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.

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Acknowledgements

We are grateful to K. Costello, M. Markl, M. Movshev and J. Stasheff for illuminating discussions. We are indebted to the valuable comments of the referee. We would like to express our gratitude to the wonderful environment of Simons Summer Workshops where this work was partially done. A.M.Z. is grateful to A.N. Fedorova for careful reading of the manuscript.

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Correspondence to Anton M. Zeitlin.

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].

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 \(\mathbf{Q}^2=0\).

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 \(\mu _n\).

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

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 \(D^2=C^{\alpha \beta }D_{\alpha }D_{\beta }\), \(\bar{D}^2=C^{\dot{\alpha }\dot{\beta }}D_{\dot{\alpha }}D_{\dot{\beta }}\).

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 (VW).

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 \(D^2=C^{\alpha \beta }D_{\alpha }D_{\beta }\), \(\bar{D}^2=C^{\alpha \beta }\bar{D}_{\alpha }\bar{D}_{\beta }\).

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Rocek, M., Zeitlin, A.M. Homotopy algebras of differential (super)forms in three and four dimensions. Lett Math Phys 108, 2669–2694 (2018). https://doi.org/10.1007/s11005-018-1109-5

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Keywords

  • Supersymmetric field theories
  • Homotopical algebra
  • Supergeometry

Mathematics Subject Classification

  • 18G55
  • 81T60
  • 83E30