{'fraction_non_alphanumeric': 0.051622641509433964, 'fraction_numerical': 0.013509433962264152, 'mean_word_length': 4.553646269907795, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Super Self-Duality as Analyticity in Harmonic Superspace

A twistor correspondence for the self-duality equations for supersymmetric Yang-Mills theories is developed. Their solutions are shown to be encoded in analytic harmonic superfields satisfying appropriate generalised Cauchy-Riemann conditions. An action principle yielding these conditions is presented.

1. There has recently been a revival of interest in self-duality equations, arising from numerous confirmations of a remarkable suggestion [1] that all integrable systems are obtainable by dimensional reduction from 4D self-dual theories. The purpose of this letter is to show that for supersymmetric Yang-Mills theories [2] the self-duality equations can be written in a form in which their integrability becomes manifest and their solutions can be constructed in terms of superfields which are in a definite sense holomorphic. In other words, we shall establish the so-called "twistor correspondence" for the supersymmetric self-duality equations analogous to that for the ordinary (N = 0) self-duality equations [3], which in the harmonic space language [4][5][6] involves a splitting of the coordinate x αα into x +α = u + β x αβ and x −α = u − β x αβ , where u ± β are harmonics on the two-sphere, which appears in the harmonisation of the rotation group [4][5][6], and α andβ are two-spinor indices. The gist of ordinary self-duality is the Cauchy-Riemann-like equation where ∇ + α is the covariant derivative in x −α . We shall show that this construction can be extended naturally to supersymmetric gauge theories. In the N = 1 case we shall use the harmonic superspace with coordinates Now super-self-duality is the condition for the integrability of the equation whereD + is the gauge-covariant spinorial derivative with respect to the variableθ − . N = 1 supersymmetric gauge theories respect chirality, so we may, without loss of generality, take the field φ in (3) to be a chiral superfield φ(x +α , x −α ,θ + ,θ − ) independent of ϑ α , i.e.
where D α is the covariant spinorial derivative with respect to the variables ϑ α , which may be made flat by definition; and we shall set, throughout this letter, the corresponding connections to zero, Of importance is the fact that consistency of (3) and (4) implies (1) in virtue of the algebra of spinorial derivatives, so N = 1 self-duality implies the usual N = 0 self-duality. We shall demonstrate that all solutions of the supersymmetric self-duality conditions are encoded in a "holomorphic" chiral superfield which satisfies generalised Cauchy-Riemann conditions; in other words, we shall establish the famous twistor correspondence [3] for the supersymmetric self-duality equations. By holomorphicity we mean that there is a basis in which this superfield is independent of x −α ,θ − and ϑ α . The resulting formulation greatly simplifies the problem of constructing gauge superpotentials A αβ , Aβ solving the super-self-duality equations. It also helps in the search for an action principle for superself-duality. Although our considerations are rigorous only for 4D Euclidean space, we shall remain in the complexified picture with Lorentz group SL(2, I C) L × SL(2, I C) R , with α andα labelling fundamental representations of SL(2, I C) L and SL(2, I C) R , respectively. In [6] it was shown that consideration of the complexified picture is required by conformal invariance of the self-duality equations. Reality conditions appropriate for the required signature of four dimensional real space may be imposed; e.g. by identifying undotted and dotted spinors as representations of the two different SU (2) ′ s in the 4D Lorentz group SU (2) L × SU (2) R corresponding to a Euclidean signature; or by identifying them as representations of two different SL(2,IR)'s, with Lorentz group SL(2, IR) L × SL(2, IR) R corresponding to a (2,2) signature. In the latter case we expect intriguing peculiarities due to the non-compactness of SL(2, IR) and an appropriate harmonic space needs to be considered (see e.g. [7]).
