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Topological Chern-Simons Sigma Model

We consider topological twisting of recently constructed Chern-Simons-matter theories in three dimensions with N=4 or higher supersymmetry. We enumerate physically inequivalent twistings for each N, and find two different twistings for N=4, one for N=5,6, and four for N=8. We construct the two types of N=4 topological theories, which we call A/B-models, in full detail. The A-model has been recently studied by Kapustin and Saulina. The B-model is new and it consists solely of a Chern-Simons term of a complex gauge field up to BRST-exact terms. We also compare the new theories with topological Yang-Mills theories and find some interesting connections. In particular, the A-model seems to offer a new perspective on Casson invariant and its relation to Rozansky-Witten theory.


Introduction
For the past two years, a large class of new Chern-Simons-matter theories has been discovered. Since the seminal work of Bagger and Lambert [1] and Gustavsson [2] (BLG), where a theory with maximal N = 8 was first constructed, the list of supersymmetric Chern-Simons theories has expanded quickly.
Gaiotto and Witten [3] (GW) gave a general prescription for coupling Chern-Simons theory to hyper-multiplets, allowing for construction of a large class of new theories at once. It was soon augmented [4] by adding twisted hyper-multiplets, so that all N ≥ 4 theories can be understood in a unified way in the GW framework. 1 The scalar fields of N = 4 theories can describe a sigma model with a hyper-Kähler target space. The N ≥ 5 theories have much less freedom for their target spaces: flat spaces and their orbifolds. An exhaustive list of N ≥ 5 theories can be found in [9][10][11].
The main goal of this paper is to study topological field theories obtained by twisting the new Chern-Simons-matter theories. There are two well-known topological theories in three dimensions: pure Chern-Simons theory [12] and Rozansky-Witten theory [13]. The latter is a twisted version of N = 4 sigma model without gauge symmetry. Clearly, topological twisting of Chern-Simons-matter theories would lead to a mixture of pure Chern-Simons and Rozansky-Witten theories. It is conceivable that the mixed theory may shed light on relations between the two seemingly different theories. In this paper, we take a first step toward understanding the new topological theories.
In section 2, we begin by writing down the physical N = 4 theories in full generality. It was already done in [9] for flat target space. For non-linear hyper-Kähler target spaces, the recipe was given in [3], but the explicit form of the Lagrangian and supersymmetry transformation rules were not available in the literature.
We then proceed to topological twisting in section 3. First, we enumerate all possible inequivalent twistings for N ≥ 4 theories, 2   For N = 4 theories with SO(4) ≃ SU(2) L × SU(2) R R-symmetry, we can use either of the two SU (2) factors for twisting with the Lorentz group SU(2) E . When the N = 4 theory contains hyper-multiplets but no twisted hyper-multiplets, the two choices lead to different topological theories, which we call A and B models. The distinction disappears when the theory contains both types of hyper-multiplets, which is automatically true of N = 5, 6 theories. We call the result AB-model. For N = 8 theories, the triality of SO (8) offers three new possibilities aside from the AB-model, which we call C/D/E models. The C-twisting of the BLG theory has been considered in [14], while the A-model has been studied in a recent paper [15] which have some overlap with the current work.
In the remainder of section 3, we explicitly construct the A/B/AB-models in the N = 4 notation and study their properties. The scalar super-charges of the A-model are nilpotent up to a gauge transformation, while nilpotency in the B-model holds up to the equations of motion for the fermions. A BRST gauge fixing and introduction of auxiliary fields renders the super-charges fully nilpotent. Topological invariance of the resulting Lagrangian is verified in the usual manner by splitting the Lagrangian into a manifestly metric-independent part and a Q-exact part.
In section 4, we take a preliminary step toward the computation of topological invariants from the new theories. Inspired by the Mukhi-Papageorgakis (MP) map [16] relating Chern-Simons and Yang-Mills theories, we compare our new theories with more well-known topological theories in the literature, and argue that the linear A-model with gauge group SU (2) × SU (2) should capture the Casson invariant. The MP map also suggests that A-model is naturally related to the Rozansky-Witten theory [13] as well. We also speculate briefly on how the A and B-models may make contact with the pure Chern-Simons theory [12].

