A one-dimensional theory for Higgs branch operators

Journal of High Energy Physics, Mar 2018

Mykola Dedushenko, Silviu S. Pufu, Ran Yacoby

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A one-dimensional theory for Higgs branch operators

Accepted: March one-dimensional theory for Higgs branch operators Mykola Dedushenko 0 1 2 3 Silviu S. Pufu 0 1 2 Ran Yacoby 0 1 2 4 0 Rehovot 76100 , Israel 1 452-48 1200 E California Blvd , Pasadena CA 91125 , U.S.A 2 Princeton , NJ 08544 , U.S.A 3 Walter Burke Institute for Theoretical Physics, California Institute of Technology , USA 4 Department of Physics and Astrophysics, Weizmann Institute of Science We use supersymmetric localization to calculate correlation functions of halfBPS local operators in 3d N = 4 superconformal eld theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d N = 4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal eld theory from three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d N = 2 subalgebra of the N = 4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge- xed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from Extended Supersymmetry; Supersymmetric Gauge Theory; Conformal Field - A the vectormultiplet scalars. Theory ArXiv ePrint: 1610.00740 Actions with vectormultiplets and hypermultiplets Closure of the supersymmetry transformations Non-conformal supersymmetry algebra on S3 Central extension of non-conformal supersymmetry algebra 5.5 Integration cycle from Morse theory 6 Properties of twisted Higgs branch theory Brief summary Topological gauged quantum mechanics The conformal case Non-vanishing mass and FI parameters 6.3 Correlators of twisted Higgs branch operators 6.3.1 The conformal case Twisted operators on S3 by stereographic map Interpretation in terms of su(2j1)` 5.2 3d Gaussian theory coupled to a matrix model 5.3 1d Gaussian theory for twisted Higgs branch operators 5.4 1d theory from localization of hypermultiplet Localizing with QKWY Localizing with Q Localizing with Q C H O -shell closure 1d action from BPS equations Hypermultiplet 1-loop determinant Integration cycle from localization 1 Introduction 2 3d N = 4 theories on S3 3 Cohomology in SCFTs SCFT in at space SCFT on the sphere 4 5 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 6.1 6.2 2.3.1 2.3.2 2.3.3 3.2.1 3.2.2 5.2.1 5.2.2 5.2.3 5.4.1 5.4.2 5.4.3 5.4.4 6.2.1 6.2.2 { i { 2- and 3-point correlators of Higgs branch operators of the SCFT Star product, Higgs branch chiral ring, and deformation quantization Operator mixing SQED with N charged hypermultiplet avors N -node quiver U(2) with adjoint hypermultiplet and fundamental hypermultiplet 8.2 Introducing mass parameters SQED with non-zero FI parameter N -node quiver with non-zero FI parameters Mass-deformed N -node quiver Mass-deformed SQED 50 51 rems, such as in [1, 2] who showed that the non-perturbative answer for 3-point functions result.1 The conformal bootstrap approach also allows one to calculate certain correlators of BPS operators in some particular supersymmetric conformal eld theories (SCFTs) with 8 Poincare supersymmetries in various dimensions [4{8]. Other examples use the technique of supersymmetric localization (for recent reviews, see [9{25] and references therein) allowing for the calculation of two-point functions of conserved avor or R-symmetry currents in 3d N = 2 superconformal eld theories (SCFTs) [26, 27], and of Coulomb branch operators in 4d N = 2 SCFTs [28]. (See also [29{32].) Our goal here is to provide more instances of such exact computations of correlation functions of local operators. We focus on 3d quantum eld theories with N = 4 functions of certain 12 -BPS operators. The derivation of these formulas also relies on supersymmetric localization, albeit using a di erent supercharge from the one used in previous supersymmetric localization studies of such theories. The 12 -BPS operators whose correlation functions we compute fall within two classes of more general operators. The rst class, referred to as Higgs branch operators, consists of gauge invariant operators constructed from the scalar elds in the hypermultiplet, while the second class, referred to as Coulomb branch operators, contains operators constructed from the scalars in the vectormultiplet as well as from 12 -BPS scalar monopole operators. The naming of the two classes re ects that, when the QFT is de ned on R3, these operators acquire non-zero expectation values on either the Higgs or Coulomb branch of the moduli space of supersymmetric vacua. We restrict our attention to the origin of the moduli space, where the N = 4 theories we consider ow in the IR to SCFTs. Let us denote the gauge group of such a theory by G and assume, without loss of generality, that there is only one hypermultiplet transforming in a unitary representation R of G, where R may be reducible.3 In a nutshell, we have three main results: We present a relatively simple matrix model coupled to a 1d Gaussian theory, ZHiggs = 1 Z jWj Cartan )] Z ` Z d det 0adj [2 sinh( DQ DQe exp 1See also [3] for a proof, and [4] for a generalization to 4d N = 2 theories. 