Homotopy Algebras of Differential (Super)forms in Three and Four Dimensions
aa r X i v : . [ m a t h - ph ] F e b HOMOTOPY ALGEBRAS OF DIFFERENTIAL (SUPER)FORMSIN THREE AND FOUR DIMENSIONS
MARTIN ROCEK AND ANTON M. ZEITLIN
Abstract.
We consider various A ∞ -algebras of differential (super)forms, whichare related to gauge theories and demonstrate explicitly how certain reformula-tions of gauge theories lead to the transfer of the corresponding A ∞ -structures.In N = 2 3D space we construct the homotopic counterpart of the de Rhamcomplex, which is related to the superfield formulation of the N = 2 Chern-Simons theory. Introduction
Homotopy associative algebra, or A ∞ -algebra is a certain generalization of adifferential graded associative algebra. For the first time this concept appeared inthe 1960s in the works of Jim Stasheff [20], [16] in the context of Algebraic Topology.The theory continued developing both in the 1970s and the 1980s, but it wasn’tuntil the 1990s that people realized that the same structures appeared in StringPhysics and Geometry, e.g. in the context of String Field Theory [4] and MirrorSymmetry [13]. In this article, we are studying several examples of such algebraicobjects related to the basic gauge (Super) Field Theories, following previous resultsobtained in [24], [25], [26], and related articles devoted to L ∞ -algebras, e.g. [23],[21], [22], see also [18] and recent paper [9].First, let us give an idea of what A ∞ -algebra is and present one of the reasons whyit is a natural object from a mathematical point of view. Suppose, F is a differentialgraded algebra (DGA), i.e. F is a chain complex that has an associative bilinearoperation satisfying the Leibniz rule with a differential. Assume, there is anothercomplex K , homotopically equivalent to F . Therefore, we can deduce that thecohomology of K is isomorophic to F , and both have the structure of an associativealgebra. The natural question is the following: what happens on the level of chaincomplexes? It appears that we have a transfer of the algebraic structure from F to K but in general the transferred product will not be associative. The structureinduced from the bilinear operation and the homotopical equivalence is what we callan A ∞ -algebra [8], [17],[13]. The induced bilinear operation µ on F will satisfy anassociativity condition up to homotopy that is described by a trilinear operation µ .Moreover, (see e.g. [17],[13],[15],[10],[12]) one can continue and construct operations µ n , satisfying “higher” bilinear associativity relations. Thus, if we are given theoperations µ n , n = (1 , , , ... ) satisfying such bilinear relations, we say that thereis the structure of A ∞ -algebra on K . In fact, all these relations can be summarizedin one: ∂ = 0 (see Appendix A for details).An important feature of the A ∞ -algebras is that one can define a generalizationof the generalized Maurer-Cartan (GMC) equation: Q Φ + µ (Φ , Φ) + µ (Φ , Φ , Φ) + ... = 0 , (1) here Φ has degree 1. This equation possesses the following infinitesimal symmetrytransformation: Φ → Q Λ + µ (Φ , Λ) − µ (Λ , Φ) + µ (Λ , Φ , Φ) − µ (Φ , Λ , Φ) + µ (Φ , Φ , Λ) ..., (2)where Λ is infinitesimal and has degree 0.In physics, particularly in String Field Theory, A ∞ -algebras appear in the con-text of the Batalin-Vilkovisky (BV) formalism (see e.g. [11]). Namely, for a givenaction functional satisfying the BV master equation, one can construct the nilpotentnoncommutative odd vector field Q encoding all the operations of the A ∞ -algebra(see Appendix A).In this article, we will be studying homotopy algebras related to (super)fieldtheories, as well as their transfers.In Section 2, as an invitation to the subject, we examine the N = 2 D=3 Chern-Simons theory in the superfield formulation (this theory appears to be physicallyrelevant, see e.g. [1],[2]). Using the corresponding BV action, we construct thesuperfield counterpart of the de Rham Complex in three dimensions that turns outto be a homotopy algebra instead of the usual DGA.In Section 3, we consider the Yang-Mills (YM) theory in its usual second and firstorder formulations. We describe the corresponding A ∞ -algebras, for which GMCcoincides with YM equations of motion, and show how they are related to theBatalin-Vilkovisky (BV) formalism. In this case, the first order formulation givesus just a differential graded algebra, while the second order formulation leads to an A ∞ -algebra where all the operations starting from the quadrilinear one vanish.In the end of Section 3, we are showing that these two A ∞ -algebras are related bythe transfer formula. This is one of the simplest non-artificial examples of a transferformula. This way, we observe that some of the basic field theory reformulations,like the transfer from the second order action to the first order action, can beexplained purely on the level of homotopical algebra (see also [23]).In Section 4, we generalize these results to the supersymmetric case. We usethe superfield formulation of N = 1 supersymmetric (SUSY) YM theory and studythe homotopical algebras corresponding to the second and first order formulations.In this case, A ∞ -algebras are “richer”, i.e. we have an infinite chain of polylinearoperations in both first and second order formulations. As in the previous exam-ple, they are related by the transfer formula. In the end, we discuss some furtherdevelopments and future tasks, in particular the meaning of the representation ofreal superfield via complex ones from the homotopical algebra point of view. Acknowledgements.
We are grateful to K. Costello, M. Markl, M. Movshev andJ. Stasheff for illuminating discussions. We would like to express our gratitudeto the wonderful environment of Simons Summer Workshops where this work waspartially done. A.M.Z. is supported by AMS Simons travel grant.2.
Homotopic de Rham complex in N = 2
3D superspace.
