Topological twisting of conformal supercharges

Putting a twisted version of N=4 super Yang-Mills on a curved four-dimensional manifold generically breaks all conformal supersymmetries. In the special case where the four-manifold is a cone, we show that exactly two conformal supercharges remain unbroken. We construct an off-shell formulation of the theory such that the two unbroken conformal supercharges combine into a family of topological charges parameterized by CP^1. The resulting theory is topological in the sense that it is independent of the metric on the three-dimensional base of the cone.


Introduction
Topological quantum field theory (TQFT) provides an intriguing link between physics and mathematics. The study of TQFT was initiated with [1], wherein a quantum field-theoretical representation of Donaldson invariants was given. A quantum field theory is called topological if all vacuum expectation values (VEVs) of a certain set of operators ('observables') are metric-independent. In particular, TQFTs of cohomological type are constructed as follows. Let us assume there is a nilpotent symmetry of the action Q, such that Q 2 = 0. It follows that, at least formally, one can deform the Lagrangian by adding an arbitrary Q-exact term without affecting the partition function or the VEVs of observables (which are defined as elements in the cohomology of Q). Since Q is a symmetry of the action, the Lagrangian can be expressed as a sum of a Q-exact and a Q-closed piece. The theory is therefore independent of any coupling constant in the Qexact piece. Moreover, if the energy momentum tensor is Q-exact all VEVs of observables are metric-independent and the theory is topological.
One way of constructing TQFTs is by 'twisting' theories with extended supersymmetry in Euclidean space. Roughly-speaking, twisting can be thought of as an embedding of the rotation group in the global R-symmetry group, thereby changing the spins of the fields in the parent theory. For the resulting theory to be topological on a general manifold, there should exist at least one scalar among the twisted supersymmetry generators. The TQFT of [1] was obtained, by twisting, from N = 2 supersymmetric Yang-Mills theory in four dimensions. Similarly, N = 4 SYM in four dimensions can be twisted in three inequivalent ways to obtain a TQFT [2][3][4]. It has been subsequently noted that all three twisted theories can be thought of as world-volume theories on D3-branes wrapping supersymmetric cycles in string-theory compactifications [5]. Two of the twists have been extensively studied in [2], in connection to instanton invariants. The third twist, which is the focus of the present paper, has been studied in a series of papers [4,[6][7][8][9][10], and has been recently found to be relevant to the geometric Langlands program [11].
Four-dimensional N = 4 SYM in flat space is superconformal: in addition to sixteen ordinary supercharges it possesses sixteen conformal supercharges. The aim of the present paper is to explore the effect of the twisting on the conformal supercharges. On a general four-manifold all conformal supercharges will be broken. However, in the special case where the manifold is a cone over a three-dimensional base, we find that the twisting leads to exactly two unbroken conformal supercharges, S 0 , S 5 . We construct an off-shell formulation of the theory such that the two twisted conformal supercharges combine into a family of topological charges S z := z 1 S 0 + z 2 S 5 , z := z 2 /z 1 , parameterized by CP 1 . In this formalism the components of the energy-momentum tensor along the directions of the base of the cone are manifestly S z -exact -since S z is nilpotent off-shell and independent of the metric on the base of the cone. It follows that (at least formally) the theory is topological in the sense that it is independent of the metric on the base of the cone.
There are two important caveats in this statement. Firstly, as is usually the case in the pathintegral formulation of TQFTs, we shall assume that the path-integral measure is well-defined and S z -invariant. Secondly, in order to define the functional integration on a cone we have to impose boundary conditions on the fields. As we explain in more detail in sections 3.5, 3.6, there is a certain freedom in the choice of boundary conditions. Generic boundary conditions will break superconformal invariance. Requiring that the scalar fields vanish at infinity is sufficient for the superconformal invariance to be preserved.
The aim of this paper is to furnish a new tool for the study of topological properties of threemanifolds (viewed as bases of four-dimensional cones). We believe that the topological off-shell theory we construct reveals a rich enough structure to warrant further study. Of course the construction of the off-shell formalism should be viewed as just the first step; eventually we would like to identify the set of observables of the theory (for this, a more complete study of the boundary conditions will be necessary) and their relevance to three-dimensional topological invariants. We will leave this second nontrivial step for future investigation.
The organization of this paper is as follows. In section 2 we review some well-known facts about ten-dimensional SYM and its reduction to N = 4 SYM in four dimensions. In particular, we point out that in flat space the conformal supersymmetries can be viewed as similarity transformations of the ordinary supersymmetries, with respect to a certain inversion map.
Section 3 explains the twisting of the N = 4 super Yang-Mills on which we focus in this paper. We obtain the twisted Lagrangian as well as the ordinary and conformal scalar supersymmetry transformations. The superconformal algebra of the twisted SYM in flat space contains two ordinary scalar supercharges and two conformal scalar supercharges. We show that the conformal scalar supersymmetries survive on a curved four-manifold, if and only if the manifold is (locally) a cone over an arbitrary three-dimensional base. We identify the most general linear combination of the four scalar supercharges which is nilpotent, and thus can be used to define a cohomological structure. Some important 'vanishing theorems' are presented in subsections 3.5 and 3.6.
In section 4, we show that there exists an off-shell formulation of the theory such that the two twisted conformal supercharges combine into a family of nilpotent operators parameterized by CP 1 . Moreover we show that (at least formally) the theory is topological in the sense that it is independent of the metric on the base of the cone.
We conclude with a discussion of open questions in section 5. The appendices contain some useful proofs and formulae.