2Such theories have been extensively studied in the literature starting with refs. [33, 34]. For string theory constructions of such theories, see, for instance, [35{38]. 3Let NH be the total number of hypermultiplets in the absence of gauging. The 2NH complex scalars and 2NH complex two component fermions transform in the pseudoreal fundamental representation of the avor symmetry USp(2NH ). We consider the U(NH ) subgroup of USp(2NH ) under which the fundamental of USp(2NH ) decomposes as NH NH , and we take the gauge group G to be a subgroup of U(NH ). At the level of Lie algebras, we have that a subalgebra g of u(NH ) is gauged, and we de ne the representation map R : g ! u(NH ) from the gauge algebra into NH xed point. In addition, (1.1) can be used to calculate n-point correlators of certain twisted Higgs branch operators of the SCFT. These twisted operators are speci c position-dependent linear combinations of Higgs branch operators, to be de ned precisely in section 3, obtained by contracting the various R-symmetry components of Higgs branch operators with position-dependent polarization vectors. Let us describe (1.1). jWj is the order of the Weyl group of G. The variable is the matrix degree of freedom and takes values in the Cartan of the Lie algebra g = Lie(G).4 It is coupled to a 1d Gaussian theory de ned on a circle parameterized ; ) whose degrees of freedom are the anti-periodic scalar elds Q(') and Qe(') which transform in the representations R and R of g, respectively. The path integral over Q and Qe is over a middle-dimensional integration cycle in the space of complex-valued elds Q and Qe. We will discuss this integration cycle in more detail in section 5. Lastly, ` is a parameter with dimensions of length whose meaning we will explain momentarily. The twisted Higgs branch operators of the 3d SCFT whose correlators can be calculated using (1.1), can be inserted anywhere along a line in R3 or, equivalently, along a great circle on S3 that maps to this line under the stereographic projection. In (1.1), the angle ' parameterizes the great circle on S3, and ` 4 r is proportional to the radius of S3. Moreover, the twisted Higgs branch operators are represented in the 1d model (1.1) by gauge-invariant polynomials in Q(') and Qe('). From the 2- and 3-point functions of the twisted Higgs branch operators, one can extract in a simple way the 2- and 3-point functions of the most general Higgs branch operators. The 1d sector consisting of the twisted Higgs branch operators of a general 3d N = 4 SCFT was previously studied in [4, 7, 8] abstractly, using only properties of the superconformal algebra.5 These properties imply that the correlation functions of the twisted Higgs branch operators are topological, in the sense that they do not depend on the relative separation between the insertion points, but do depend on the ordering of the insertions. Moreover, the 1d OPE algebra of this sector is an associative noncommutative algebra obeying certain very special properties. In some cases, refs. [7, 8] used bootstrap-type arguments to show that these properties determine the 1d OPE algebra uniquely up to a nite number of parameters. The model (1.1) provides a complementary approach to the analysis in [7, 8] whereby (1.1) can be used to calculate explicitly the structure constants of the 1d OPE algebra. We provide a partial result toward a similar computation of correlation functions of Coulomb branch operators. In particular, we only consider Coulomb branch operators 4Very roughly, the theory (1.1) can be interpreted as a 1d gauged quantum mechanics with gauge group G, in the gauge where A' = . The determinant factor in (1.1) is precisely the Fadeev-Popov determinant corresponding to the gauge 5At this abstract level, there is no di erence between the 1d sector associated with the Higgs branch and that associated with the Coulomb branch. { 3 { that are not monopole operators. Such non-monopole operators are given by gaugeinvariant polynomials in the vectormultiplet scalars. At the SCFT xed point, the 2- and 3-point correlators of non-monopole Coulomb branch operators, as well as npoint functions of their twisted analogs, can be calculated by inserting gauge-invariant polynomials in into the matrix model ZCoulomb = 1 Z as a result of a supersymmetric localization computation that uses only N supersymmetry, and with the goal of calculating expectation values of BPS Wilson loop operators. Its relation to the Higgs branch theory (1.1) is that one obtains (1.2) after integrating out Q and Qe in (1.1). As was the case with Higgs branch operators, the