2. In complexified superspace M 4|4 of complex dimension (4|4) with coordinates (x αβ , ϑ α ,θα), the N = 1 super Yang-Mills theory is conventionally described in terms of two spinorial field strengths w α ,wα defined by where the gauge-covariant derivatives and the supertranslations realise the free superalgebra The self-duality equations for the superconnection A A take the form of the following further (in comparison with (6)) constraints which say thatwα vanishes. That this is the supersymmetrisation of the usual (N = 0) self-duality condition is evident from the dimension (1|1) Jacobi identity which yields the following superfield equations for the self-and anti-dual vector field strengths fαβ, f αβ appearing in the definition The dimension ( 3 2 |1) and ( 1 2 | 3 2 ) Jacobi identities are then satisfied, respectively, if w α and f αβ satisfy the (anti-) chirality equations All other Jacobi identities are then automatic, requiring the introduction of no further superfield strengths. Equations (8) are the superfield self-duality equations. They indeed imply the equations of motion ǫ γα D γγ f αβ = ǫ γα D γγ w α = 0. We have therefore shown that the constraints (7), (8) for A A imply the superfield equations (9) for the superfieldstrength w α and superfield vector-potential A αβ . These in turn imply the ordinary space supersymmetric self-duality equations for the component fields (which we denote by the same symbols as the superfields of which they are the leading components) on eliminating all gauge degrees of freedom depending on the anticommuting superspace coordinates. The converse, that given a set of component fields satisfying these component equations, one can reconstruct superfields satisfying (9), and in turn the superconnection A A satisfying (7), (8) is also true. The proof closely follows the methods of [8].
3. Our main purpose here, however, is to introduce yet another piece of data corresponding to the above three: a 'holomorphic' prepotential in harmonic superspace. We shall show that the constraints (7), (8) imply generalised Cauchy-Riemann (CR) conditions for a prepotential in harmonic superspace and that any superconnection satisfying (7), (8) may be expressed in terms of such holomorphic prepotentials. (We shall use 'holomorphic' in this generalised sense; to describe solutions of these generalised CR conditions) . This construction is a realisation of the twistor construction [3] for supersymmetric self-dual systems in the harmonic superspace framework.
In the present complex setting, the harmonics u ± α remain, as usual, S 2 harmonics, the 2-sphere being a coset of the SL(2, I C) R part of the Lorentz group with its maximal parabolic subgroup. In this setting, u + and u − are not complex conjugates of each other; and are defined up to parabolic subgroup transformations. They obey the usual constraints u +α u − α = 1. Details of this construction, the role of conformal invariance, as well as appropriate reality conditions may be found in [6]. For our N = 1 harmonic superspace the derivatives ∂ ± α ≡ u ±α ∂ αα ,D + = u ±αDα , D α , together with harmonic ones realise the free superalgebra with all other commutators vanishing. Now, in terms of the gauge-covariant derivatives (recall that A α = 0) obeying the commutation relations the constraints (7), (8) are equivalent to the Cauchy-Riemann system Remarkably, these are precisely the integrability conditions for equations (1), (3) and (4). We therefore have the following pure-gauge-like expressions for A + α andĀ + 4. Equations (11), (12) are therefore equivalent to the constraints (7), (8). Let us now choose a coordinate basis, which we shall call the analytic frame, in which the derivatives take the forms In this basis the covariant derivatives D+ and the ∇ + α become flat, losing their connections ( D α remains flat), while the harmonic derivatives D ±± clearly acquire the connections In order to preserve the operator D 0 as a charge counting operator, we have used (as usual, see e.g. [3]) the conventional gauge in which it does not acquire a connection: In this basis the dynamical content is contained entirely in (11), the rest of the equations being kinematical. Using the identity for arbitrary differential operators A and B, eqs. (11) take the form of generalised CR conditions V ++ is therefore holomorphic: it depends on x +α , ϑ + and u ± , being independent of x −α ,θ − and ϑ α ; both V ++ and V −− are chiral. Note that the third equation is a consequence of the other two. We have shown that to any solution of the super self-duality constraints (8), there corresponds a chiral holomorphic superfield V ++ taking values in the gauge algebra and having component expansion The superfield V ++ is defined modulo gauge transformations where λ is an arbitrary holomorphic superfield. Note that due to the presence of the fermion mode there is an important difference with the N = 0 case: whereas the N = 0 connection was encoded in one function on the 2-sphere, in the present case we have two functions, v ++ and χ + , instead.

