Homotopy algebras of differential (super)forms in three and four dimensions

Abstract

We consider various $A_{\infty }$-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding $A_{\infty }$-structures. In addition, for $N=2$ 3D space, we construct the homotopic counterpart of the de Rham complex, which is related to the superfield formulation of the $N=2$ Chern–Simons theory.

Appendices

Appendix A: $A_{\infty }$-algebras and the BV formalism

In this appendix, we summarize all necessary information about $A_{\infty }$-algebras. For more details, see, for example, [11, 14, 15].

1.1 Appendix A1: $A_{\infty }$-algebras

The $A_{\infty }$-algebra is a generalization of an associative algebra with a differential. Namely, consider a graded vector space $V=\oplus _k V_k$ with a differential Q. Consider the multilinear operations $\mu _r: V^{\otimes r}\rightarrow V$ of the degree $2-r$, such that $\mu _1=Q$.

Definition

The space V is an $A_{\infty }$-algebra if $\mu _n$ satisfy the following bilinear identity:

$$\begin{aligned} \sum ^{n-1}_{i=1}(-1)^{i}M_i\circ M_{n-i+1}=0 \end{aligned}$$

(61)

on $V^{\otimes n}$. Here, $M_s$ acts on $V^{\otimes m}$ ($m\ge s$) as the sum of all possible operators of the form $\mathbf{1}^{\otimes ^l}\otimes \mu _s\otimes \mathbf{1}^{\otimes ^{m-s-l}}$ taken with appropriate signs. In other words, $M_s:V^{\otimes ^m}\rightarrow V^{\otimes ^{m-s+1}}$ and

$$\begin{aligned} M_s=\sum ^{m-s}_{l=0}(-1)^{l(s+1)}\mathbf{1}^{\otimes ^l}\otimes \mu _s\otimes \mathbf{1}^{\otimes ^{m-s-l}}. \end{aligned}$$

(62)

Let’s write several relations which are satisfied by Q, $\mu _1$, $\mu _2$, $\mu _3$:

$$\begin{aligned}&Q^2=0,\nonumber \\&Q\mu _2(a_1,a_2)=\mu _2(Q a_1,a_2)+(-1)^{n_{a_1}}\mu _2(a_1,Q a_2),\nonumber \\&Q\mu _3(a_1,a_2, a_3)+\mu _3(Q a_1,a_2, a_3)+(-1)^{n_{a_1}}\mu _3(a_1,Q a_2, a_3)_h\nonumber \\&\quad +(-1)^{n_{a_1}+n_{a_2}}\mu _3( a_1, a_2, Q a_3)=\mu _2(\mu _2(a_1,a_2),a_3)-\mu _2(a_1,\mu _2(a_2,a_3)).\nonumber \\ \end{aligned}$$

(63)

In such a way, we see that if $\mu _n=0$, $n\ge 3$ , then we have just graded differential associative algebra. It appears that relations (61) can be encoded into one equation $\partial ^2=0$. To see this, one applies the desuspension operation (the operation which shifts the grading $s^{-1}: V_{k}\rightarrow V_{k-1}$) to $\mu _n$; i.e., one can define operations of degree 1: $\nu _n=s\nu _n {s^{-1}}^{\otimes ^n}$. More explicitly,

$$\begin{aligned} \nu _n(s^{-1} a_1,\ldots ,s^{-1} a_n)=(-1)^{s(a)}s^{-1}\mu _n(a_1,\ldots ,a_n), \end{aligned}$$

(64)

such that $s(a)=(1-n)n_{a_1}+(2-n)n_{a_2}+\cdots +a_{n-1}$. The relations between $\nu _n$ operations can be summarized in the following simple equations:

$$\begin{aligned} \sum ^n_{i=1}N_i\circ N_{n+1-i}=0. \end{aligned}$$

(65)

on $V^{\otimes n}$. Here, each $N_s$ acts on $V^{\otimes ^m}$ ($m\ge s$) as the sum of all operators $\mathbf{1}^{\otimes ^l}\otimes \nu _s\otimes \mathbf{1}^{\otimes ^k}$, such that $l+s+k=m$. Combining them into one operator $\partial =\sum _n\nu _n$, acting on a space $\oplus _kV^{\otimes ^k}$, relations (61) can be combined into one equation $\partial ^2=0$.