twisted Coulomb branch operators whose correlation functions can be computed from (1.2) are part of the 1d topological Coulomb branch sector studied abstractly in [4, 7, 8]. In terms of the elds of the 3d SCFT, the twisted Coulomb branch operators represented by gauge-invariant polynomials in correspond to position-dependent linear combinations of polynomials in the vectormultiplet scalars. A more complete analysis that includes monopole operators is left for future work. The above results can be generalized to non-conformal N = 4 QFTs on S3 that are obtained by introducing real mass and Fayet-Iliopolous (FI) parameters. The real mass parameters are introduced in (1.1){(1.2) by shifting exponent of (1.1) and denominator of (1.2), where m is a real mass matrix taking value in the Cartan of the avor symmetry algebra of the hypermultiplet.6 For each abelian factor of the gauge group, one can introduce an FI parameter a by including ! + mr in the in (1.1){(1.2) an additional factor (1.2) HJEP03(218) e 8 2ir tr : (1.3) In (1.3), tr P a a a with the sum taken over all abelian factors in g, while a are the corresponding FI parameters and a are the components of that take values in those abelian factors. The correlators of twisted Higgs (Coulomb) branch operators that are not charged under the avor (topological) symmetries associated with non-zero real mass (FI) 6As in footnote 3, let NH be the total number of hypermultiplets in the absence of gauging. The full avor symmetry group GbF of the NH hypermultiplets is de ned as the normalizer of the gauge group inside USp(2NH ) modulo the gauge group [40]. However, we take our gauge group G to be contained in a U(NH ) subgroup of USp(2NH ), and when de ning the avor symmetry we also consider GF = GbF \ U(NH ). We sometimes refer to GF as the avor symmetry of the hypermultiplet. The embedding of GF into U(NH ) induces a map F : gF ! u(NH ), where gF = Lie(GF ). By + mr in the main text we then mean R( ) + rF(m), where R : g ! u(NH ) is the representation map from the gauge algebra into NH parameters are still topological. We will show that when a real mass parameter ow between two SCFTs, the correlation functions in the 1d theory interpolate between topological correlators in the UV and the IR, even though in the intermediate regime these correlation functions may be position-dependent. Let us explain in more detail the procedure by which one arrives at the aforementioned a Q results. While our results hold in SCFTs de ned on any conformally at space, we nd it convenient to rst consider more general non-conformal 3d N = 4 QFTs on a round S3 of radius r. Due to the large amount of supersymmetry, placing the N = 4 theories on S3 is unambiguous. The S3 Lagrangians we consider are curved space generalizations of the usual at space ones: they contain kinetic terms for the hypermultiplets, a YangMills term for the vectormultiplet with Yang-Mills coupling gYM, and, optionally, real mass and FI parameters, all containing certain curvature couplings that vanish in the limit r ! 1. Setting to zero the real mass and FI parameters and taking both gYM; r ! 1, the correlation functions computed in an N = 4 theory on S3 approach those of the deep infrared limit of the same N = 4 theory de ned on R3. In the examples we consider, such a deep infrared limit is a non-trivial interacting SCFT. Alternatively, we can rst take gYM ! 1 at xed r and then conformally map from S3 to R3. As we will see, the S3 correlators we study are independent of gYM, so the limit gYM ! 1 is taken trivially. After placing the theories of interest on S3, we perform supersymmetric localization with an appropriately chosen supercharge. The choice of supercharge is guided by the SCFTs on R3 in [7, 8]. In particular, the authors of [7, 8] identi ed a supercharge Q cohomological construction of [4], which was elaborated upon in the context of 3d N = 4 H in the N = 4 superconformal algebra whose cohomology classes are represented by the twisted Higgs branch operators mentioned in the rst bullet point above. A similar construction for Coulomb branch operators involves the cohomology of a di erent supercharge QC . We perform supersymmetric localization using precisely the supercharge Q mapped to S3 using the stereographic map. At rst, this statement may seem puzzling for H or QC , the following reason. In performing supersymmetric localization, one adds to the action H;C -exact localizing term. The standard localizing term for the vectormultiplet is usually constructed by acting with two supercharges on fermion bilinears, and it thus has scaling dimension 4 and breaks conformal invariance. However, both supercharges Q C were constructed in [7, 8] as speci c linear combinations of Poincare and conformal supercharges on R3, so conformal symmetry seemed important. Consequently it seems H and confusing why the standard vectormultiplet localizing term could even be invariant under will show, however, that the supercharges Q H;C belong to an N = 4 supersymmetry algebra su(2j1)r, which, upon mapping to S3, can be seen to contain only the isometries of S3 and a U(1)2 R-symmetry, without any conformal generators.7 Thus, Q H;C -invariant 7We stress that upon contraction r ! 