Coupled free fermion conformal field theories

The study of formal properties of CFTs was initiated in the late 1960s MACK1969174 ; polyakov1974nonhamiltonian ; Ferrara:1971vh ; Parisi:1972zm ; Caianiello1978NonPerturbativeMI . Early work was done in general dimension $d\geq 2$, where the Lie algebra of the group of conformal transformations is almost always finite-dimensional, except when $d=2$ where the group is infinite-dimensional. The infinite-dimensional symmetry group in 2D CFTs allows us to define the theories in a more abstract way via operator algebras and their representation theory.

With continuous developments since then, CFT has become an active area of research beyond its origins in statistical physics and attracted much attention for its intrinsic mathematical interest. The quest to classify and solve CFTs is a major goal of modern theoretical physics. A complete understanding of the space of CFTs is not yet available. It is only the rational CFT subclass that is reasonably well-understood.

Starting from the minimal models FQSminmodel ; Belavin:1984vu , various other constructions of rational CFTs, including the lattice construction DONG1993245 , WZW models KZWZW ; Witten1984WZW ; GepnerWitten , GKO coset construction GKOcoset , and the orbifold construction Dixon1985strorb ; Dixon1988strorb ; dixon1987orb ; DGM1990orb ; Dolan1996 , have been intensively investigated. Those theories that contain fields with spin-1 and spin-2 have drawn more attention from physicists, while the theories that contain fields of fractional spin, such as parafermionic CFTs, have not yet been fully explored.

Notable families of parafermion CFTs includes the FKF-type parafermions FRADKIN19801 , and the ZF-type parafermions, which relate to the WZW models as cosets in the form of $\widehat{\mathfrak{sl}(2)}_{k}/\widehat{\mathfrak{u}(1)}$ ZF1985 . A more general version of coset parafermion CFTs was introduced by Gepner in GEPNER87 , which can be thought of as cosets of WZW models in the form of $\hat{\mathfrak{g}}_{k}/\widehat{\mathfrak{u}(1)}^{n}$, where $n$ is the rank of $\mathfrak{g}$, and $k$ to so-called level GEPNER89 ; Bratchikov_2000 .

We notice that when $\mathfrak{g}$ is simply-laced and $k=2$, Gepner’s parafermions are spin-$\frac{1}{2}$ free fermions. Moreover, when $\mathfrak{g}\neq\widehat{\mathfrak{sl}(2)}$, these fermions are coupled to each other.

In Section 2 we study the coupled free fermions arising from the coset CFT $\hat{\mathfrak{g}}_{k}/\widehat{\mathfrak{u}(1)}^{n}$, in particular for $\mathfrak{g}=\mathfrak{sl}(3)$ and $\mathfrak{g}=\mathfrak{sl}(4)$. We construct a basis for the modules of this CFT in terms of the coupled free fermions so that their characters can be computed in terms of so-called universal chiral partition functions (‘fermionic partition functions’), mathematically giving alternate explicit expressions for the string functions of affine Lie algebra modules. In Section 3 we give another construction of coupled free fermions, associated to (rescaled) lattice CFTs, and observe an intriguing relation between coset construction for $\mathfrak{sl}(4)$ and the lattice CFT for $\mathfrak{sl}(3)$

Some details regarding the proof for the bases are given in Appendix A, while the proofs of the character identities can be found in a companion paper BCH .

Given an affine Lie algebra $\hat{\mathfrak{g}}$ at level $k$, one can construct a parafermionic conformal field theory containing fields which are non-local, of fractional spin, and chiral, as described in GEPNER87 . For simplicity, we only focus on the holomorphic part and assume that $\mathfrak{g}$ is simply-laced.

We associate a parafermionic field $\psi^{(\alpha)}$ to each $\alpha\in Q$, where $Q$ is the root lattice of $\mathfrak{g}$, and identify $\psi^{(\alpha)}$ and $\psi^{(\beta)}$ if $\alpha\equiv\beta\mod kQ$. Among these parafermionic fields, we are particularly interested in the so-called generating parafermions $\{\psi^{(\alpha)}\mid\alpha\in\Delta\}$, where $\Delta$ is the set of roots of $\mathfrak{g}$. The OPEs of generating parafermions are determined so that the following fields

$\displaystyle J^{\alpha}(z)=\sqrt{\dfrac{2k}{\alpha^{2}}}c_{\alpha}\psi_{\alpha}e^{i\alpha\cdot\phi(z)/\sqrt{k}},\;\;\alpha\in\Delta,$

$\displaystyle J^{i}(z)=\dfrac{2i\sqrt{k}}{\alpha_{i}^{2}}\alpha_{i}\cdot\partial_{z}\phi(z)\;,\;\;\alpha_{i}\in\Delta_{0},$

where $c_{\alpha}$ are cocycles, $\phi(z)$ is the vector of $n$ free bosons and $\Delta_{0}$ is the set of simple roots of $\mathfrak{g}$, obey the current algebra commutation relations

$$J^{a}(z)J^{b}(w)\sim\dfrac{k\kappa^{ab}}{(z-w)^{2}}+\dfrac{f^{ab}{}_{c}\,J^{c}(w)}{z-w},$$

where $J^{a}(z)$ stands for either $J^{\alpha}(z)$ or $J^{i}(z)$ defined above, $\kappa^{ab}$ is the Killing form and $f^{ab}{}_{c}$ are structure constants of $\mathfrak{g}$ in the Chevalley basis.