Chern-Simons sigma model
In this section, we write down the most general form of Chern-Simons sigma model in (2 + 1) dimensions. We first review the linear model constructed in [3] and extended in [4]. Then we write down the non-linear model following the prescription given in [3]. We will mostly follow notations of [4], except for an overall rescaling of matter fields.

Linear model
We start with an Sp(2n) group and let A, B indices run over a 2n-dimensional representation. We denote the anti-symmetric invariant tensor of Sp(2n) by ω AB and choose all the generators t A B to be anti-Hermitian (2n × 2n) matrices, such that t AB ≡ ω AC t C B are symmetric matrices. We consider a Chern-Simons gauge theory whose gauge group is a subgroup of Sp(2n) and we denote the anti-Hermitian generators of the gauge group by (t m ) A B which satisfy the commutation relations, Gauge fields are denoted by (A m ) µ and the adjoint indices are raised or lowered by an invariant quadratic form k mn or its inverse k mn of the gauge group.
The necessary and sufficient condition for N = 4 supersymmetry [3] is that t m AB satisfy the "fundamental identity", where the indices A, B, C are symmetrized over cyclic permutations. This identity can be understood [3] as the Jacobi identity for three fermionic generators of a Lie super-algebra, This turns out to be a rather strong constraint on the field content of the theory. Namely, the gauge group and the matter content should be such that the gauge symmetry algebra can be extended to a Lie super-algebra by adding fermionic generators in one-to-one correspondence with hyper-multiplets.
To write down the Lagrangian in a manifestly N = 4 covariant form, it is useful to introduce the "moment map" multiplet, As for the Chern-Simons term in the Lagrangian, we use the notation As our discussion in this paper will be mostly classical, we will suppress an overall coefficient of the Lagrangian, which should satisfy an integrality condition to make the quantum theory well-defined.
Collecting all notations, we can summarize the Lagrangian of the Gaiotto-Witten model, and its supersymmetry transformation rules The supersymmetry parameter η transforms in the (2, 2) representation of SU(2) L × SU(2) R and satisfies the reality condition 3 To obtain the most general N = 4 Chern-Simons (linear) sigma model, one should add twisted hyper-multiplets (q Ȧ α ,ψ A α ) to the Gaiotto-Witten model [4]. The gauge generatorst m AB also satisfy the fundamental identity (2.3) and define the twisted moment map multiplet similar to (2.5). It is also useful to introduce yet another notation, The full Lagrangian is given by (2.11) 3 In [9], the same reality condition was stated with a wrong sign.
The supersymmetry transformation rules read Mass deformation The N = 4 superconformal Chern-Simons theories allow a massdeformation which preserves all of the Poincaré supersymmetry [4] (See also [9,17,18]). For the Gaiotto-Witten model, the deformation amounts to adding the mass terms and a quartic interaction term to the Lagrangian, One show that the mass-deformed Lagrangian still preserves the N = 4 supersymmetry, provided that the supersymmetry transformation rule for the fermion is also modified by an additional term, 14) The deformed supersymmetry algebra contains a non-central extension, where R αβ , R˙αβ denote the generators of SU(2) L × SU(2) R .
It follows from (2.15) that for the mass deformation of the general theory with both types of hyper-multiplets, the mass parameters of the hyper-and twisted hyper-multiplets should be equal. The mass-deformed term in the Lagrangian of the general theory turns out to be the sum of the contributions from the two types of multiplets.
The supersymmetry transformation rules for fermion fields are again modified as

Non-linear model
The linear sigma model explained above can be generalized to a non-linear model [3] whose target space is a hyper-Kähler manifold X. The scalar fields are now local coordinates q i on X (i = 1, · · · , 4n = dimX). We begin this subsection with a brief review of hyper-Kähler geometry, closely following [13], and move on to describe the sigma model as explained in [3].