1 the supercharges QH;C 2 su(2j1)` su(2j1)r we de ne on S3 reduce to ordinary Poincare supercharges on R3, and not to the supercharges constructed in [7, 8]. The latter were also denoted by QH;C above, in a slight abuse of notation. { 5 { theories on S3 are not necessarily conformal invariant; they include the more general nonconformal QFTs on S3 mentioned above. It is worth commenting on the relation between the localization computation using H;C and that preformed by Kapustin, Willett and Yaakov (KWY) in [39] for N theories that yielded the matrix model (1.2). This computation was later generalized to 3 N = 2 theories in [41, 42]. The supercharge Q also resides in an N = 2 sub-algebra, namely su(2j1) KWY used for localization in [39] thus su(2), of the full N = 4 algebra localizing with Q described above. C or Q C do not reside in any such N = 2 subalgebra, but are instead di erent linear combinations of supercharges in the two su(2j1) factors of the N = 4 superalgebra. In spite of these di erences, we nd that the results of H are very much related to the KWY matrix model, as was brie y Concretely, our calculation proceeds as follows. Just as in [42], we nd that the YangMills action is Q-exact (with respect to Q H or QC ) and can be added with a large coe cient, thus localizing the N = 4 vectormultiplet in precisely the same way as in [39, 42]. In other words, the localization of the vectormultiplet is realized by taking the theory to small gauge coupling. At any point on the vectormultiplet localization locus, the hypermultiplet is thus free, but massive, with its mass matrix depending on the precise location on the localization locus. In this weakly coupled theory, the correlation functions of the hypermultiplet can be computed by rst using Wick's theorem at a xed point on the vectormultiplet localization locus, and then integrating over it with a measure given by the determinant of uctuations of all the elds. If we focus our attention on Q are twisted Higgs (or Coulomb) branch operators inserted along a great circle of S3, a standard argument shows that these correlation functions are independent of the YangH;C -closed operators, which as we show Mills coupling. We conclude that once the vectormultiplet has been localized, calculating correlators of twisted Higgs branch operators does not require a further localization of the hypermultiplet. Indeed, the hypermultiplet action in the background of the localized vectormultiplet is Gaussian, and thus the remaining path integral is trivially solvable. For localization with QH , however, it is instructive to also localize the hypermultiplet, which leads to the explicit description (1.1) of the correlators in terms of the 1d Gaussian theory coupled to the matrix model obtained from localizing the vectormultiplet. Localization of the hypermultiplet with Q H has several features that are worth mentioning. Unlike the N = 4 vectormultiplet, the supersymmetry algebra does not close o -shell on the hypermultiplet. Since the supersymmetric localization arguments require a supercharge that does close o -shell, the rst step is to add a number of auxiliary elds and modify the supersymmetry transformation rules such that the algebra generated by H does close o -shell on the hypermultiplet elds.8 The next step is to add to the action a QH -exact term whose bosonic part is positive-de nite. To describe the localization locus, let us think of S3 as a circle bered over a disk with the circle shrinking at the boundary 8A similar construction in the case of N = 4 supersymmetric Yang-Mills theory in 4d was used in [43, 44] following the work of [48]. { 6 { of the disk. We nd that the hypermultiplet localizes on eld con gurations that are independent of the coordinate parameterizing the circle and that obey additional di erential constraints in the disk directions. These constraints imply that the hypermultiplet action, when evaluated on the localization locus, becomes a total derivative on the disk and reduces to a boundary term. This boundary term, living on the boundary of the disk, is the Gaussian action for our localized theory. We will argue that the one-loop determinant of uctuations around this con guration equals 1, so there is no additional determinant factor coming from the hypermultiplet. The situation presented here for the localization of the hypermultiplet is very similar to the one encountered in [44] in the case of 4d N = 4 We apply (1.1){(1.2) in a few examples where we calculate explicitly several