It turns out that the cocycles and OPEs are given by

	$\displaystyle c_{\alpha}c_{-\alpha}=1,\;\;\;\;c_{\alpha}c_{\beta}=S_{\alpha,\beta}c_{\alpha+\beta},$
	$\displaystyle\psi^{(\alpha)}(z)\psi^{(-\alpha)}(w)\sim\dfrac{1}{(z-w)^{2-|\alpha|^{2}/k}},$		(1)
	$\displaystyle\psi^{(\alpha)}(z)\psi^{(\beta)}(w)\sim\dfrac{c_{\alpha,\beta}\psi^{(\alpha+\beta)}(w)}{(z-w)^{1+(\alpha,\beta)/k}},$		(2)

where $S_{\alpha,\beta}$ are some roots of unity and $c_{\alpha,\beta}$ are some constants to be fixed by the Borcherds identity (2).

Denote the energy-momentum tensor of the current algebra in the Sugawara form by $L^{(c)}(z)$, and that of the (uncoupled free) bosonic systems by $L^{(b)}(z)$. The energy-momentum tensor $L(z)$ of the parafermionic system is then $L(z)=L^{(c)}(z)-L^{(b)}(z)$ and the central charge of the parafermionic system is

$$c\left(\dfrac{\hat{\mathfrak{g}}_{k}}{\widehat{\mathfrak{u}(1)}^{n}}\right)=c\left(\hat{\mathfrak{g}}_{k}\right)-c\left(\widehat{\mathfrak{u}(1)}^{n}\right)=\dfrac{k\dim\mathfrak{g}}{k+h^{\vee}}-n.$$

(3)

where $h^{\vee}$ is the dual Coxeter number of $\mathfrak{g}$.

The conformal dimension of a parafermionic field $\psi^{(\alpha)}$ with respect to $L(z)$ is given by

$$h(\alpha)=-\dfrac{|\alpha|^{2}}{2k}+n(\alpha),$$

(4)

where $n(\alpha)$ is the minimal number of roots of $\mathfrak{g}$ from which $\alpha$ is composed. The mode expansions of parafermionic fields $\psi^{(\alpha)}$ are of the form

$$\psi^{(\alpha)}(z)=\sum_{n\in\mathbb{Z}+\epsilon(\alpha)}\psi^{(\alpha)}_{n}z^{-n-h(\alpha)},$$

(5)

where the twisting $\epsilon\in\mathbb{R}/\mathbb{Z}$ depends on relevant OPEs. Unitarity is defined by the conjugation relation

$$\left(\psi^{(\alpha)}_{n}\right)^{{\dagger}}=\psi^{(\alpha)}_{-n},$$

which is consistent with the generalised commutation relations (14).

One can derive generalised commutation relations between the modes of parafermionic fields from the Borcherds identity for parafermionic fields

		$\displaystyle\sum_{j\geq 0}\binom{m}{j}[[AB]_{n+1+j}C]_{m+k+1-j}$
	$\displaystyle=$	$\displaystyle(-1)^{j}\binom{n}{j}\sum_{j\geq 0}[A[BC]_{k+1+j}]_{m+n+1-j}-\mu_{AB}(-1)^{\alpha_{AB}-n}[B[AC]_{m+1+j}]_{n+k+1-j},$		(6)

where $n\equiv\alpha_{AB}\mod\mathbb{Z}$, $m\equiv\alpha_{AC}\mod\mathbb{Z}$, $k\equiv\alpha_{BC}\mod\mathbb{Z}$ and the commutation factors $\mu_{AB}$ and singularity $\alpha_{AB}$ are given by the axiom

$$A(z)B(w)(z-w)^{\alpha_{AB}}=\mu_{AB}B(w)A(z)(w-z)^{\alpha_{AB}}.$$

(7)

The parafermionic primary fields of $\hat{\mathfrak{g}}_{k}/\widehat{\mathfrak{u}(1)}^{n}$, denoted by $\Phi^{\Lambda}_{\lambda}$, are labelled by $\Lambda\in\hat{P}^{(k)}_{+}$, a dominant integral weight at level $k$, and an integral weight $\lambda\in\hat{P}$ such that $\Lambda-\lambda\in\hat{Q}$, where $\hat{Q}$ is the root lattice of $\hat{\mathfrak{g}}$. We notice that $\Phi^{\Lambda}_{\lambda}$ and $\Phi^{\Lambda^{\prime}}_{\lambda^{\prime}}$ are identified if

$$\Lambda^{\prime}=\Lambda\;\;\text{and}\;\;\lambda^{\prime}\equiv\lambda\mod k\hat{Q}$$

or

$$\Lambda^{\prime}=\sigma\Lambda\;\;\text{and}\;\;\lambda^{\prime}=\sigma\lambda,$$

where $\sigma\in\text{Aut}(\text{Dyn}(\mathfrak{g}))$. The modules of $\hat{\mathfrak{g}}_{k}/\widehat{\mathfrak{u}(1)}^{n}$, denoted by $\mathcal{L}^{\Lambda}_{\lambda}$, are therefore labelled and identified in the same manner.

