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The first one involves susy QM on [itex] \mathbb{C}^2 [/itex]. He says the following:

"Let us first consider the quantum mechanics of a supersymmetric particle on [itex] \mathbb{C}^2 [/itex], parametrized by [itex] (z,w) [/itex]. Let the supersymmetry be such that z and w are invariant and [itex] (\bar{z},\psi_{\bar{z}}) [/itex] and [itex] (\bar{w}, \psi_{\bar{w}})[/itex] are paired. This system has global symmetries [itex]J_1 [/itex] and [itex] J_2 [/itex] such that [itex] (J_1,J_2) = (1,0) [/itex] for z and [itex](J_1,J_2) = (0,1) [/itex] for w.

Let us consider its supersymmetric partition function

[tex] Z ( \beta; \epsilon_1, \epsilon_2) = tr_{H} (-1)^F e^{\beta \epsilon_1 J_1 } e^{\beta \epsilon_2 J_2} [/tex]

where H is the total Hilbert space. "He then proceeds to evaluate this. I have further questions but for now, I am trying to understand these two global symmetries. They are just assumed, guessed or they follow from the theory? And what is the meaning of the epsilons? Usually, in a partition function we sum over the exponential of beta times the energy of each state, so here, it looks as if the states are labelled by the values of J1 and J2, with corresponding energies epsilon1 and epsilon2. But what does that mean, exactly? Why are global symmetries showing up in this way in the partition function?

Later on, he adds that the contribution from the the pairs [itex] (\bar{z},\psi_{\bar{z}} [/itex] and [itex] (\bar{w}, \psi_{\bar{w}})[/itex]cancel out in the sum over states because they are susy partners. I can buy that. But then he adds that this leaves the sum over the supersymmetric states

[tex] H_{susy} \simeq \sum_{m,n \leq 0} \mathbb{C} z^m z^n [/tex]

How is it that these are "states"?

I hope someone can give a hand. Thanks in advance.