Consider the DGArelated to the de Rham complex in three dimensions:0 → Ω d −→ Ω d −→ Ω d −→ Ω → , (3) here Ω i is the space of differential forms of degree i . Moreover, this complexpossesses a natural pairing given by the integral of the wedge product of forms andtrace, giving a cyclic structure of the de Rham DGA. The same DGA structure canbe constructed for the forms with values in U ( g ) where g is some semisimple Liealgebra. Then the formal action functional, associated with this cyclic DGA (seeAppendix A), corresponding to the element Φ = c + A + A ∗ + c ∗ , where c ∈ Ω [1] ⊗ g , A ∈ Ω ⊗ g , A ∗ ∈ Ω [ − ⊗ g , c ∗ ∈ Ω [ − ⊗ g has the following form: S [Φ] = S CS [ A ] + Z d x Tr( d A c ∧ A ∗ + [ c, c ] c ∗ ) , (4)where S CS [ A ] = R d x ( A ∧ d A + [ A , A ] ∧ A ) is the usual 3D Chern-Simons action. Let us considerN=2 3D Euclidean superspace. This is a superspace where in addition to three evencoordinates there are 4 odd coordinates that are the components of Weyl spinors θ α , ¯ θ α . As in the N=1 4D case, one can consruct the superderivatives D α , ¯ D α .The difference is that one can make new Lorentz scalars out of them, like D α ¯ D α (see Appendix B.3 for details). The relations between superderivatives allow toconstruct the following complex:0 −→ Θ id −→ Σ ¯ D α D α −−−−→ ˜Σ ¯ D −−→ ˜Θ → . (5)Here Θ ∼ = ˜Θ is the space of chiral scalar fields (if Λ ∈ Θ, then ¯ D α Λ = 0), andΣ ∼ = ˜Σ is the space of complex scalar fields. Afterwards, we will denote this complex( SdR · , d S ). It is not hard to show that the 3D de Rham complex can be embeddedin (5). Moreover, one can define a nondegenerate pairing on (5) similar to h· , ·i F : h· , ·i : SdR i ⊗ SdR − i → C , h Λ , ˜Λ i = Z d xd θ Λ ˜Λ , h V, ˜ V i = Z d xd θV ˜ V , where Λ ∈ SdR , V ∈ SdR , ˜ V ∈ SdR , ˜Λ ∈ SdR . This pairing satisfies thefamiliar property (42). Therefore, we can hope for the existence of a cyclic A ∞ -algebra related to the complex (5). In order to construct it, we can consider thefollowing action giving the superfield formulation to the N=2 Chern-Simons theory[27]: S susyCS = Z d xd θ Z dt ( V ¯ D α ( e − tV D α e tV )) , (6)where V ∈ Σ ⊗ g . This action has the symmetry that is the same for all thesupersymmetric theories we are considering here, namely e V → e Λ e V e ¯Λ , (7)where Λ , ¯Λ are respectively chiral and atichiral scalar fields with values in Lie alge-bra g . The corresponding infinitesimal version of the symmetry is: V → V + δ Λ , ¯Λ V = V + 12 L V (Λ − ¯Λ + coth( 12 L V )(Λ + ¯Λ)) , (8)where L V · = [ V, · ]. Let us restrict the symmetry to the chiral transformations: e V → e Λ e V . Then, the BV modification of the action (6) is (cf. (55)): S BVsusyCS = S susyCS + Z d xd θ ( δ C ( V ) V ∗ ) + Z d xd θ ([ C, C ] C ∗ ) , (9) here C ∈ SdR [+1], V ∈ SdR , V ∗ ∈ SdR [ − C ∗ ∈ SdR [ − A ∞ -algebra. It is obvious that the corresponding chain complex, where the A ∞ operations act, is the one from (5). Therefore, we have the following proposition. Proposition 2.1.
Complex ( SdR · , d S ) possesses a nontrivial A ∞ structure (withall the operations nonvanishing) provided by the action functional S BVsusyCS .Let us express the bilinear operation explicitly: ν ( f , f )= ❍❍❍❍❍ f f Λ V ˜ V ˜Λ Λ Λ Λ V Λ ˜ V Λ ˜Λ Λ V Λ V ( V , V ) h − ¯ D ( ˜ V V ) 0˜ V Λ ˜ V − ¯ D ( V ˜ V ) 0 0˜Λ Λ ˜Λ , Λ ∈ SdR , V , V ∈ SdR , ˜ V , ˜ V ∈ SdR , ˜Λ , ˜Λ ∈ SdR , and( V , V ) h = − D α V ¯ D α V −
12 ¯ D α V D α V . (10) Corollary 2.1.
The operation ν is homotopy associative on ( SdR · , d S ).The equations of motion for S susyCS are [27]: f ( L V ) ¯ D α ( e − V D α e V ) = 0 , (11)where f ( x ) = e x − x . This equation is the GMC equation for the resulting A ∞ -algebra, which is the superfield generalization of the Maurer-Cartan equation in3D: d A + A ∧ A = 0. Remark 2.1.
Also, we note that in principle, one can extend the complex (5) bymeans of the space of antichiral scalars (compare to the SUSY Yang-Mills case insubsection 3.4.). The resulting complex will be0 / / Θ id / / Σ ¯ D α D α / / ˜Σ ¯ D / / D % % ❏❏❏❏❏❏❏❏❏❏ ˜Φ / / ⊕ ⊕ ¯Θ − id tttttttttt ˜¯ΘThe resulting A ∞ -algebra, generated by the action S fullBVsusyCS = S susyCS + Z d xd θ ( δ C, ¯ C ( V ) V ∗ ) + Z d xd θ ([ C, C ] C ∗ + [ ¯ C, ¯ C ] ¯ C ∗ ) . (12)will describe the full symmetry of the action (6). . Homotopy algebras of Yang-Mills theory. A ∞ -algebra of the Maxwell complex [24] . We consider Maxwellcomplex 0 → F Q −→ F Q −→ F Q −→ F → , (13)where the spaces F i and the action of Q are as follows:0 −→ Ω ( M ) d −→ Ω ( M ) d ∗ d −−→ Ω ( M ) d −→ Ω ( M ) → , (14)where M stands for any four dimensional (pseudo)Riemannian manifold (chaincomplex (14) was studied also in [7], [6] in a different context). This complex has astructure of homotopy commutative associative algebra [24]. Let us introduce thecorresponding multilinear operations:( · , · ) h : F i ⊗ F j → F i + j , ( · , · , · ) h : F i ⊗ F j ⊗ F k → F i + j + k − . (15)The bilinear operation is defined by means of the following table:( f , f ) h = ❍❍❍❍❍ f f v A V aw vw A w V w aw B v B ( A , B ) B ∧ V W v W A ∧ W b vb f takes values in the set { v, A , V , a } from the first row, and f takes valuesin the set { w, B , W , b } from the first column. Other elements in the table representthe value of bilinear operation ( f , f ) h for appropriately chosen f and f . In thetable above v, w ∈ F ; A , B ∈ F ; V , W ∈ F ; a, b ∈ F . The bilinear operation( A , B ) is defined as follows:( A , B ) = A ∧ ( ∗ d B ) − ( ∗ d A ) ∧ B + d ∗ ( A ∧ B ) . (16)The operation ( · , · , · ) h is defined to be nonzero only when all the arguments belongto F . For A , B , C ∈ F we have:( A , B , C ) h = A ∧ ∗ ( B ∧ C ) − ( ∗ ( A ∧ B )) ∧ C . (17)By the direct calculations (see [24]), one can show that these operations satisfy thefollowing relations providing the structure of homotopy commutative A ∞ -algebra. Proposition 3.1.
