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2

The operator $\delta(\gamma(z))$ is meant to be the operator dual to the state $|0\rangle_{NS}$ according to the state-operator correspondence. One could call it $E(z)$ or anything like that. But Polchinski uses the notation $\delta(\gamma(z))$ with this "nested" structure because the operator described in the previous paragraph may also be interpreted as ...


0

I would expect a theory of open strings, with Chan-Paton charges at the ends, to be interpreted very much like a theory of preons. It could be claimed that the points at the end of the string are only "mathematical preons".


1

It can be rather involved. A lot of technical progress as been on this subject leading up to the modern conformal bootstrap work. Something you can exploit is that these functions should behave like correlation functions and thus are eigenfunctions of the conformal Casimir. That gives you differential equations which in some cases, especially in $D=2$ and ...


5

Applying $\partial_1\overline{\partial}_1$, we have the following. First term:$$\partial_1\overline\partial_1 G_1 = -\pi\alpha' \eta^{\mu_1\mu_2} \delta^2 (z_1-z_2,\overline z_1-\overline z_2) X^{\mu_3} (z_3,\overline z_3) X^{\mu_4} (z_4,\overline z_4) $$$$+\text{ } {\rm permutations~of~indices~} (2,3,4).$$In the second term, $z_1$ can be the logarithm or ...


0

I just want to add to Lumo's answer: The paper by vafa and Strominger instigated a lot of work in determining the statistical formulation of entropy in black holes. Although it must be pointed out that most of these are for cases with supersymmtry and (near) extremal conditions at small couplings. There has also been work in trying to address the microscopic ...


0

I would recommend S.T. Yau's book on Mathematical Aspects of String Theory, following @Tomas Smith. There is also a two volume set based on lectures given at Princeton. The books can be found on Amazon at http://www.amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820125 and ...


0

Type II string theory also contains open strings. The statement on Wikipedia is misleading. The GSO projection removes part of the open string spectrum. Most notably, it removes tachyonic modes from the theory, making it stable. You can read this up in detail in Polchinski, Volume II, Chapter 10.


1

A priori, it is hard to know without having any experience with dimensional reductions. One has to get a feeling for how certain quantities change under the process, e.g. how components of the higher-dimensional gauge fields may turn into adjoint scalars, how spinors behave. An interesting thing to note is that symmetries of the lower-dimensional theory ...


0

This is a very profound question in physics. Given that a black hole has an entropy which scales as $$S_{BH} \sim \frac{A}{4}, $$ the question is how does this relate to $S_{Boltzmann} = K_B \ln W$. As in, what are the microstates of the theory which hold the information in the black hole. This was answered in part by a series of papers by Vafa, Strominger, ...


2

The limit $\alpha'\rightarrow 0$ is understood in terms of an expansion in $\alpha'$, where the leading order term is given by supergravity. This does not mean that one cannot have solutions with curved spacetime. As $\alpha'=\ell_s^2$, where $\ell_s$ is the string length, this limit tells us what happens if we remove "stringiness", and as it turns out, we ...


1

Some years ago, Gerard 't Hooft posted "How to Become a Good Theoretical Physicist", which is more inclusive than just string theory but which you'll probably still find a valuable list. Here's what he recommends for mathematics: "Primary Mathematics": Natural numbers: 1, 2, 3, … Integers: …, -3, -2, -1, 0, 1, 2, … Rational numbers (fractions): ...


0

It really depends on what you want to research within string theory, but it's one of most mathematically intensive areas within physics. List a mathematical discipline, and chances are you can apply it within string theory. At a bare minimum, you'll need everything through quantum field theory and general relativity, which includes calculus of variations, ...


1

As Wikipedia explains, $E_7$ refers to several, closely related real and complex Lie groups and Lie algebras. All the various $E_7$ Lie groups (algebras) are Lie subgroups (subalgebras) of the complex Lie group $E_7$ (algebra $e_7$), respectively. The latter has complex dimension $133$ and rank $7$. Specifically, $E_{7(7)}\equiv E_{7(+7)}\equiv E_{7,7}$ ...