Review of super Yang-Mills
In order to establish our conventions and notation, we now give a brief review of super Yang-Mills in ten Euclidean dimensions and its reduction to four-dimensional N = 4 super Yang-Mills.
By Wick-rotating the ordinary super Yang-Mills in ten dimensions, one obtains a Euclidean theory which is still formally supersymmetric in that it is invariant under the same set of fermionic transformations as the original theory. However, the minimal spinor representation in ten Euclidean dimensions is no longer Majorana, resulting in a complex action. Of course this phenomenon is not new: it occurs routinely in string theory in a variety of different contexts.
There are essentially two ways to deal with this problem. The point-of-view we take here is to simply ignore the fact that the action is complex, as for example in [11]. This would naively seem to double the fermionic degrees of freedom. However, the complex-conjugate fermions do not appear in the Lagrangian. Therefore in the path-integral one integrates over only half of the (complexified) fermionic degrees of freedom -resulting in the same counting as for the Minkowskian theory. Another way to state this is that the action is not required to be real -it is only required to be holomorphic. The other way to handle the Wick rotation is by imposing reality conditions on the fields. These reality conditions are sometimes introduced in an ad hoc manner. A systematic way to derive them is by time-like reduction of a higher-dimensional theory, as demonstrated in [12].

Super Yang-Mills in ten dimensions
Consider the 32 × 32 gamma matrices in ten-dimensional Euclidean space These are Hermitian Γ M † = Γ M and satisfy We define the ten-dimensional chirality operator by The main Fierz identity is whereα,β,γ,δ are the 32 component spinorial indices, and P ± are the chiral/anti-chiral projection matrices P + = 1 2 (1 + Γ (11) ) , Super Yang-Mills in ten Euclidean dimensions is given by: where the gaugino Ψ is a chiral spinor andΨ is defined by 1Ψ := Ψ T C ,ΨΓ (11) = −Ψ . (2.8) In our conventions the gauge field strength and the gauge-covariant derivative read so that gauge transformations are The sixteen ordinary supersymmetries are given by where the supersymmetry parameter ε + is a constant Weyl spinor, and the subscript denotes positive chirality in ten dimensions.

Dimensional reduction and superconformal symmetry
We now consider the dimensional reduction to four-dimensional N = 4 super Yang-Mills. Let us denote the four-dimensional coordinates by x µ , µ = 1, 2, 3, 4. We also set Φ I := A I , 5 ≤ I ≤ 10. Four-dimensional N = 4 super Yang-Mills is then given by In addition to the sixteen ordinary supersymmetries (2.11), there are sixteen conformal supersymmetries where X := x µ Γ µ and the conformal supersymmetry parameter ε − is a Weyl spinor of negative chirality, Γ (11) Both ordinary and conformal supersymmetries can be collected in terms of a single 32component spinor (2.14) All 32 supersymmetries are then given by Accordingly, the Lagrangian (2.12) transforms into a total derivative leaving the action invariant.
From the corresponding Noether current we obtain a 32-component supercurrent

19)
2 For simplicity the spinors are given in ten-dimensional notation.
where we have set The chiral and anti-chiral parts P + Q µ , P − Q µ , correspond to the conformal and ordinary supersymmetries respectively. It is worthwhile to note the existence of an inversion map I defined in flat background [13][14][15] x µ The inversion map flips the chirality of the fermion, which is consistent with the fact that the two supersymmetry parameters ε + and ε − have opposite chiralities. Note, however, that this symmetry is broken in a generic curved background.

