DAILY PHYSICS

 Holonomy. Given a manifold, M, and a point p on M. We append a vector,v  at p, and subject it to parallel transport around a loop. The number of times we revolve around said loop to match the original orientation of v at p is the holonomy. Take a loop over $I=[0,1]  i.e f(0) = f(1)$ We can write the group of parallel transport maps around a loop  $Hol_p(g)$. Naturally $Hol_p(g) \rightarrow GL(T_p M)$

-Recall a manifold -Construct a Kahler manifold: complex manifold with Reimann metric -Force Ricci flat -make compact -Two topologically distinct Calabi Yau can have the same stringy physics  TO FORCE ME TO SHOW THE MATH LEAVE A COMMENT OR CALL ME 818-310-5698.  Actually, maybe read this article by Brian Green first: https://arxiv.org/pdf/hep-th/9702155.pdf

 Chiral anomaly: Current not conserved!

 Consider: $X(\tau,\sigma + 2 \pi) = X(\tau,\sigma) + m(2\pi R)$ Set  $X(\tau,\sigma + 2\pi) = X(\tau, \sigma) + 2\pi \alpha$ Follow through as usual.  - Do Fourier mode expansion -Compute the momentum, and write the Hamiltonian -Use creation and annihilation operators to compute the spectrum At the level of the spectrum: "the closed string spectrum for a compactification with radius R is identical to the closed string spectrum for a compactification with radius $\bar{R} = \frac{\alpha}{R}$."

 No time to write trivial mathematical details. Might update this with details in the future.  Recipe: Consider first the Free boson first, apply a similar technique to Majorana, then get Dirac directly     - Recall:            - Two point function  equations:  $G^{2} = \langle \phi(x) \phi(y) \rangle$ $ -g \partial_x^2 G^{(2)} (x,y) = \delta( x-y)$ $G^{2} = \langle \phi(x) \phi(y) \rangle =  \frac{1}{4 pi g} ln^2( x-y)$ The energy-momentum tensor can be computed: $T = g(\partial_\mu \phi \partial_nu \phi - \frac{1}{2} \eta_{\mu \nu} \partial_\rho \phi \partial^\rho \phi )$ OPE is calculated as $T(z)T(w) = (2 \pi  g)^2:\partial \phi(z) \partial \phi(z)::\partial \phi(w) \partial \phi(w):$ Read the central charge from the general form! $T(z)T(w) = \frac{c/2}{ (z-w)^4} + \frac{2}{(z-w)^2} T(w) + \frac{\partial  T(w)}{z-w} +  . . $ Read the conformal dimension! $T(z) \Phi(w, \bar(w)) =  (\frac{h}{(z-w)^2} + \frac{\partial}{z -w}) \phi(w,\bar(w))$ For free fermion - Write Majorana action -

 -- Guide: -Couple field EM to exotic scalar field -Compute -Call this dark matter --Mathematical details are obvious

 Take graviton apply susy transform get spin 3/2 fermion. Write Rarita -Schwinger lagrangian density: $\mathcal{L} = \bar{\psi}^{\mu \nu \rho} \partial_\mu \psi_\rho$ Play the game: This is the gravitino