On Twisted Modules for N=2 Supersymmetric Vertex Operator Superalgebras

Abstract

The classification of twisted modules for N = 2 supersymmetric vertex operator superalgebras with twisting given by vertex operator superalgebra automorphisms which are lifts of a finite automorphism of the N = 2 Neveu–Schwarz Lie superalgebra representation is presented. These twisted modules include the Ramond-twisted sectors and mirror-twisted sectors for N = 2 vertex operator super algebras, as well as twisted modules related to more general “spectral flow” representations of the N = 2 Neveu–Schwarz algebra.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

We give an expository presentation following [6] of the classification of twisted modules for N=2 superconformal vertex operator superalgebras for the case of vertex operator superalgebra automorphisms that arise from finite Virasoro-preserving automorphisms of the underlying N=2 Neveu–Schwarz algebra.

For g an automorphism of a vertex operator superalgebra (VOSA), V, we have the notion of “g-twisted V -module”. Twisted vertex operators were discovered and used in [24], and twisted modules arose in [19] in the course of the construction of the moonshine module vertex operator algebra. This structure came to be understood as an “orbifold model” in the sense of conformal field theory and string theory. Twisted modules are the mathematical counterpart of “twisted sectors”, which are the basic building blocks of orbifold models in conformal field theory and string theory. The notion of twisted module for VOSAs was developed in [25]. In general, it is an open problem as to how to construct a g-twisted V -module.

An automorphism g of a VOSA, in particular, fixes the Virasoro vector, and thus also fixes the corresponding endomorphisms giving the representation of the Virasoro algebra. A VOSA is said to be “N=2 supersymmetric”, if in addition to being a positive energy representation for the Virasoro algebra, it is a representation of the N=2 Neveu–Schwarz Lie superalgebra, an extension of the Virasoro algebra; see, for instance, [11,  5]. The group of automorphisms of the N=2 Neveu–Schwarz algebra over ℂ, which preserve the Virasoro algebra, is isomorphic to ℂ ^× ×ℤ ₂. It is generated by a continuous family of automorphisms, denoted by σ_ξ for ξ ∈ ℂ ^×, and an order two automorphism κ called the “mirror map”. If ξ is a root of unity, then σ_ξ is of finite order.

Given an N=2 supersymmetric VOSA, V, some questions naturally arise: When does κ or σ_ξ lift to an automorphism of V, and when is this lift unique? When such an automorphism of the N=2 Neveu–Schwarz algebra does lift to an automorphism g of V, what is the structure of a g-twisted V module? In this paper, we present the answer to the second question, and in [6] we provide the details of this study and answer the first question for free and lattice N=2 VOSAs.

If the mirror automorphism κ of the N=2 Neveu–Schwarz algebra lifts to a VOSA automorphism of an N=2 VOSA, V, then a “mirror-twisted V -module” is naturally a representation of what we call the “mirror-twisted N=2 superconformal algebra”, which is also referred to as the “twisted N=2 superconformal algebra” [29,  16,  26], or the “topological N=2 superconformal algebra” [21]. If the automorphism σ_ξ of the N=2 Neveu–Schwarz algebra, for ξ a root of unity, lifts to a VOSA automorphism of V, then we show that a “σ_ξ-twisted V -module” is naturally a representation of one of the algebras in the one-parameter family of Lie superalgebras we call “shifted N=2 superconformal algebras”. If ξ =  − 1, then σ_ξ is the parity map, and such a shifted N=2 superconformal algebra is the N=2 Ramond algebra. The N=2 Ramond algebra and the other shifted N=2 algebras are isomorphic, as Lie superalgebras, to the N=2 Neveu–Schwarz algebra via the “spectral flow” operators, as was first realized in [29]. The mirror-twisted N=2 algebra is not isomorphic to the N=2 Neveu–Schwarz algebra.