The generating parafermions $\psi^{(\alpha)}$ act on modules $\mathcal{L}^{\Lambda}_{\lambda}$ as chiral vertex operators (CVOs). Consider a generic CVO

$$\phi^{\Lambda^{(i)}}_{\lambda^{(i)}}\binom{i}{k\;j}(z):\mathcal{L}^{\Lambda^{(j)}}_{\lambda^{(j)}}\to\mathcal{L}^{\Lambda^{(k)}}_{\lambda^{(k)}},$$

whose mode expansion is

$$\phi^{\Lambda^{(i)}}_{\lambda^{(i)}}\binom{i}{k\;j}(z)=\sum_{n\in\mathbb{Z}}\phi^{\Lambda^{(i)}}_{\lambda^{(i)}}\binom{i}{k\;j}_{n-(h^{\Lambda^{(k)}}_{\lambda^{(k)}}-h^{\Lambda^{(j)}}_{\lambda^{(j)}})}z^{-n+(h^{\Lambda^{(k)}}_{\lambda^{(k)}}-h^{\Lambda^{(j)}}_{\lambda^{(j)}}-h^{\Lambda^{(i)}}_{\lambda^{(i)}})},$$

(8)

where $h^{\Lambda}_{\lambda}$ is the conformal dimension of the field $\Phi^{\Lambda}_{\lambda}$, given by

$$h^{\Lambda}_{\lambda}=\dfrac{(\Lambda,\Lambda+2\rho)}{2(k+h^{\vee})}-\dfrac{|\lambda|^{2}}{2k}+n^{\Lambda}_{\lambda},$$

(9)

where $n^{\Lambda}_{\lambda}$ is some integer. (If $\Lambda=k\Lambda_{0}$, then $n^{\Lambda}_{\lambda}$ may be described as the minimal number of finite roots from which the finite part of $\lambda$ is composed. If $\lambda$ is a weight in the representation $\Lambda$, then $n^{\Lambda}_{\lambda}=0$.)

The number of such CVOs are determined by the fusion rules of $\hat{\mathfrak{g}}_{k}/\widehat{\mathfrak{u}(1)}^{n}$, which are given by

$$\Phi^{\Lambda^{(i)}}_{\lambda^{(i)}}\times\Phi^{\Lambda^{(j)}}_{\lambda^{(j)}}=\sum_{k}\mathcal{N}_{ij}^{k}\,\Phi^{\Lambda^{(k)}}_{\lambda^{(i)}+\lambda^{(j)}\mod k\hat{Q}}$$

(10)

where $\mathcal{N}_{ij}^{k}$ are fusion rules of $\hat{\mathfrak{g}}_{k}$.

The characters of $\mathcal{L}^{\Lambda}_{\lambda}$, denoted by $b^{\Lambda}_{\lambda}$, are defined in the standard way, that is,

$$b^{\Lambda}_{\lambda}(\tau):=\text{Tr}_{\big{|}{\mathcal{L}^{\Lambda}_{\lambda}}}\ q^{\left(L_{0}-c/24\right)}\,,\qquad q=e^{2\pi i\tau}$$

where $L_{0}$ is the zeroth mode of the energy-momentum tensor $L(z)$ of the parafermionic system. Recalling that $L(z)=L^{(c)}(z)-L^{(b)}(z)$, we have, as shown in GEPNER87 ,

$$b^{\Lambda}_{\lambda}(\tau)=\dfrac{\text{Tr}_{\big{|}\mathcal{L}^{\Lambda}_{\lambda}}\;q^{\left(L_{0}^{(c)}-c(\hat{\mathfrak{g}}_{k})/24\right)}}{\text{Tr}_{\big{|}\mathcal{L}^{\Lambda}_{\lambda}}\;q^{\left(L_{0}^{(b)}-n/24\right)}}=\eta(\tau)^{n}c^{\Lambda}_{\lambda}(\tau),$$

(11)

where $\eta(\tau)$ is the Dedekind eta function and $c^{\Lambda}_{\lambda}(\tau)$ are the string functions of $\hat{\mathfrak{g}}_{k}$ KACPETERSON .

We notice that when $\mathfrak{g}$ is simply-laced and $k=2$, this construction is of particular interest because then the OPEs (1) and (2) for generating parafermions reduce to

	$\displaystyle\psi^{(\alpha)}(z)\psi^{(-\alpha)}(w)\sim\dfrac{1}{(z-w)},$		(12)
	$\displaystyle\psi^{(\alpha)}(z)\psi^{(\beta)}(w)\sim\dfrac{c_{\alpha,\beta}\psi^{(\alpha+\beta)}(w)}{(z-w)^{1/2}},\;\;\text{if}\;\alpha+\beta\in\Delta.$		(13)

Keeping in mind that $\alpha\equiv-\alpha\mod 2Q$ and the conformal dimension of any generating parafermion $\psi^{(\alpha)}$ is $\frac{1}{2}$ according to (4), we see that (12) tells us that we have a set of real free fermions $\{\psi^{(\alpha)}\mid\alpha\in\Delta_{+}\}$ and (13) tells us that these fermions are coupled to each other according to the root structure of $\mathfrak{g}$. Therefore, we will call such a system a “coupled free fermion CFT” hereafter.