correlation functions of twisted Higgs and Coulomb branch operators, with the main focus on the Higgs branch. As in [8], we interpret the 1d OPE algebra as a non-commutative star product on the Higgs branch chiral ring,9 and we compute explicitly the values of the parameters that the bootstrap analysis of [8] left undetermined. As we will discuss, in calculating correlation functions using (1.1){(1.2), one should be aware of the possibility of operator mixing on S3. The mixing can be removed by diagonalizing the matrix of 2-point functions using the Gram-Schmidt procedure. A similar approach was taken in [28] for the Coulomb branch operators of 4d N = 2 theories. The rest of this paper is organized as follows. In section 2 we introduce the N = 4 QFTs on S3 we will study. In section 3 we review the cohomological construction of [4, 7, 8] in the case of N = 4 SCFTs in In section 4 we generalize this construction to QFTs on S3 that do not necessarily possess conformal symmetry. Section 5 contains a description of the localization computation that leads to the results (1.1){(1.2) summarized above. In section 6 we describe in general terms the various properties of the 1d theory (1.1) and its applications. Sections 7 and 8 contain applications of our results to speci c theories. We end with a brief discussion in section 9. at space and explain how it is mapped stereographically to S3. 2 3d N = 4 theories on S3 In this section we will review the construction of N = 4 supersymmetric Lagrangians using vectormultiplets and hypermultiplets on S3. We rst provide the supersymmetry transformation rules and the supersymmetric actions. We then discuss the supersymmetry algebras preserved by these actions. 2.1 Actions with vectormultiplets and hypermultiplets The components of the vectormultiplets and hypermultiplets carry Lorentz indices as well as su(2)C su(2)H R-symmetry indices. Explicitly, the components of the vectormultiplet 9The twisted Higgs branch operators are in 1-to-1 correspondence with chiral Higgs branch operators. While a generic Higgs branch operator corresponds to a function on the Higgs branch, a chiral Higgs branch These operators form the Higgs branch chiral ring. Similar statements hold for Coulomb branch operators. V transform in the adjoint representation of the gauge group G and will be denoted by The vector A in (2.1) is the gauge eld, the spinor aa_ is the gaugino, and a_ b_ and Dab are scalars, transforming in the (1; 1), (2; 2), (3; 1) and (1; 3) irreps of su(2)C respectively.10 The hypermultiplet H transforms in some unitary representation R of G and has components V = (A ; aa_ ; a_ b_ ; Dab) : H = (qa; qea; a_ ; e a_ ) ; where qa, qea are complex scalars transforming in (1; 2) and (1; 2) irreps of the R-symmetry and representations R and R of G, and a_ , e ;a_ are their spinor superpartners, which transform, respectively, in the (2; 1) and (2; 1) irreps of the R-symmetry, and in the R and R representations of G. The multiplets V and H also have \twisted" versions, in which the roles of su(2)C and su(2)H are interchanged, though we will not consider them in this paper. On S3, superconformal transformations are generated by spinors aa_ in the (2; 2) irrep of the R-symmetry, which satisfy the conformal Killing spinor equations r aa_ = a0a_ ; r particular, the transformation rules for the vectormultiplet elds (2.1) are11 A = ab_ = a_ b_ = c i ab_ 2 i 2 " Dab = (a_ jcjb_) ; iD ( (a c_ ab_ ; ab_ F and those of the hypermultiplet (2.2) are Dac cb_ i c_ a D c_b_ + 2i b_ c_ a0c_ + i _ 2 ad_[ b_ c_; c_d] ; b)c_) 2i (0ac_ b)c_ + i[ (ac_ b)d_; c_d_] ; qa = ab_ _ ; b e qa = ab_ eb_ ; a_ = i ea_ = i a a_D qa + i a0a_ qa i ac_ c_a_ qa ; a a_D qea + iqea a0a_ + i ac_qe a c_a_ : One can check that acting twice with the transformation rules presented above realizes the superconformal algebra osp(4j4) up to gauge transformations and fermionic equations of motion. We will return to this point with more details shortly. 10We label the doublet irrep of su(2)rot. frame rotations by indices ; ; : : : = 1; 2, of su(2)H by a; b; : : : = 1; 2, and of su(2)C by a_ ; b_; : : : = 1; 2. See appendix A for more details on our conventions. 