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4

I guess you have to read this in the context of intersection theory. On a manifold of dimension $2n$ we have the $n$-th homology group, which can informally be thought of as being generated by equivalence classes of $n$-dimensional submanifolds. Intersecting two of these should generally give a discrete set of points. Intuitively the intersection product can ...


2

You can either vary the action directly, or apply the classical field theory Euler-Lagrange equations. The latter for a Lagrangian $\mathcal{L}(\phi^{\alpha}, \partial_{\mu}\phi^{\alpha})$ read $$\frac{\partial \mathcal{L}}{\partial \phi^{\alpha}} - \partial_{\mu}\Big(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi^{\alpha})}\Big) = 0.$$ (Note that ...


2

Let's do this explicitly for both cases. For these examples, the classical formula for the geodesic curvature $k_g$ suffices. Let $\gamma(t)$ be a curve in a surface $S \subset \mathbb{R}^3$, and let $n(t)$ be the unit normal to $S$ at the point $\gamma(t)$. Then $$ k_g = \frac{\ddot{\gamma}(t) .(n(t) \times \dot{\gamma}(t))}{|\dot{\gamma}(t)|^3} $$ First ...


2

I know some derivations in which one can track the emergence of the concrete value, without having to buy that the second order contribution in the Euler-MacLaurin formula (see other answer) is $-\frac{1}{2!}$ times the second Bernoulli number $B_2$. The limit $\lim_{z\to 1}$ of the sum $0+1\,z^1+2\,z^2+3\,z^3+\dots$ diverges, because of the pole in ...


1

What equation do we use to measure the energy level of a string, to determine it's “particle correlation” We have already measured the particles. We have studied their properties and "measured" their quantum numbers as expressed in this table Measured within ( using the tools of) the standard theory of quantum mechanics and special relativity. ...


1

For any particle, we can define a continuous quantum number - its momentum $k^\mu$ and a discrete internal quantum number - for instance, its spin or charge under some symmetry group (note spin is also the charge under Lorentz transformations). The continuous quantum number defines the mass of the particle via $k^2 = -m^2$. In any actual theory, $m^2$ is ...


1

Equation (2.4.6): $T(z)X^\mu(0)\sim \frac{1}{z}\partial X^\mu(0)$ means that the RHS is the most singular term of the LHS. $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} \partial X_{\mu}:\tag{2.4.4}$ So \begin{align*} T(z)X^{\mu}(0) & =-\frac{1}{\alpha'}:\partial X^{\nu}(z)\partial X_{\nu}(z):X^{\mu}(0)\\ & =-\frac{2:\partial ...


1

You don't need to use the metric of the hemisphere. This is because the pullback of arbitrary forms onto the submanifold is the trivial pullback operator. All you need to do is apply the projection operator. Therefore, the extrinsic curvature tensor is just $K_{ab} = - \gamma_{a}{}^{c}\gamma_{b}{}^{d}\nabla_{c}n_{d}$, where $\gamma_{ab}$ is the metric of ...


1

One answer to your question, which laws of physics should be reproduced in any string compactification, could be the following: If the Weak Gravity Conjecture, see http://arxiv.org/abs/hep-th/0601001, holds true, then gravity should come out as the weakest force in any string compactification. (Notice that there are - as far as I know - only arguments for ...


1

Comments to the question (v3): The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, ...



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