It is not hard to see that when $\mathfrak{g}=\mathfrak{sl}_{2}$, the coupled free fermion CFT simply recovers the free fermion CFT. Thus, coupled free fermion CFTs are natural generalisations of the free fermion CFT.

For a coupled free fermion CFT $\widehat{\mathfrak{sl}(n+1)}_{2}/\widehat{\mathfrak{u}(1)}^{n}$, the twisting $\epsilon$ in the mode expansion (5) can only be $0$ or $\frac{1}{2}$, corresponding to NS- or R-sector respectively. The generalised commutation relations¹¹1Also known as $Z$-algebra relations Lepowsky1984 . between the modes of coupled free fermions derived from (2) are given in DING1994 ; BorisPF as follows:

	$\displaystyle\psi^{(\alpha)}_{n}\psi^{(\alpha)}_{m}+\psi^{(\alpha)}_{m}\psi^{(\alpha)}_{n}$	$\displaystyle=\delta_{m+n,0},$			(14)
	$\displaystyle\psi^{(\alpha)}_{n}\psi^{(\beta)}_{m}+\mu_{\alpha,\beta}\psi^{(\beta)}_{m}\psi^{(\alpha)}_{n}$	$\displaystyle=0,$	$\displaystyle\text{if}\;\alpha+\beta\not\in\Delta,$
	$\displaystyle\sum_{l\geq 0}\binom{l-\frac{1}{2}}{l}\left(\psi^{(\alpha)}_{m-\frac{1}{2}-l}\psi^{(\beta)}_{n+\frac{1}{2}+l}+\mu_{\alpha,\beta}\psi^{(\beta)}_{n-l}\psi^{(\alpha)}_{m+l}\right)$	$\displaystyle=c_{\alpha,\beta}\psi^{(\alpha+\beta)}_{m+n},$	$\displaystyle\text{if}\;\alpha+\beta\in\Delta,$

where the commutator factors $\mu_{\alpha,\beta}$ are some roots of unity that satisfy

$\displaystyle\mu_{\alpha,\beta}\mu_{\beta,\alpha}=1,$

$\displaystyle\mu_{\alpha,\beta}c_{\alpha,\beta}=c_{\beta,\alpha}.$

It is also calculated in DING1994 ; BorisPF using (2) that

$$L(z)=\frac{4}{n+3}\sum_{\alpha\in\Delta_{+}}L^{(\alpha)}(z),$$

(15)

where $L^{(\alpha)}(z)$ is the energy-momentum tensor of a free fermion subalgebra:

$$L^{(\alpha)}(z):=-\frac{1}{2}:\psi^{(\alpha)}(z)\partial\psi^{(\alpha)}(z):.$$

The mode expansion of $L(z)$ is then given by

$$L(z)=\sum_{n}L_{n}z^{-n-2},$$

with

$$L_{n}=\frac{4}{n+3}\sum_{\alpha\in\Delta_{+}}L_{n}^{(\alpha)},$$

where

$$L_{n}^{(\alpha)}=\left\{\begin{aligned} &\frac{1}{2}\sum_{r\in\mathbb{Z}+\frac{1}{2}}\left(r+\frac{1}{2}n\right):\psi^{(\alpha)}_{-r}\psi^{(\alpha)}_{n+r}:&(\text{NS}),\\ &\frac{1}{2}\sum_{r\in\mathbb{Z}}\left(r+\frac{1}{2}n\right):\psi^{(\alpha)}_{-r}\psi^{(\alpha)}_{n+r}:+\frac{1}{16}\delta_{n,0}&(\text{R}).\end{aligned}\right.$$

Here, whether the mode numbers are integers or half-integers is determined by (8) depending on which particular intertwiner we are considering.

It is easily checked that the modes of $L(z)$ form the Virasoro algebra with central charge

$$c\left(\dfrac{\widehat{\mathfrak{sl}(n+1)}_{2}}{\widehat{\mathfrak{u}(1)}^{n}}\right)=\dfrac{n(n+1)}{n+3},$$

(16)

which agrees with (3). We notice that this number is the same as the sum of the first $n$ central charges of the discrete series of unitary minimal models, that is,

$$c\left(\dfrac{\widehat{\mathfrak{sl}(n+1)}_{2}}{\widehat{\mathfrak{u}(1)}^{n}}\right)=\sum_{m=1}^{n}\left(1-\dfrac{6}{(m+2)(m+3)}\right).$$

(17)

Indeed, it has been verified in BelGep that the coset parafermionic CFT $\widehat{\mathfrak{sl}(n+1)}_{2}/\widehat{\mathfrak{u}(1)}^{n}$ can be decomposed into minimal models, taking these cosets as special cases of the AGT correspondence. Detailed analysis and examples of such decompositions can be found in BHphDthesis .

