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1. What the question is asking (in plain language)

When we compactify a ten‑dimensional superstring on a circle (S^{1}) we obtain a lower‑dimensional theory (for definiteness we will go to nine dimensions).
The massless fields in ten dimensions come either from the NS (Neveu–Schwarz) sector or from the R (Ramond) sector of the world‑sheet theory.

Empirically one finds the following pattern:

10‑d field 9‑d field after compactification Why?
NS field with no index along the circle → stays NS in 9‑d a 9‑d field of the same type  
NS field with one index along the circle → becomes an R field in 9‑d the internal index “flips’’ the sector  
The same pattern holds for fermions, but the spinor bundle behaves as a tensor product rather than a direct sum.    

The student wants a clear, step‑by‑step explanation of why the presence of an index along the compact direction changes the sector, and how the splitting of spinor bundles explains the fermionic case.

2. Detailed solution

Below we work in the RNS formulation of the superstring. The same logic applies to any supersymmetric theory that is reduced on a circle.

2.1 Geometry of the compactification

Take the ten‑dimensional spacetime to be

[ \mathcal{M}{10}= \mathcal{M}{9}\times S^{1}_{R}, \qquad y\sim y+2\pi R . ]

The Lorentz group factorises as

[ SO(1,9)\;\longrightarrow\;SO(1,8)\times SO(1) ]

and any tensor (or spinor) of (SO(1,9)) can be decomposed into representations of the 9‑d Lorentz group together with representations of the “internal’’ group (SO(1)) (the rotation group of a line). For a line the only non‑trivial representation is the sign representation, i.e. a field can be periodic or anti‑periodic when we go once around the circle.

2.2 Bosonic fields (NS sector)

The NS sector of the world‑sheet contains the spacetime bosons: the metric (G_{MN}), the Kalb‑Ramond 2‑form (B_{MN}) and the dilaton (\Phi).
All of them are periodic on the world‑sheet, so in spacetime they are ordinary (periodic) fields on (S^{1}).

When we expand a generic ten‑dimensional bosonic field (\Phi(x^{\mu},y)) in Kaluza–Klein (KK) modes we write

[ \Phi(x^{\mu},y)=\sum_{n\in\mathbb Z} \phi_{n}(x^{\mu})\,e^{i n y/R}\; . \tag{2.1} ]

Because the field is periodic in (y) we only get integer Fourier modes (n\in\mathbb Z). The zero mode (n=0) is a genuine nine‑dimensional field; the non‑zero modes are massive excitations with mass ( n /R).

Now split the 10‑d index (M) as ((\mu,9)) where (\mu=0,\dots ,8) runs over (\mathcal{M}_{9}) and (9) denotes the circle direction. Two kinds of components appear:

Component 9‑d interpretation KK expansion
(G_{\mu\nu}, B_{\mu\nu}) (no 9‑index) 9‑d tensors (graviton, 2‑form) (2.1) with integer (n) – NS
(G_{\mu 9}, B_{\mu 9}) (one 9‑index) 9‑d vectors also (2.1) with integer (n). Why do they look like R?

The key point is that a vector component along the circle carries one unit of charge under the internal (U(1)) isometry (the momentum around the circle). Under a shift (y\to y+2\pi R) the component picks up a factor

[ G_{\mu 9}(x,y+2\pi R)= G_{\mu 9}(x,y)\;, ]

so the field itself is still periodic. However, from the world‑sheet point of view the internal index behaves like a world‑sheet fermion with different moding, as we now explain.

2.3 World‑sheet fermions and the NS/R sectors

In the RNS formalism the ten spacetime coordinates are accompanied by ten world‑sheet fermions (\psi^{M}(\sigma,\tau)) (left‑moving) and (\tilde\psi^{M}) (right‑moving). Their mode expansions depend on the spin structure on the world‑sheet circle (\sigma\sim\sigma+2\pi):

Sector Boundary condition on (\psi^{M}) Mode expansion
NS (Neveu–Schwarz) anti‑periodic (\psi^{M}(\sigma+2\pi)=-\psi^{M}(\sigma)) (\displaystyle \psi^{M}(\sigma)=\sum_{r\in\mathbb Z+1/2}\psi^{M}_{r}e^{-ir\sigma})
R (Ramond) periodic (\psi^{M}(\sigma+2\pi)=+\psi^{M}(\sigma)) (\displaystyle \psi^{M}(\sigma)=\sum_{n\in\mathbb Z}\psi^{M}_{n}e^{-in\sigma})

Thus the moding (half‑integer vs. integer) is the distinguishing feature of the two sectors.

When we compactify one spacetime direction on a circle we also have to specify how the world‑sheet fermions transform under translations in the internal direction. The ten‑dimensional fields are functions of (y). A Fourier mode (e^{i n y/R}) carries momentum (p_{9}=n/R). Because the world‑sheet fermion carries a spin index, parallel transport around the internal circle multiplies it by a factor ((-1)^{n}). Concretely,

[ \psi^{9}(\sigma,\tau, y+2\pi R) = (-1)^{n}\,\psi^{9}(\sigma,\tau, y), \qquad n \;\text{the KK number of the mode}. \tag{2.2} ]

  • If (n) is even ((n\in 2\mathbb Z)), the sign is (+1); the field is periodic on the world‑sheet and belongs to the R sector.
  • If (n) is odd ((n\in 2\mathbb Z+1)), the sign is (-1); the field is anti‑periodic and belongs to the NS sector.

For a pure NS field (no internal index) the Fourier modes are always integer, so (2.2) never flips the sign: all KK excitations stay in the NS sector.

For a field with one index along the circle, the factor ((-1)^{n}) appears because the internal index transforms as a vector under the little group (SO(1)) of the circle. Consequently the overall boundary condition for the world‑sheet fermion associated with that component becomes periodic (R) for even KK number and anti‑periodic (NS) for odd KK number. In other words, the sector of the mode is shifted by one unit of KK momentum.

Since the lowest non‑zero KK momentum is (n=\pm 1), the first massive mode of a component with a circle index lands in the R sector, exactly what the empirical rule states.

2.4 Spinor bundle viewpoint (fermions)

The above story can be rephrased geometrically in terms of spin bundles.
For a product manifold (M^{9}\times S^{1}) the spin bundle factorises as

[ S\bigl(M^{9}\times S^{1}\bigr)= S(M^{9})\;\otimes\; S(S^{1}) . \tag{2.3} ]

  • (S(M^{9})) is the ordinary 9‑d spinor bundle.
  • (S(S^{1})) is a two‑dimensional bundle whose fibre consists of the two possible spin structures on the circle: periodic (Ramond) and anti‑periodic (Neveu–Schwarz).

A ten‑dimensional Majorana–Weyl spinor (\Psi(x,y)) can therefore be

Original question: Compactifying on a circle and the exchange of R and NS sectors on Physics Stack Exchange, licensed CC BY-SA.