Another way to represent relations (61) as the nilpotency condition of some operator is by using the differential operators on a noncommutative manifold. Let the set $\{e_i\}$ be the homogeneous elements, which form a basis of V. Introduce noncommutative supercoordinates $x^i$ such that $X=x^ie_i$ has degree 1 on V. One can write a noncommutative vector field, such that

$$\begin{aligned} \mathbf{Q}x^i=\nu _{n;j_1,\ldots ,j_n}x^{j_1} \ldots x^{j_n}, \end{aligned}$$

(66)

where $\nu _{n;j_1, \ldots ,j_n}=\nu _n(e_{j_1}, \ldots ,e_{j_n})$. Then, relations (65) can be formulated as $\mathbf{Q}^2=0$.

1.2 Appendix A2: Cyclic structures, BV formalism and the generalized Maurer–Cartan equation

First, we define what cyclic structure is.

Definition

The $A_{\infty }$-algebra on a space V is called cyclic if there exists a nondegenerate pairing $\langle \cdot , \cdot \rangle$, such that it is graded symmetric

$$\begin{aligned} \langle a, b \rangle =-(-1)^{(n_a+1)(n_b+1)}\langle b,a \rangle \end{aligned}$$

(67)

and satisfies the following conditions for $\mu _n$:

$$\begin{aligned}&\langle a_1,\mu _{n-1}(a_2, \ldots ,a_n)\rangle \nonumber \\&\quad =(-1)^{(n-1)(a_1+a_2+1)+a_1(a_2+ \cdots +a_n)}\langle a_2,\mu _{n-1}(a_3, \ldots ,a_n,a_1)\rangle . \end{aligned}$$

(68)

It makes sense to define $\psi _n(a_1, \ldots ,a_n)=\langle a_1, \mu _{n-1}(a_2, \ldots , a_n)\rangle$. Consider again the element of degree 1 $X=x^ie_i$, where $x^i$ are noncommutative supercoordinates, and define the formal action functional:

$$\begin{aligned} S[X]=\sum ^{\infty }_{n=2}\frac{1}{n}\Psi _n(X, \ldots ,X). \end{aligned}$$

(69)

Let $\omega _{ij}=\langle e_i, e_j\rangle$. Then one can define a BV bracket:

$$\begin{aligned} (\alpha ,\beta )_{BV}=\frac{{\overleftarrow{\partial }}\alpha }{\partial x^i}\omega ^{ij}\frac{\overrightarrow{\partial }\beta }{\partial x^j} \end{aligned}$$

(70)

on the space of polynomials of $x^i$, such that S satisfies classical Master equation:

$$\begin{aligned} (S,S)_{BV}=0 \end{aligned}$$

(71)

Using this condition, one can find that S defines an odd vector field such that

$$\begin{aligned} (S,x^i)_{BV}=\nu _{n;j_1, \ldots ,j_n}x^{j_1} \ldots x^{j_n}, \end{aligned}$$

(72)

which coincides with odd vector field $\mathbf{Q}$ was defined in (66). Moreover, for a given action

$$\begin{aligned} S=v_{i_1i_2}x^{i_1}x^{i_2}+\sum _{n\ge 3}v_{i_1 \ldots i_n}x^{i_1} \ldots x^{i_n}, \end{aligned}$$

(73)

which is cyclic in $x^i$ and satisfies Master equation, we obtain the cyclic $A_{\infty }$-algebra (see, e.g., [4, 11]).

Varying action (69) with respect to X, one finds the equation of motion which is known as generalized Maurer–Cartan equation:

$$\begin{aligned} QX+\sum _{n\ge 2}\mu _n(X, \ldots ,X)=0. \end{aligned}$$

(74)

This equation is known to have the following symmetry

$$\begin{aligned} X\rightarrow Q\alpha +\sum _{n\ge 2,k}(-1)^{n-k}\mu _n(X, \ldots ,\alpha , \ldots ,X), \end{aligned}$$

(75)

where $\alpha$ is an element of degree 0 and k means the position of $\alpha$ in $\mu _n$.