Let a , a , a , a , b, c ∈ F . Then the following relations hold: Q ( a , a ) h = ( Q a , a ) h + ( − n a ( a , Q a ) h , ( a , a ) h = ( − n a n a ( a , a ) h , Q ( a , a , a ) h + ( Q a , a , a ) h + ( − n a ( a , Q a , a ) h +( − n a + n a ( a , a , Q a ) h = (( a , a ) h , a ) h − ( a , ( a , a ) h ) h , ( − n a ( a , ( a , a , a ) h ) h + (( a , a , a ) h , a ) h =(( a , a ) h , a , a ) h − ( a , ( a , a ) h , a ) h + ( a , a , ( a , a ) h ) h , (( a , a , a ) h , b, c ) h = 0 . (18) f we tensor complex ( F · , Q ) with some universal enveloping algebra for somereductive Lie algebra g , one obtains that inherited operations ( · , · ) h and ( · , · , · ) h satisfy the relations of A ∞ -algebra on the resulting complex ( F · g , Q ).If the manifold M is compact, or the fields are with the compact support, onecan show that the Maxwell complex possesses a pairing h f , f i = Z M T r ( f ∧ f ) , (19)which makes the A ∞ -algebra introduced above to be cyclic. Namely, one can define(20) {· , ..., ·} h : F g ⊗ ... ⊗ F g → C in the following way: { f , f } h = hQ f , f i , { f , f , f } h = h ( f , f ) h , f i , { f , f , f , f } h = h ( f , f , f ) h , f i . (21)For more details about the cyclic structures see Appendix A. One of the mostimportant applications of the cyclic structure is that one can write the followingaction functional in the form that is the “homotopy” generalization of the Chern-Simons action functional S HCS [ f ] = 12 { f, f } + 13 { f, f, f } + 14 { f, f, f, f } (22)such that f ∈ F g and the variation of this functional with respect to f leads tothe generalized Maurer-Cartan equation for f . The Maurer-Cartan equation andits symmetries in the case of the A ∞ -algebra considered above Q A + ( A , A ) h + ( A , A , A ) h = 0 A → A + ǫ ( Q u + ( u, A ) h − ( A , u ) h ) , (23)which lead to Yang-Mills equations and the gauge symmetry if f = A , where A ∈ F g and u ∈ F g .In the case when f = c + A + A ∗ + c ∗ , where c ∈ F g [1], A ∈ F g , A ∗ ∈ F g [ − c ∗ ∈ F [ − F i g [ j ] is i + j , the actionabove leads to the BV Yang-Mills action: S BV Y M = Z M Tr( F ∧ ∗ F + A ∗ ∧ d A c + [ c, c ] c ∗ ) . (24)It is well-known that this action satisfies the so-called BV master equation that,according to the general principle, leads to the A ∞ -algebra on the complex ( F · , Q )(see Appendix A).In the next section, we will construct the associative algebra related to the firstorder Yang-Mills theory. [3] . Let us consider the following complex:0 −→ K Q −→ K Q −→ K Q −→ K −→ uch that K = Ω ( M ), K = Ω ( M ) ⊕ Ω ( M ), K = Ω ( M ) ⊕ Ω ( M ), K = Ω ( M )and the differential ˜ Q acts as follows:0 / / Ω ( M ) d / / Ω ( M ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ Ω ( M ) d / / Ω ( M ) / / − d L L d + Ω ( M ) − Id / / ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Ω ( M )where d + = d + ∗ d and Ω ( M ) is the space of self-dual 2-forms on the manifold M . One can define a bilinear operation on the resulting complex: µ ( f , f )= ❍❍❍❍❍ f f v ( A , F ) ( W , G ) a v v v ( v A , v F ) ( v W , v G ) v a ( A , F ) ( v A , v F ) ( A ∧ F − − A ∧ W − A ∧ F , P + ( A ∧ A ) F ∧ G ( W , G ) v W − A ∧ W − F ∧ G a v a P + = ∗ is the projection operator on Ω ( M ). Here f , f take values inthe set of variables with prime and double prime correspondingly. Other elementsin the table correspond to the appropriate values of µ ( f , f ). In the table above v , v ∈ K ; ( A , F ) , ( A , F ) ∈ K = Ω ( M ) ⊕ Ω ( M ); ( W , G ) , ( W , G ) ∈ K = Ω ( M ) ⊕ Ω ( M ); a , a ∈ K . It is not hard to see that this operation givesto the complex above the structure of differential graded abelian algebra. If wetensor it with some Lie algebra g , we will find that it possesses the cyclic structure,where the corresponding pairing on ( K , ˜ Q ) is defined by the same formula as in theprevious section. One can obtain that it is related to the following action: S = Z M Tr (cid:16) F ∧ F + F ∧ (d A + A ∧ A ) (cid:17) , (26)where F ∈ Ω + ( M ), which is equivalent to the usual YM theory. Here we note thatwe could choose the space of anti self-dual 2-forms Ω − ( M ) instead of Ω ( M ) in thecomplex above, and the resulting complex will also have the structure of differentialgraded abelian algebra. A ∞ structure. Suppose, F is a differential graded algebra(DGA), i.e. it is a chain complex with a differential Q ′ , and m is an associativebilinear operation on K . Suppose we have a chain complex K with a differential Q ′ ,which is homotopically equivalent to F , i.e. there are chain maps f : F → K and g : K → F , such that f g − id = Q ′ H + HQ ′ , where H is of degree −
1. Then F and K are quasiisomorhic, i.e. we have isomorphism on the level of cohomology: H ∗ Q ′ ( K ) ∼ = H ∗ Q ( F ) and µ ( · , · ) = g ◦ m ( f ( · ) , f ( · )) is an associative bilinear operationon H ∗ Q ′ ( K ). Therefore, we have a transf er of the associative multiplication on thelevel of cohomology. As we explained in the introduction, there is a transfer on he level of complexes, but not of the associative algebra. The structure inducedfrom the operation m and the homotopical equivalence is the A ∞ -algebra. Anelementary calculation shows that µ ( µ ( a, b ) , c ) − µ ( a, µ ( b, c )) =(27) Qµ ( a, b, c ) + µ ( Qa, b, c ) + ( − | a | µ ( a, Qb, c ) + ( − | a | + | b | µ ( a, b, Qc ) , where µ ( a, b, c ) =(28) g ◦ m ( H ◦ m ( f ( a ) , f ( b )) , f ( c )) − ( − | a | g ◦ m ( f ( a ) , H ◦ m ( f ( a ) , f ( b ))is a trilinear operation of degree −