Hyper-Kähler geometry
The hyper-Kähler structure of X can be described by the existence of anti-symmetric inner products ω AB and ǫ αβ , which leads to three symplectic forms We introduced the hyper-Kähler vielbein satisfying where Γ A B = Γ i A B dq i is the metric connection in the Sp(2n) holonomy group. The curvature tensor is given by The hyper-Kähler structure and the identity R i [jkl] = 0 further implies that where Ω ABCD is totally symmetric. (2.24) The hyper-Kähler version of the identity can be written as The moment maps are defined by The other two components of the moment map multiplet can be defined by In terms of µ m αβ , the fundamental identity for the non-linear model can be written as In the non-conformal cases, a weaker condition, which is a second descendant of the fundamental identity, is sufficient to ensure N = 4 supersymmetry, Gauging the isometry The target space isometry can be gauged by imposing the following transformation rules on the fields: The covariant derivatives for the matter fields are defined by They transform homogeneously under the gauge symmetry, The moment map multiplet also transforms as expected, To verify (2.37) and (2.38), one needs to use the identities (2.25), (2.26) and (2.28).

Adding twisted hypers
To obtain the most general model, one should also add twisted hyper-multiplets. One simply introduces another target spaceX for the twisted hypers and define the corresponding moment map multiplet and so on. One also defines The minus sign is required for (2.39) to reduce to (2.10) in the linear case.
Lagrangian and supersymmetry Using the notations introduced so far, we can write down the Lagrangian for the most general Chern-Simons gauged non-linear sigma model: The supersymmetry transformation rules read (2.41) Here, we used some short-hand notations such as Dq Aα ≡ e Aα i Dq i and (δ η q) Aα ≡ e Aα i (δ η q) i . The parameter η satisfies the reality condition (η αα ) * = +ǫ αβ ǫ˙αβη ββ . (2.42) Mass deformation Unlike the linear model of the previous subsection, the non-linear model does not admit a mass deformation. For the linear model, the SU(2) L × SU(2) R Rsymmetry acts on a target space as a tri-holomorphic Killing vector. The mass deformed Lagrangian (2.13) and supersymmetry algebra (2.15) can be associated to this Killing vector in a standard way [19]. In the non-linear model, however, the R-symmetry is no longer an isometry of the target space, so the mass deformation is not allowed.
Examples Gaiotto-Witten [3] gave a classification of linear model in terms of Lie superalgebra; see (2.4). The same classification can be used even after adding twisted hypers, and the resulting theory is typically a linear quiver with product gauge group and alternating series of hypers and twisted hypers [4].
Such a classification for non-linear model is not known, but a method for generating non-linear models from linear models was given in [3]. The idea is to use a linear quiver allowed by the Lie super-algebra conditions. The linear quiver has two open ends. The non-linear model is obtained by taking the usual hyper-Käler quotient [20,21] with all the gauge groups except those at the end points.
Another class of non-linear models was given by Kapustin-Saulina [15]. The gauge group G is any compact simple Lie group and the target space is the cotangent bundle of the flag manifold, T * (G/T r ), where T r is the maximal torus of G.
We do not have any new example to offer here. Instead, as an illustration, we present the simplest example T * (SU(2)/U(1)) = T * (CP 1 ) from Kapustin-Saulina [15] in our notation. This space is endowed with the famous Eguchi-Hanson metric [22], where σ a are left-invariant one-forms on S 3 satisfying dσ 1 = σ 2 ∧ σ 3 , etc. Explicitly, in the Euler-angle coordinate, The vielbeins are written in terms of the invariant one-forms as where τ a are the Pauli matrices. The three symplectic forms are given by The gauge group G = SU(2) acts on the target space as isometries V m leaving σ a invariant, The moment maps can be computed by solving (2.27): Checking the fundamental identity, we find Removing the trace part, we see that the strong version of fundamental identity (2.30) does not hold. However, since the discrepancy is a constant, the weaker version (2.31) holds and the theory is well-defined.
All known examples so far, those of Gaiotto-Witten [3] and of Kapustin-Saulina [15] are non-compact. It is not clear (to us) whether there exists any compact hyper-Kähler manifold satisfying either the strong or the weak version of fundamental identity.