11The eld strength is F [A ; A ], and D = r iA is the space and gauge covariant derivative. Brackets () enclosing indices denote an average over their permutations. { 8 { (2.1) (2.2) With the supersymmetry transformation rules in hand, one can construct supersymmetric actions. The action of a hypermultiplet coupled to a vectormultiplet is This action is invariant under the full osp(4j4) algebra. Indeed, one can check that it is invariant under the transformations (2.4){(2.9) provided that (2.3) is obeyed. This action could have been deduced from the analogous at space action by simply covariantizing all derivatives and introducing the conformal mass term 43r2 qeaqa for the hypermultiplet Multiplets V and H as de ned above have too many bosonic components and, in the path integral, have to be integrated over the middle-dimensional cycle determined by the following reality conditions on bosons: HJEP03(218) e qa = (qa) ; (AI T I ) = AI (T I ) ; ( Ia_ b_ T I ) = (DaIbT I ) = Ia_ b_ (T I ) ; DIab(T I ) ; where, for the vectormultiplet elds, we made the representation matrices T I by which they act on R explicit. In Lorentzian signature, the fermions would also obey reality constraints, namely ea_ would be the hermitian conjugate of a_ and ab_ would be the hermitian conjugate of ab_ , but in the Euclidean signature, the representations R and R of G respectively, and a_ and e a_ are independent spinors in ab_ do not obey any constraints either. As far as we know, it is not possible to write down other superconformal actions on S3 with just vectormultiplets and hypermultiplets.12 It is possible, however, to write down actions that are invariant under half the supersymmetries in osp(4j4), which anti-commute to the isometries of S3 without any conformal transformations. Such actions provide curved space analogs of the Yang-Mills action or of the actions corresponding to real masses and FI terms, all of which are not conformally invariant on R3, and therefore cannot be mapped to S3 using the stereographic map. The projection from 16 supersymmetries in osp(4j4) to the 8 under which these actions on S3 are invariant is described in terms of two su(2) matrices, hab and ha_ b_ , hab 2 su(2)H ; a_ h b_ 2 su(2)C ; (2.12) normalized such that hachcb = ab and ha_ c_hc_b_ = a_ b_ , and obeying the tracelessness condition haa = ha_ a_ = 0. We interpret these matrices as representing the Cartan elements of 12A Chern-Simons action for the vectormultiplet would be conformal, but preserving N = 4 supersymmetry would require the presence of twisted hypermultiplets which we do not consider here. { 9 { a_ b_ qa (2.10) (2.11) this fashion. The Yang-Mills action is given by The non-conformal supersymmetric actions depend explicitly on the matrices (2.12). Under a decomposition of N = 4 into an N = 2 sub-algebra, one can show that (2.14) is nothing but the S3 action of an N = 2 vectormultiplet plus an adjoint chiral of R-charge 1. For each U(1) factor in G we can introduce an FI-term: SFI[V] = i Z d3xpg habDba r 1 a_ _ h b_ ba_ : su(2)H su(2)C . Then one can restrict the S3 Killing spinors (2.3) to those obeying the further condition This condition reduces the number of independent aa_ by a factor of two. We will interpret it shortly in terms of generating a subalgebra of osp(4j4), but let us actions on S3 that are invariant under the 8 supersymmetry transformations restricted in rst present the (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) Note that while on R3 the FI parameters ab take value in the (1; 3) irrep of su(2)C only the single component = hab ab invariant under the Cartan of su(2)H survives on S3. Finally, one can introduce mass terms for the hypermultiplets by coupling them to background vectormultiplets Vb.g. in the Cartan of the avor symmetry. In order to preserve supersymmetry all the components of Vb.g. are set to zero except for 2 1 ha_ b_ ( b.g.)ba_ = _ r 2 hab(Db.g.)ba ; and the supersymmetry variations (2.8) and (2.9) have to be deformed accordingly to account for the masses. As happened with the FI terms, out of the su(2)C -triplet of mass parameters that exist on R3 for each avor group Cartan element, only one survives on S3. In the remainder of this paper, in order to conform with the conventions of [7] we will sometimes choose hab = 2 ; ha_ b_ = 3 : 2.2 Closure of the supersymmetry transformations Irrespective of the actions presented above, the superconformal transformations (2.4){(2.7) of V close o -shell into f ; ~gV = where G^ is a gauge transformation with parameter = ( ~ca_ cb_ ) a_ b_ i( ~aa_ de ned as aa_ )A ; while K^ ;~ generates bosonic symmetries in osp(4j4) and is written explicitly in terms of and ~ in appendix B. The transformations of the scalars qa in H also close o -shell as in (2.18), but those of the fermions a_ do not. Instead, one nds f ; ~g a_ = f ; ~g ea_ = where the equations of motion operators are given by eom a_ eom e a_ h i D= a_ + _ a_ b b_ + aa_ q ai ; h i D= ea_ _ eb_ ba_ ea a a_ : q i These are precisely the equations of motion following from the hypermultiplet action (2.10). That the supersymmetry algebra closes only up to the fermion equations of motion will be important in section 5.4.1, since the supercharge used for localization of the hypermultiplet has to be closed o -shell. 