5.
The converse statement, that any chiral analytic superfield prepotential V ++ encodes a superconnection A A satisfying (7), (8) also holds. To reconstruct the superconnection A A there are two options. a) We can start with the chiral superfield φ. In this case we need to solve (15a) for φ; V ++ being given. Equations on the 2-sphere of this kind are not too easy to solve and they appear in many applications of the harmonic-twistor approach, see, e.g. [4]. In order to determine the corresponding superconnection solving (7), (8), we need to insert the φ thus obtained into the following formulae which follow immediately from (13); and A α is of course zero. b) Instead of φ, we can start with the harmonic connection V −− . It follows from (16) that For a given holomorphic V ++ taking values in the gauge algebra, it contains a set of coupled first-order linear equations for the gauge algebra components of V −− . These may be solved [9] somewhat more easily than (15a). Now, as further consequences of gauge-covariantising the harmonic derivatives (14), we have, from (10), from which we obtain superconnections in terms of solutions of (20). Now, superconnections satisfying (7),(8) may be recovered by harmonic integration similar to (19). In fact, from V −− we may also directly construct the superfield strength w α satisfying the superfield equations (9); namely, from (6), An alternative to equation (20) follows from the following commutators contained in the superalgebra (10) (in the analytic frame) These yield the alternative equations for V −− : The latter equation is in fact the one introduced for the N = 0 case in [10]. We note the following interesting interrelations amongst the left-hand-sides of equations (20), (24) and (25):D 6. We now present an action for super self-duality. Since all we need to have is the generalised CR conditions (17), with V ++ expressed in terms of the field φ, as a variational equation, we plug this condition into an action functional with the help of a Lagrange multiplier-type auxiliary field. The latter does not propagate if it contains only gauge degrees of freedom. For the chiral superfield φ, the action functional yields, on varying the auxiliary field ζ −3 , the CR conditionD + V ++ = 0, V ++ is chiral by definition; and the final condition ∂ ∂x −α V ++ = 0 is a consequence. Now, on varying φ, we obtain It follows (cf. [11]) thatD All solutions of this equation have the form However ζ −3 enters the action viaD + ζ −3 , so it is only defined modulo the addition of D + y −4 . ζ −3 therefore does not represent any additional physical degree of freedom. For N = 0 an analogous action was discussed in [11,12]. Alternatively, we may choose V −− (instead of φ) as the dynamical field, express V ++ in terms V −− with the help of (20), and construct an action having analyticity conditions for the functional V ++ [V −− ] as variational equations.
There also exists the possibility (analogous to the N = 0 action considered in [10]) of writing an action for eq.(25) trilinear in V −− . It explicitly contains a constant harmonic factor of charge +4, say (u + 1 u + 2 ) 2 , and is consequently not Lorentz invariant. 7. To conclude we generalise our construction to N-extended harmonic-superspace with coordinates x ± α , ϑ αi ,θ ± j ≡ u ± αθα j , u ± α , where i, j = 1, ..., N . Solutions of the generalised CR conditions encode N-extended self-dual superconnections. It may be verified that the superconnection components with φ satisfying (29), automatically satisfy the self-dual restrictions of the conventional extended superconnection constraints That integrability conditions for these equations yield (29) follows from reasoning parallel to that for the N=1 case above.
For the full (non-self-dual) N = 2 and 3 theories, for which harmonic superspace formulations (harmonising the internal automorphism group) exist [4,6], the self-duality conditions are also equivalent to "double" analyticity conditions arising from Lorentz as well as internal automorphism group harmonisation. We intend to return to these theories elsewhere.