Witten Index, 4-D N = 1 Gauge Theories, and Riemann Surfaces

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

\tilde Z = \hat A\exp \int_{{\Sigma _g}} {\left[ {\hat BF + c\wp {\eta ^\dagger } \wedge \eta } \right]}

is topological on {\Sigma _g}

Quantum Foam, Spacetime and Non-Linear Multigravity Theory

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

{S_{tot}} = \sum\limits_{i = 1}^N {S\left[ {{g_i}} \right]} + \lambda {S_{{\mathop{\rm int}} }}\left( {{g_1},...,{g_N}} \right)

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

Frobenius Structures, the Total Descendent Potential and Gromov – Witten Theory

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

{\bar \psi _i}: = {\pi ^ * }\left( {{\psi _i}} \right)

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

\pi :{X_{g,\,m + l,\,d}} \to {\bar M_{g,\,m + l,\,d}} \to {\bar M_{g,\,m}}