The representation theory of the N=2 Neveu–Schwarz algebra has been studied in, for instance, [11,  10,  9,  28,  13,  27,  23,  14,  30,  18,  17,  15,  31] and from a VOSA theoretic point of view in [1]. The representation theory of the N=2 Ramond algebra has been studied in, e.g, [13,  30,  18,  21,  20], and of the mirror-twisted N=2 superconformal algebra in, e.g, [13,  16,  21,  22,  26].

The realization of the N=2 Ramond algebra (i.e. the $\frac{1} {2}$-shifted N=2 superconformal algebra) and the mirror-twisted N=2 superconformal algebra as arising from twisting an N=2 VOSA (or comparable structure) has long been known, e.g. [29,  9,  11]. However to our knowledge, the other algebras related to the N=2 Neveu–Schwarz algebra—the shifted N=2 superconformal algebras other than the N=2 Ramond algebra—have only been studied through the spectral flow operators (which do not preserve the Virasoro algebra). We believe that the realization of these algebras as arising naturally as twisted modules for an N=2 VOSA is new.

Thus this complete classification of the twisted modules for an N=2 VOSA for finite automorphisms arising from Virasoro-preserving automorphisms of the N=2 Neveu–Schwarz algebra provides a uniform way of understanding and studying all of the N=2 superconformal algebras—the continuous one-parameter family of shifted N=2 Neveu–Schwarz algebras and the mirror-twisted N=2 superconformal algebra—in the context of the theory of VOSAs and their twisted modules.

2 The N=1 and N=2 Superconformal Algebras and Their Virasoro-Preserving Automorphisms

The N=1 Neveu–Schwarz algebra or N=1 superconformal algebra is the Lie superalgebra with basis consisting of the central element d, even elements L _n for n ∈ ℤ, and odd elements G _r for $r \in \mathbb{Z} + \frac{1} {2}$, and supercommutation relations

$$\displaystyle\begin{array}{rcl} \left [L_{m},L_{n}\right ]& =& (m - n)L_{m+n} + \frac{1} {12}({m}^{3} - m)\delta _{ m+n,0}\;d, \end{array}$$

(1)

$$\displaystyle\begin{array}{rcl}\left [L_{m},G_{r}\right ]& =& \left (\frac{m} {2} - r\right )G_{m+r},\ \ \ \left [G_{r},G_{s}\right ]\ =\ 2L_{r+s} + \frac{1} {3}\left ({r}^{2} -\frac{1} {4}\right )\delta _{r+s,0}\;d,\ \ \end{array}$$

(2)

for m, n ∈ ℤ, and $r,s \in \mathbb{Z} + \frac{1} {2}$. The N=1 Ramond algebra is the Lie superalgebra with basis the central element d, even elements L _n for n ∈ ℤ, and odd elements G _r for r ∈ ℤ, and supercommutation relations given by (1)–(2), where now r, s ∈ ℤ.

Note that the only nontrivial Lie superalgebra automorphism of the N=1 Neveu–Schwarz algebra which preserves the Virasoro algebra is the parity automorphism which is the identity on the even subspace (the Virasoro Lie algebra) and acts as − 1 on the odd subspace (the subspace spanned by G _r for $r \in \mathbb{Z} + \frac{1} {2}$).

The N=2 Neveu–Schwarz Lie superalgebra or N=2 superconformal algebra is the Lie superalgebra with basis consisting of the central element d, even elements L _n and J _n for n ∈ ℤ, and odd elements G _r ^(j) for j = 1, 2 and $r \in \mathbb{Z} + \frac{1} {2}$, and such that the supercommutation relations are given as follows: L _n , d and G _r ^(j) satisfy the supercommutation relations for the N=1 Neveu–Schwarz algebra (1)–(2) for both G _r  = G _r ⁽¹⁾ and for G _r  = G _r ⁽²⁾; the remaining relations are given by

$$\displaystyle\begin{array}{rcl} \left [L_{m},J_{n}\right ]& =& -nJ_{m+n},\quad \ \left [J_{m},J_{n}\right ]\ =\ \frac{1} {3}m\delta _{m+n,0}d, \end{array}$$