Survey of possible twistings
Before performing the topological twisting of the Chern-Simons sigma model described in the last section, we pause to enumerate physically inequivalent twistings for N = 4, 5, 6, 8 theories. A similar discussion is well-known in four dimensions, where there is only one twisting for N = 2 super-Yang-Mills [23] and three different twistings for N = 4 super-Yang-Mills [24][25][26] Recall also how the matter fields behave under the R-symmetry: Consider a theory with hyper-multiplets (q, ψ) only. Twisting with SU(2) R gives scalar supercharges in doublet of SU(2) L and matter fields transforming in SU (2) We will call this "A-twisting." Switching the roles of SU(2) L and SU(2) R leads to a different twisting, which we call "B-twisting." Equivalently, we can continue to use SU(2) R for twisting and consider a theory with twisted hyper-multiplets only. In the latter convention, the resulting theory contains matter fields transforming in SU (2) We will continue to use SU(2) R for twisting even when both types of hyper-multiplets are present; the other twisting amounts to exchanging the roles of hyper and twisted hyper-multiplets. We will call this broader class of theories "AB-models," to distinguish them from the two extreme cases. In general, N = 4 theories are not symmetric under the "mirror reflection" between hyper and twisted hyper-multiplets, while N > 4 theories are automatically symmetric. N = 5 . The supercharges transform as vector 5 under the SO(5) R-symmetry group, which have the following decomposition: The case 1 is the same as the AB-twisting of N = 4 theories. In the case 2, the supercharges transform as 3 0 ⊕ 1 + ⊕ 1 − and do not yield any scalar supercharge upon twisting. N = 6 . The supercharges transform as vector 6 under the SO(6) R-symmetry group, which have the following decomposition: The case 1 can be regarded as a refinement of the AB-twisting of N = 4 theories in the sense that there is a left-over U(1) R-symmetry group. The scalar supercharge is neutral under this U(1), but the theory contains charged matter fields. The case 2 can be regarded as a refinement of the case 2 of N = 5 theories considered above. Again, we find that there is no possible twisting to have scalar supercharges. N = 8 . The N = 8 BLG theories have the R-symmetry group SO (8). If we keep the supercharges in the vector 8 v representation, we will only obtain refinements of the ABtwisting of N = 4 theories. New possibilities may arise if we use the triality of SO (8) to let the supercharges transform in the spinor 8 s representation.
Consider the following decompositions: Notice that the cases 1 and 2 can be enhanced to the cases 3 and 4, respectively. So, it is sufficient to examine the latter two cases.
There is a novelty here. Unlike all other cases considered so far, we can now use more than one SU(2) factors for twisting. Up to permutations, we have four candidates for different twistings. (2)   • E-twisting : Note that the C' and D' twistings are not really new as they can be obtained from the C and D twistings, respectively, by breaking USp(4) down to SU(2) × SU (2).
The C-twisting of the BLG theory has been studied in [14]. In the rest of this section, we will construct the A/B/AB-twisted Chern-Simons sigma models, leaving the other twistings for a future work.

