One-field N=(2,2) Landau–Ginzburg model and A-series minimal-model conjecture

Given: Let $k\in {\mathbb{Z}}_{\ge 1}$, let $\lambda \in \mathbb{C}\setminus \{0\}$ be a fixed neutral coupling of mass dimension one, and use $\mathrm{\hslash }=1$. Work in flat two-dimensional Minkowski spacetime with metric $(+,-)$, coordinates of length dimension one, a dimensionless scalar, Weyl fermions of mass dimension $1\mathrm{/}2$, and auxiliary fields of mass dimension one.

Full local two-dimensional CFT data: The full-local-CFT item defines a local state-space/OPE package with specified convergent OPEs, locality and crossing; listing an action or its predicted spectrum does not prove those data exist.

$N=2$ superconformal mode algebra in NS and Ramond sectors: The $N=2$ algebra has generators $L_n,J_n,G_r^{\pm }$ with central charge $c$ and independent NS/R choices in the left and right copies.

The one-field $N=2$ Landau–Ginzburg Jacobian algebra has dimension k+1: For $W(X)=\lambda X^{k+2}\mathrm{/}(k+2)$ with $k\ge 1$ and $\lambda \mathrm{\ne }0$, $\mathrm{Jac}(W)=\mathbb{C}[X]\mathrm{/}(X^{k+1})$ with basis $[1],\dots ,[X^k]$ and dimension $k+1$.

The flat two-dimensional N=(2,2) superspace used here has coordinates $x^0,x^1\in \mathbb{R}$ and independent Grassmann-odd coordinates ${\theta }^{\pm },{\bar{\theta }}^{\pm }$, each of mass dimension $-1\mathrm{/}2$; the odd coordinates square to zero and mutually anticommute. Set ${\mathrm{\partial }}_{\pm }={\mathrm{\partial }}_0\pm {\mathrm{\partial }}_1$ and, using left Grassmann derivatives, define $D_{\pm }={\mathrm{\partial }}_{{\theta }^{\pm }}-i{\bar{\theta }}^{\pm }{\mathrm{\partial }}_{\pm }$ and ${\bar{D}}_{\pm }=-{\mathrm{\partial }}_{{\bar{\theta }}^{\pm }}+i{\theta }^{\pm }{\mathrm{\partial }}_{\pm }$. These satisfy $\{D_{\pm },{\bar{D}}_{\pm }\}=2i{\mathrm{\partial }}_{\pm }$, with same-type squares zero and cross-chirality anticommutators zero. A chiral superfield means a superfunction $\mathcal{X}$ satisfying ${\bar{D}}_{+}\mathcal{X}={\bar{D}}_{-}\mathcal{X}=0$. Let $\mathrm{\Lambda }$ be a complex Grassmann algebra, so its odd elements anticommute. The superfield's components are a smooth complex scalar $x:{\mathbb{R}}^{1,1}\to \mathbb{C}$, smooth one-component Grassmann-odd fields ${\psi }_{\pm }:{\mathbb{R}}^{1,1}\to {\mathrm{\Lambda }}_{\bar{1}}$ of opposite chiralities (with ${\bar{\psi }}_{\pm }$ their conjugate fields), and a smooth complex auxiliary scalar $F:{\mathbb{R}}^{1,1}\to \mathbb{C}$. The classical action below is evaluated on smooth compactly supported profiles, so its spacetime integrals are finite. Berezin integration is normalized by $\int d\theta 1=0$ and $\int d\theta \theta =1$, with the order of odd integrations as displayed below. Extend complex conjugation to an antilinear parity-preserving order-reversing involution $\ast$: $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$, $i^{\ast }=-i$, $({\theta }^{\pm }{)}^{\ast }={\bar{\theta }}^{\pm }$, $x^{\ast }=\bar{x}$, ${\psi }_{\pm }^{\ast }={\bar{\psi }}_{\pm }$ and $F^{\ast }=\bar{F}$. The coordinate generators also anticommute with the odd coefficient fields. No extra sign is inserted in the involution; signs appear when odd factors are put back in the displayed order. Spacetime derivatives commute with $\ast$. Iterated Berezin integrals apply the rightmost integration first, using the left odd derivatives and then setting the integrated odd variables to zero. In particular, $\int d^4\theta P$ is the coefficient of ${\theta }^{+}{\bar{\theta }}^{+}{\theta }^{-}{\bar{\theta }}^{-}$, while $\int d{\theta }^{+}d{\theta }^{-}P$ is minus the coefficient of ${\theta }^{+}{\theta }^{-}$ at $\bar{\theta }=0$. The conjugate chiral measure is $d^2\bar{\theta }=d{\bar{\theta }}^{-}d{\bar{\theta }}^{+}$, so conjugating the chiral integral gives the antichiral integral with that order. Spacetime integration and integration by parts are understood coefficientwise in the Grassmann algebra for the compactly supported smooth profiles above.

