Free two-dimensional fermion CFT with spin sectors

A spin structure on a connected oriented Riemann surface $\mathrm{\Sigma }$ is a holomorphic line bundle $S$ together with an isomorphism $S^{\otimes 2}\simeq K_{\mathrm{\Sigma }}$, where $K_{\mathrm{\Sigma }}$ is the holomorphic cotangent line bundle. On the flat torus ${\mathrm{\Sigma }}_{\tau }=\mathbb{C}\mathrm{/}(\mathbb{Z}+\tau \mathbb{Z})$, the four spin structures are recorded by the signs acquired around the spatial and temporal cycles. Antiperiodic boundary condition is called Neveu–Schwarz (NS); periodic boundary condition is called Ramond (R).

Take two real Majorana fields ${\psi }^1(z),{\psi }^2(z)$ of weights $(1\mathrm{/}2,0)$ and their right-moving partners ${\bar{\psi }}^1(\bar{z}),{\bar{\psi }}^2(\bar{z})$ of weights $(0,1\mathrm{/}2)$. Their CAR convention is ${\psi }^a(z){\psi }^b(w)\sim {\delta }^{ab}\mathrm{/}(z-w)$, ${\bar{\psi }}^a(\bar{z}){\bar{\psi }}^b(\bar{w})\sim {\delta }^{ab}\mathrm{/}(\bar{z}-\bar{w})$, with mixed left/right anticommutators zero. The real fields satisfy $({\psi }^a{)}^{\ast }={\psi }^a$ and $({\bar{\psi }}^a{)}^{\ast }={\bar{\psi }}^a$. The two real species and their right-moving partners form one nonchiral complex Dirac spinor, with $c_L=c_R=1$; they are not two independent complex Dirac species. Define the complex chiral fields ${\psi }^{\pm }=\frac{{\psi }^1\pm i{\psi }^2}{\sqrt{2}},{\bar{\psi }}^{\pm }=\frac{{\bar{\psi }}^1\mp i{\bar{\psi }}^2}{\sqrt{2}}\mathrm{.}$ These two real Majorana species form one nonchiral complex Dirac spinor; they are not two independent complex Dirac species.

For NS modes the indices are $s\in \mathbb{Z}+\frac{1}{2}$; for R modes they are $s\in \mathbb{Z}$. They satisfy $\{{\psi }_s^a,{\psi }_t^b\}={\delta }^{ab}{\delta }_{s+t,0}I$, and the right modes satisfy the same relation. The NS ground space is $\mathbb{C}\mathrm{\mid }0{⟩}_{NS}$, annihilated by every positive mode; its algebraic Fock space is the exterior algebra generated by the negative half-integer modes. In the R sector the zero modes obey $\{{\psi }_0^a,{\psi }_0^b\}={\delta }^{ab}I$ and act in a chosen irreducible complex Clifford module $S_R\cong {\mathbb{C}}^2$ per chirality. Every positive integer mode annihilates $S_R$, and the algebraic Ramond Fock space is the exterior Fock space of the negative nonzero integer modes over $S_R$. The left and right Fock factors are independent. Complete the mode Fock spaces using the positive inner product for which the ordered occupation basis is orthonormal and creation and annihilation modes are adjoints. The local spin-field algebra is generated from the NS vacuum by the four Majorana fields; the Ramond Fock spaces are the corresponding spin-twisted sectors used for torus traces and twist fields.

The stress tensors are $T(z)=-\frac{1}{2}{\sum }_{a=1}^2:{\psi }^a(z)\mathrm{\partial }{\psi }^a(z):$ and $\bar{T}(\bar{z})=-\frac{1}{2}{\sum }_{a=1}^2:{\bar{\psi }}^a(\bar{z})\bar{\mathrm{\partial }}{\bar{\psi }}^a(\bar{z}):$, in the Virasoro convention of the earlier mathematical definition. The NS state space has discrete spectrum and a unique NS/NS vacuum. For the local fields take the common smooth-energy domain ${\bigcap }_{k\ge 0}\mathrm{Dom}(1+L_0+{\bar{L}}_0{)}^k$ with the graph seminorms, and give $C_c^{\mathrm{\infty }}(U)$ its usual compact-support smooth topology. Weak smearings, covariance, convergence, crossing, and spin locality are part of the data and must satisfy the earlier full-local-CFT definition; no existence theorem follows merely from the formal mode algebra. This package makes no higher-genus sewing claim.