In the case of $\widehat{\mathfrak{sl}(3)}_{2}/\widehat{\mathfrak{u}(1)}^{2}$, let $\alpha_{1},\alpha_{2}$ denote the simple roots of $\mathfrak{sl}(3)$. Then the positive roots of $\mathfrak{sl}(3)$ are $\{\alpha_{1},\alpha_{2},\alpha_{3}:=\alpha_{1}+\alpha_{2}\}$ and we have 3 coupled free fermions $\psi^{(1)}:=\psi^{(\alpha_{1})},\psi^{(2)}:=\psi^{(\alpha_{2})}$ and $\psi^{(3)}:=\psi^{(\alpha_{3})}$. There are 8 inequivalent primary fields in this model.

The algebraic structure of this model was studied in BorisPF , where the author noted that a kind of Poincaré-Birkhoff-Witt (PBW) theorem should hold for the model, but they failed to determine one. Despite the fact that, in Ard2002 , the author had found a basis involving (positive and negative) modes of parafermions associated with simple roots, with our understanding of coupled free fermions in terms of chiral vertex operators together with the explicit generalised commutation relations, we have found that it is straightforward to state a basis of PBW type in terms of non-positive modes of all coupled free fermions.

Let $\mathcal{B}_{1}$ be set of states in the form of

	$\displaystyle\psi^{(3)}_{-N_{3}-\frac{N_{1}+N_{2}}{2}+\frac{1}{2}-s^{(3)}_{N_{3}}}\dots\psi^{(3)}_{-\frac{N_{1}+N_{2}}{2}-\frac{1}{2}-s^{(3)}_{1}}$	$\displaystyle\psi^{(2)}_{-N_{2}-\frac{N_{1}}{2}+\frac{1}{2}-s^{(2)}_{N_{2}}}\dots$		(18)
		$\displaystyle\dots\psi^{(2)}_{-\frac{N_{1}}{2}-\frac{1}{2}-s^{(2)}_{1}}\psi^{(1)}_{-N_{1}+\frac{1}{2}-s^{(1)}_{N_{1}}}\dots\psi^{(1)}_{-\frac{1}{2}-s^{(1)}_{1}}\ket{0}$

with $N_{1},N_{2},N_{3}\geq 0$ and $s^{(i)}_{N}\geq\dots s^{(i)}_{2}\geq s^{(i)}_{1}\geq 0$ for $i=1,2,3$.

Let $\mathcal{B}_{2}$ be the set of states in the form of

	$\displaystyle\psi^{(3)}_{-N_{3}-\frac{N_{1}+N_{2}}{2}+1-s^{(3)}_{N_{3}}}\dots\psi^{(3)}_{-\frac{N_{1}+N_{2}}{2}-s^{(3)}_{1}}$	$\displaystyle\psi^{(2)}_{-N_{2}-\frac{N_{1}}{2}+1-s^{(2)}_{N_{2}}}\dots$		(19)
		$\displaystyle\dots\psi^{(2)}_{-\frac{N_{1}}{2}-s^{(2)}_{1}}\psi^{(1)}_{-N_{1}+\frac{1}{2}-s^{(1)}_{N_{1}}}\dots\psi^{(1)}_{-\frac{1}{2}-s^{(1)}_{1}}\ket{1^{\prime}}$

with $N_{1},N_{2},N_{3}\geq 0$ and $s^{(i)}_{N}\geq\dots s^{(i)}_{2}\geq s^{(i)}_{1}\geq 0$ for $i=1,2,3$.

We claim that $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are the bases of the untwisted sector and twisted sector of $\widehat{\mathfrak{sl}(3)}_{2}/\widehat{\mathfrak{u}(1)}^{2}$ respectively. One can inductively prove both the independence and the spanning property of our proposed basis with extensive manipulations of generalised commutation relations (14). Instead, in this paper we prove the spanning property by an inductive proof (see Appendix A) and conclude the independence from the equality of dimensions. The latter can be proven by a combinatorial argument as given in Ard2002 or by number theory techniques as given in BCH using Universal Chiral Partition Functions (UCPFs) KKMM1992 ; KKMM1993 ; berkovich1999universal ; BouwknegtUCPF .

In the case of $\widehat{\mathfrak{sl}(4)}_{2}/\widehat{\mathfrak{u}(1)}^{3}$, let $\alpha_{1},\alpha_{2},\alpha_{3}$ denote the simple roots of $\mathfrak{sl}(4)$ such that $(\alpha_{1},\alpha_{3})=0$. Then the positive roots are $\Delta_{+}=\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}:=\alpha_{1}+\alpha_{2},\alpha_{5}:=\alpha_{2}+\alpha_{3},\alpha_{6}:=\alpha_{1}+\alpha_{2}+\alpha_{3}\}$ and therefore we have 6 coupled free fermions $\{\psi^{(i)}:=\psi^{(\alpha_{i})}\mid\alpha_{i}\in\Delta_{+}\}$. There are 24 inequivalent primary fields in this model.