(3)

$$\displaystyle\begin{array}{rcl}\left [J_{m},G_{r}^{(1)}\right ]& =& -iG_{ m+r}^{(2)},\left [J_{ m},G_{r}^{(2)}\right ]\ =\ iG_{ m+r}^{(1)},\left [G_{ r}^{(1)},G_{ s}^{(2)}\right ]\ =\ -i(r - s)J_{ r+s}.\ \ \ \ \ \end{array}$$

(4)

The N=2 Ramond algebra is the Lie superalgebra with basis consisting of the central element d, even elements L _n and J _n for n ∈ ℤ, and odd elements G _r ^(j) for r ∈ ℤ and j = 1, 2, and supercommutation relations given by those of the N=2 Neveu–Schwarz algebra but with r, s ∈ ℤ, instead of $r,s \in \mathbb{Z} + \frac{1} {2}$.

More generally, there is an infinite family of algebras which includes the N=2 Neveu–Schwarz and Ramond algebras. However these are easier to express if we make a change of basis which is ubiquitous in superconformal field theory. So consider the substitutions $G_{r}^{(1)} = \frac{1} {\sqrt{2}}\left (G_{r}^{+} + G_{r}^{-}\right )$, and $G_{r}^{(2)} = \frac{i} {\sqrt{2}}\left (G_{r}^{+} - G_{r}^{-}\right )$, or equivalently $G_{r}^{\pm } = \frac{1} {\sqrt{2}}(G_{r}^{(1)} \mp iG_{r}^{(2)})$. This substitution is equivalent to the change of variables ${\varphi }^{\pm } = \frac{1} {\sqrt{2}}({\varphi }^{(1)} \pm i{\varphi }^{(2)})$ in the variables (x, φ⁽¹⁾, φ⁽²⁾) representing the one even and two odd local coordinates on an N=2 superconformal worldsheet representing superstrings propagating in space-time in N=2 superconformal field theory, cf. [4]. In terms of this basis (called the homogeneous basis), the N=2 Neveu–Schwarz (or Ramond) algebra supercommutation relations are given by (1), (3) and

$$\displaystyle\begin{array}{rcl} \left [L_{m},G_{r}^{\pm }\right ]& =& \left (\frac{m} {2} - r\right )G_{m+r}^{\pm },\quad \left [J_{ m},G_{r}^{\pm }\right ]\ =\ \pm G_{ m+r}^{\pm },\quad \left [G_{ r}^{\pm },G_{ s}^{\pm }\right ]\ =\ 0,\ \ \ \ \ \end{array}$$

(5)

$$\displaystyle\begin{array}{rcl}\qquad \left [G_{r}^{+},G_{ s}^{-}\right ]& =& 2L_{ r+s} + (r - s)J_{r+s} + \frac{1} {3}\left ({r}^{2} -\frac{1} {4}\right )\delta _{r+s,0}\;d, \end{array}$$

(6)

for m, n ∈ ℤ, and $r,s \in \mathbb{Z} + \frac{1} {2}$, or r, s ∈ ℤ, respectively.

Observe then that there is also the notion of a Lie superalgebra generated by even elements L _n and J _n for n ∈ ℤ and by odd elements G _r ± t ^±, for $r \in \mathbb{Z} + \frac{1} {2}$ and for t ∈ ℂ. We shall call this algebra the t-shifted N=2 superconformal algebra or t-shifted N=2 Neveu–Schwarz algebra. Thus the t-shifted N=2 Neveu–Schwarz algebra is the N=2 Neveu–Schwarz algebra if t ∈ ℤ, and is the N=2 Ramond algebra if $t \in \mathbb{Z} + \frac{1} {2}$. As was first shown in [29], the t-shifted N=2 Neveu–Schwarz algebras are all isomorphic under the continuous family of spectral flow maps, denoted $\mathcal{D}(t)$, for t ∈ ℂ, but which fix the Virasoro algebra only for t = 0. These are given by

$$\mathcal{D}(t) : \quad \begin{array}{ll} L_{n}\ \mapsto \ L_{n} + tJ_{n} + \frac{{t}^{2}} {6} \delta _{n,o}d,&\ \ d\ \mapsto \ d, \\ J_{n}\ \,\mapsto \ J_{n} + \frac{t} {3}\delta _{n,0}d, &\ G_{r}^{\pm }\ \mapsto \ G_{r\pm t}^{\pm }.\end{array}$$