Twisted N = 4 super Yang-Mills in four dimensions
We now come to the description of the twist. In the following subsections, we also discuss the unbroken conformal supercharges and derive certain important 'vanishing theorems'.

Taking (2.2) and (2.3) into account, it follows that
so that the ten-dimensional chirality coincides with the chirality in the Spin(2) part. The fermions Ψ ±αβ carry three indices for Spin(2)×Spin(4)×Spin (4). The focus of the present paper is the twist considered in [3,4,[6][7][8][9]11]: it amounts to replacing the four-dimensional rotation group by the diagonal subgroup of Spin(4) × Spin(4). Accordingly, the twisted ten-dimensional chiral fermion (2.7) admits the following expansion It follows that the fermions decompose into a pair of anticommuting scalars η, ζ, a pair of vectors ψ µ , ω µ and a two-form χ µν = −χ νµ . The chiral and anti-chiral supersymmetry parameters ε + , ε − in (2.14) admit similar expansions, as we shall see later in (3.20). In particular, there will be two ordinary scalar and two conformal scalar supercharges. For the bosons we set These obey the following reality conditions: while there is no analogous constraint for fermions 3 .

Twisted Lagrangian
Taking (3.1), (3.5), (3.6) into account, it is straightforward to rewrite the N = 4 super Yang-Mills (2.12) in terms of the anticommuting fields η, ζ, ψ µ , ω µ , χ µν and the bosons A ± µ , φ µ , ϕ,φ. The resulting action defines our twisted N = 4 super Yang-Mills in four-dimensions: 4 (3.9) In the above, we have introduced an auxiliary bosonic scalar h. We have coupled the action to a generic curved background with metric g µν . 5 Moreover, ǫ κλµν is the totally antisymmetric tensor density, such that ǫ 1234 = 1 and ǫ 1234 = g. All derivatives are covariant with respect to both diffeomorphisms and gauge transformations: (3.10) Note that L top is manifestly metric-independent, as follows from the fact that the Christoffel connection is torsion-free. Moreover, we have defined the field strengths Note that the superscript ± above does not denote (anti) self-duality. It is also useful to define a covariant tensor H µν by such that and (3.14) The field strengths can be seen to obey the following generalized Bianchi identities It is important to note that the identities above are valid on any curved manifold. Moreover, taking the following identities into account 6 the bosonic part of the Lagrangian containing F µν and φ µ can be rewritten as The Ricci term on the right-hand side is necessary to ensure the invariance of the action under the unbroken scalar supersymmetries. 6 In our conventions the Riemann tensor is given by The equations of motion are Finally, the energy-momentum tensor is given by 7 where the brackets denote symmetrization with weight one.