The algebraic structure of this model is also briefly mentioned in BorisPF . Although the author noted that the commutation factors between the orthogonal fermions are equal $\pm 1$, they did not specify any constraint to determine the sign. We have found that the constraints for $\widehat{\mathfrak{sl}(4)}_{2}/\widehat{\mathfrak{u}(1)}^{3}$ are

$$\mu_{54}=\dfrac{c_{14}c_{21}}{\mu_{26}c_{61}c_{15}}=\dfrac{c_{12}c_{64}}{c_{16}c_{25}},$$

(20)

where $\mu_{ij}:=\mu_{\alpha_{i},\alpha_{j}}$ and $c_{ij}:=c_{\alpha_{i},\alpha_{j}}$. Explicitly, a compatible choice of parameters could be

		$\displaystyle c_{ij}=\mu_{ij}c_{ji},\;\mu_{ij}\mu_{ji}=1,$		(21)
		$\displaystyle\mu_{13}=\mu_{26}=1,\;\mu_{54}=-1,$
	$\displaystyle\mu_{12}=\mu_{24}=\mu_{41}=\mu_{15}=$	$\displaystyle\mu_{56}=\mu_{61}=\mu_{23}=\mu_{35}=\mu_{52}=\mu_{43}=\mu_{64}=\mu_{36}=x^{2},$
	$\displaystyle c_{12}=c_{24}=c_{41}=c_{15}=$	$\displaystyle c_{56}=c_{61}=c_{23}=c_{35}=c_{52}=c_{43}=c_{64}=c_{36}=\dfrac{x}{\sqrt{2}}.$

where $x$ is again an 8th root of unity which is chosen to be $e^{-\frac{i\pi}{4}}$.

Let $\mathcal{B}_{3}$ be the set of states in the form of

		$\displaystyle\psi^{(6)}_{-N_{6}-\frac{N_{1}+N_{3}+N_{4}+N_{5}}{2}+\frac{1}{2}-s^{(6)}_{N_{6}}}\dots\psi^{(6)}_{-\frac{N_{1}+N_{3}+N_{4}+N_{5}}{2}-\frac{1}{2}-s^{(6)}_{1}}\psi^{(5)}_{-N_{5}-N_{4}-\frac{N_{1}+N_{2}+N_{3}}{2}+\frac{1}{2}-s^{(5)}_{N_{5}}}\dots$		(22)
		$\displaystyle\dots\psi^{(5)}_{-N_{4}-\frac{N_{1}+N_{2}+N_{3}}{2}-\frac{1}{2}-s^{(5)}_{1}}\psi^{(4)}_{-N_{4}-\frac{N_{1}+N_{2}+N_{3}}{2}+\frac{1}{2}-s^{(4)}_{N_{4}}}\dots\psi^{(4)}_{-\frac{N_{1}+N_{2}+N_{3}}{2}-\frac{1}{2}-s^{(4)}_{1}}\psi^{(3)}_{-N_{3}-\frac{N_{2}}{2}+\frac{1}{2}-s^{(3)}_{N_{3}}}\dots$
		$\displaystyle\dots\psi^{(3)}_{-\frac{N_{2}}{2}-\frac{1}{2}-s^{(3)}_{1}}\psi^{(2)}_{-N_{2}-\frac{N_{1}}{2}+\frac{1}{2}-s^{(2)}_{N_{2}}}\dots\psi^{(2)}_{-\frac{N_{1}}{2}-\frac{1}{2}-s^{(2)}_{1}}\psi^{(1)}_{-N_{1}+\frac{1}{2}-s^{(1)}_{N_{1}}}\dots\psi^{(1)}_{-\frac{1}{2}-s^{(1)}_{1}}\ket{0}$

with $N_{1},N_{2},\dots,N_{6}\geq 0$ and $s^{(i)}_{N}\geq\dots s^{(i)}_{2}\geq s^{(i)}_{1}\geq 0$ for $i=1,2,\dots,6$.

We claim that $\mathcal{B}_{3}$ is a basis of PBW-type of the untwisted sector of $\widehat{\mathfrak{sl}(4)}_{2}/\widehat{\mathfrak{u}(1)}^{3}$. Analogous results for twisted sectors are included in BHphDthesis . In order to prove these results, we need explicit expressions of the coset characters of $\widehat{\mathfrak{sl}(4)}_{2}/\widehat{\mathfrak{u}(1)}^{3}$ or, in other words, the string functions of $\widehat{\mathfrak{sl}(4)}_{2}$, according to (11). The expressions are as follows:

$$\begin{array}[]{lp{1.6cm}l}\lx@intercol b^{2000}_{2000}(\tau)+b^{2000}_{0020}(\tau)=\dfrac{\eta(4\tau)^{4}\eta(6\tau)^{8}}{\eta(\tau)\eta(2\tau)^{4}\eta(3\tau)^{3}\eta(12\tau)^{4}}+\dfrac{\eta(2\tau)^{8}\eta(3\tau)\eta(12\tau)^{4}}{\eta(\tau)^{5}\eta(4\tau)^{4}\eta(6\tau)^{4}},\hfil\lx@intercol\\ b^{2000}_{2000}(\tau)-b^{2000}_{0020}(\tau)=\dfrac{\eta(\tau)^{2}}{\eta(2\tau)^{2}},&&b^{2000}_{0101}(\tau)=\dfrac{\eta(2\tau)^{2}\eta(6\tau)^{2}}{\eta(\tau)^{3}\eta(3\tau)},\\ b^{0101}_{2000}(\tau)=b^{0101}_{0002}(\tau)=\dfrac{3\eta(6\tau)^{3}}{\eta(\tau)^{2}\eta(2\tau)},&&b^{0101}_{0101}(\tau)=\dfrac{\eta(2\tau)^{3}\eta(3\tau)^{2}}{\eta(\tau)^{4}\eta(6\tau)},\\ b^{1100}_{1100}(\tau)=\dfrac{\eta(4\tau)^{5}}{\eta(\tau)^{3}\eta(8\tau)^{2}},&&b^{1100}_{0011}(\tau)=\dfrac{2\eta(2\tau)^{2}\eta(8\tau)^{2}}{\eta(\tau)^{3}\eta(4\tau)},\end{array}$$

(23)

where affine weights are given in terms of Dynkin labels. The complete proofs of these expressions and the basis are given in BCH ; BHphDthesis .

Motivated by recent investigations of the scaled lattice CFTs $V_{\sqrt{2}A_{n}}$ Bae2021 ; LAM2004614 ; KITAZUME2000893 , where $A_{n}$ denotes the root lattice of $\mathfrak{sl}(n+1)$, we construct another family of coupled free fermions based on the lattice $\frac{1}{\sqrt{2}}A_{n}$. In this CFT, each root of $A_{n}$ contributes a field of conformal dimension $\frac{1}{2}$, i.e. a fermion. For each $\alpha\in A_{n}$, define

$$\psi^{\alpha}(z)=:e^{i\frac{\alpha}{\sqrt{2}}\cdot\Phi(z)}:$$

and then the non-trivial OPEs between the generating fields $\psi^{\alpha}$ with $\alpha\in\Delta$ read as follows:

	$\displaystyle\psi^{\alpha}(z)\overline{\psi^{\alpha}}(w)$	$\displaystyle\sim\dfrac{1}{z-w},$		(24)
	$\displaystyle\psi^{\alpha}(z)\psi^{\beta}(w)$	$\displaystyle\sim\dfrac{c_{\alpha,\beta}\psi^{\alpha+\beta}(w)}{(z-w)^{1/2}},\;\text{if}\;\;\alpha+\beta\in\Delta,$		(25)

where we used the fact that $\psi^{-\alpha}(z)=\overline{\psi^{\alpha}}(z)$. Now, (24) suggests that each $\psi^{\alpha}$ is a free complex fermion and (25) suggests that the complex fermions are coupled to each other, so we indeed have coupled free fermions in $V_{\frac{1}{\sqrt{2}}A_{n}}$ as desired.

We shall derive an energy-momentum tensor $L(z)$ in terms of coupled free fermions as in (15). First, by expanding $e^{\frac{i}{\sqrt{2}}\alpha\cdot\Phi(z)}e^{-\frac{i}{\sqrt{2}}\alpha\cdot\Phi(z)}$, we see that

$$:\psi^{\alpha}(z)\overline{\psi^{\alpha}}(z):=\frac{i}{\sqrt{2}}\alpha\cdot\partial\Phi(z).$$

Then we have

$$L^{\alpha}(z):=-\frac{1}{2}:\psi^{\alpha}(z)\partial\overline{\psi^{\alpha}}(z):=\frac{1}{2}(\frac{i}{\sqrt{2}}\alpha\cdot\partial\Phi)(z)(\frac{i}{\sqrt{2}}\alpha\cdot\partial\Phi)(z).$$

We construct the total energy-momentum tensor as

$$L(z)=\frac{2}{n+1}\sum_{\alpha\in\Delta_{+}}L^{\alpha}(z)=-\frac{1}{n+1}\sum_{\alpha\in\Delta_{+}}:\psi^{\alpha}(z)\partial\overline{\psi^{\alpha}}(z):$$

so that the conformal weight of the field $\psi^{\alpha}$ for any $\alpha\in A_{n}$ is indeed $\frac{1}{2}$.