The local field grading is $p({\psi }^a)=p({\bar{\psi }}^a)=1$ and is extended additively modulo $2$ to products and descendants. Require $e^{2\pi i(h-\bar{h})}=(-1{)}^p$ on each local joint-weight component and supercommutation for disjointly supported smearings. The Ramond ground fields $\sigma ,\mu$ are spin-twisted insertions, not additional NS-local generators.

The classification flags are: unitary, yes, with the stated positive Fock inner product, creation/annihilation adjoints, $L_n^{\ast }=L_{-n}$, ${\bar{L}}_n^{\ast }={\bar{L}}_{-n}$, real Majorana fields self-adjoint, and $\sigma ,\mu$ self-adjoint; discrete spectrum, yes; continuous spectrum, no. Reflection positivity and rationality are not asserted by this item.

Define the complex current $J(z)=i:{\psi }^1(z){\psi }^2(z):={\sum }_{n\in \mathbb{Z}}{J}_n{z}^{-n-1}$ and its independent right-moving copy $\bar{J}(\bar{z})=i:{\bar{\psi }}^1(\bar{z}){\bar{\psi }}^2(\bar{z}):={\sum }_{n\in \mathbb{Z}}{\bar{J}}_n{\bar{z}}^{-n-1}$. The current charges are $Q_L=J_0$ and $Q_R={\bar{J}}_0$. In NS they lie in $\mathbb{Z}$; in R they lie in $\mathbb{Z}+\frac{1}{2}$, with the two zero-mode ground charges $\pm \frac{1}{2}$. The charge-$Q$ chiral Fock character uses conformal weight $Q^2\mathrm{/}2$ above its Heisenberg oscillator levels.

For each torus spin structure, let $Z_{NS,NS},Z_{NS,R},Z_{R,NS},Z_{R,R}$ be the corresponding left/right Fock traces, with a temporal periodic condition represented by insertion of $(-1{)}^F$. Each chiral trace is $\mathrm{Tr}(q^{L_0-c\mathrm{/}24})$ or its right-moving analogue, with $c=1\mathrm{/}2$ per real Majorana component; the two components give $c_L=c_R=1$. With $q=e^{2\pi i\tau }$, $\bar{q}=e^{-2\pi i\bar{\tau }}$, and $\mathrm{Im}\tau >0$, define ${\theta }_3(\tau )={\sum }_{k\in \mathbb{Z}}{q}^{k^2\mathrm{/}2},{\theta }_4(\tau )={\sum }_{k\in \mathbb{Z}}(-1{)}^k{q}^{k^2\mathrm{/}2},{\theta }_2(\tau )={\sum }_{k\in \mathbb{Z}}{q}^{(k+1\mathrm{/}2{)}^2\mathrm{/}2},{\theta }_1(\tau )={\sum }_{k\in \mathbb{Z}}(-1{)}^k{q}^{(k+1\mathrm{/}2{)}^2\mathrm{/}2}\mathrm{.}$ Define the Dedekind eta function by $\eta (\tau )=e^{\pi i\tau \mathrm{/}12}{\prod }_{k=1}^{\mathrm{\infty }}(1-e^{2\pi ik\tau }),\mathrm{Im}\tau >0.$ The four amplitudes are $Z_{NS,NS}=\mathrm{\mid }{\theta }_3(\tau )\mathrm{/}\eta (\tau ){\mathrm{\mid }}^2$, $Z_{NS,R}=\mathrm{\mid }{\theta }_4(\tau )\mathrm{/}\eta (\tau ){\mathrm{\mid }}^2$, $Z_{R,NS}=\mathrm{\mid }{\theta }_2(\tau )\mathrm{/}\eta (\tau ){\mathrm{\mid }}^2$, and $Z_{R,R}=\mathrm{\mid }{\theta }_1(\tau )\mathrm{/}\eta (\tau ){\mathrm{\mid }}^2=0$. The last vanishes because the periodic-periodic trace has fermion zero modes.