1.3 Appendix A3: Transfer of the $A_{\infty }$ structure

Suppose you have two complexes which are quasiisomorphic and moreover homotopically equivalent. Moreover, suppose that on one of them there exists the structure of $A_{\infty }$-algebra. Then, there exists an $A_{\infty }$-algebra on another complex. The explicit formulas are given in [14] following the results of [13, 16].

In fact, in [14], there is even a more general statement. Let us formulate it in a precise way. Consider two complexes $(\mathcal {F},Q)$ and $(\mathfrak {K}, Q')$ such that there are maps $f:(\mathcal {F},Q)\rightarrow (\mathfrak {K}, Q')$, $g:(\mathfrak {K},Q')\rightarrow (\mathcal {F},Q)$ such that fg is chain homotopic to identity. In other words, there exists a map $H: (\mathfrak {K},Q')\rightarrow (\mathfrak {K}, Q')$ of degree ${-1}$, such that $fg=id+Q' H+H Q'$. Then, given an $A_{\infty }$-algebra structure $\hat{\mu }_{\{n\}}$ on $\mathfrak {K}$, such that $\hat{\mu }_1=Q'$, one can construct an $A_{\infty }$-algebra on $(\mathcal {F},Q)$ by means of the formula

$$\begin{aligned} \mu _n=g\circ p_n \circ f^{\otimes ^n}, \end{aligned}$$

(76)

where $p_n$ is obtained by means of consecutive recurrent procedure of applying the homotopy operators to $\tilde{\mu }_n$. The explicit formula is:

$$\begin{aligned} p_n=\sum _B(-1)^{\theta (r_1, \ldots ,r_k)}\hat{\mu }_k(H\circ p_{r_1}, \ldots , H\circ p_{r_k}), \end{aligned}$$

(77)

where $B=\{k, r_1, \ldots , r_k | 2 \le k \le n, r_1, \ldots , r_k \ge 1, r_1 + \cdots + r_k = n\}$, $\theta (r_1, \ldots ,r_k)=\sum _{1\le \alpha \le \beta \le _r}r_{\alpha }(r_{\beta }+1)$ and $p_2\equiv \hat{\mu }_2$, $p_1\circ H\equiv id$.

Appendix B: Explicit calculations for $A_{\infty }$ superalgebras of superforms

1.1 Appendix B1: Notations for $N=1$ 4D superspace

We keep the notations from [5]. The world-volume metric in the 4D space with the coordinates $x^{\mu }$ is $\eta _{\mu \nu }=\mathrm {diag}(-1,+1,+1,+1)$. The $N=1$ four-dimensional space has two anticommutative Weyl spinor coordinates $\theta ^{\alpha }, \theta ^{\dot{\alpha }}$. The spinor indices are raised and lowered by means of $C_{\alpha \beta }, C_{\dot{\alpha }\dot{\beta }}$, such that $C_{21}=-C_{12}=i$. One can define superderivatives:

$$\begin{aligned} D_{\alpha }=\partial _{\alpha }+\frac{i}{2}\bar{\theta }^{\dot{\beta }}\partial _{\alpha \dot{\beta }}, \quad \bar{D}_{\dot{\alpha }}=\bar{\partial }_{\dot{\alpha }}+\frac{i}{2}{\theta }^{\beta }\partial _{\beta \dot{\alpha }}, \end{aligned}$$

(78)

where $\partial _{\alpha \dot{\beta }}=\sigma _{\alpha \dot{\beta }}^{\mu }\partial _{\mu }$. Therefore, one can introduce the (anti)chiral scalar superfields, which are defined by the nilpotency of certain antiderivatives $D_{\dot{\alpha }}\Lambda (x,\theta )=0$ ($D_{\alpha }\bar{\Lambda }(x,\theta )=0$) The relations between superderivatives are:

$$\begin{aligned} \{D_{\alpha },D_{\dot{\beta }}\}=i\partial _{\alpha \dot{\beta }}, \quad \{D_{\alpha },D_{{\beta }}\}=0, \quad \{\bar{D}_{\dot{\alpha }},\bar{D}_{\dot{\beta }}\}=0. \end{aligned}$$

(79)

For calculations, it is also useful to introduce operators $D^2=C^{\alpha \beta }D_{\alpha }D_{\beta }$, $\bar{D}^2=C^{\dot{\alpha }\dot{\beta }}D_{\dot{\alpha }}D_{\dot{\beta }}$.