A-model
We begin with the Euclidean version of the Lagrangian of the linear-model, This Euclidean Lagrangian differs from the Lorentzian one (2.7) by the factor of (−i) for the Chern-Simons term and an overall sign for the matter terms. Besides, the fermions no longer satisfy the reality condition, but the Lagrangian depends holomorphically on the fermions. The supersymmetry transformation rules remain formally the same as in the Lorentzian theory, Twisting The twisting is done by taking the diagonal subgroup of the Lorentz group SU(2) E and the SU(2) R part of the R-symmetry group as the new Lorentz group: For instance, we make the replacements, where σ denote the Lorentz indices, while (α,α) denotes two SU(2) R-symmetries indices. We suppressed the SU(2) ′ E -triplet components of the parameter η. Plugging these into the transformation rules (2.8), we find the following twisted transformation rules: From the definition, one can show that (3.14) Introducing the twisted supercharges by δ η X = [η α Q α , X], we can rewrite (3.12) as Nilpotency The twisted supercharges Q α are nilpotent up to a gauge transformation. Concretely, the following relations hold, with the gauge parameter The fundamental identity k mn t m (AB t n C)D = 0 is needed to verify (3.16). See appendix A for details.

Lagrangian and Invariance
The twisted Lagrangian can be divided into two pieces, The two parts L 1 and L 2 are Q-invariant separately. The Q-invariance of L 1 can be checked explicitly, where · · · denotes a part which vanishes due to the fundamental identity. L 2 is also Qinvariant because it is Q-exact in the sense that Mass deformation For the mass deformed linear model, the A-twisting leads to the following super-algebra for the scalar super-charges Q α , where R αβ denote the generators of SU(2) L ; see (2.15). Without nilpotent scalar supercharges, we cannot define a topological field theory. Note also that (3.23) somewhat resembles, but clearly differs from, the defining relation of equivariant cohomology which states for some parameters k αβ . As a side remark, we note that N = 2 super Yang-Mills theory in four dimensions admit a mass deformation of a different kind if the world-volume is a Kähler manifold [28].

Non-linear A-model
We now consider applying the A-twist to the non-linear model discussed in section 2.2. The twisted transformation rules are as follows, where the definitions of H A α , (j m µ ) α are appropriately covariantized in terms of the Killing vectors of the target space, The main difference from the linear model is the four fermion term inherited from the physical theory, The Q-exact part, L 2 , becomes a covariantized version of (3.20).
Let us remark on differences from the Rozansky-Witten theory in [13]. Firstly, the derivatives become gauge covariant, thus the variation of the fermion kinetic term can cancel the variation of the Chern-Simons term. Secondly, δ η λ A = 0 in the RW theory, while δ η λ A = 0 in (3.25) due to the non-trivial bosonic potential. It suggests that the variation of λ of the curvature term shall be canceled by the variation of the gauge boson in the fermionic kinetic term. The cancellation has been shown in detail in [15] for a holomorphic supercharge.

B-model
Next, we consider a theory only with twisted hyper multiplets. The physical Lagrangian in the Euclidean signature is Twistingq Ȧ α becomes a bosonic spinor in (2; 1) representation of su(2) ′ E ⊕ su(2) L . For later convenience, we rescaleψ A α , which is in (2; 2), as The supersymmetry variations become Note that spinor indices of su(2) ′ E are implicit. The quantitiesj m µ andH A are defined in the same way as (3.13).
Nilpotency The twisted supercharges Q α are nilpotent up to the equations of motion of the fermion. The results read Note that the variation of the Lagrangian with respect to the fermion is, up to a surface term, An aspect of (3.30) different from the result of A-model (3.16) is the absence of the gauge transformation on the right hand side. One can expect it from the symmetry, sine if a field-dependent gauge transformation parameter exists, it should be in (1; 3) representation of su(2) ′ E ⊕ su(2) L with conformal dimension 1. A short computation shows that such a composite field cannot be constructed from the matter fields of the B-model.

Off-shell Supersymmetry Algebra
One can construct an off-shell formalism by introducing an auxiliary field, a bosonic spinorh A . The off-shell variations are defined as follows The variations ofq A and A m i are unchanged from (3.30). The variation of the auxiliary field in (3.33) Given the off-shell supersymmetry algebra, the following replacement of L V leads to the off-shell supersymmetric Lagrangian , Again, the twisted Lagrangian can be split into two pieces, L = L 1 + L 2 , (3.37) where the covariant derivative D + µ now involves A + µ rather than A µ . The Q-closed Lagrangian L 1 is simply the Chern-Simons action, except that the gauge boson is shifted by a bi-linear product of the boson fields, A similar redefinition of the gauge field has been noticed [14] for the C-twisting of BLG theory discussed in section 3.1 and even earlier in topological Yang-Mills theories in [26,27,38]. The meaning of this shift will be discussed in section 4.
As in the A-model, L 1 and L 2 are separately Q-invariant. Q-invariance of L 1 is trivial To show that the Lagrangian in (3.37) indeed results from the topological twisting of the physical Lagrangian (3.27), one needs to use rather nontrivial identity See appendix A for a proof.