0

The solution to this problem, which Szabo gives in the appendix, is somewhat misleading since it invokes the regularization of the $\zeta$ function, whereas the central term arises from ordinary sums as follows. For a start, we can re-write the Virasoro operators by making the normal ordering explicit: $$ L_n=\frac{1}{2}:\!\left( \sum_{k\in\mathbb ...


1

Firstly, string theory is a mathematical hypothesis that is currently speculative. It is a possible candidate for a quantum theory of gravity - a unification of QM with Einstein's theory of gravity - General Relativity. The idea of superstring theory emerged from another theory called supergravity. Supergravity was an attempt at a supersymmetric theory of ...


1

Higher dimensional dynamical objects in string theory exist, and are called branes. They arise, for example, as the objects the endpoints of open strings lie on.


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I'm not sure anyone has tried (except possibly Smolin in some of his older papers). But it wouldn't be that hard: take a simple loop-quantum-gravity spin-foam-analog model in 25+1 dimensions (which is going to be rather more complex than the usual 3+1 dimensional spin foam), pick a ground-state-like solution for it that looks something like an extended ...


2

This seems to be a simple matter of confusion regarding which variables are held constant. His notation $$\frac{\partial X}{\partial\tau}(\tau,\sigma_*)=0$$ is misleading. What he really means is $$\frac{\partial}{\partial\tau}\left(X(\sigma_*)\right)(\tau)=0$$ In other words, we fix $\sigma$ to be one of the end points and look at how it changes with ...


1

I) The action principle of a theory is the usually taken as the first principle of a theory, and therefore it can strictly speaking not be derived. Nevertheless, the Nambu-Goto action is a natural a generalization of the following line of thought: In a Riemannian space $(M,g)$ [with Euclidean signature], a geodesics are (locally) the shortest path between ...


-2

I think physically it is very clear. If the string endpoint ($\sigma=\sigma_1$) is fixed, the variations must vanish there i.e. $\delta X(\tau,\sigma_1)=0$.


0

We are assuming that this is the case. In general, one can think that every Lagrangian corresponds to some equations of motion, so the NG action corresponds to some sort of motion of a string. The powerful justification for this action is that it is manifestly Lorentz covariant, and one can show that the equations of motion that it leads to are those one ...


1

The action principle holds by assumption. It is assumed that all equations of motion follow from this principle with the appropriate action. By introducing an auxiliary tensor field $h_{\alpha\beta}$, one may write down the so-called Polyakov action $$S_\mathrm{Poly}=-\frac{T}{2}\int_\Sigma ...


1

Details have to be filled in, but I think the general idea went along these lines: the variation of the action with respect to the metric $g_{\mu\nu}$ is given by $$ \delta_gS \sim \int T^{\mu\nu}\delta g_{\mu\nu}\,. $$ Now specialize to particular variations of $g_{\mu\nu}$, the diffeomorphisms. For an infinitesimal diffeomorphism of the form $x^\mu\to ...


0

A key point to make is that people think that a solution to Einstein's equations is a particular space-time geometry, a particular background geometry, when really a solution is an equivalence class of distinct geometries related to each other through (what mathematicians call) diffeomorphisms. And if you want to talk about observables and hence physics, you ...


0

Maxwell's theory in Minkowski spacetime is background-dependent because the Minkowski metric - a fixed geometric structure - is PART of the FORMULATION of the theory. The Minkowski metric appears in the action principle for example. General relativity is profoundly different because there is no fixed background geometric structure in the Einstein-Hilbert ...


15

Neutrinos are weakly interacting quantum mechanical point particles, with very small mass. Refraction is a classical mechanics phenomenon, happens to waves traveling in a medium and it is a collective synergy of many photons impinging on the field of the atoms and molecules of the medium. Individual photons are not refracted but are scattered. In synergy ...


0

When people talk about $\mathcal{N}=2$ QED in 4d I think they normally mean a $U(1)$ gauge theory (one $\mathcal{N}=2$ vector multiplet) coupled to one or more hypermultiplets (usually all with the same $U(1)$ charge). As an example of this usage see Witten's discussion of $\mathcal{N}=4$ QED in 3d (which can be obtained by dimensional reduction from the ...



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