2.3 Non-conformal supersymmetry algebra on S3 Let us now return to the projection condition (2.13) and interpret it from the point of view of which supersymmetry algebra it is that the actions (2.14){(2.15) as well as the mass terms introduced via (2.16) are invariant under. Let us assume for now that we have not introduced any mass terms and that we set all possible FI parameters to zero. The anti-commutator of two supersymmetries restricted by (2.13) does not produce all the bosonic generators of the superconformal algebra osp(4j4), but only a subset of them. This was to be expected, because we have argued that the Yang-Mills action (2.14) is invariant under supersymmetries obeying (2.13), and since the Yang-Mills action is not conformal, it must be that the anti-commutator of supersymmetries (2.13) does not generate any conformal transformations. Judiciously working out all possible (anti-)commutators, one can check that (2.13) parameterize the 8 supersymmetry transformations of the algebra (See also [49].) The bosonic generators of this algebra consist of the so(4) = su(2)` su(2)r isometries of S3 as well as two u(1) R-symmetries that we will denote by u(1)` and u(1)r, re ecting which su(2j1) factor they belong to. The u(1)` u(1)r is a subalgebra of su(2)H su(2)C . That (2.24) contains 8 supersymmetries means it is an N = 4 supersymmetry algebra. The algebra (2.24) will be central in our work, so let us describe it in more detail. Let us denote the generators by J (`), R`, and Q (` ) for su(2j1)` and J (r), Rr, and Q (r ) for su(2j1)r. Abstractly, the algebra obeyed by J (`), R`, and Q (` ) is [Ji(`); Jj(`)] = i ijkJk(`) ; [J (`); Q(` )] = Q (` ) + " Q where The generators of su(2j1)r obey the same relations with ` ! r. To be more concrete, let us explain how these generators act on the various operators in the theory. We will take this opportunity to set up some of the notation we will use later. 2.3.1 Action of S3 isometries The commutators with S3 isometries act on a gauge-invariant operator O as the Lie derivative J (`) (J1(`) + iJ2(`)) J (`) 3 J (`) 3 J (`) 1 iJ2(`) ! : we can use the parameterization in terms of the coordinates vectors in (2.28) are v1` = v2` = v3` = v1r = v2r = v3r = i 2 i 2 i 2 i 2 i 2 i 2 ( sin( tan( ) sin( + ')@ + cot( ) sin( + ')@') ; 2 [0; 2 ], ; ' 2 [ ; ]. In this parameterization, the Killing with respect to the Killing vectors vi` and vir that obey the su(2) algebra, [vi`; vj`] = i"ijkvk`, In an explicit description where the three-sphere of radius r is embedded in R4 via [Ji(`); O] = Lvi` O ; [Ji(r); O] = Lvir O X12 + X22 + X32 + X42 = r2 ; X1 + iX2 = r cos ei ; X3 + iX4 = r sin ei' which manifests S3 as a U(1)- bration over D2, with the bers being \warped" by w and HJEP03(218) shrinking to zero size at the boundary of the disk. 2.3.2 Action of R-symmetries The action of R` and Rr on the elds of the previous section depends on the precise embedding of u(1)` and u(1)r into su(2)C su(2)H given in terms of the matrices h and h in (2.12) as follows. Let us rst de ne the operators RH = 1 2 habRba ; RC = 2 1 ha_ b_ Rb_ a_ ; where Rba and Rb_ a_ are the generators of su(2)H and su(2)C respectively. In our conventions, we then have R` = RH + RC ; Rr = RH RC : This equation provides an identi cation of R` and Rr with linear combination of the Cartan elements RH and RC of the R-symmetry of the superconformal algebra. In terms of their action on elds, it is su cient to describe how they act on su(2)H and su(2)C fundamental operators. We have13 (2.32) (2.33) (2.34) (2.35) (2.36) We will make signi cant use of the parameterization (2.30) in the remainder of this paper. The metric in these coordinates is ds2(S3) = r2(d 2 + cos2( )d 2 + sin2( )d'2): Coordinates and ' parametrize a disk with the metric ds2(D2) = r2(d 2 + sin2( )d'2), where is the radial coordinate of the disk. The sphere metric then becomes: ds2(S3) = ds2(D2) + w2d 2; w = r cos ; [RH ; Oa] = 1 2 habOb ; [RC ; Oa_ ] = 1 b_ 2 h a_ Ob_ ; with a straightforward generalization to operators with multiple su(2)H su(2)C indices. For instance, [RH ; Oabc] = 12 hadOdbc + 12 hbdOad c 12 hdcOabd. The action of R` and Rr on operators can then be inferred from simply combining (2.35) and (2.36). 2.3.3 Action of supersymmetries The action of the odd generators of su(2j1)l su(2j1)r on operators in general multiplets can be quite complicated. As mentioned above, on the vector and hypermultiplet operators their action is just a particular subset of the transformation rules (2.4){(2.9). The precise (r ) is given in (C.9) and (C.21). correspondence between the various aa_ obeying (2.13) and the supercharges Q (` ) and 13Note that in our conventions hab = hba, and similarly for h. 2.4 The discussion in section 2.3 was restricted to the case of vanishing mass and FI parameters. Introducing these parameters amounts to central extensions of the algebra (2.24), as we will now describe. It is not hard to see, using Jacobi identity, that one cannot introduce central charges in (anti-)commutators between left and right algebras, so one can only separately centrally extend su(2j1)` and