Wilson-loop observable
The BRST invariance of A +m µ implies that the Wilson-loop operators W(C) are good observables of the B-model, (3.42) We will see in section 4 that, not coincidentally, the B-twisted N = 4 super Yang-Mills theory also carries a shifted gauge field and Wilson loop observables.

Non-linear B-model
In the B-model, the bosonic matter fields become world-volume spinors. As such, it seems difficult, if not impossible, to generalize the model to non-flat hyper-Kähler target space. A similar problem arise from the topological Seiberg-Witten theory [29] in four dimensions, in which the monopole fields are bosonic spinors. In this case, a generalization to non-flat hyper-Kähler target space was proven possible [30] by coupling the hyper-Kähler structure of the four-dimensional world-volume to that of the target space. Our B-model does not seem to allow for such a construction.

AB-model
Finally, we move on to theories containing both hyper and twisted hyper multiplets. In addition to the linear combination of A and B-model, non-trivial mixing terms arise. The supersymmetry transformation laws are unchanged from (2.41).
Twisting We begin with implementing the features of the B-model to the AB-model, namely, the shift in the gauge field (3.38) and the introduction of auxiliary fields (3.33).
Using the notations introduced in the previous subsections, we can summarize the twisted supersymmetry variations as follows, Again, the Q-variation of the auxiliary fieldh A is chosen to guarantee the nilpotency oñ ζ A α , in this case up to a gauge transformation to be discussed next.
Nilpotency Now we deal with the problem inherited from the A-model; the twisted super-charges are nilpotent up to a gauge transformation. For (q A γ , λ A , χ A µ ) in the hyper multiplet, the results in (3.16) still hold, with the gauge parameter Λ m αβ in (3.17). Consistently, Q 2 acts on the twisted hyper-multiplet (q A ,ζ A γ ,h A ) as well as the modified gauge field A + µ m as a gauge transformation by the same gauge parameter, (3.45)

Lagrangian and Invariance
The Lagrangian still admits the usual splitting, L = L 1 + L 2 . The L 1 term is almost the same as in the A-model, except for the shift from A m to A m+ we noticed in the B-model. The cubic term of the shift S m ≡ A m+ − A m in the Chern-Simons term comes from the bosonic potentialμ 3 in the physical Lagrangian, while the change in the covariant derivative Dχ → D + χ has its origin in the Yukawa term,μ m ρ m . It is straightforward to show that L 1 is Q-invariant, where f + β includes the sum of the terms which appeared in the Q-exact parts of the Aand B-models, The other two terms in (3.48) are novel in the AB-model.

Gauge fixing
Gauge fixing Let us consider the BRST quantization of the AB-model. The BRST quantization of A-model has been considered in [15]. In there, it has been noticed that imposition of a non-trivial variation of the ghost c m with respect to the holomorphic super-charge, Q + in our notation, can result in a nilpotent scalar charge. We will see that the same prescription also works for the AB-model. Reduction to the A-or B-models can be done trivially.
For the quantization, we introduce fermionic ghost and anti-ghost, c m ,c m , and bosonic ghost B m . Recall the standard Fadeev-Popov BRST variations for all fields, Q 2 is nilpotent as in the standard BRST quantization. Now, following [15], we impose the following super-symmetry variations of ghost fields, where Q ≡ Q + ,

