# A few simple questions about Susy

#### nrqed

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The Wikipedia entry on susy says

"Typically the number of copies of a supersymmetry is a power of 2, i.e. 1, 2, 4, 8. In four dimensions, a spinor has four degrees of freedom and thus the minimal number of supersymmetry generators is four in four dimensions and having eight copies of supersymmetry means that there are 32 supersymmetry generators."

Why is that we consider only powers of 2? I don't see why this follows from the basic definition of the algebra.

Also, a simple question: does anyone know the origin of the term "R symmetry"? Why is it called this way? A wild guess would be that the R stands for "rotation" as it implements a kind of rotation between the fields but I am only guessing here, I would appreciate knowing for sure where it comes from.

Thanks

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#### fresh_42

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Why is that we consider only powers of 2? I don't see why this follows from the basic definition of the algebra.
Isn't 2 just the grading that makes the Lie algebras super?

#### nrqed

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Isn't 2 just the grading that makes the Lie algebras super?
Well, the supercharges are $Q_\alpha^I$ plus their dagger, where alpha takes two values and I runs from 1 to N, the number of supersymmetry. For N=2, that makes a total of 8 supercharges which can be organized into 4 creation operators and 4 annihilation operators. I don't see why we could not take N=3.

#### fresh_42

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I'm no physicist and have a rather mathematical point of view. (My curiosity led me here.) But won't you loose then all those tools like the quaternions and Graßmann algebras in this context? I always thought these were the main technical reasons to consider SUSY at all because they deliver the framework.

#### fzero

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The Wikipedia entry on susy says

"Typically the number of copies of a supersymmetry is a power of 2, i.e. 1, 2, 4, 8. In four dimensions, a spinor has four degrees of freedom and thus the minimal number of supersymmetry generators is four in four dimensions and having eight copies of supersymmetry means that there are 32 supersymmetry generators."

Why is that we consider only powers of 2? I don't see why this follows from the basic definition of the algebra.
It is not really the case that we only consider powers of 2, it is that spinors in various dimensions have properties such that, when all requirements are considered, it might be that there are additional SUSYs present. For instance, there is the possibility of rigid $N=3$ in 4d, but once you add the CPT conjugate of the simplest multiplet, you fill a vector multiplet of $N=4$ SUSY. I believe that this example is discussed in Weinberg's vol. 3. However, there is a rigid $N=3$ multiplet in 3d, so the statement in the wiki is not a hard rule.

I don't know a reference off-hand for these uncommon SUSYs, but I would suspect that the relatively recent text of van Proeyen would give the completest discussion.

Also, a simple question: does anyone know the origin of the term "R symmetry"? Why is it called this way? A wild guess would be that the R stands for "rotation" as it implements a kind of rotation between the fields but I am only guessing here, I would appreciate knowing for sure where it comes from.
Yes, you are probably right about the origin. In superspace, it is most explicitly the analogue of the rotation group for the Grassman directions. Otherwise it can be viewed as internal rotation of the SUSY charges.

• nrqed

#### fresh_42

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• MacRudi

#### nrqed

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It is not really the case that we only consider powers of 2, it is that spinors in various dimensions have properties such that, when all requirements are considered, it might be that there are additional SUSYs present. For instance, there is the possibility of rigid $N=3$ in 4d, but once you add the CPT conjugate of the simplest multiplet, you fill a vector multiplet of $N=4$ SUSY. I believe that this example is discussed in Weinberg's vol. 3. However, there is a rigid $N=3$ multiplet in 3d, so the statement in the wiki is not a hard rule.

I don't know a reference off-hand for these uncommon SUSYs, but I would suspect that the relatively recent text of van Proeyen would give the completest discussion.

Yes, you are probably right about the origin. In superspace, it is most explicitly the analogue of the rotation group for the Grassman directions. Otherwise it can be viewed as internal rotation of the SUSY charges.
Ah, very informative. Thank you for the reply, fzero.

#### nrqed

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I have another related question: Let's consider N=2, then there is a single central charge Z. Is this quantity complex or real? Some expressions make it look like Z is complex (for example they will use Z* in the anti-commutation relations of the Q dagger) but the BPS bound is (in some normalization at least) $M \geq \sqrt{2} Z$, which seems to imply that Z is real. So is it real or complex?