1. One can find that µ also satisfies bilinear“higher associativity” relations with µ and µ ≡ Q ′ and leads to next operation µ . One can continue the process and in general obtain the infinite amount ofoperations satisfying certain billinear relations. The general construction of such µ n can be described by certain sum over the expressions parametrized by tree graphswith vertices corresponding to operation m and edges corresponding to homotopy H [17], [13]. In fact, there is the more general statement: the differential gradedalgebra structure can be replaced by A ∞ structure, and it again will be transferredto the A ∞ -algebra (see Appendix A and e.g. [15]).Now we will show that one can transfer the A ∞ structure from the first ordercomplex to the Maxwell one, according to [15]. Firstly, we will show that theMaxwell complex and the first order complex are quasiisomorphic. Namely let usconstruct the maps g : ( K · , ˜ Q ) → ( F · , Q ), f : ( F · , Q ) → ( K · , ˜ Q ) such that theircomposition is homotopic to identity. The explicit expression for f and g are: f ( u ) = u, f ( A ) = ( A , P + d A ) , f ( V ) = ( V , , f ( v ) = v,g ( u ) = u, g (( A , F )) = A , g (( V , G )) = d G + V , g ( v ) = v. (29)Here u ∈ Ω , A ∈ Ω , F , G ∈ Ω , V ∈ Ω , v ∈ Ω . Therefore, g ◦ f = id, f ◦ g = id + ˜ Q H + H ˜ Q (30)Here H is the homotopy on the complex ( K · , ˜ Q ), i.e. the map H : K i → K i − , andit is nonzero only on K . The explicit formula is: H (( V , G )) = (0 , G )(31)Hence, f ◦ g (( A , F )) = ( A , P + d A ) , f ◦ g (( V , G )) = ( d G + V , , ˜ Q H (( V , G )) = ( d G , − G ) , H ˜ Q (( A , F )) = (0 , P + d A − F ) . (32)Therefore we have a Proposition. Proposition 3.2. i)The complexes ( F · , Q ) , ( K · , ˜ Q ) are quasiisomorphic and ho-motopically equivalent provided by the maps f, g .ii)Under the homotopy equivalence, the structure of DGA (from subsection 2.2) on ( K · , ˜ Q ) is transferred to the described above A ∞ structure (from subsection 2.1)on ( F · , Q ) . roof. We have proved ( i ) above. Let us prove ( ii ). Namely, consider for examplehow the A ∞ structure is transferred from ( K · , ˜ Q ) to ( F · , Q ). The bilinear andtrilinear operations that are transferred have the following form (28):( a , a ) ′ h = g ◦ µ ( f ( a ) , f ( a )) , ( a , a , a ) ′ h = g ◦ µ ( f ( a ) , H ◦ µ ( f ( a ) , f ( a )) − ( − | a | µ ◦ ( H ◦ µ ( f ( a ) , f ( a )) , f ( a )) , (33)where a i ∈ ( F · , Q ) ( i = 1 , , a , a ) ′ h ,( a , a , a ) ′ h coincide with the defined earlier ( a , a ) h , ( a , a , a ) h . (cid:4) The example we have considered in this subsection shows that some of the fieldtheory reformulations can be described in terms of homological algebra only. Inthe next section we will see the same picture in the case of supersymmetric four-dimensional Yang-Mills theory and its first order reformulation.4. A ∞ -algebras of superforms in 4D. We claim that theMaxwell complex(34) 0 −→ Ω ( M ) d −→ Ω ( M ) d ∗ d −−→ Ω ( M ) d −→ Ω ( M ) → / / Ω ( M ) d / / Ω ( M ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ Ω ( M ) d / / Ω ( M ) / / − d L L d + Ω ( M ) − Id / / ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Ω ( M )(35)where M = R , , can be naturally embedded into certain complex in N=1 super-space. This is a superspace where in addition to four even spacetime coordinates,there are two odd Weyl spinor coordinates θ α , θ ˙ α . We keep the notation close to [5],see also Appendix B. Denote the space of complex scalar superfields V ( x, θ, ¯ θ ) as Σand the space of chiral spinor superfields W α ( x, θ ) (corresponding to field strength)as Θ. Finally, we denote the space of chiral superfields corresponding to gaugetransformations Λ( x, θ ) as Φ. The following complex:0 −→ Ξ id −→ Σ D α ¯ D D α −−−−−−→ ˜Σ ¯ D −−→ ˜Ξ → . (36)where Ξ ∼ = ˜Ξ and Σ ∼ = ˜Σ is the supersymmetric generalization of the Maxwell com-plex, it is clear that the Maxwell complex (34) naturally embeds into the complex / / Ξ id / / Σ (cid:28) (cid:28) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ ˜Σ ¯ D / / ˜Ξ / / c div L L ¯ D D α Θ Id / / B B ☎☎☎☎☎☎☎☎☎☎☎☎☎☎☎ ˜Θ(37)where c divW = D α W α , W is a chiral spinor field from Σ and ˜Θ ∼ = Θ. One canshow that (35) can be embedded in (37). As we noted in the section 2.2, one couldconsider the first order Maxwell complex with antiselfdual 2-forms. This complexwill be embedded into antichiral version of the (37) (i.e. the spaces Ξ and Θ wouldbe replaced by their antichiral counterparts).Afterwards, we will denote the supersymmetric Maxwell complex as ( F · susy , D )and the supersymmetric complex for the first order theory as ( K · susy , D ). As onemay suspect, these complexes are homotopy equivalent as in the previous case. Onecan construct the supersymmetric version of maps f, g from the previous subsection,i.e. maps g s : ( K · susy , D ) → ( F · susy , D ), f s : ( F · susy , D ) → ( K · susy , D ): f s (Λ) = Λ , f s ( V ) = ( − V, ¯ D D α V ) , f s ( ˜ V ) = ( ˜ V , , f s (˜Λ) = ˜Λ , (38) g s (Λ) = Λ , g s (( V, W )) = − V, g s (( ˜ V , ˜ W α ) = ˜ V − D α ˜ W α , g s (˜Λ) = ˜Λ . Moreover, g s ◦ f s = id and f s ◦ g s = id + D H + H D where the homotopy operatoris nonzero only on K susy and is defined by the formula:(39) H (( V, W )) = ((0 , − W )) . The precise statement is the following.
Proposition 4.1.
The complexes ( K · susy , D ) , ( F · susy , D ) are quasiisomorphic andhomotopically equivalent where the corresponding maps g s : ( K · susy , D ) → ( F · susy , D ) , f s : ( F · susy , Q ) → ( K · susy , D ) are given by (38). Another issue about two complexes above is that one can define graded antisym-metric bilinear forms on F susy , K susy such that(40) h· , ·i K : K isusy ⊗ K − isusy → C , h· , ·i F : F isusy ⊗ F − isusy → C . On K susy they are defined by the following formulas: h Λ , ˜Λ i K = Z d xd θ Λ ˜Λ , h V, ˜ V i K = Z d xd θV ˜ V , h W, ˜ W i K = 12 Z d xd θW α ˜ W α . (41) Proposition 4.2.