Relation to other topological theories
We have carried out the (A-and B-) topological twisting of general N = 4 supersymmetric Chern-Simons gauged sigma models of [3,4]. The next task is to evaluate the partition function and correlation functions of the quantum theory, which should provide (hopefully new) topological invariants of three manifolds.
The computation involves roughly three steps; see, for instance, [31]. First, one should identify observables, which are elements of Q-cohomology. Second, the path integral often localizes onto the moduli space of Q-invariant configurations ("instantons"). Third, perturbation theory is used to compute the partition function unless a more powerful tool is available.
In this section, we point out some interesting connections between our new theories and previously well studied theories in the literature. We believe that our findings will serve as a useful guide in taking each of the three step of computation, most of which we leave for a future work.
For the readers' convenience, we begin with a brief review of well-known topological theories in three dimensions and how they are related to each other. Next, we review a useful fact about (physical) Chern-Simons-matter theories which inspired our main observation. It is the generalized Higgs mechanism of Mukhi-Papageorgakis (MP) [16] which, to some extent, transforms Chern-Simons theories to Yang-Mills theories. We will see that the MP map is compatible with the topological twisting and that some main results of the previous sections have dual interpretations on the Yang-Mills side.
In the last subsection, we will combine everything and try to obtain some clues as to what topological invariants our new theories may compute. In particular, we will argue that the A-model should capture the Casson invariant and speculate on the role of a complexified gauge group in the B-model.

Review of old results
There are largely three well-known three dimensional TFTs in the literature; pure Chern-Simons theory [12], A-twisted N = 4 super Yang-Mills theory [32,33], and A-twisted ungauged N = 4 sigma model of Rozansky-Witten [13]. Let us review some features of these theories relevant for our discussion below.
The A-twisted Yang-Mills theory is known to compute Casson invariant which is, roughly speaking, a signed sum over flat connections (F µν = 0). An important fact for our discussion is that Casson invariant admits an alternative field theory description [32]. It is a sort of super-BF theory whose Lagrangian is, in a differential form notation, Here, A is the gauge connection and F A the curvature. B is a bosonic one-form and χ, ψ fermionic one-forms, all of which are Lie-algebra-valued. The Yang-Mills and BF descriptions look very different at first sight, but it was shown [33] that the Yang-Mills theory can be deformed in a topologically invariant way to the BF theory.
Rozansky-Witten theory [13] is not a gauge theory. Nevertheless, it is intimately related to both pure Chern-Simons and Yang-Mills theories.
First, it computes the SU(2) Casson invariant if one chooses the target space to be the Atiyah-Hitchin space. The physical explanation is that the low energy limit of the (physical) Yang-Mills theory is a sigma model on the moduli space of vacua. Taking account of loop and instanton effects, the quantum moduli space of vacua of the SU(2) super Yang-Mills theory has been shown to be the Atiyah-Hitchin space [35]. 5 Second, the relation to pure Chern-Simons theory can be seen by comparing the perturbation theory of Rozansky-Witten theory to that of Chern-Simons theory. Topologi-cal invariance of amplitudes in perturbation theory is verified by the so-called IHX relation [37]. The essence of the IHX relation for Chern-Simons theory is the Jacobi identity of Lie algebra, while a similar role is played by the Bianchi identity for the Riemann curvature in Rozansky-Witten theory. With this formal similarity in mind, a detailed comparison of Feynman diagrams in the two theories reveals certain relations between Chern-Simons and Casson invariants.
Relatively less known, but equally important for our discussion, is the B-twisted N = 4 super Yang-Mills theory [38]. We will review some aspects of this theory in the next subsection.
In the N = 2 super-field notation, the scalar fields split into two charged chiral superfields Z α = (A, B † ). The N = 4 vector multiplets decomposes into vector super-fields V B (V C ) combined with and two neutral chiral super-fields Φ B , Φ C .
For simplicity, let us discuss the abelian (Maxwell) case first. The N = 4 Lagrangian of the Gaiotto-Witten model in the N = 2 notation is where Σ B,C are the field strength super-fields for V B,C , defined as They satisfy the defining relations of a linear superfield The equations of motion for auxiliary superfields V B and Φ B are Integrating out the auxiliary fields, we recover the Lagrangian in the main text with Yukawa-like term and sextic bosonic potential. Instead, we may choose to eliminate the matter multiplets by solving for e V B , and inserting it back into (4.3). Taking account of the chirality of A, A † and the properties of linear superfields Σ C (4.5), one can obtain the dual vector description of the model, whose bosonic parts take the form Here, φ and q are the lowest components of Σ C and Φ C , and ω µ is the pull-back of the Dirac vector potential in the target space (φ, q,q) ∈ R 3 : Restoring the dependence of Chern-Simons level k, one can conclude that the Coulomb branch of (4.9) is the orbifold C 2 /Z k , the same as the moduli space of the vacua of the U(1) × U(1) Gaiotto-Witten model.
To complete the MP map, we give a vev to Σ C by setting Σ C = v 2 + δΣ C . The leading behavior of (4.8) with respect to 1/v 2 is with 1/g 2 = 1/2v 2 . It is precisely the N = 4 supersymmetric Maxwell theory.
The same idea can be applied to the U(N) × U(N) theories, but the computations are more involved. We jump directly to the final identification between the hyper-multiplets of the Chern-Simons side and the vector-multiplets of the Yang-Mills side, up to the leading order of 1/v 2 expansion. The MP map can be understood as a nonabelian version of the vector-scalar duality in three dimensions.