Superconformal symmetry
Upon twisting, as in the case of the Yang-Mills gaugino (3.5), the chiral ordinary supersymmetry and the anti-chiral conformal supersymmetry parameters, ε + and ε − respectively, decompose as (3.20) The corresponding supercharges Q − , S + have the opposite chiralities from (2.20), (3.4), and decompose similarly Altogether, there are (1 + 4 + 6 + 4 + 1) × 2 = 32 components, all of which are unbroken in a flat background. In curved backgrounds, however, in order to have unbroken supersymmetries it is necessary that the corresponding supersymmetry parameters should be covariantly constant. Generically, this requirement can only be met for the scalar parameters ε 0 , ε 5 ,ε 0 ,ε 5 . Hence, in a generic curved background all the non-scalar supersymmetries Q µ ,Q µ , Q µν , S µ ,S µ , S µν are broken 8 .
In addition, one has to check that in putting the theory on a curved background the action remains supersymmetric. This is not obvious: in passing from flat to curved background there may be curvature terms arising from the commutator of two covariant derivatives [∇ µ , ∇ ν ], spoiling the invariance. In the case at hand, one can verify that the two ordinary scalar supercharges Q 0 and Q 5 of (3.22) indeed give rise to the two unbroken topological symmetries of the twisted super Yang-Mills (3.9), as shown in [3,4,[6][7][8][9]11]. Explicitly the action of Q 0 , Q 5 is given by: In contrast to the above ordinary scalar supercharges, the conformal scalar supercharges S 0 and S 5 are not straightforwardly realized in the twisted super Yang-Mills as Noether symmetries, because the conformal supersymmetry transformations in flat space involve the space coordinate x µ explicitly, see (2.13). In a curved background, x µ should be replaced by a vector field v µ satisfying the following relation: Indeed, the condition ∂ µ x ν = δ µ ν is the only property of x µ which is required for the invariance of the action in a flat background. Moreover, it can be checked explicitly that the condition (3.23) suffices for the conformal scalar supercharges S 0 and S 5 to survive as symmetries of the action in a curved background, provided the vector field v µ exists. It is straightforward to see that any cone ds 2 = dr 2 + r 2 dŝ 2 , admits such a vector field: v = r ∂ ∂r . (3.25) Conversely, as we show in Appendix A, any manifold admitting such a vector field is a cone, at least locally. Hence, the conformal supercharges S 0 and S 5 remain unbroken in a cone background and, similarly to the case of the ordinary supercharges Q 0 and Q 5 , they give rise to a family of TQFTs, as we shall demonstrate explicitly in section 4.
Using the vector field (3.23), the conformal scalar supersymmetries are given by It is worth noting that there exists a discrete symmetry Z given by 10 under which the Lagrangian (3.9) is invariant. Furthermore Z acts on the scalar supercharges as Apart from the four scalar supersymmetries, the discrete symmetry Z and the diffeomorphisms of the three-dimensional base, there exist another two (bosonic) symmetries of the twisted N = 4 super Yang-Mills in a generic cone background. One is the U(1) R-symmetry or the SO(2) rotation on the (Φ 9 , Φ 10 ) plane (3.29) 10 Note the similarity with the 'Hermitian-conjugation' defined in [4].
The other is a 'dilatation' symmetry 11 Since in flat space the inversion map (2.21) transforms the conformal supersymmetries to the ordinary supersymmetries and vice versa (2.22), one might wonder whether the theory defined by the conformal supercharges S 0 , S 5 is truly inequivalent to the one given by the ordinary supercharges Q 0 , Q 5 . In a flat background, a straightforward computation shows that the conformal scalar supercharges are related by the inversion map to the ordinary vector supercharges. Schematically 12 , Hence the conformal scalar supercharges are not equivalent to the ordinary scalar supercharges.

Superconformal algebra
All scalar supercharges are nilpotent off-shell Moreover, among the scalar supercharges there are two pairs which anticommute off-shell: (3.34) 11 Considering the asymmetric scaling dimensions of ϕ andφ in (3.30), one may wish to regard D + R as the dilation operator instead. 12 More precisely: with all other commutators identically zero.
All other pairs of supercharges only anticommute on-shell. More specifically, up to the equations of motion (3.18) we have where L v·A ± denotes infinitesimal gauge transformations generated by v µ A ± µ , while D corresponds to the dilatation (3.30). In the present paper ' ≡ ' denotes an on-shell equality, i.e. an equality up to equations of motion. The dilation operator commutes with all the scalar supercharges off-shell The U(1) charges of the supercharges can be read off of the following commutation relations From the superconformal algebra above, we conclude that the most general linear combination of the four scalar supercharges which is nilpotent, is parameterized by four complex numbers subject to one constraint: In particular, there are four possible pairs of supercharges which are nilpotent: In the above, z 1,2 are arbitrary complex numbers. As in [11] we note that an overall constant is unimportant, hence each pair of supercharges corresponds to a family of topological theories parameterized by CP 1 . For simplicity, henceforth we work in the patch z 1 = 0. 13 The first linear combination in (3.40) was recently investigated by Kapustin and Witten [11] and was shown to be relevant to the Langlands program. The second linear combination is the main focus of the present paper. For completeness, in Appendix B we give all the supersymmetric conditions, i.e. the BPS equations, arising from each linear combination in (3.40).