The graded symmetric bilinear forms h· , ·i F , h· , ·i K obey therelation h a , Qa i + ( − a + a + a a h a , Qa i = 0 , (42) when a , a ∈ F r or a , a ∈ K r and Q stands for D or D . he bilinear form on F susy is defined by the restriction of the form on K susy .Therefore, from now on we suppress the index notation for the bilinear forms.One of the immediate consequences of the Proposition above is that if one canconstruct the A ∞ -algebra on the complex ( K · susy , D ), then the A ∞ structure on( F · susy , D ) can be obtained by the transfer procedure as in the previous section. Let us consider the action of the N=1SUSY YM theory in 4 dimensions: S SY M [ V ] = 12 Z d xd θT r ( e − V D α e V ¯ D ( e − V D α e V )) , (43)where V is the real superfield, taking values in some Lie algebra g , i.e. V ∈ Σ ⊗ g .It is invariant under the following transformations: e V → e ¯Λ e V e Λ , (44)where Λ is a chiral scalar superfield. It is useful to define the supercurvature, i.e.the spinor chiral superfield W ∈ Θ ⊗ g W α = − ¯ D ( e − V D α e V ) . (45)Using the global transformation formula (44), one can find the infinitesimal trans-formations of V and W : V → V + δ Λ , ¯Λ V = V + 12 L V (Λ − ¯Λ + coth( 12 L V )(Λ + ¯Λ)) W α → W α + [ W α , Λ](46)where Λ , ¯Λ are infinitesimal and L V · = [ V, · ]. Varying the action S SY M with respectto V , one finds the following expression: f ( L V )[ ∇ α , W α ] = 0 , (47)where ∇ α = e − V D α e V , and f ( x ) = e x − x . One can see that this equation isequivalent to the ”physically covariant” equation [ ∇ α , W α ] = 0 because operator f ( L V ) is invertible. Here we note (this is important for the next subsection) thatthe equation (47) is the equation of motion for the functional (43) even when V isnot real. The reality condition gives the constraint:[ ∇ α , W α ] = [ ∇ ˙ α , W ˙ α ] . (48)Below we are showing that the equation (47) is covariant from the homologicalpoint of view, i.e. we are observing that it arises as GMC equation for the certaincyclic A ∞ -algebra. A ∞ -algebra. One can also rewrite the action (43) in the first orderform the way we we did in the case of usual Yang-Mills: S fo [ V, W ] = 12 Z d xd θ T r ( W α W α + ¯ D ( e − V D α e V ) W α ) . (49)We see that both actions S SY M and S fo in the case of abelian Lie algebra can bewritten as h ψ, Qψ i , where ψ is an element of degree 1 and Q stands either for D or D . Let us forget about ¯Λ-symmetry in this section and consider only the one, enerated by chiral scalar superfields Λ. Then the BV generalization of the lastaction looks as follows: S BVfo = S fo + Z d xd θT r ( δ C ( V ) V ∗ ) + Z d xd θT r ([ W α , C ] W ∗ α + 12 [ C, C ] C ∗ ) . (50)Here, as usual, ghosts and antifields are c + ∈ Φ[1] , V ∗ ∈ ˜Σ[ −
1] and is a real super-field, W ∗ ∈ ˜Σ[ − c ∈ ˜Θ[ − δ c ( V ) = L V ( c +coth( L V )( c )).It is feasible to check that these actions satisfy the Master equation. Therefore, anodd vector field Q that is defined by means of the formula Q · = ( S BVfo , · ) BV actson the fields in the following way: Q c = 12 [ C, C ] , Q V = δ C ( V ) , Q V ∗ = f ( L V )[ e − V D α e V , W α ] + δδV δ C ( V ) V ∗ , Q W α = [ W α , C ] , Q W ∗ α = [ W ∗ α , C ] , Q C ∗ = [ C, C ∗ ] − L V ( V ∗ − coth( 12 L V )( V ∗ )) + [ W α , W α ∗ ] . (51)Then we have the following Proposition. Proposition 4.3.
The odd vector field Q satisfies the nilpotency condition Q = 0 and therefore determines the A ∞ structure on the complex (35). Therefore, the action S BVfo has the form of the homotopy Chern-Simons theory, i.e.:12 h ψ, D ψ i + X n ≥ n + 1 h µ n ( ψ, ..., ψ ) , ψ i , (52)where µ n stand for bilinear and higher operations in the A ∞ -algebra. One of thespecific features of the corresponding A ∞ -algebra is that the corresponding L ∞ -algebra vanishes in the case of abelian Lie algebra g . One can see that directlyfrom (51) or from the fact that the BV action in this case becomes bilinear infields. Therefore, for example, the bilinear operation for this A ∞ -algebra in thecase of abelian g is commutative, like in the usual Yang-Mills case. Let us writethe expression for this bilinear operation explicitly. µ ( f , f )= ❍❍❍❍❍ f f Λ ( V , W ) ( ˜ V , ˜ W ) ˜Λ Λ Λ Λ ( V Λ , W Λ ) ( ˜ V Λ , ˜ W Λ ) ˜Λ Λ ( V , W ) ( Λ V , Λ W ) (( V , W ) , ( V , W )) h ˜ W ,α W α − ¯ D ( ˜ V V )( ˜ V , ˜ W ) ( Λ ˜ V , Λ ˜ W ) − W α ˜ W ,α − ¯ D ( V ˜ V )˜Λ Λ ˜Λ ere f , f take values in the set of variables with indices 1 and 2 correspondingly.In the table above Λ i ∈ K susy , ( V i , W i ) ∈ K susy ( i = 1 , V i , ˜ W i ) ∈ K susy ( i = 1 , i ∈ K susy , and(( V , W ) , ( V , W )) h =(53)( D α V W α + W α D α V + 12 ( V D α W α − D α W α V ) , −
12 ¯ D ( V D α V − D α V V ) . Corollary 4.1.
The operation µ on the complex (37) is homotopy associative on ( K · , ˜ D ).In Appendix B, we explicitly prove this proposition.Knowing all the operations of A ∞ -algebra based on complex (37), one can con-struct ones on the complex (36). The resulting bilinear operation is: m ( f , f )= ❍❍❍❍❍ f f Λ V ˜ V ˜Λ Λ Λ Λ V Λ ˜ V Λ ˜Λ Λ V Λ V ( V , V ) h − ¯ D ( ˜ V V ) 0˜ V Λ ˜ V − ¯ D ( V ˜ V ) 0 0˜Λ Λ ˜Λ V , V ) h = 12 D α ¯ D ( V D α V − V D α V ) − D α V ¯ D D α V − D α V ¯ D D α V D α V + 12 ( V D α ¯ D D α V − V D α ¯ D D α V ) . (54)Therefore, we have another proposition. Proposition 4.4.