(7)

The group of automorphisms of the N=2 Neveu–Schwarz algebra (or more generally the t-shifted N=2 superconformal algebras) which preserve the Lie subalgebra generated by L _n and J _n for n ∈ ℤ are given by:

$$\sigma _{\xi } : \quad G_{r}^{\pm }\mapsto {\xi }^{\pm 1}G_{ r}^{\pm },\quad J_{ n}\mapsto J_{n},\quad L_{n}\mapsto L_{n},\quad d\mapsto d,$$

(8)

for ξ ∈ ℂ ^×. In addition, we have the mirror map, given by:

$$\kappa : \quad G_{r}^{\pm }\mapsto G_{ r}^{\mp },\quad J_{ n}\mapsto - J_{n},\quad L_{n}\mapsto L_{n},\quad d\mapsto d.$$

(9)

The family σ_ξ along with κ generate the Virasoro-preserving automorphisms of the N=2 Neveu–Schwarz algebra, and thus this group is isomorphic to ℤ ₂ ×ℂ ^×, cf. [5].

The mirror-twisted N=2 Neveu–Schwarz algebra is defined to be the Lie superalgebra with basis consisting of even elements L _n , and J _r and central element d, odd elements G _r ⁽¹⁾ and G _n ⁽²⁾, for n ∈ ℤ and $r \in \mathbb{Z} + \frac{1} {2}$, and supercommutation relations given as follows: The L _n and G _r ⁽¹⁾ satisfy the supercommutation relations for the N=1 Neveu–Schwarz algebra with central charge d; the L _n and G _n ⁽²⁾ satisfy the supercommutation relations for the N=1 Ramond algebra with central charge d; and the remaining supercommutation relations are

$$\displaystyle\begin{array}{rcl} \left [L_{n},J_{r}\right ]& =& -rJ_{n+r},\quad \left [J_{r},J_{s}\right ]\ =\ \frac{1} {3}r\delta _{r+s,0}d \end{array}$$

(10)

$$\displaystyle\begin{array}{rcl} & & \\ \left [J_{r},G_{s}^{(1)}\right ]& =& -iG_{ r+s}^{(2)},\quad \left [J_{ r},G_{n}^{(2)}\right ]\ =\ iG_{ r+n}^{(1)},\quad \left [G_{ r}^{(1)},G_{ n}^{(2)}\right ]\ =\ -i(r - n)J_{ r+n}.\ \ \ \ \ \ \\ & & \end{array}$$

(11)

Note that this mirror-twisted N=2 Neveu–Schwarz algebra is not isomorphic to the ordinary N=2 Neveu–Schwarz algebra [29].

3 The Notions of VOSA, Supersymmetric VOSA and Twisted Module

In this section, we recall the notions of VOSA, and N=1 or N=2 Neveu–Schwarz VOSA, following the notation and terminology of [2,  3] and [5]. We also recall the notion of g-twisted V -module for a VOSA, V, and an automorphism g of V of finite order following the notation of, e.g. [25,  12,  7,  8].

Let x, x ₀, x ₁, x ₂, denote commuting independent formal variables. Let $\delta (x) =\sum _{n\in \mathbb{Z}}{x}^{n}$. Expressions such as (x ₁ − x ₂)ⁿ for n ∈ ℂ are to be understood as formal power series expansions in nonnegative integral powers of the second variable.

Definition 3.1.