Vanishing theorems -weak version
For the rest of this section we shall focus on the conformal supersymmetry We shall derive certain important 'vanishing theorems', which follow from a single algebraic identity. Let us define From the on-shell conformal supersymmetry transformations of the fermions (3.26), we see that an S z -supersymmetric configuration must satisfy, by definition, the following conditions (BPS equations) Or, in compact notation, |V(z) = 0. Disregarding the boundary terms we obtain the following algebraic identity involving two arbitrary complex numbers z, w : twisted is the bosonic part of our twisted N = 4 super Yang-Mills Lagrangian (3.9), after eliminating all the auxiliary fields: Note that L boson twisted is positive semi-definite. Eq.(3.44) is an algebraic identity which holds for field configurations which are smooth enough at the origin of the cone and fall off fast enough at spatial infinity, so that we can neglect the boundary terms.
For w = z, the identity (3.44) reduces to where we have set T (z) := 1 Note that T (z) is real and topological, in that it does not depend on the metric. On the other hand, the choice w * = z −1 gives another identity for the bosonic action:  Hence, given a complex number z, a supersymmetric configuration (for which |V(z) = 0) minimizes the action within a given topological sector. For an arbitrary bosonic configuration, regarding T (z) as a function of z, the minimum and the maximum of T (z) are located generically 15 at two antipodal points on the unit circle: Since the left hand side of (3.46) is independent of z, the above analysis implies that, for any given bosonic configuration, V(z)|V(z) /(1 + |z| 2 ) is bounded both from above and below: (3.54) In particular, if V(z min )|V(z min ) = 0, V(z)|V(z) cannot vanish for all z. As in [11], this leads to certain vanishing theorems.

• Vanishing theorem for generic configurations:
A configuration is supersymmetric if and only if V(z min )|V(z min ) = 0.
Furthermore, from the identities (3.44), (3.46) we obtain the following vanishing theorems for supersymmetric configurations.
• Vanishing theorems for BPS configurations: 1. The choice w * = −z −1 in (3.44) shows that any BPS state, for which |V(z) = 0, should satisfy Hence, by complex conjugation, we obtain 4. Any state which satisfies the BPS equations |V(z) = 0 for some z such that |z| = 1, satisfies the BPS equations for all z.
5. Any state which satisfies the BPS equations |V(z) = 0 for some z such that |z| = 1, satisfies the BPS equations for all z iff T (z) = 0.
As a corollary we obtain an alternative criterion for a generic configuration to be supersymmetric.

• Corollary:
A configuration is supersymmetric if and only if |V(z) = 0 for some |z| = 1.
Using explicitly the cone coordinates (3.24), the BPS equations read for |z| = 1: In particular if the BPS configuration has a trivial topology T (z) = 0 so that L boson twisted = 0, it follows from (3.45) that the supersymmetric configuration satisfies In the special case where the base of the cone is the round sphere, the cone reduces to R 4 . From (3.17) it then follows that a BPS configuration with trivial topology satisfies F µν = 0, ϕ = φ µ = 0. These conditions are more restrictive than the usual vacuum equations F µν = 0, D µ Φ I = 0, [Φ I , Φ J ] = 0, where Φ I := (φ µ , ϕ,φ). This is due to our requirement that the fields should decay fast enough at the boundary, and is consistent with the fact that the conformal symmetry remains unbroken if the VEVs of the scalars vanish.

Vanishing theorems -strong version
Taking equations (3.11), (3.16) and the Bianchi identity for the Riemann tensor into account, we obtain (3.60) This implies that the topological term is real and it reduces to the usual instanton number provided we can neglect the boundary terms in (3.60). This requirement is automatically satisfied in the case where the VEVs of the scalars vanish at infinity. The latter condition is also sufficient for the conformal symmetry to be unbroken. 16 Here, we shall not discuss the issue of the boundary conditions any further; instead we shall assume that (3.61) holds and examine the consequences. In this case the minimum and maximum of T (z) occur at z = ±1, so that (3.62) Consequently we have the following strong version of the vanishing theorems of the previous subsections.

Off-shell formulation
The conformal supersymmetry transformations (3.26) given in the previous section have two unsatisfactory features. Firstly, the anti-commutator relation {S 0 , S 5 } ≡ 2L v 2φ only holds on-shell. Secondly, the transformations depend explicitly on the metric on the base of the cone,ĝ ij (y). In this section, we shall construct an off-shell formalism for the superconformal symmetry which is manifestly independent of the metric on the base. Specifically, we shall introduce additional auxiliary fields and off-shell superconformal transformations generated by a new conformal supercharge S ′ z , such that the following relation holds: i.e. the supercharge S ′ z reduces on-shell to the supercharge S z defined in (3.41). In addition, S ′ z will be shown to be nilpotent off-shell. For simplicity of notation, in the following we suppress the prime in S ′ z .