Operation m is homotopy associative on the supersymmetricgeneralization of the Maxwell complex (36). As one could expect, the operation m comes from the BV functional S BVSY M [ V ] = 12 Z d xd θT r ( e − V D α e V ¯ D ( e − V D α e V )) + Z d xd θT r ( δ C ( V ) V ∗ ) + Z d xd θ T r ([ C, C ] C ∗ ) . (55)Again, as in the case of nonsupersymmetric Yang-Mills, we see that the A ∞ -algebras of the second and the first order theories are related by means of transferformula. So far we neglected the fullsymmetry (44) of the action (43). Namely, we considered only the chiral part ofthe symmetry. In order to obtain full symmetry on the homological level there aretwo options. The first one is to consider the following extension of the complex / / Θ id / / Σ D α ¯ D D α / / ˜Σ ¯ D / / D % % ❏❏❏❏❏❏❏❏❏❏ ˜Θ / / ⊕ ⊕ ¯Θ − id tttttttttt ˜¯ΘHere ¯Θ ∼ = ¯˜Θ is a space of antichiral superfields. From now on, we denote it as( F · full , D ). To reproduce this complex, one should consider the full BV action: S BV,fullSY M [ V ] =12 Z d xd θT r ( e − V D α e V ¯ D ( e − V D α e V )) + Z d xd θT r ( δ C, ¯ C ( V ) V ∗ ) + Z d xd θ T r ([ C, C ] C ∗ ) + Z d xd ¯ θ T r ([ ¯ C, ¯ C ] ¯ C ∗ ) . (56)Here δ C, ¯ C ( V ) = L V ( C − ¯ C + coth ( L V )( C + ¯ C )). The corresponding bilinearoperation should be modified in this way: m full ( f , f )= ❍❍❍❍❍ f f (Λ , ¯Λ ) V ˜ V (˜Λ , ¯˜Λ )(Λ , ¯Λ ) (Λ Λ , ¯Λ ¯Λ ) V (Λ + ¯Λ ) ˜ V (Λ + ¯Λ ) (˜Λ Λ , ¯˜Λ ¯˜Λ ) V (Λ + ¯Λ ) V ( V , V ) h ( − ¯ D ( ˜ V V ) , D ( ˜ V V )) 0˜ V (Λ + ¯Λ ) ˜ V − ¯ D ( V ˜ V ) 0 0(˜Λ , ¯˜Λ ) (Λ ˜Λ , ¯Λ ¯˜Λ ) 0 0 0Using the transfer formula, one can find the modification for the bilinear operationof the first order model complex.Sometimes it is useful to define the following representation of the field V : e V = e ¯Ω e Ω , (57)such that under (44), the transformation of Ω and ¯Ω is: e Ω → e K e Ω e Λ , e ¯Ω → e ¯Λ e ¯Ω e − K , (58)where the new K -symmetry appears. Therefore, another approach to get full sym-metry into the picture is to represent V in terms of Ω and ¯Ω and treat them asseparate varibles in the BV action. The resulting A ∞ -algebra is based on the fol-lowing complex:Θ id / / Σ c (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾✾✾✾✾✾ D α ¯ D D α / / ˜Σ c id % % ❑❑❑❑❑❑❑❑❑❑ ¯ D / / ˜Θ ⊕ ⊕ / / Υ id ssssssssss − id % % ❏❏❏❏❏❏❏❏❏❏ ⊕ ⊕ ˜Υ / / ⊕ ⊕ ¯Θ − id / / ¯Σ c B B ✆✆✆✆✆✆✆✆✆✆✆✆✆✆✆ D α ¯ D D α / / ¯˜Σ c − id tttttttttt D / / ˜¯ΘHere all maps in the middle of the diagram are given by the action of the opera-tor D α ¯ D D α , and the spaces Σ r , ˜Σ r , ¯Σ r , ¯˜Σ r , Υ are isomorphic to the space Σ (seeabove). We will denote the resulting complex as ( F · real , D ). The immediately arising nteresting question is: how the A ∞ structure on this complex is related to A ∞ -algebra on ( F · full , D )? Below we will show that they are homotopically equivalent.Let us explicitly construct the chain maps f : F · full → F · real and g : F · real → F · full : f ((Λ , ¯Λ)) = (Λ , −
12 (Λ + ¯Λ) , ¯Λ) , f ( V ) = ( 12 V, V ) ,f ( ˜ V ) = ( ˜ V , ˜ V ) , f ((Λ , ¯Λ)) = (Λ , , ¯Λ) g ((Λ , K, ¯Λ)) = (Λ , ¯Λ) , g ((Ω , ¯Ω)) = Ω + ¯Ω g (( ˜Ω , ˜¯Ω)) = 12 ( ˜Ω + ˜¯Ω) , g ((Λ , K, ¯Λ)) = (Λ −
12 ¯ D K, ¯Λ − D K )(59)It is feasible to check that gf = id and f g = id + [ Q, H ] where H is an operator ofdegree − F · chiral which is nonzero on F real and F real , such that H ((Ω , ¯Ω)) = (0 ,
12 ( ¯Ω − Ω) , ,H ((˜Λ , ˜ K, ˜¯Λ)) = ( − K, K )(60)Therefore, one can formulate the following Proposition. Proposition 4.5.
Complex ( F · full , D ) is homotopically equivalent to F · real . This allows us to transfer the homotopy algebra structure from F · real to F · full .Here we also note the following. If we look at ”Baker-Campbell-Hausdorff” changeof variables e V = e ¯Ω e Ω in the corresponding equation of motion (Maurer-Cartanequation), we find out that it becomes part of the transfer of the A ∞ -algebra. Appendix A: A ∞ -algebras and the BV formalism. In this appendix we summarize all necessary information about A ∞ -algebras.For more details see e.g. [16], [11], [15]. A1. A ∞ -algebras. The A ∞ -algebra is a generalization of an associative algebrawith a differential. Namely, consider a graded vector space V = ⊕ k V k with a dif-ferential Q . Consider the multilinear operations µ r : V ⊗ r → V of the degree 2 − r ,such that µ = Q . Definition.
The space V is an A ∞ -algebra if µ n satisfy the following bilinear iden-tity: n − X i =1 ( − i M i ◦ M n − i +1 = 0(61) on V ⊗ n . Here M s acts on V ⊗ m ( m ≥ s ) as the sum of all possible operatorsof the form ⊗ l ⊗ µ s ⊗ ⊗ m − s − l taken with appropriate signs. In other words, M s : V ⊗ m → V ⊗ m − s +1 and M s = m − s X l =0 ( − l ( s +1) ⊗ l ⊗ µ s ⊗ ⊗ m − s − l . (62) et’s write several relations which are satisfied by Q , µ , µ , µ : Q = 0 , (63) Qµ ( a , a ) = µ ( Qa , a ) + ( − n a µ ( a , Qa ) ,Qµ ( a , a , a ) + µ ( Qa , a , a ) + ( − n a µ ( a , Qa , a ) h +( − n a + n a µ ( a , a , Qa ) = µ ( µ ( a , a ) , a ) − µ ( a , µ ( a , a )) . In such a way we see that if µ n = 0, n ≥ ∂ = 0. To see this, one applies the desuspension operation (the operationwhich shifts the grading s − : V k → V k − ) to µ n , i.e. one can define operations ofdegree 1: ν n = sν n s − ⊗ n . More explicitly, ν n ( s − a , ..., s − a n ) = ( − s ( a ) s − µ n ( a , ..., a n ) , (64)such that s ( a ) = (1 − n ) n a + (2 − n ) n a + ... + a n − . The relations between ν n operations can be summarized in the following simple equations: n X i =1 N i ◦ N n +1 − i = 0 . (65)on V ⊗ n . Here each N s acts on V ⊗ m ( m ≥ s ) as the sum of all operators ⊗ l ⊗ ν s ⊗ ⊗ k , such that l + s + k = m . Combining them into one operator ∂ = P n ν n , actingon a space ⊕ k V ⊗ k the relations (61) can be combined into one equation ∂ = 0.Another way to represent the relations (61) as the nilpotency condition of someoperator is by using the differential operators on a noncommutative manifold. Letthe set { e i } be the homogeneous elements, which form a basis of V . Introducenoncommutative supercoordinates x i such that X = x i e i has degree degree 1 on V . One can write a noncommutative vector field, such that Q x i = ν n ; j ,...,j n x j ...x j n , (66)where ν n ; j ,...,j n = ν n ( e j , ..., e j n ). Then the relations (65) can be formulated as Q = 0. A2. Cyclic structures, BV formalism and the generalized Maurer-Cartanequation.