#### fzero

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I have another related question: Let's consider N=2, then there is a single central charge Z. Is this quantity complex or real? Some expressions make it look like Z is complex (for example they will use Z* in the anti-commutation relations of the Q dagger) but the BPS bound is (in some normalization at least) $M \geq \sqrt{2} Z$, which seems to imply that Z is real. So is it real or complex?
$Z$ is generally complex, so the correct BPS bound is $M\geq \sqrt{2} |Z|$. For the $N=2$ pure vector multiplet, if you explicitly construct the supercurrents, you can work out an expression
$$Z = a ( n_e + \tau n_m),$$
where $a$ is the VEV of the complex scalar in the vector multiplet, $\tau = \theta/(2\pi) + 4\pi i/g^2$ is the complexified coupling and $(n_e,n_m)$ are integer electric and magnetic charges for the state used to compute the charge. There are some details culminating on pg. 40 of http://arxiv.org/abs/hep-th/9701069. We can see that $Z$ can be real under certain conditions, including $n_m=0$ and $a$ real.

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#### nrqed

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$Z$ is generally complex, so the correct BPS bound is $M\geq \sqrt{2} |Z|$. For the $N=2$ pure vector multiplet, if you explicitly construct the supercurrents, you can work out an expression
$$Z = a ( n_e + \tau n_m),$$
where $a$ is the VEV of the complex scalar in the vector multiplet, $\tau = \theta/(2\pi) + 4\pi i/g^2$ is the complexified coupling and $(n_e,n_m)$ are integer electric and magnetic charges for the state used to compute the charge. There are some details culminating on pg. 40 of http://arxiv.org/abs/hep-th/9701069. We can see that $Z$ can be real under certain conditions, including $n_m=0$ and $a$ real.
Great, this is very useful. Thank you for your time!

#### nrqed

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$Z$ is generally complex, so the correct BPS bound is $M\geq \sqrt{2} |Z|$. For the $N=2$ pure vector multiplet, if you explicitly construct the supercurrents, you can work out an expression
$$Z = a ( n_e + \tau n_m),$$
where $a$ is the VEV of the complex scalar in the vector multiplet, $\tau = \theta/(2\pi) + 4\pi i/g^2$ is the complexified coupling and $(n_e,n_m)$ are integer electric and magnetic charges for the state used to compute the charge. There are some details culminating on pg. 40 of http://arxiv.org/abs/hep-th/9701069. We can see that $Z$ can be real under certain conditions, including $n_m=0$ and $a$ real.
But in that paper, on page 25, they write
$$\{b_\alpha, b_\beta^\dagger \} = \delta_{\alpha,\beta} (M- \sqrt{2} \, Z )$$

If I take alpha = beta and I sandwiched this between $|\psi\rangle$ and $\langle \psi |$, it seems to me that I get that $M- \sqrt{2} \, Z$ must be a great or equal to zero, real quantity. And yet, just below they say that they get $M \geq \sqrt{2} \, |Z|$. I don't see where the absolute value comes from.

#### fzero

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But in that paper, on page 25, they write
$$\{b_\alpha, b_\beta^\dagger \} = \delta_{\alpha,\beta} (M- \sqrt{2} \, Z )$$

If I take alpha = beta and I sandwiched this between $|\psi\rangle$ and $\langle \psi |$, it seems to me that I get that $M- \sqrt{2} \, Z$ must be a great or equal to zero, real quantity. And yet, just below they say that they get $M \geq \sqrt{2} \, |Z|$. I don't see where the absolute value comes from.
I had forgotten something that led to this confusion. I said that $Z$ was in general complex, but I believe that we can use an R-symmetry transformation to rotate the $Q^A_\alpha$ to make the central charge real. This is a unitary transformation, so it wouldn't make the $\{Q,\bar{Q}\}$ commutator nonreal. If we didn't do this, in any given massive state, we can absorb the phase of $Z$ into the normalization of the state, so the central charge eigenvalue on that state can be considered real. Then the norm of any state will be real and things make better sense. Now since our positivity condition is $M\pm \sqrt{2} Z \geq 0$, we see that we must have $M\geq \sqrt{2} |Z|$, where $|Z|$ is absolute value.

• nrqed

#### nrqed

I had forgotten something that led to this confusion. I said that $Z$ was in general complex, but I believe that we can use an R-symmetry transformation to rotate the $Q^A_\alpha$ to make the central charge real. This is a unitary transformation, so it wouldn't make the $\{Q,\bar{Q}\}$ commutator nonreal. If we didn't do this, in any given massive state, we can absorb the phase of $Z$ into the normalization of the state, so the central charge eigenvalue on that state can be considered real. Then the norm of any state will be real and things make better sense. Now since our positivity condition is $M\pm \sqrt{2} Z \geq 0$, we see that we must have $M\geq \sqrt{2} |Z|$, where $|Z|$ is absolute value.