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I'm trying to get my head around part of my course in string theory from last term, specifically the nature of Neveu-Schwarz (NS) and Ramond (R) sectors.
When constructing states in NS you start with a groundstate |0,p\rangle and use the bosonic and fermionic operators \alpha_{-n}^{i} and \psi_{-\frac{1}{2}-m}^{j} (m,n integer). These states are then truncated by the GSO projector P_{NS} = \big( 1 + (-1)^{F} \big) where F = \sum_{r=\frac{1}{2}}^{\infty}\psi_{r}^{\dagger}. \psi_{r} so that only the states satisfying P_{NS}|\phi\rangle =0 are taken to be physical (thus getting rid of the old tachyon ground state). All well and good, I can get my head around that
However, in R you don't start with the same kind of ground state, but instead something of the form |0,p,A\rangle u^{A} where u^{A} is a polarisation spinor (which can also be written in the form \left(u_{+},u_{-}\right)^{T}). Perhaps it's a very stupid question, but can someone explain why this spinor is there please?
The GSO projector is also defined in a different way, using the d=10 gamma matrices,
P_{R}^{\pm} = \big( 1 \pm \Gamma_{11}(-1)^{F} \big)
with F = \sum_{r=0}^{\infty}\psi_{r}^{\dagger}. \psi_{r} and \Gamma_{11} = \Gamma_{0}...\Gamma_{9}
I can see the plus/minus projector relating to the chirality of the spinor, but why is there a gamma matrix in R but not in NS (I would guess it's the same reason there's a spinor in R and not in NS?)
Probably related to it all is the Dirac equation \alpha_{0}. \Gamma u_{+} = 0 taking u_{+}^{A} (16 components) to u_{+}^{a} ('a' is a spinor index and u_{+}^{a} an 8_{s} rep of SO(8)) and similarly u_{-}^{A} \to u_{-}^{\dot{a}} (an 8_{c} of SO(8)), though having not taken the supersymmetry course and only passing familiarity with the dot index notation from a group symmetry course last autumn I'm also a little hazy about this bit too. Does it mean the Dirac equation puts constraints on the u spinor such that it behaves as an 8 component spinor instead of 16, with the components being different depending on the chirality of the initial spinor the Dirac equation/operator is applied on?
Having read through my notes a fair few times and looking in a textbook or two (such as 'Superstring Theory' by Witten, Schwarz and Green) it's got a bit clearer (I now get torus invariance under SL(2,Z) ) but if anyone could shed a bit more light on some of the bits I've mentioned I'd be very very grateful. If (as I suspect) I've asked a bunch of somewhat trivial questions I can get answers to in certain books or papers, just name them (and where in the book I'd find the relevant part) and I'll check my department library rather than waste anyone's time on here.
Thanks
When constructing states in NS you start with a groundstate |0,p\rangle and use the bosonic and fermionic operators \alpha_{-n}^{i} and \psi_{-\frac{1}{2}-m}^{j} (m,n integer). These states are then truncated by the GSO projector P_{NS} = \big( 1 + (-1)^{F} \big) where F = \sum_{r=\frac{1}{2}}^{\infty}\psi_{r}^{\dagger}. \psi_{r} so that only the states satisfying P_{NS}|\phi\rangle =0 are taken to be physical (thus getting rid of the old tachyon ground state). All well and good, I can get my head around that
However, in R you don't start with the same kind of ground state, but instead something of the form |0,p,A\rangle u^{A} where u^{A} is a polarisation spinor (which can also be written in the form \left(u_{+},u_{-}\right)^{T}). Perhaps it's a very stupid question, but can someone explain why this spinor is there please?
The GSO projector is also defined in a different way, using the d=10 gamma matrices,
P_{R}^{\pm} = \big( 1 \pm \Gamma_{11}(-1)^{F} \big)
with F = \sum_{r=0}^{\infty}\psi_{r}^{\dagger}. \psi_{r} and \Gamma_{11} = \Gamma_{0}...\Gamma_{9}
I can see the plus/minus projector relating to the chirality of the spinor, but why is there a gamma matrix in R but not in NS (I would guess it's the same reason there's a spinor in R and not in NS?)
Probably related to it all is the Dirac equation \alpha_{0}. \Gamma u_{+} = 0 taking u_{+}^{A} (16 components) to u_{+}^{a} ('a' is a spinor index and u_{+}^{a} an 8_{s} rep of SO(8)) and similarly u_{-}^{A} \to u_{-}^{\dot{a}} (an 8_{c} of SO(8)), though having not taken the supersymmetry course and only passing familiarity with the dot index notation from a group symmetry course last autumn I'm also a little hazy about this bit too. Does it mean the Dirac equation puts constraints on the u spinor such that it behaves as an 8 component spinor instead of 16, with the components being different depending on the chirality of the initial spinor the Dirac equation/operator is applied on?
Having read through my notes a fair few times and looking in a textbook or two (such as 'Superstring Theory' by Witten, Schwarz and Green) it's got a bit clearer (I now get torus invariance under SL(2,Z)
) but if anyone could shed a bit more light on some of the bits I've mentioned I'd be very very grateful. If (as I suspect) I've asked a bunch of somewhat trivial questions I can get answers to in certain books or papers, just name them (and where in the book I'd find the relevant part) and I'll check my department library rather than waste anyone's time on here. Thanks
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