A vertex operator superalgebra is a $\frac{1} {2}\mathbb{Z}$-graded vector space $V =\bigoplus _{n\in \frac{1} {2} \mathbb{Z}}V _{n}$, satisfying dim V < ∞ and V _n  = 0 for n sufficiently negative, that is also ℤ ₂-graded by sign, V = V ⁽⁰⁾ ⊕ V ⁽¹⁾, and equipped with a linear map

$$V \rightarrow (\mbox{ End}\,V )[[x,{x}^{-1}]],\quad v\mapsto Y (v,x) =\displaystyle\sum _{ n\in \mathbb{Z}}v_{n}{x}^{-n-1},$$

(12)

and with two distinguished vectors 1 ∈ V ₀, (the vacuum vector) and ω ∈ V ₂ (the conformal element) satisfying the following conditions for u, v ∈ V : u _n v = 0 for n sufficiently large; Y (1, x)v = v; Y (v, x)1 ∈ V [[x]], and lim _x → 0 Y (v, x)1 = v;

$$\displaystyle\begin{array}{rcl} & & x_{0}^{-1}\delta \left (\frac{x_{1} - x_{2}} {x_{0}} \right )Y (u,x_{1})Y (v,x_{2}) - {(-1)}^{\vert u\vert \vert v\vert }x_{ 0}^{-1}\delta \left (\frac{x_{2} - x_{1}} {-x_{0}} \right )Y (v,x_{2})Y (u,x_{1})\ \ \ \ \ \\ & & \quad = x_{2}^{-1}\delta \left (\frac{x_{1} - x_{0}} {x_{2}} \right )Y (Y (u,x_{0})v,x_{2})\ \ \ \ \ \end{array}$$

(13)

(the Jacobi identity), where | v |  = j if v ∈ V ^(j) for j ∈ ℤ ₂; writing $Y (\omega,x) =\sum _{n\in \mathbb{Z}}L(n){x}^{-n-2}$, i.e. L(n) = ω _n + 1, for n ∈ ℤ, then the L(n) give a representation of the Virasoro algebra with central charge c ∈ ℂ (the central charge of V ); for $n \in \frac{1} {2}\mathbb{Z}$ and v ∈ V _n , L(0)v = nv = (wt v)v; and the L(−1)-derivative property holds: $\frac{d} {dx}Y (v,x) = Y (L(-1)v,x)$.

If a VOSA, (V, Y, 1, ω), contains an element τ ∈ V _3 ∕ 2 such that writing $Y (\tau,z) =\sum _{n\in \mathbb{Z}}\tau _{n}{x}^{-n-1} =\sum _{n\in \mathbb{Z}}G(n + 1/2){x}^{-n-2}$, the G(n + 1 ∕ 2) = τ _n + 1 ∈ (End V )⁽¹⁾ generate a representation of the N=1 Neveu–Schwarz Lie superalgebra, then we call (V, Y, 1, τ) an N=1 Neveu–Schwarz VOSA, or an N=1 supersymmetric VOSA, or just an N=1 VOSA for short.

If a VOSA (V, Y, 1, ω) has two vectors τ⁽¹⁾ and τ⁽²⁾ such that (V, Y, 1, τ^(j)) is an N=1 VOSA for both j = 1 and j = 2, and the τ _n + 1 ^(j) = G ^(j)(n + 1 ∕ 2) generate a representation of the N=2 Neveu–Schwarz Lie superalgebra, then we call such a VOSA an N=2 Neveu–Schwarz VOSA or an N=2 supersymmetric VOSA, or for short, an N=2 VOSA. If V is an N=2 VOSA, then there exists a vector $\mu = \frac{i} {2}{G}^{(1)}(1/2){\tau }^{(2)} = -\frac{i} {2}{G}^{(2)}(1/2){\tau }^{(1)} \in V _{ (1)}$ such that writing Y (μ, x) = $\sum _{n\in \mathbb{Z}}\mu _{n}{x}^{-n-1} =\sum _{n\in \mathbb{Z}}J(n){x}^{-n-1}$, we have that the J(n) ∈ (End V )⁰ along with the G ^(j)(n + 1 ∕ 2) and L(n) = ω _n + 1 for $\omega = \frac{1} {2}{G}^{(j)}(-1/2){\tau }^{(j)}$ satisfy the supercommutation relations for the N=2 Neveu–Schwarz Lie superalgebra.