First we define what cyclic structure is.
Definition.
The A ∞ -algebra on a space V is called cyclic if there exists a nonde-genenerate pairing h· , ·i , such that it is graded symmetric h a, b i = − ( − ( n a +1)( n b +1) h b, a i (67) and satisfies the following conditions for µ n : h a , µ n − ( a , ..., a n ) i =( − ( n − a + a +1)+ a ( a + ... + a n ) h a , µ n − ( a , ..., a n , a ) i . (68)It makes sense to define ψ n ( a , ..., a n ) = h a , µ n − ( a , ..., a n ) i . Consider again theelement of degree 1 X = x i e i , where x i are noncommutative supercoordinates anddefine the formal action f unctional : S [ X ] = ∞ X n =2 n Ψ n ( X, ..., X ) . (69) et ω ij = h e i , e j i . Then one can define a BV bracket:( α, β ) BV = ←− ∂ α∂x i ω ij −→ ∂ β∂x j (70)on the space of polynomials of x i , such that S satisfies classical Master equation:( S, S ) BV = 0(71)Using this condition, one can find that S defines an odd vector field such that( S, x i ) BV = ν n ; j ,...,j n x j ...x j n , (72)which coincides with odd vector field Q was defined in (66). Moreover, for a givenaction S = v i i x i x i + X n ≥ v i ...i n x i ....x i n , (73)which is cyclic in x i and satisfies Master equation, we obtain the cyclic A ∞ -algebra(see e.g. [11],[4]).Varying the action (69) with respect to X , one finds the equation of motionwhich are known as generalized Maurer-Cartan equation: QX + X n ≥ µ n ( X, ..., X ) = 0 . (74)This equation is known to have the following symmetry X → Qα + X n ≥ ,k ( − n − k µ n ( X, ..., α, ..., X ) , (75)where α is an element of degree 0 and k means the position of α in µ n . A3. Transfer of the A ∞ structure. Suppose you have two complexes whichare quasiisomorphic and moreover homotopically equivalent. Moreover, supposethat on one of them there exists the structure of A ∞ -algebra. Then there exists an A ∞ -algebra on another complex. The explicit formulas are given in [15] followingthe results of [13] and [17].In fact, in [15] there is even a more general statement. Let us formulate it in aprecise way. Consider two complexes ( F , Q ) and ( K , Q ′ ) such that there are maps f : ( F , Q ) → ( K , Q ′ ), g : ( K , Q ′ ) → ( F , Q ) such that f g is chain homotopic toidentity. In other words, there exists a map H : ( K , Q ′ ) → ( K , Q ′ ) of degree − f g = id + Q ′ H + HQ ′ . Then, given an A ∞ -algebra structure ˆ µ { n } on K ,such that ˆ µ = Q ′ , one can construct an A ∞ -algebra on ( F , Q ) by means of theformula µ n = g ◦ p n ◦ f ⊗ n , (76)where p n is obtained by means of consequtive recurrent procedure of applying thehomotopy operators to ˜ µ n . The explicit formula is: p n = X B ( − θ ( r ,...,r k ) ˆ µ k ( H ◦ p r , ..., H ◦ p r k ) , (77)where B = { k, r , ..., r k | ≤ k ≤ n, r , ..., r k ≥ , r + ... + r k = n } , θ ( r , ..., r k ) = P ≤ α ≤ β ≤ r r α ( r β + 1) and p ≡ ˆ µ , p ◦ H ≡ id . ppendix B: Explicit calculations for A ∞ superalgebras ofsuperforms. 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 µ is η µν =diag( − , +1 , +1 , +1). The N = 1 four-dimensional space has two anticommutativeWeyl spinor coordinates θ α , θ ˙ α . The spinor indices are raised and lowered by meansof C αβ , C ˙ α ˙ β , such that C = − C = i . One can define superderivatives: D α = ∂ α + i θ ˙ β ∂ α ˙ β , ¯ D ˙ α = ¯ ∂ α + i θ β ∂ β ˙ α , (78)where ∂ α ˙ β = σ µα ˙ β ∂ µ . Therefore, one can introduce the (anti)chiral scalar super-fields, which are defined by the nilpotency of certain antiderivatives D ˙ α Λ( x, θ ) = 0( D α ¯Λ( x, θ ) = 0) The relations between superderivatives are: { D α , D ˙ β } = i∂ α ˙ β , { D α , D ˙ β } = 0 , { D ˙ α , ¯ D ˙ β } = 0 . (79)For calculations it is also useful to introduce operators D = C αβ D α D β , ¯ D = C ˙ α ˙ β D ˙ α D ˙ β . B2. Explicit calculations for the A ∞ -algebra of N = 1 SUSY Yang-Millstheory.
