# Supercharges commutation rules

• L0r3n20

#### L0r3n20

Hi all I'm new on this forum. I'm here since I'm working with n-extended susy and R-sym and I don't know how to calculate a commutator. First of all I introsuce my notation:

$$\mu_A T^A$$ is a potential cupled to R-sym generator

$$\mu_{\alpha i}$$ is a superpotential cupled to supercharges

$$(-1)^F = e^{2 \pi i J_z}$$ counts number of fermions

$$\{ Q_{\alpha}^i ; \bar{Q}_{\dot{\alpha}j} \} = 2 \sigma_{ \alpha \dot{\aplha}}^\mu P_\mu \delta^i_j$$

(I hope you see it right since on my pc I don't see curly bracket of anticommutator and just one alpha dot under sigma, but this is just the super algebra)
and spinorial indices are raised and lowered with

$$\varepsilon^{12}=\varepsilon_{21}=\varepsilon^{\dot{1}\dot{2}}=\varepsilon_{\dot{2}\dot{1}}=1$$

Now what I have to compute is

$$\left[(-1)^F ; e^{\mu_A T^A + \mu^{\alpha i} Q_{\alpha}^i + \bar{\mu}_{\dot{\alpha}j} \bar{Q}^{\dot{\alpha}j} } \right]$$

How can I do? Thanks for advices!

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Hello L0r3n20. I need this practice so I will try to help. But can you tell me why this commutator? Also, can you provide more information on the potential and superpotential?

Hi! I'm trying to calculate partition function of super-hydrodynamycs so I need to know wheter it depends on superpotential cupled to supercharges. I do know this is true in the case N=1 (See Kovtun and Yaffe http://arxiv.org/PS_cache/hep-th/pdf/0303/0303010v1.pdf) but I don't know if it is in general N.
Superpotential cupled to supercharges are fermionic objects used to make superalgebras work with usual commutator. The potential cupled to R-symmetry generator is bosonic since it does commute with everything. Last, but not least, I'm assuming no central charges. I hope I've been clear. Tell me if you need further information.

It sounds like it's connected to "elliptic flow" and other exciting topics!

You will probably solve your problem by yourself before I can solve it. I never did a super-calculation in my life before, that's why I have to attempt it; but I will need to think it through from fundamentals, and who knows how long that will take.

I do have another question, though. The mathematical expression you are computing does not look algebraically mysterious - it's just a commutator of two exponentiated operators. Naively, if you expand those as power series, you can reexpress your commutator as an infinite sum of commutators of terms from those series. There might be technicalities to do with operator ordering and spinor indices. My question is: Are you just wanting to check that you get the algebra technically right, or are there also larger conceptual issues to discuss?

My question is just about algebra involved in calculus. I think I can reexpress it in a power series using BCH forumula but I just don't know how does J_z commute with supercharges. I know how does $$e^{2 \pi i J_z}$$ act. Do you think it's possible to expand in a series and get the rule to commute J_z and Qs?

My question is just about algebra involved in calculus. I think I can reexpress it in a power series using BCH forumula but I just don't know how does J_z commute with supercharges. I know how does $$e^{2 \pi i J_z}$$ act. Do you think it's possible to expand in a series and get the rule to commute J_z and Qs?
See, http://www.pha.jhu.edu/~gbruhn/IntroSUSY.html [Broken] for the commutation rules. What surprises me is that you have to calculate the commutator between group elements, usually that doesn't make any sense. Or do you mean the group commutator: [A,B] = ABA^{-1}B^{-1} ?

Careful

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Sorry about the delay, I wasn't at home.

I think I didn't get your argument, I'll try to show you my entire problem.
What I'd like to show is that

$$Tr\left[(-1)^F e^{\mu_A T^A + \mu^{\alpha i} Q_{\alpha}^i + \bar{\mu}_{\dot{\alpha}j} \bar{Q}^{\dot{\alpha}j} } \right]$$

is $$\mu^\alpha_i$$ indipendent (same for barred). I know this is true if N=1 (see article cited before) I'm just trying to generalize.
I thought to derivate and then using cyclicity of the trace and (anti)commuting rules but I'm in a dead end... Any ideas?

On page 20, after the proof for N=1, Kovtun and Yaffe say "if a non-zero chemical potential μR for the R-charge is present, then the generalized partition function does acquire dependence on the fermionic chemical potentials". So maybe it's just not true for N>1.