Basically, I would really like it if somebody just explained to me what is going on here. Please use any physics lingo you feel is necessary, but explain what you mean. I am just having trouble finding a nice explanation without having to know tons of physics. ( I have a fifth grade physics level)

So the reference I have states that the symmetric stress-energy tensor is given by $T=a_1(dx)^2+2a_2dxdy+a_3dy^2$, which I understand as being a section of $S^2(T^{\ast}\mathbb{R}^2)$, the 2nd-symmetric power of the cotangent space to $\mathbb{R}^2$. Why does this have anything to do with energy or momentum?

If we change to $\mathbb{C}$ rather than $\mathbb{R}^2$, then $T=(a_1-a_3-2ia_2)\,dz^2+2(a_1+a_3) dz d\bar{z}$. Conformal invariance is implied by the condition $\mbox{tr}\,T=a_1+a_3=0$. (What does this sentence mean?)

Lastly, we have some kind of expansion (in a conformal field theory with central charge $c=0$, whatever that means): $$T(z)=\sum_{n\in\mathbb{Z}}L_n z^{-n}\, \big(\frac{dz}{z}\big)^2,$$ where I take it that the $L_n$ are the generators for the centerless Virasoro algebra. What type of object is $T(z)$? And how is obtained from $T$?.


I'm going to try to get at the crux of your questions without worrying too much about mathematical rigor/details (as is the physicist's way), but hopefully there are enough details so that the answer is clear.

Why does this have anything to do with energy or momentum?

First, a bit of background. In physics, a theory of fields $\phi$ on a manifold $M$ is often specified by an action $S$; a functional which maps a given field configuration $\phi$ to a number (often the target set of the action is either $\mathbb R$ or $\mathbb C$). For, concreteness, consider a field theory on $\mathbb R^d$. As it often turns out, the action of such a field theory is translation invariant. This means that if we define the action of the group of translations of $\mathbb R^d$ on the fields $\phi$ of the theory by $\phi\to\phi_\epsilon$ where $$ \phi_\epsilon(x) = \phi(x-\epsilon) $$ then $$ S[\phi] = S[\phi_\epsilon] $$ In such cases, a theorem in field theory called Noether's theorem guarantees the existence of a conserved tensor $T^{\mu\nu}$ associated with this invariance, namely one for which $$ \partial_\mu T^{\mu\nu} = 0 $$ This conserved tensor associated with translation invariance of the action is what we call the energy-momentum tensor, and this is essentially the tensor we're talking about in the context of conformal field theory.

So what the heck does this object having anything to do with energy and/or momentum? Well, we can motivate this physically through examples. If you take, as an example of a field theory, electromagnetism, then you find that the components $T^{\mu\nu}$ of the energy-momentum tensor physically represent quantities like the energy density stored in the fields. One finds, for example, that the $00$ component of the electromagnetic energy-momentum tensor has the expression $$ T^{00} = \frac{1}{8\pi}(\mathbf E^2 + \mathbf B^2) $$ which one can show, by other means, is precisely the physical energy density stored in the electromagnetic fields.

Conformal invariance is implied by the condition $\mathrm{tr} \,T=a_1+a_3=0$. (What does this sentence mean?)

One can show that under a coordinate transformation $x\to x+\epsilon(x)$, the action of a sufficiently generic field theory transforms as $$ S \to S+\frac{1}{2}\int d^dx \,T^{\mu\nu}(\partial_\mu\epsilon_\nu + \partial_\nu \epsilon_\mu) + O(\epsilon^2) $$ A conformal transformation has the property that $$ \partial_\mu\epsilon_\nu + \partial_\nu \epsilon_\mu = \frac{2}{d}\partial_\rho \epsilon^\rho \,\delta_{\mu\nu} $$ which gives $$ S \to S+ \frac{1}{d}\int d^dx\, T^\mu_{\phantom\mu\mu}\partial_\rho\epsilon^\rho + O(\epsilon^2) $$ Notice that the integrand contains the trace $T^\mu_{\phantom\mu\mu}$ of the energy-momentum tensor, and we see that if this trace vanishes, then the action has the property $$ S \to S+ O(\epsilon^2) $$ It is invariant to first order in $\epsilon$. This is a sort of "infinitesimal invariance" as a physicist might call it, and it is what the statement is referring to in this context.

What type of object is $T(z)$? And how is obtained from $T$?.

For a conformal field theory on $\mathbb R^2$, after going to complex coordinates $z,\bar z$ it is possible to show that $\partial_{\bar z} T_{zz}(z,\bar z) = 0$, so for the sake of notational compactness, one often writes $T_{zz}(z, \bar z) = T(z)$


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