For an N=2 VOSA, it follows from the definition that ω = L( − 2)1, τ^(j) = G ^(j)( − 3 ∕ 2)1, for j = 1, 2, μ = J( − 1)1, and

$$L(n)\mathbf{1} = {G}^{(j)}(n + 1/2)\mathbf{1} = J(n + 1)\mathbf{1} = 0,\quad \mbox{ for $n \geq -1$, $j = 1,2$}.$$

(14)

If V is an N=2 VOSA such that V is not only $\frac{1} {2}\mathbb{Z}$ graded by L(0) but also ℤ-graded by J(0) such that J(0)v = nv with n ≡ j mod 2, for v ∈ V ^(j) for j = 0, 1, then we say that V is J(0)-graded or graded by charge.

An automorphism of a VOSA, V, is a linear map g from V to itself, preserving 1 and ω such that the actions of g and Y (v, x) on V are compatible in the sense that gY (v, x)g ^− 1 = Y (gv, x), for v ∈ V. Then gV _n  ⊂ V _n for $n \in \frac{1} {2}\mathbb{Z}$.

If g has finite order, V is a direct sum of the eigenspaces V ^j of g, i.e., V =  _{j ∈ ℤ ∕ kℤ} V ^j, where k ∈ ℤ ₊ is a period of g (i.e., g ^k = 1 but k is not necessarily the order of g) and V ^j = { v ∈ V   |  gv = η^j v}, for η a fixed primitive k-th root of unity.

Definition 3.2.

Let (V, Y, 1, ω) be a VOSA and g an automorphism of V of period k ∈ ℤ ₊. A weak g-twisted V -module is a vector space M equipped with a linear map

$$V \rightarrow (\mbox{ End}\,M)[[{x}^{1/k},{x}^{-1/k}]],\quad v\mapsto {Y }^{g}(v,x) =\displaystyle\sum _{ n\in \frac{1} {k} \mathbb{Z}}v_{n}^{g}{x}^{-n-1},$$

(15)

satisfying the following conditions for u, v ∈ V and w ∈ M: v _n ^g w = 0 for n sufficiently large; Y ^g(1, x)w = w; ${Y }^{g}(v,x) =\sum _{n\in \mathbb{Z}+ \frac{j} {k} }v_{n}^{g}{x}^{-n-1}$ for j ∈ ℤ ∕ kℤ and v ∈ V ^j;

$$\displaystyle\begin{array}{rcl} & & x_{0}^{-1}\delta \left (\frac{x_{1} - x_{2}} {x_{0}} \right ){Y }^{g}(u,x_{ 1}){Y }^{g}(v,x_{ 2}) - {(-1)}^{\vert u\vert \vert v\vert }x_{ 0}^{-1}\delta \left (\frac{x_{2} - x_{1}} {-x_{0}} \right ){Y }^{g}(v,x_{ 2}){Y }^{g}(u,x_{ 1})\ \ \\ & & \quad = x_{2}^{-1} \frac{1} {k}\displaystyle\sum _{j\in \mathbb{Z}/k\mathbb{Z}}\delta \left ({\eta }^{j}\frac{{(x_{1} - x_{0})}^{1/k}} {x_{2}^{1/k}} \right ){Y }^{g}(Y ({g}^{j}u,x_{ 0})v,x_{2})\ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}$$

(16)

(the twisted Jacobi identity) where η is a fixed primitive k-th root of unity.

If we take g = 1, then we obtain the notion of weak V -module. The term “weak” means we are making no assumptions about a grading on M.