1.2 Appendix B2: Explicit calculations for the $A_{\infty }$-algebra of $N=1$ SUSY Yang–Mills theory

In this subsection, we show by explicit calculation that the bilinear operation we defined on complex (35) is homotopy associative and the differential acts on it as a derivation.

For simplicity, we denote $\mu _n(\cdot , \ldots , \cdot )$ as $(\cdot , \ldots , \cdot )_h$. We remind that we defined the graded symmetric bilinear operation $(\cdot ,\cdot )_h$, i.e., $(a_1,a_2)_h=(-1)^{|a_1||a_2|}(a_2,a_1)_h$ on complex (37) in the following way:

$$\begin{aligned}&(\Lambda _1,\Lambda _2)_h=\Lambda _1\Lambda _2,\nonumber \\&(\Lambda ,V)_h=\frac{1}{2}\Lambda V,\nonumber \\&(\Lambda ,W_{\alpha })_h=\Lambda W_{\alpha },\nonumber \\&(\Lambda ,\tilde{W}_{\alpha })_h=\Lambda \tilde{W}_{\alpha },\nonumber \\&(\Lambda ,\tilde{V})_h=\frac{1}{2}\Lambda \tilde{V},\nonumber \\&(\Lambda ,\tilde{\Lambda })_h=\Lambda \tilde{\Lambda },\nonumber \\&(V_1,V_2)_{h,{\alpha }}=-\frac{1}{2}\bar{D}^2(V_1D_{\alpha }V_2-D_{\alpha }V_1V_2),\nonumber \\&(V,W)_h=D_{\alpha }VW^{\alpha }+\frac{1}{2}D^{\alpha }W_{\alpha }V,\nonumber \\&(\tilde{W},W)_h=\tilde{W}^{\alpha }W_{\alpha },\nonumber \\&(\tilde{V}, V)=-\frac{1}{2}{\bar{D}}^2(\tilde{V}V),\nonumber \\&(W_1,W_2)_h=0,\quad (V, \tilde{W})_h=0, \end{aligned}$$

(80)

such that $\Lambda _1,\Lambda _2\in \Phi$; $V,V_1,V_2\in \Sigma$; $W,W_1,W_2\in \Theta$, $\tilde{W}\in \tilde{\Theta }$; $\tilde{V}\in \tilde{\Sigma }$; $\tilde{\Lambda }\in \tilde{\Xi }$.

Now we start checking that this operation satisfies all necessary relations of homotopy associative algebra. The first relation is the Leibniz rule, i.e.,

$$\begin{aligned} {\mathbb {D}}(a_1,a_2)_h=({\mathbb {D}}a_1,a_2)_h+(-1)^{|a_1|}(a_1,{\mathbb {D}}a_2)_h. \end{aligned}$$

(81)

Let us check it step by step:

$$\begin{aligned}&{\mathbb {D}}(\Lambda ,V)_h=\frac{1}{2}\bar{D}^2D_{\alpha }(\Lambda V)\nonumber \\&\quad =\frac{1}{2}\bar{D}^2(D_{\alpha }\Lambda V-\Lambda D_{\alpha }V)+\Lambda {\bar{D}}^2 D_{\alpha }V\nonumber \\&\quad =({\mathbb {D}}\Lambda ,V)_h+(\Lambda ,{\mathbb {D}}V)_h, \end{aligned}$$

(82)

$$\begin{aligned}&{\mathbb {D}}(\Lambda ,W)_h=D^{\alpha }(\Lambda W_{\alpha })+\Lambda W_{\alpha }\nonumber \\&\quad = D_{\alpha }\Lambda W^{\alpha }+\frac{1}{2}\Lambda D^{\alpha }W_{\alpha }+\Lambda W_{\alpha }+ \frac{1}{2}\Lambda D^{\alpha }W_{\alpha }\nonumber \\&\quad =({\mathbb {D}}\Lambda ,W)_h+(\Lambda ,{\mathbb {D}}W)_h, \end{aligned}$$

