$, i = 1, ..., m, from ${\bar M_{g,m}}$ relative to the composition