As a consequence of the definition, we have the following supercommutator relation on M for u ∈ V ^j:

$$\displaystyle\begin{array}{rcl} [{Y }^{g}(u,x_{ 1}),{Y }^{g}(v,x_{ 2})]\ \ \ && \\ & & \quad =\mathrm{ Res}_{x_{0}}x_{2}^{-1}\delta \left (\frac{x_{1} - x_{0}} {x_{2}} \right ){\left (\frac{x_{1} - x_{0}} {x_{2}} \right )}^{-j/k}{Y }^{g}(Y (u,x_{ 0})v,x_{2}).\end{array}$$

(17)

It also follows that writing ${Y }^{g}(\omega,x) =\sum _{n\in \mathbb{Z}}{L}^{g}(n){x}^{-n-2}$, i.e., setting L ^g(n) = ω _n + 1 ^g, for n ∈ ℤ, then the L ^g(n) satisfy the relations for the Virasoro algebra with central charge c the central charge of V . And the L( − 1)-derivative property for the twisted vertex operators holds:

$$\frac{d} {dx}{Y }^{g}(u,x) = {Y }^{g}(L(-1)u,x).$$

(18)

4 Twisting by Automorphisms Arising from Virasoro-Preserving Automorphisms of the N=2 Neveu–Schwarz Algebra

In this section, we give the structure of weak g-twisted V -modules for V an N=2 VOSA and g any automorphism of V which is a lift of a Virasoro-preserving automorphism of the N=2 Neveu–Schwarz algebra of finite order.

We first consider the mirror map κ. If an N=2 VOSA, V, with central charge c, has a VOSA automorphism κ such that κ(μ) =  − μ, κ(τ⁽¹⁾) = τ⁽¹⁾ and κ(τ⁽²⁾) =  − τ⁽²⁾, then such a VOSA automorphism of V is called an N=2 VOSA mirror map. If such a map exists for V, and M is a weak κ-twisted module for V, then write Y ^κ for the κ-twisted operators, and

$$\begin{array}{llllll} {Y }^{\kappa }(\omega,x)\ \ \ & = \displaystyle\sum _{n\in \mathbb{Z}}{L}^{\kappa }(n){x}^{-n-2}, &{Y }^{\kappa }({\tau }^{(1)},x)\ \ \ & = \displaystyle\sum _{ r\in \mathbb{Z}+\frac{1} {2} }{G}^{(1),\kappa }(r){x}^{-r-\frac{3} {2} } \\ {Y }^{\kappa }(\mu,x)\ \ \ & =\ \displaystyle\sum _{r\in \mathbb{Z}+\frac{1} {2} }{J}^{\kappa }(r){x}^{-r-1},\ \ \ &{Y }^{\kappa }({\tau }^{(2)},x)\ \ \ & = \displaystyle\sum _{ n\in \mathbb{Z}}{G}^{(2),\kappa }(n){x}^{-n-\frac{3} {2} }.\end{array}$$

(19)

That is, define J ^κ(n) = μ _n ^κ and G ^(2), κ(n − 1 ∕ 2) = τ _n ^(2), κ, for $n \in \mathbb{Z} + \frac{1} {2}$. Then, using the supercommutator relation (17) for the κ-twisted vertex operators acting on M, using the L( − 1)-derivative property (18), using the N=2 Neveu–Schwarz supercommutation relations on V, and using (14), we have that the supercommutation relations for the κ-twisted modes of ω, μ, τ⁽¹⁾ and τ⁽²⁾, given by L ^κ(n), G ^(2), κ(n), for n ∈ ℤ, and J ^κ(r), G ^(1), κ(r), for $r \in \mathbb{Z} + \frac{1} {2}$, satisfy the relations of the mirror-twisted N=2 Neveu–Schwarz algebra given by (10)–(11) with central charge c.

In particular, a weak κ-twisted module, M, for an N=2 VOSA reduces the N=2 Neveu–Schwarz algebra representation to an N=1 Neveu–Schwarz algebra representation coupled with an N=1 Ramond algebra representation.