(83)

$$\begin{aligned}&{\mathbb {D}}(V,W)_h=\bar{D}^2(D_{\alpha }VW^{\alpha }+\frac{1}{2}VD^{\alpha }W_{\alpha })\nonumber \\&\quad =\bar{D}^2 D^{\alpha }VW_{\alpha }+\frac{1}{2}\bar{D}^2VD^{\alpha }W_{\alpha }\nonumber \\&\quad =({\mathbb {D}}V,W)_h-(V,{\mathbb {D}}W)_h, \end{aligned}$$

(84)

$$\begin{aligned}&{\mathbb {D}}(\Lambda ,\tilde{V})_h=\frac{1}{2}\bar{D}^2(\Lambda \tilde{V})=-\frac{1}{2}\Lambda \bar{D}^2\tilde{V}+ \Lambda \bar{D}^2\tilde{V}= ({\mathbb {D}}\Lambda ,\tilde{V})_h+(\Lambda ,{\mathbb {D}}\tilde{V})_h.\nonumber \\ \end{aligned}$$

(85)

Our next task is to verify that $(\cdot ,\cdot )_h$ satisfies homotopy associativity relation:

$$\begin{aligned}&(a_1,(a_2,a_3)_h)_h-((a_1,a_2)_h,a_3)_h+{\mathbb {D}}(a_1,a_2,a_3)_h+ ({\mathbb {D}}a_1,a_2,a_3)_h\nonumber \\&\quad +(-1)^{|a_1|}(a_1,{\mathbb {D}}a_2,a_3)_h+(-1)^{|a_1|+|a_2|}(a_1,a_2,{\mathbb {D}}a_3)_h=0, \end{aligned}$$

(86)

where $(\cdot ,\cdot ,\cdot )_h$ is the trilinear operation we will derive below.

Let us proceed as above, checking associativity step by step:

$$\begin{aligned}&(\Lambda ,(V_1,V_2)_h)_h-((\Lambda ,V_1)_h,V_2)_h\nonumber \\&\quad =-\frac{1}{2}\bar{D}^2(\Lambda V_1D_{\alpha }V_2-\Lambda D_{\alpha }V_1 V_2)+ \frac{1}{4}\bar{D}^2(\Lambda V_1D_{\alpha }V_2-V_1D_{\alpha }(\Lambda V_2))\nonumber \\&\quad =-\frac{1}{4}\bar{D}^2(\Lambda (V_1D_{\alpha }V_2- D_{\alpha }V_1 V_2)+D_{\alpha }\Lambda V_1V_2)\nonumber \\&\quad =-{\mathbb {D}}(\Lambda ,V_1,V_2)_h-({\mathbb {D}}\Lambda ,V_1,V_2)_h-(\Lambda ,{\mathbb {D}}V_1,V_2)_h+(\Lambda ,V_1,{\mathbb {D}}V_2)_h, \end{aligned}$$

(87)

$$\begin{aligned}&(V_1,(\Lambda ,V_2)_h)_h-((V_1,\Lambda )_h,V_2)_h\nonumber \\&\quad =-\frac{1}{4}\bar{D}^2 (V_1D_{\alpha }(\Lambda V_2)-D_{\alpha }V\Lambda V_2)+\frac{1}{4}\bar{D}^2 (V_1\Lambda D_{\alpha }V_2-D_{\alpha }(V_1\Lambda V_2)\nonumber \\&\quad =-\frac{1}{2}\bar{D}^2(V_1D_{\alpha }\Lambda V_2)\nonumber \\&=-{\mathbb {D}}(V_1,\Lambda ,V_2)_h-({\mathbb {D}}V_1,\Lambda ,V_2)_h+(V_1,{\mathbb {D}}\Lambda ,V_2)_h+(V_1,\Lambda ,{\mathbb {D}}V_2)_h, \end{aligned}$$

(88)

$$\begin{aligned}&(\Lambda _1,(\Lambda _2,V)_h)_h-((\Lambda _1,\Lambda _2)_h,V)_h=\frac{1}{4}(\Lambda _1\Lambda _2 V)-\frac{1}{2}(\Lambda _1\Lambda _2V)\nonumber \\&\quad =-\frac{1}{4}\Lambda _1\Lambda _2 V\nonumber \\&\quad =-{\mathbb {D}}(\Lambda _1,\Lambda _2,V)_h-({\mathbb {D}}\Lambda _1,\Lambda _2,V)_h-(\Lambda _1,{\mathbb {D}}\Lambda _2,V)_h-(\Lambda _1,\Lambda _2,{\mathbb {D}}V)_h.\nonumber \\ \end{aligned}$$