For the terms below, a left-moving $N=2$ superconformal primary is a nonzero joint eigenvector $v$ of $L_0,J_0$ annihilated by $L_n,J_n$ for all integers $n>0$ and by $G_r^{\pm }$ for all positive NS half-integers $r$. A left chiral primary further satisfies $G_{-1\mathrm{/}2}^{+}v=0$ and $h=Q\mathrm{/}2$, where $L_{0}v=hv$ and $J_{0}v=Qv$; an anti-chiral primary satisfies $G_{-1\mathrm{/}2}^{-}v=0$ and $h=-Q\mathrm{/}2$. Define the right-moving conditions with the barred algebra; the $(c,c)$ sector consists of operators whose left and right states are chiral primaries. For polynomial variables $X_i$ assigned weights $q_i$, a monomial ${\prod }_i{X}_i^{n_i}$ has weight ${\sum }_i{n}_i{q}_i$; a polynomial is quasi-homogeneous of weight one when each nonzero monomial has that weight. For $W$ in one complex field, the origin is an isolated critical point when $W^{\mathrm{\prime }}(x)=0$ has no other solution in some neighborhood of $x=0$.

Adopt the one-field N=(2,2) Landau–Ginzburg chiral-ring prescription reported in the linked February 1989 Lerche–Vafa–Warner preprint, §3, equation (3.3), printed p. 22. The ring product in the $(c,c)$ sector is induced by the local operator product. The D-term is the full-superspace kinetic term and the F-term is the chiral-superspace superpotential term in the action below. For a quasi-homogeneous superpotential with an isolated critical point, assume a D-term choice for which the action is conformally invariant; the physical prescription sets the equation-of-motion insertion ${\mathrm{\partial }}_{X}W$ to zero in the chiral ring. Thus the protected chiral ring is the polynomial ring modulo $({\mathrm{\partial }}_{X}W)$. This is a physical model input; the algebraic quotient theorem does not prove it.

Adopt the $N=2$ Landau–Ginzburg quasi-homogeneous central-charge prescription, reported in that February 1989 Lerche–Vafa–Warner preprint, §3, equation (3.7), printed p. 23: if $W$ has polynomial weight one and chiral fields have weights $q_i$, then the proposed infrared $N=2$ fixed point has $c=3{\sum }_i(1-2q_i)$. This is a fixed-point prediction conditional on that prescription, not a theorem that the flow reaches the fixed point.

Witten, §§1–3, describes the identification of this one-field model with the A-series minimal model as a conjecture; elliptic-genus or character agreement alone does not establish full local-CFT equivalence.

Witten, §2.2, equation (2.14), fixes the left-moving R charges: $x$ and ${\psi }_{+}$ have charge $q_X$, while ${\psi }_{-}$ has charge $-(k+1)q_X$. Use the independent chirality-swapped assignment for the right-moving R current.