Since $\frac{1}{\sqrt{2}}A_{n}$ is not integral, $\frac{1}{\sqrt{2}}A_{n}\not\subset\left(\frac{1}{\sqrt{2}}A_{n}\right)^{*}$ and we cannot expect to classify the modules by the cosets of $\frac{1}{\sqrt{2}}A_{n}$ in its dual as one does for $V_{\sqrt{2}A_{n}}$. However, we notice that $\frac{1}{\sqrt{2}}A_{n}\subset\frac{1}{2}\left(\frac{1}{\sqrt{2}}A_{n}\right)^{*}=\frac{1}{\sqrt{2}}A^{*}_{n}$, so we may expect the modules correspond to the coset $\left(\frac{1}{\sqrt{2}}A^{*}_{n}\right)/\left(\frac{1}{\sqrt{2}}A_{n}\right)$. If this is the case, for $\gamma\in\left(\frac{1}{\sqrt{2}}A^{*}_{n}\right)/\left(\frac{1}{\sqrt{2}}A_{n}\right)$, the character would be given by

$$\dfrac{1}{\eta(\tau)^{n}}\sum_{\nu\in\gamma+\frac{1}{\sqrt{2}}A_{n}}q^{\frac{1}{2}|\nu|^{2}}.$$

(26)

We can first test our conjecture on $V_{\frac{1}{\sqrt{2}}A_{1}}$. Let $\alpha$ denote the root of $A_{1}$. It is obvious that there is only one complex free fermion $\psi^{\alpha}$ in $V_{\frac{1}{\sqrt{2}}A_{1}}$. We can rewrite the complex free fermion $\psi^{\alpha}$ in terms of two real free fermions by setting

$\displaystyle\chi^{\alpha}(z)=\frac{1}{\sqrt{2}}\left(\psi^{\alpha}(z)+\overline{\psi^{\alpha}}(z)\right),$

$\displaystyle\xi^{\alpha}(z)=\frac{1}{i\sqrt{2}}\left(\psi^{\alpha}(z)-\overline{\psi^{\alpha}}(z)\right),$

for which it can be seen that

$\chi^{\alpha}(z)\chi^{\alpha}(w)\sim\dfrac{1}{z-w}$, $\xi^{\alpha}(z)\xi^{\alpha}(w)\sim\dfrac{1}{z-w}$, $\chi^{\alpha}(z)\xi^{\alpha}(w)\sim 0$.

Therefore we should expect $V_{\frac{1}{\sqrt{2}}A_{1}}$-modules corresponding to the NS and R sectors of two (uncoupled) free fermions.

On the other hand, we know $\frac{1}{\sqrt{2}}A_{1}^{*}=\mathbb{Z}\left\{\frac{1}{2\sqrt{2}}\alpha\right\}$ and therefore we expect two modules $V_{\frac{1}{\sqrt{2}}A_{1}}$ and $V_{\frac{1}{2\sqrt{2}}\alpha+\frac{1}{\sqrt{2}}A_{1}}$, whose characters are, according to (26),

	$\displaystyle\mathrm{ch}\left[V_{\frac{1}{\sqrt{2}}A_{1}}\right](\tau)$	$\displaystyle=\dfrac{1}{\eta(\tau)}\sum_{n\in\mathbb{Z}}q^{\frac{1}{4}|n\alpha|^{2}}=\dfrac{1}{\eta(\tau)}\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}n^{2}},$		(27)
	$\displaystyle\mathrm{ch}\left[V_{\frac{1}{2\sqrt{2}}\alpha+\frac{1}{\sqrt{2}}A_{1}}\right](\tau)$	$\displaystyle=\dfrac{1}{\eta(\tau)}\sum_{n\in\mathbb{Z}}q^{\frac{1}{4}\left|\left(n+\frac{1}{2}\right)\alpha\right|^{2}}=\dfrac{1}{\eta(\tau)}\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}\left(n+\frac{1}{2}\right)^{2}}.$

Comparing (27) and characters of fermions $\chi_{h}$, with $h$ the conformal dimension of the corresponding representation, given in SchCFT , we note that

$\displaystyle\textstyle\mathrm{ch}\left[V_{\frac{1}{\sqrt{2}}A_{1}}\right]=\left(\chi_{0}+\chi_{\frac{1}{2}}\right)^{2},$

$\displaystyle\textstyle\mathrm{ch}\left[V_{\frac{1}{\sqrt{2}}A_{1}}\right]=\left(\chi_{\frac{1}{16}}\right)^{2},$

which can be proved by the Jacobi triple identity

$$\prod_{k=1}^{\infty}(1-q^{k})(1-z^{-1}q^{k-\frac{1}{2}})(1-zq^{k-\frac{1}{2}})=\sum_{k\in\mathbb{Z}}(-1)^{k}z^{k}q^{\frac{1}{2}k^{2}}.$$

We shall proceed to investigate $V_{\frac{1}{\sqrt{2}}A_{2}}$. Let $\{\alpha_{1},\alpha_{2}\}$ denote the set of simple roots of $A_{2}$ and set $\alpha_{3}=\alpha_{1}+\alpha_{2}$. Then we have three complex fermions $\psi^{\alpha_{1}}$, $\psi^{\alpha_{2}}$, $\psi^{\alpha_{3}}$ in $V_{\frac{1}{\sqrt{2}}A_{2}}$. We can also rewrite them in terms of real fermions by setting, for $i=1,2,3$,

$\displaystyle\chi^{i}(z)=\frac{1}{\sqrt{2}}\left(\psi^{\alpha_{i}}(z)+\overline{\psi^{\alpha_{i}}}(z)\right),$

$\displaystyle\xi^{i}(z)=\frac{1}{i\sqrt{2}}\left(\psi^{\alpha_{i}}(z)-\overline{\psi^{\alpha_{i}}}(z)\right).$

(28)