(89)

Since we have

$$\begin{aligned} ((\Lambda _1,V)_h,\Lambda _2)_h=(\Lambda _1,(V,\Lambda _2)_h)_h, \end{aligned}$$

(90)

we obtain

$$\begin{aligned}&(V_1,V_2,V_3)_h=\frac{1}{6}\bar{D}^2(V_1V_2D_{\alpha }V_3-2V_1(D_{\alpha }V_2)V_3+(D_{\alpha }V_1)V_2V_3)\nonumber \\&(\Lambda ,V_1,V_2)_h=(V_1,V_2,\Lambda )_h=\frac{1}{12}\Lambda V_1 V_2\nonumber \\&(V_1,\Lambda ,V_2)_h=\frac{1}{6}\Lambda V_1V_2 \end{aligned}$$

(91)

One can check that these expressions fit the formulas above. Then, we need to study the associativity relation involving W-terms, i.e.,

$$\begin{aligned}&(\Lambda , (V,W)_h)_h-((\Lambda ,V)_h,W)_h\nonumber \\&\quad =\frac{1}{2}\Lambda D_{\alpha }VW^{\alpha }+\frac{1}{4}D^{\alpha }W_{\alpha }V\Lambda -\frac{1}{2} D_{\alpha }(\Lambda V)W^{\alpha }-\frac{1}{4}\Lambda VD^{\alpha }W_{\alpha }\nonumber \\&\quad = -\frac{1}{2}D_{\alpha }\Lambda VW^{\alpha }\nonumber \\&\quad = -{\mathbb {D}}(\Lambda ,V,W)_h-({\mathbb {D}}\Lambda ,V,W)_h-(\Lambda ,{\mathbb {D}}V,W)_h+(\Lambda ,V,{\mathbb {D}}W)_h, \end{aligned}$$

(92)

where $(V_1,V_2, W)_h= \frac{1}{2}D_{\alpha }V_1V_2W^{\alpha }+\frac{1}{6}V_1V_2D_{\alpha }W^{\alpha }$ and $(\Lambda , V, \tilde{V})=\frac{1}{6}V_1V_2\tilde{V}$. Another terms involving W are

$$\begin{aligned}&(V_1,(V_2,W)_h)_h-((V_1,V_2)_h,W)_h\nonumber \\&\quad =-\frac{1}{2}\bar{D}^2\left( \frac{1}{2}V_1\left( D_{\alpha }V_2W^{\alpha }+\frac{1}{2}D^{\alpha }W_{\alpha }V_2\right) +\frac{1}{2}\bar{D}^2(V_1D_{\alpha }V_2-D_{\alpha }V_1V_2)W^{\alpha }\right) \nonumber \\&\quad =-\bar{D}^2\left( \frac{1}{2}D_{\alpha }V_1V_2W^{\alpha }+\frac{1}{4}V_1D^{\alpha }W_{\alpha }V_2\right) \nonumber \\&\qquad -{\mathbb {D}}(V_1,V_2,W)_h-({\mathbb {D}}V_1,V_2,W)_h+(V_1,{\mathbb {D}}V_2,W)_h-(V_1,V_2,{\mathbb {D}}W)_h, \end{aligned}$$

(93)

$$\begin{aligned}&(V_1, (W,V_2)_h)_h-((V_1, W)_h,V_2)_h)=\frac{1}{2}{\bar{D}^2}(V_1(D^{\alpha }V_2W_{\alpha }+\frac{1}{2}D^{\alpha }W_{\alpha }V_2)\nonumber \\&\quad +V_2\left( D^{\alpha }V_1W_{\alpha }+\frac{1}{2}D^{\alpha }W_{\alpha }V_1\right) \nonumber \\&\quad =-{\mathbb {D}}(V_1,W,V_2)_h-({\mathbb {D}}V_1,W,V_2)_h+(V_1,{\mathbb {D}}W,V_2)_h-(V_1,W,{\mathbb {D}}V_2)_h. \end{aligned}$$