Off-shell algebra for conformal supersymmetries
In addition to h, let us introduce the following new auxiliary fields where all tensor fields are anti-symmetric We require all the auxiliary fields to be orthogonal to the vector v Furthermore, the bosonic auxiliary fields are required to satisfy reality conditions similar to (3.7) We also introduce a second fermionic two form fieldχ µν = −χ νµ . Under the discrete symmetry (3.27) we have (4.6) Let us also define the projection operator which projects onto subspaces orthogonal to the vector v.
We are now ready to give the off-shell transformations of all fields: (4.8)

Off-shell Lagrangian
Let us define (4.14) Our main formula is then where L twisted is the original Lagrangian given in (3.8) and L auxiliary consists of auxiliary terms Here h + µ on , h ± µν on , ξ ± µν on denote the on-shell expressions of the auxiliary fields given in (4.12), (4.13), and we have set We also define which corresponds to the equation of motion for χ µν (3.18). Thus all the terms in L auxiliary vanish on-shell. The equations of motion for the auxiliary fields ξ ± µν , h ± µ , h ± µν give the on-shell relations (4.11) and (4.12). In particular, a nice feature is that, due to the orthogonal property (4.4), we need to integrate out both ξ + µν and ξ − µν in order to arrive at the on-shell relationχ µν ≡ − (⋆ χ) µν (4.11). This is important for a nontrivial path integral measure for the fermionic auxiliary fields. All other equations of motion then consist of those coming from the original Lagrangian (3.18) together with the additional equation which is of course consistent with the on-shell relation (4.13). From the above construction it follows that the components of the energy-momentum tensor along the base are manifestly S z -exact. Hence the theory is topological, at least formally, in the sense that it is independent of the metric on the base.

Discussion
In this paper we focused on the twisting of four-dimensional N = 4 SYM studied in [4,[6][7][8][9]11]. We have found that putting the theory on a four-dimensional cone leaves two conformal supercharges unbroken. The resulting theory is topological in that it is independent of the metric on the base of the cone.
A few questions still remain open. Firstly one would like to identify the set of observables of the theory, i.e. the set of operators in the cohomology of the topological charge, and their potential relevance to topological invariants of the three-dimensional base of the cone. The role of all possible linear combinations of nilpotent scalar supercharges identified in section 3, also deserves further study.
The S-duality of N = 4 SYM has significant implications for the twisted version considered in the present paper, and this played a central role in the recent discussion of [11]. It would be interesting to examine this issue further. Another possible direction would be to consider reductions of the theory to lower-dimensional cones. A more thorough discussion of the different possible boundary conditions is postponed for future work.

A Cone geometry
This section contains, among some other useful facts about cone geometry, a proof of the statement that the existence of a vector field v satisfying (3.23) implies that, at least locally, the manifold is a cone: and the vector field is given by

Proof
Let us define coordinates x µ = r, y i such that Under diffeomorphisms of the form r −→ r ′ = r , y i −→ y ′i = f i (r, y) , (A.5) the g ri (x) component of the metric transforms as whereḡ kl is the inverse of g ij ,ḡ ij g jk = δ i k . Given initial conditions f k (0, y) at r = 0, one can uniquely fix the evolution along the r-direction by demanding ∂ r f k (r, y) = −ḡ kl (r, f (r, y))g rl (r, f (r, y)) .
This determines all the higher-order derivatives of f k (r, y) with respect to r. Thus, it is possible to choose a gauge such that [16] g ri = 0 .
(A. 10) In particular, by taking the ∇ i v j component, we see thatĝ ij is independent of the radial coordinate r. Hence the geometry is a cone and the vector field is given by (A.3). All other components of (A.10) can also be seen to hold, since: il (∂ jĝlk + ∂ kĝjl − ∂ lĝjk ) , Γ r rr = Γ r ri = Γ i rr = 0 . (A.11) This completes the proof.

C Useful identities
Any two-form tensors A µν , B µν satisfy which implies in particular Some other useful identities are