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The R-R sector of IIA and IIB are respectively given as,

$8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$

$8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$

Now looking at this Polchinski in his lecture notes states about developing the vertex operators for the R-R states as,

"..This will involve a product of spin fields $e^{-\frac{\phi}{2} - \frac{\bar{\phi} }{2}} S_\alpha \tilde{S}_\beta$. This again decomposes into anti-symmetric tensors of $SO(9,1)$ as, $V = e^{-\frac{\phi}{2} - \frac{\bar{\phi} }{2}} S_\alpha \tilde{S}_\beta (\Gamma^{[\mu_1}...\Gamma^{\mu_n]} C)_{\alpha \beta}G_{[\mu_1 ... \mu_n]}(X) $ with $C$ as the charge conjugation matrix. In IIA theory it is $16\otimes16'$ giving even $n$ ($n \sim 10-n$) and in IIB it is $16\otimes16$ giving odd $n$..."

I have absolutely no clue as what happened here in the above argument! What are these spin-fields from nowhere? What is this $G$? How is this $V$ conceived? Where did these $16$ and $16'$ representations come from?

  • Can someone kindly split this up into conceptual chunks and may be reference/explain what happened here? How does this lead to the conclusion about there being only $0,2,4,6,8$-branes in IIA and $-1,1,3,5,7,9$ branes in IIB?
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1 Answer 1

First, the tensor product of two Dirac spinors is the direct sum of differential $p$-forms for all values of $p$. It's because this tensor product is the same thing as the space of all matrices acting on the Dirac spinor space. But all matrices (e.g. $4\times 4$ Dirac-like matrices in 4 dimensions) may be written as linear combinations of $1$, $\gamma_\mu$, $\gamma_{\mu\nu}$, and so on up to "$\gamma_5$", i.e. the product of all $d$ matrices.

Second, if the word "Dirac" is replaced by a "Weyl" i.e. a chiral spinor in the previous paragraph, the tensor product will still include differential forms but only "all even $p$" or "all odd $p$" differential forms, depending on whether the chiralities of the two Weyl spinors are the same or opposite ones. That explains the decomposition of the two $8\otimes 8$ tensor products relevant for the Ramond-Ramond sectors of type IIB. The $p$-forms only go up to $d/2$, with the middle (five) form being self-dual, if we consider "Weyl times Weyl".

Third, the ground state of periodic NSR fermions on the string is degenerate and transforms as the spinor (or the tensor product of two such spinors). Why? Because the ground state is a representation of the zero mode operators $\psi^\mu_0$ which don't change the energy. But their anticommutators form the same algebra as the Dirac matrices $\Gamma^\mu$, so the representation has to be the same as a spinor, too. The GSO projections imply that only the chiral/Weyl spinor of the same chirality is left in the physical spectrum. The even/odd $p$ arises because the sandwiched products $\bar s_1 \gamma_\mu\dots s_2$ may be shown to be zero if $s_1,s_2$ are some particular $\gamma_5$ (chirality) eigenstates, by the fact that $\gamma_5$ anticommutes with $\gamma_\mu$ etc. (that's why either all even or all odd $p$ forms may be shown to vanish).

Fourth, the operator associated with these spinor-like ground state is the spin field. This may be seen by realizing that the spin fields are the lowest-dimension operators transforming as spinors, and the ground state in the periodic R (or RR) sector is the lowest energy state transforming as a spinor (or the tensor product of two of them).

Fifth, the appearance of $\exp(-\phi/2)$ is just a bosonization of the operator fully analogous to the spin fields that lives in the $\beta\gamma$ conformal field theory. If the superconformal ghosts were fermions, the spin field could be bosonized to $\exp(\pm \phi/2)$ of some sort, and this rule is still true for "rebosonization" of $\beta\gamma$ into the $\phi$ fields. Why the rebosonization is an equivalence is a complex problem in CFT but it may be proven.

Sixth, $G_{\mu\dots}$ are just the differential forms of polarization parameters for the given state/operator, analogously to the polarization vector $\vec\epsilon$ of a photon. Their indices are contracted with the same indices of the spin fields so that the vertex operator has no free indices left. Alternatively, you may just omit the $G$ parameters and talk about vertex operators with free, uncontracted Lorentz indices.

Seventh, the translation between states and operators – the way how $V$ was derived – is a straightforward task but requires work. You must understand why the "state-operator correspondence" exists and know some CFT methods to determine the dictionary. All these things are discussed in Polchinski's book so that a reader should be able to learn it from scratch.

Eighth, $16$ and $16'$ don't come from nowhere. They are just the two real inequivalent chiral spinor representations of $Spin(9.1)$. The $2^{10/2}=32$-dimensional Dirac spinor is split into two "halves" in every even spacetime dimensions. You need to learn some basic representation theory (especially spinors) to understand these things but the appendices of Polchinski's book are sufficiently self-contained so that one should be able to learn these things from scratch, too, at least if he has been exposed to the basic Dirac spinors in 4D in a generic quantum field theory course.

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