The situation I am describing is a $10D$ heterotic string theory which is compactified on a Calabi-Yau to get a $N=1$, $4D$ effective theory. It is mentioned in Ashoke Sen's notes on string compactification that the difference between the number of complex structure moduli and the Kahler moduli, ...
isn't there are factor of two in the Euler characteristic, or this is a convention (or are "complex structure moduli" and "Kahler moduli" different from the Hodge numbers)?
@NiharKarve not sure off the top of my head, but from the general form of the argument the number of generations would just be $\lvert h^2- h^1\vert$ if that can happen
@ACuriousMind AFAIR the only restrictions on the moduli is that $h^{1,1}\ge1$ and $h^{2,1}\ge0$. Do you think $n_G = \lvert h^{2,1}- h^{1,1}\vert$ then has something to do with mirror symmetry?
@NiharKarve No, it's just that a negative number for $h^2 - h^1$ means that the surviving effective generations come from the "anti-generations", not the "generations"
given any case where the difference is non-zero, you could always swap to the case where it has the opposite sign simply by renaming what you call the generation and what the "anti-"generation, right?
If $g$ is a symmetry of the Hamiltonian $H$ and $\psi$ is an eigenstate, then $g\psi$ must be an eigenstate as well. I know this to be true, but is it formalized in some famous theorem?
show that (i) the necessary and sufficient condition for the vector function $$ \mathbf u (t) = u1(t)i + u2(t)j + u3(t)k$$ to be a constant is $d\mathbf{u}/dt=0$
how do we even show this?
Isnt this just.....trivial? the definition of a derivative?
I have a function $x(t)=Ae^{-\gamma t/2}e^{i\omega_1 t}=f(t)e^{i\omega_1 t}$ for $t\geq 0$ and $0$ otherwise. I'm looking for the Fourier transform (FT) of this function, that is $X\left(\omega\right) = \int_{-\infty}^{\infty}{x\left(t\right)e^{-i\omega t}dt}$. I have not studied a lot of FT. Any ideas on how to tackle this? I've been looking at 103 here (en.m.wikipedia.org/wiki/…).
@ACuriousMind Actually, I'm calculating a power spectrum, so $S\left(\omega\right) = \left|\int_{-\infty}^{\infty}{x\left(t\right)e^{-i\omega t}dt}\right|^2$. Do you know what is meant by the width of $S$ denoted $\Delta \omega$?
I apologize if I am delusional and imagined this, but I believe that in a not-so-distant past it was possible to display questions sorted by date so as to get the most recent ones on top of the list. And to achieve this, I selected "new" as opposed to "active" after clicking on the hyperlink "active