(94)

Here,

$$\begin{aligned}&(V_1,V_2,W)_h=(W,V_1,V_2)_h =\frac{1}{2}D_{\alpha }V_1V_2W^{\alpha }+\frac{1}{6}V_1V_2D_{\alpha }W^{\alpha },\nonumber \\&(V_1,W,V_2)_h=-\frac{1}{2}(V_1D_{\alpha }V_2+D_{\alpha }V_1V_2)W^{\alpha }-\frac{1}{3}V_1D^{\alpha }W_{\alpha }V_2,\nonumber \\&(V_1,V_2,\tilde{V})_h=(\tilde{V}, V_1,V_2)=\frac{1}{12}\bar{D}^2(V_1V_2\tilde{V}),\nonumber \\&(V_1,\tilde{V}, V_2)=\frac{1}{6}\bar{D}^2 (V_1\tilde{V} V_2). \end{aligned}$$

(95)

Therefore, from the calculations above, one can obtain that algebraically up to the third order these equations look as follows:

$$\begin{aligned}&{\mathbb {D}}U+(U,U)_h+(U,U,U)_h+ \cdots =0,\nonumber \\&U\rightarrow U+{\mathbb {D}}\Lambda +(U,\Lambda )_h+\nonumber \\&(U,U,\Lambda )_h-(U,\Lambda ,U)_h+(\Lambda ,U,U)_h+ \cdots , \end{aligned}$$

(96)

where U is the element of the grading 1 from complex (35) corresponding to the pair (V, W).

1.3 Appendix B.3: Notations for $N=2$ 3D superspace

We work with three-dimensional space with coordinates $x^{\mu }$ and euclidean metric $\eta _{\mu \nu }=\mathrm {diag}(+1,+1,+1)$. The Dirac matrices are $(\gamma ^{\mu })^{\beta }_{\alpha }=i(\sigma _1,\sigma _2,\sigma _3)$. We can raise and lower the corresponding spinor indices via $C^{\alpha \beta }$, i.e., $\xi ^{\alpha }=C^{\alpha \beta }\xi _{\beta }$ and $\xi _{\alpha }=C_{\alpha \beta }\xi ^{\beta }$, such that $C^{12}=-C_{12}=i$.

The $N=2$ 3D superspace has two two-component anticommuting coordinates $\theta ^{\alpha }_1$ and $\theta ^{\alpha }_2$. Therefore (this is similar to $N=1$ superspace), one can define complex coordinates $\theta ^{\alpha }=\theta ^{\alpha }_1-i\theta ^{\alpha }_2$ and $\bar{\theta }^{\alpha }=\theta ^{\alpha }_1+i\theta ^{\alpha }_2$. This allows to define superderivatives (cf. $N=1$ D=4 case):

$$\begin{aligned} D_{\alpha }=\partial _{\alpha }+\frac{i}{2}\bar{\theta }^{\beta }\partial _{\alpha \beta }, \quad \bar{D}_{\alpha }=\bar{\partial }_{\alpha }+\frac{i}{2}{\theta }^{\beta }\partial _{\alpha \beta } \end{aligned}$$

(97)

where $\partial _{\alpha \beta }=\gamma ^{\mu }_{\alpha \beta }\partial _{\mu }$. One defines the (anti)chiral scalar fields via the familiar equation: $\bar{D}_{\alpha }\Lambda (x,\theta )=0$, $D_{\alpha }\bar{\Lambda }(x,\theta )=0$. For the calculations, we will need the following commutation relations between superderivatives:

$$\begin{aligned} \{D_{\alpha },\bar{D}_{\beta }\}=i\gamma ^{\mu }_{\alpha \beta }\partial _{\mu }=i\partial _{\alpha \beta }, \quad \{D_{\alpha },D_{\beta }\}=0, \quad \{\bar{D}_{\alpha },\bar{D}_{\beta }\}=0. \end{aligned}$$

(98)

As well as in $N=1\,D=4$ case, it is useful to introduce operators $D^2=C^{\alpha \beta }D_{\alpha }D_{\beta }$, $\bar{D}^2=C^{\alpha \beta }\bar{D}_{\alpha }\bar{D}_{\beta }$.