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A bosonic symmetry that acts differently on the different components of a supermultiplet is an R-symmetry. Such a symmetry does not commute with the supercharges. Since the commutator between an R-symmetry and a supercharge gives something Grassmann odd, it has to given another supercharge. Schematically $$ [R,Q] = Q $$ or in terms of variations acting on ...


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The ten-dimensional case is explained in some detail in appendix 4.A of volume one of Green-Schwarz-Witten. Let me therefore consider the 4d case here. The calculation works essentially the same in 3d, 6d and 10d, though you can sometimes make use of Majorana and/or Weyl conditions of the spinors to simplify things. We want to check that the expression ...


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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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Yes, you are correct. $\mathcal{N}=2$ supersymmetry in 3d can be obtained by dimensionally reducing 4d $\mathcal{N}=1$. The 4d chiral superfield contains a complex scalar, a Weyl fermion, and a complex auxiliary field. Reducing this to 3d we get a compex scalar, a Dirac fermion, and the auxiliary. You can also impose reality conditions to rewrite the 4d ...


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Hint: Think of way to write the Hamiltonian as $$\hat{H}~=~ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x)~=~\hat{B}^{\dagger}\hat{B} ~\geq~0 $$ for some first-order differential operator $$\hat{B}~=~a(x)\frac{d}{dx} +b(x) ,$$ with suitable functions $a(x)$ and $b(x)$. Here the potential $V(x)$ is given by formula (1). Can you see what the operator ...


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It is important to remember that Gravity is not a force. The search for gravitons or gravitinos is misguided. As explained by Einstein, Gravity is an affect caused by space being curved, where the curvature is a linear combination of the curvature caused by the sum of the energy density provided by each object in the Universe. Einstein's GR equations lead ...


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I figured it out. $\mathcal{V}^{0}$ is the vertex operator integrated over $\theta$. Integrating $\phi$ over $\theta$ gives us $\Psi$, and we can use equations (12.3.15) to rewrite $\Psi$ as $G_{-1/2}\cdot \mathcal{O}$. In order to get the $\tilde G_{-1/2}$ factor for the $\mathcal{V}^{0,0}$ operator we must expand $\phi$ further, $$ \phi = \mathcal{O} ...


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Weyl Invariance in 8d, Majorana invariance in 9D and no reductions are possible in 7D. So the minimum number of supercharges in 8,9D = $2^{8/2}/2$ complex components = 16 real supercharges. And in 7D that number is $2^{6/2}$ complex components = 16 real supercharges For further details, refer to the appendix of Polchinski vol 2


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One way to see the expression vanishes (maybe the one you already know how to do?) is to use the fact that there can not exist a 2*2*2 totally antisymmetric matrix. First, write everything using indices: $$\begin{align} A &= \chi_\alpha (\xi \eta) + \xi_\alpha (\eta \chi) + \eta_\alpha (\chi \xi) \\ &= \chi_\alpha ...



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