The selected GSO projection is the common-spin-structure sum $\frac{1}{2}(Z_{NS,NS}+Z_{NS,R}+Z_{R,NS}+Z_{R,R})$. Equivalently it projects the spatial NS Hilbert space onto $(-1{)}^F=+1$ and the spatial R Hilbert space onto $(-1{)}^F=+1$, using the Ramond zero-mode convention for which the even R charge pairs have $Q_L+Q_R$ odd. Here $Q_L,Q_R$ are the eigenvalues of the complex-fermion currents $J_0,{\bar{J}}_0$; they are integers in NS and half-integers in R. Thus the projection keeps $Q_L+Q_R$ even in NS and odd in R. This zero-mode shift is part of the convention. It makes the retained pairs exactly $m=Q_L+Q_R\in \mathbb{Z}$, $n=(Q_L-Q_R)\mathrm{/}2\in \mathbb{Z}$; the complementary pairs with $n\in \mathbb{Z}+\frac{1}{2}$ are outside the projected bosonic local algebra. If each real Majorana is spin-summed independently, its nonchiral torus sum is $\frac{1}{2}(a_3+a_4+a_2)$, where $a_j=\mathrm{\mid }{\theta }_j\mathrm{/}\eta \mathrm{\mid }$ for $j=2,3,4$; the two independent Majoranas therefore have product partition function $\frac{1}{4}(a_2+a_3+a_4{)}^2$, not the common-spin-structure Dirac sum.

Before parity projection, the nonchiral Ramond zero-mode space of one real Majorana is two-dimensional. Denote its two parity-eigenfield insertions by $\sigma$ and $\mu$; each has weights $(1\mathrm{/}16,1\mathrm{/}16)$, and taking the fermion once around either insertion changes its sign. The diagonal Ising spin sum retains the parity-selected spin field, while the other insertion records the opposite Ramond parity sector. For two independently spin-summed Majoranas use superscripts ${\sigma }^a,{\mu }^a$ for species $a$. These are spin-twisted fields, not additional NS-local generators.

Fix the square-root branch cut from each twist insertion along the positive real ray. Normalize the twist fields so that the chiral OPEs are ${\psi }^a(z){\sigma }^a(w,\bar{w})\sim e^{i\pi \mathrm{/}4}{\mu }^a(w,\bar{w})\mathrm{/}(\sqrt{2}(z-w{)}^{1\mathrm{/}2})$ and ${\psi }^a(z){\mu }^a(w,\bar{w})\sim e^{-i\pi \mathrm{/}4}{\sigma }^a(w,\bar{w})\mathrm{/}(\sqrt{2}(z-w{)}^{1\mathrm{/}2})$; the right-moving OPEs have the conjugate phases. These branch singularities implement the stated sign monodromy.

Full local two-dimensional CFT data

Virasoro algebra in the $L_m$ convention

Ginsparg, Sections 7.2 and 8.2

Mnev–Reshetikhin, Sections 6.3.10.3–6.3.11.4

The Ginsparg Section 8.2 phrase “two Dirac fermions” refers to the two chiral $c=1\mathrm{/}2$ real components and their right-moving partners; Mnev–Reshetikhin explicitly identify two real Majoranas with one complex Dirac field. A literal pair of independent complex Dirac species would have $c_L=c_R=2$ and is not the theory in this definition. No path-integral determinant is used as a proof of the Fock-space construction.