Next we consider the automorphisms σ_ξ which are of finite order. Let η = e ^{2πi ∕ k}, for k ∈ ℤ ₊, and let ξ = η^j, for $j = 1,\ldots,k - 1$. Let σ_ξ be a VOSA automorphism of an N=2 VOSA, V, such that σ_ξ(μ) = μ and σ_ξ(τ^( ± )) = ξ^± 1τ^( ± ), for ${\tau }^{(\pm )} = \frac{1} {\sqrt{2}}({\tau }^{(1)} \mp i{\tau }^{(2)})$. Then ω, μ ∈ V ⁰ and τ^( ± ) ∈ V ^± j. If such a map exists for V, and M is a weak σ_ξ-twisted module for V, then write ${Y }^{\sigma _{\xi }}$ for the σ_ξ-twisted operators, and

$$\begin{array}{cc} {Y }^{\sigma _{\xi }}(\omega,x) = \displaystyle\sum _{ n\in \mathbb{Z}}{L}^{\sigma _{\xi } }(n){x}^{-n-2},\quad {Y }^{\sigma _{\xi } }(\mu,x) = \displaystyle\sum _{n\in \mathbb{Z}}{J}^{\sigma _{\xi } }(n){x}^{-n-1} \\ \\ \\ \\ {Y }^{\sigma _{\xi }}({\tau }^{(\pm )},x) = \ \ \ \displaystyle\sum _{ r\in \mathbb{Z}-\frac{1} {2} \pm \frac{j} {k} }\ \ \ \ {G}^{\pm,\sigma _{\xi } }(r){x}^{-r-\frac{3} {2} }.\end{array}$$

(20)

Then using the supercommutator relation (17) for the σ_ξ-twisted vertex operators acting on M, using the L( − 1)-derivative property (18), using the N=2 Neveu–Schwarz supercommutation relations on V, and using (14), we have that the supercommutation relations for the σ_ξ-twisted modes of ω, μ, and τ^( ± ), that is the ${L}^{\sigma _{\xi }}(n)$ and ${J}^{\sigma _{\xi }}(n)$ for n ∈ ℤ, and G ^±(r) for $r \in \mathbb{Z} + \frac{1} {2} \pm \frac{j} {k}$, respectively, satisfy the relations for the $\frac{j} {k}$-shifted N=2 Neveu–Schwarz algebra (1), (3), (5)–(6) with central charge c.

That is the sectors for N=2 supersymmetric VOSAs that arise under spectral flow $\mathcal{D}(t)$, for t = j ∕ k, k ∈ ℤ ₊, $j = 1,\ldots,k - 1$ are twisted sectors under the Virasoro-preserving automorphisms σ_ξ of the N=2 Neveu–Schwarz algebra.

If ξ =  − 1, then the map σ_ξ always extends to V, via the parity map σ_− 1(v) = ( − 1)^| v | v, for v ∈ V . In this case, a weak σ_− 1-twisted V -module is a representation of the N=2 Ramond algebra. If V is an N=2 VOSA which is also J(0)-graded such that the J(0) eigenvalues are integral with J(0)ω = J(0)μ = 0, and J(τ^( ± )) =  ± τ^( ± ), then setting σ_ξ(v) = ξⁿ v if J(0)v = nv gives a VOSA automorphism.

We summarize these results as follows:

Theorem 4.1.

If the Virasoro-preserving automorphisms of the N=2 Neveu–Schwarz algebra, κ, and σ _ξ , for ξ a root of unity, extend to VOSA automorphisms for an N=2 VOSA, V, then:

• (i) A weak κ-twisted V -module is a representation of the mirror-twisted N=2 superconformal algebra.

• (ii) A weak σ _ξ -twisted V -module, for ξ = e ^2jπi∕k , is a representation of the $\frac{j} {k}$ -shifted N=2 superconformal algebra. If ξ = −1, such a VOSA automorphism always exists (the parity map), and in this case, a weak σ ₋₁ -twisted V -module is a representation of the N=2 Ramond algebra.

In [6], we also show, in particular, that if V is a free or lattice N=2 VOSA, then each Virasoro-preserving automorphism of the N=2 Neveu–Schwarz algebra extends to a VOSA automorphism of V, but not uniquely in the case of the mirror map. In fact, we show that there are two distinct mirror maps for free and lattice N=2 VOSAs and these mirror maps give nonisomorphic mirror-twisted V -modules. We also construct examples of g-twisted V -modules for g = κ and g = σ_ξ of finite order.