In this subsection we show by explicit calculation that the bilinear oper-ation we defined on the complex (35) is homotopy associative and the differentialacts on it as a derivation.For simplicity we denote µ n ( · , ..., · ) as ( · , ..., · ) h . We remind that we defined thegraded symmetric bilinear operation ( · , · ) h , i.e. ( a , a ) h = ( − | a || a | ( a , a ) h onthe complex (37) in the following way:(Λ , Λ ) h = Λ Λ , (Λ , V ) h = 12 Λ V, (Λ , W α ) h = Λ W α , (Λ , ˜ W α ) h = Λ ˜ W α , (Λ , ˜ V ) h = 12 Λ ˜ V , (Λ , ˜Λ) h = Λ ˜Λ , ( V , V ) h,α = −
12 ¯ D ( V D α V − D α V V ) , ( V, W ) h = D α V W α + 12 D α W α V, ( ˜ W , W ) h = ˜ W α W α , ( ˜ V , V ) = −
12 ¯ D ( ˜ V V ) , ( W , W ) h = 0 , ( V, ˜ W ) h = 0 , (80)such that Λ , Λ ∈ Φ; V, V , V ∈ Σ; W, W , W ∈ Θ, ˜ W ∈ ˜Θ; ˜ V ∈ ˜Σ; ˜Λ ∈ ˜Ξ.Now we start checking that this operation satisfies all necessary relations ofhomotopy associative algebra. The first relation is the Leibniz rule, i.e. D ( a , a ) h = ( D a , a ) h + ( − | a | ( a , D a ) h . (81) et us check it step by step: D (Λ , V ) h = 12 ¯ D D α (Λ V ) =12 ¯ D ( D α Λ V − Λ D α V ) + Λ ¯ D D α V =( D Λ , V ) h + (Λ , D V ) h , (82) D (Λ , W ) h = D α (Λ W α ) + Λ W α = D α Λ W α + 12 Λ D α W α + Λ W α + 12 Λ D α W α =( D Λ , W ) h + (Λ , D W ) h , (83) D ( V, W ) h = ¯ D ( D α V W α + 12 V D α W α ) =¯ D D α V W α + 12 ¯ D V D α W α =( D V, W ) h − ( V, D W ) h , (84) D (Λ , ˜ V ) h = 12 ¯ D (Λ ˜ V ) = −
12 Λ ¯ D ˜ V + Λ ¯ D ˜ V = ( D Λ , ˜ V ) h + (Λ , D ˜ V ) h . (85)Our next task is to verify that ( · , · ) h satisfies homotopy associativity relation:( a , ( a , a ) h ) h − (( a , a ) h , a ) h + D ( a , a , a ) h + ( D a , a , a ) h +( − | a | ( a , D a , a ) h + ( − | a | + | a | ( a , a , D a ) h = 0 , (86)where ( · , · , · ) h is the trilinear operation we will derive below.Let us proceed as above, checking associativity step by step:(Λ , ( V , V ) h ) h − ((Λ , V ) h , V ) h = −
12 ¯ D (Λ V D α V − Λ D α V V ) + 14 ¯ D (Λ V D α V − V D α (Λ V )) = −
14 ¯ D (Λ( V D α V − D α V V ) + D α Λ V V ) = − D (Λ , V , V ) h − ( D Λ , V , V ) h − (Λ , D V , V ) h + (Λ , V , D V ) h , (87) ( V , (Λ , V ) h ) h − (( V , Λ) h , V ) h = −
14 ¯ D ( V D α (Λ V ) − D α V Λ V ) + 14 ¯ D ( V Λ D α V − D α ( V Λ V ) = −
12 ¯ D ( V D α Λ V ) = − D ( V , Λ , V ) h − ( D V , Λ , V ) h + ( V , D Λ , V ) h + ( V , Λ , D V ) h , (88) Λ , (Λ , V ) h ) h − ((Λ , Λ ) h , V ) h = 14 (Λ Λ V ) −
12 (Λ Λ V ) = −
14 Λ Λ V = − D (Λ , Λ , V ) h − ( D Λ , Λ , V ) h − (Λ , D Λ , V ) h − (Λ , Λ , D V ) h . (89)Since we have ((Λ , V ) h , Λ ) h = (Λ , ( V, Λ ) h ) h , (90)we obtain ( V , V , V ) h = 16 ¯ D ( V V D α V − V ( D α V ) V + ( D α V ) V V )(Λ , V , V ) h = ( V , V , Λ) h = 112 Λ V V ( V , Λ , V ) h = 16 Λ V V (91)One can check that these expressions fit the formulas above. Then we need tostudy the associativity relation involving W -terms, i.e.(Λ , ( V, W ) h ) h − ((Λ , V ) h , W ) h =12 Λ D α V W α + 14 D α W α V Λ − D α (Λ V ) W α −
14 Λ
V D α W α = − D α Λ V W α = − D (Λ , V, W ) h − ( D Λ , V, W ) h − (Λ , D V, W ) h + (Λ , V, D W ) h , (92)where ( V , V , W ) h = D α V V W α + V V D α W α and (Λ , V, ˜ V ) = V V ˜ V . An-other terms involving W are( V , ( V , W ) h ) h − (( V , V ) h , W ) h = −
12 ¯ D ( 12 V ( D α V W α + 12 D α W α V ) + 12 ¯ D ( V D α V − D α V V ) W α ) = − ¯ D ( 12 D α V V W α + 14 V D α W α V ) − D ( V , V , W ) h − ( D V , V , W ) h + ( V , D V , W ) h − ( V , V , D W ) h , (93) ( V , ( W, V ) h ) h − (( V , W ) h , V ) h ) = 12 ¯ D ( V ( D α V W α + 12 D α W α V ) + V ( D α V W α + 12 D α W α V ) = − D ( V , W, V ) h − ( D V , W, V ) h + ( V , D W, V ) h − ( V , W, D V ) h . (94)Here ( V , V , W ) h = ( W, V , V ) h = 12 D α V V W α + 16 V V D α W α , ( V , W, V ) h = −
12 ( V D α V + D α V V ) W α − V D α W α V , ( V , V , ˜ V ) h = ( ˜ V , V , V ) = 112 ¯ D ( V V ˜ V ) , ( V , ˜ V , V ) = 16 ¯ D ( V ˜ V V ) . (95) herefore from the calculations above one can obtain that algebraically up tothe third order these equations look as follows: D U + ( U, U ) h + ( U, U, U ) h + ... = 0 ,U → U + D Λ + ( U, Λ) h +( U, U, Λ) h − ( U, Λ , U ) h + (Λ , U, U ) h + ..., (96)where U is the element of the grading 1 from the complex (35) corresponding tothe pair ( V, W ). B.3. Notations for N=2 3D superspace.
We work with three-dimensionalspace with coordinates x µ and euclidean metric η µν = diag(+1 , +1 , +1). The Diracmatrices are ( γ µ ) βα = i ( σ , σ , σ ). We can raise and lower the corresponding spinorindices via C αβ , i.e. ξ α = C αβ ξ β and ξ α = C αβ ξ β , such that C = − C = i .The N = 2 3D superspace has two 2-component anticommuting coordinates θ α and θ α . Therefore (this is similar to N = 1 superspace) one can define complexcoordinates θ α = θ α − iθ α and ¯ θ α = θ α + iθ α . This allows to define superderivatives(cf. N = 1 D=4 case): D α = ∂ α + i θ β ∂ αβ , ¯ D α = ¯ ∂ α + i θ β ∂ αβ (97)where ∂ αβ = γ µαβ ∂ µ . One defines the (anti)chiral scalar fields via the familiarequation: ¯ D α Λ( x, θ ) = 0, D α ¯Λ( x, θ ) = 0. For the calculations we will need thefollowing commutation relations between superderivatives: { D α , ¯ D β } = iγ µαβ ∂ µ = i∂ αβ , { D α , D β } = 0 , { ¯ D α , ¯ D β } = 0 . (98)As well as in N = 1 D=4 case, it is useful to introduce operators D = C αβ D α D β ,¯ D = C αβ ¯ D α ¯ D β . References [1] O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M -branes and their gravity duals , arXiv:0806.1218.[2] M. Benna, I. Klebanov, T. Klose, M. Smedbaeck, Superconformal Chern-Simons Theoriesand
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