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The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime). Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius ...


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In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2-dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higher-dimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is ...


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When the Hamiltonian of theory is constructed out of creation and annihilation operators the S-matrix automatically satisfies the cluster decomposition principle given that in momentum space the coefficient of the interaction contains only one delta function. However this does not apply to the (first quantized) string theory because there even though the ...


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The motivation for the holographic principle is independent of string theory. It is that the entropy of a black hole increases with its surface area rather than with its volume. So thermodynamically, a quantum theory with black holes in it, behaves like an ordinary quantum theory with one less dimension of space. This became the idea that a theory of quantum ...


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There is a very direct relationship which answers your question, and I'll state it in the way I first learned about it (but you can derive a different connection by passing between dimensions): The 2-dimensional reduction of the Seiberg-Witten equations are the (abelian) vortex equations. The $SU(2)$-vortex equations on $\mathbb{R}^2$ are a ...


2

The "singleton" and "doubleton" language comes from the oscillator method of finding group representations. Given a (non-compact) group $G$ admitting lowest weight representation with maximal compact subgroup of the form $H\times\mathrm{U}(1)$ for some other compact group $H$, the oscillator method is to describe (unitary irreducible) representations by ...


1

It comes from S matrix theory, long before quarks were imagined, S,T and U characterize the type of exchange in the Feynman diagrams entering the S matrix calculation, and they are called Mandelstam variables. s channel-------------------------- t channel------------------------u channel duality meant that the sums could be done either in S ...


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The name T-duality stands for Target-space duality, see e.g. this preprint.


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Let $\Omega^2=\exp\phi/2$ so that $\gamma=\Omega^2 g$. Then we have the standard formula $$R_\gamma=\Omega^{-2}\big(R_g-2(n-1)\Delta\ln\Omega-(n-2)(n-1)[\nabla\ln\Omega]^2\big)$$ This is proven in any number of General Relativity texts, but the one in R.M. Wald, General Relativity (1984), is particularly easy to follow and is proved in the form shown here. ...


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Using the Dolbeault bigrading, the (2,0) and (0,2) components of the Kähler metric $g_{zz}=0$ and $g_{\bar{z}\bar{z}}=0$ do indeed vanish, respectively. In particular, the formula $$g_{z\bar{z}}=\partial_{z}\partial_{\bar{z}}K$$ for the mixed (1,1) components does not generalize to the (2,0) and (0,2) components.


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The electric field, rather than the associated (string) charge enters into the action. This means that the action is not the Hamiltonian, and does not give you the potential energy for a static configuration. The electric field is like a velocity $\dot q$. To get the Hamiltonian, you have to consider the Legendre transform of the Lagrangian $$ H = p \dot ...



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