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4

I) In this answer we will consider the standard Nambu-Goto string and show that the Hessian has co-rank 2. The target space metric has $(-,+,\ldots,+)$ sign convention, and $c=1=\hbar$. The Nambu-Goto Lagrangian density is $${\cal L}_{NG}~:=~-T_0\sqrt{{\cal L}_{(1)}}, $$ $$ {\cal L}_{(1)}~:=~-\det\left(\partial_{\alpha} X\cdot \partial_{\beta} ...


3

I) In this alternative answer we resolve the singular Hessian $H_{\mu\nu}$ of the Nambu-Goto string action by introducing two auxiliary variables from the onset, thereby indirectly showing that the Hessian $H_{\mu\nu}$ must have co-rank 2. The target space metric has $(-,+,\ldots,+)$ sign convention, and $c=1=\hbar$. Consider the extended Nambu-Goto ...


3

There are obviously differeing genus types according to which partition function in d-dimensional QFT. At the very outset of $0$-d QFT the index is the push forward in ordinary de Rahm Cohomology; in other words, the integration of differential forms. The genus as you put it, is non-existant here. In $1$-d QFT the index of the Dirac Operator is ...


2

Confinement cannot be rigorously shown in QCD with current techniques, because all analytic results in QCD are perturbative and the perturbative expansion breaks down at low energies where the coupling becomes strong. QCD has a negative $\beta$-function, i.e. the Yang-Mills coupling grows at lower energies and becomes weaker at high energies. But the ...


2

This extra factor arises from the analogy of the conformal factor $\alpha'\omega$ term in (6.2.16). The required $\omega$ is $$\omega = \ln \left ( \frac{2\pi}{\partial_\nu\vartheta_1}\right) $$ and substituting it to the exponential we get $$ \exp\left( -\frac{\alpha'}{2}\sum_ik_i^2 \cdot \ln \frac{2\pi}{\partial_\nu\vartheta_1} \right) = ...


2

Comments to the question (v2): To be specific, let us assume that the underlying 2D manifold is the Riemann sphere $S^2\cong \mathbb{C}\cup\{\infty\}$. The group of globally defined conformal transformations is the 6-dimensional group $PSL(2,\mathbb{C})$ of Moebius transformations. Mathematically speaking, one should consider the groupoid of locally ...


2

The variation $\delta F$ for any field (or degree of freedom) $F$, given an infinitesimal transformation, is always calculated as the commutator $$ \delta F = [ \bar\epsilon Q, F ] $$ where $\bar \epsilon$ is a parameter ("angle" or "shift" or some generalization) of the transformation and $Q$ is the generator. (Those may be replaced by other letters.) ...


1

Here is an outline of the reduction from the Nambu-Goto (NG) action to the light-cone (LC) formulation from a Hamiltonian perspective: The starting point is the Hamiltonian formulation of the Nambu-Goto string, cf. e.g. this Phys.SE post. The Hamiltonian density is of the form "Lagrange multipliers times constraints" $$ {\cal H}~=~\lambda^{\alpha} ...


1

I want to start by clarifying what M-theory is, in relation to string theory, just so the context of my answer will be understood. These days, "string theory" encompasses five ten-dimensional superstring theories, the 11-dimensional M-theory of membranes, 26-dimensional bosonic string theory, "supercritical strings" in more than 10 dimensions, and other ...


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Going to the conformal gauge is nothing but using coordinates in which the metric is diagonal (in euclidean space this is called isothermal coordinates ). Therefore in order to show that the Polyakov action is Weyl-invariant without using the conformal gauge, it is sufficient to show that the action does not dependent on the coordinate choice at all. ...



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