Compatibility I. A-twisting
To show that the MP map is compatible with the A-twisting, we begin by recalling the supersymmetry transformation rules of super Yang-Mills theory, δ Φ = η αα τ β α λ βα , δA µ = iη αα γ µ λ αα , The F-term equation also translates into Furthermore, a half-BPS instanton can be written down concretely. For a solution to keep η 2 unbroken, we have to require They imply that F-term conditions are automatically satisfied and (4.25) These are nothing but the equations the half-BPS instanton of A-model SYM should satisfy.

Instanton of the B-model
For the B-twisted super Yang-Mills, the supersymmetric configurations satisfy Once we define a twisted complex gauge field A + µ by the first equation of (4.28) can be rewritten as Clearly, this modification of the gauge field exactly parallels what we found in section 3.
In particular, as pointed out in [38], the B-twisted super Yang-Mills carries the Wilson line as a topological observable, beit with a modified gauge field; see (3.37) and (3.38). The shift of the gauge field is purely imaginary and leads to a complex gauge field. Of course, this does not imply complexification of the underlying gauge symmetry. Perhaps surprisingly, however, complexified gauge symmetry does play a role in some context.
Marcus [26] studied a twisted D = 4, N = 4 super Yang-Mills theory and showed that the instanton equation is the flat connection condition for the complex gauge field: F + µν = 0. Moreover, he showed that complexified gauge group somehow plays an important role in understanding the moduli space of flat connections. An interpretation of this observation was given by Baulieu [39] who re-interepreted the N = 4 Yang-Mills theory as a complexification of the N = 2 theory.
Whether a similar story holds for our B-model and its Yang-Mills cousin is an open question. If so, the B-model may even be related to recent developments (see, e.g., [40] and references therein) in pure Chern-Simons theories with complex gauge groups.
Quantization Finally, it may be helpful to study Batalin-Vilkovisky (BV) quantization of the new theories as the BV quantization facilitates comparison between different gauges which illuminates different aspects of the same theory. The BV quantization of Rozansky-Witten theory has been done recently [41] following the AKSZ prescription [42].
Spinor indices in the standard position will be omitted.
Nilpotency of the A-model Let us consider some details in (3.16).
In the second line, we used In checking the nilpotency on λ A , it is useful to note that In checking nilpotency ofh A in the AB-model (3.44), the following relation can be useful Proof of an identity Let us denote the left hand side of (3.41) by B. Then where we used the following Fierz identity, The second term of (A.5) becomes