su(2j1)r. Each of these algebras admits only one non-trivial central extension, so in total we have two central charges. We denote the centrally extended algebras with a tilde, so the supersymmetry algebra of our theories is s^u(2j1)` s^u(2j1)r. Denoting central charges of the left and right subalgebras by Z` and Zr respectively, the only place where they appear are the following anti-commutators: Physically, the central charges Z` and Zr correspond to turning on real masses and FI parameters. We turn on masses by coupling to background vectormultiplets in the Cartan of the avor symmetry, as explained in section 2. The only components of these background in (2.16). _ multiplets which are non-zero are ha__ ( b:g:)ba_ = b ( b:g:)1_2_ and hab(Db:g:)ba, as explained The supersymmetry algebra has a gauge transformation on the right, as written in eq. (2.18), with the gauge parameter of (2.19). In gauge theories, dynamical gauge elds force us to consider only operators which are not charged under the corresponding gauge symmetry. For such operators, the gauge transformation in the SUSY algebra vanishes. For background gauge elds, this is not so. We can have operators which are charged under the corresponding global symmetry (which would be gauged if the gauge eld were dynamical), and for them, such gauge transformations in the algebra will generate central charges. A simple computation, using the expression (2.19) for , shows that: 1 r (Z` + Zr) = i( b:g:)1_2_ = im ; (Z` Zr) = 0 : 1 r Here mb = diag(mI ), where mI are real masses for hypers qaI, I being the avor index. Analogously, FI parameters correspond to background twisted vectormultiplets in the Cartan of the gauge group. They similarly generate central charges with: 1 r (Z` + Zr) = 0 ; (Z` Zr) = ib: Here, b = I tI acts non-trivially only on operators charged under the topological symmetry, where I is the FI parameter and tI is the corresponding topological charge. Examples of such operators are monopole operators. (2.38) (2.39) b 1 r Our aim in this section and the next is to describe a procedure that generalizes the cohomological truncation of [4, 7, 8] from N = 4 SCFTs to the more general non-conformal N = 4 theories on S3 that were described in section 2. The construction of [4, 7, 8] was based on identifying two supercharges Q1H and Q2H in N = 4 SCFTs on R3, such that the OPE restricted to their cohomology gives a certain quantization of the Higgs branch. It was also possible to nd another pair of supercharges, Q1C and Q2C , whose cohomology similarly leads to a quantization of the Coulomb branch, though this second possibility was not explored in detail. We will generalize both cases to non-conformal theories on S3, but, just as in [4, 7, 8], our main focus will also be the Higgs branch. We will nd that local operators in the cohomology, both for QiH and for QiC , can only be inserted along a great circle S1 S3.14 The circle is the xed point locus of the U(1) isometry that appears in the anti-commutator fQ1H ; Q2 g or fQ1C ; Q2C g. In the case of H QiH , the operators that can be inserted on S1 will be referred to as \twisted Higgs branch operators", because, as we will see, they are in 1-to-1 correpsondence with Higgs branch chiral ring operators. Similarly, operators in QiC cohomology will be referred to as \twisted Coulomb branch operators". In section 3.1, we start by reviewing the construction of [4, 7, 8] in at space, and then in section 3.2 we translate this construction to S3. In section 4 we describe the generalization of this cohomology directly based on the su(2j1)` su(2j1)r algebra. 3.1 SCFT in at space Consider theories living on a three-dimensional Euclidean space R 3 with the standard coordinates ~x = (x1; x2; x3). The bosonic subalgebra of the osp(4j4) superconformal algebra is so(4) sp(4), where the generators of sp(4) are rotations M , translations P and special conformal transformations K , and the generators of the so(4) = su(2)H su(2)C Rsymmetry are denoted by Rab and Ra_ b_ and act on the Higgs and Coulomb branches, respectively. The fermionic generators are Q aa_ and S aa_ , denoting Poincare and conformal supercharges, respectively. The detailed description of this algebra can be found in appendix C.1. De ne the two supercharges Q1H and Q2H by Q1H = Q112_ + Q2H = Q211_ + 1 2r S121_ : are nilpotent, i.e., (Q1H )2 = (Q2H )2 = 0, and their anticommutator is given by In (3.1), r is some arbitrary parameter with dimensions of length.15 The supercharges Q1;2 H Z = ir H H 4 fQ1 ; Q2 g = M12 + R1_ 1_ : Whether we consider the cohomology of Q1H or Q2H , the above equation implies that it can be represented by elements from the Z = 0 subspace. In order to satisfy Z = 0, local operators with zero R_ 1_ charge can only be inserted at the xed point locus of the M12 1 rotation, i.e., at the line x1 = x2 = 0. 15When we map (3.1) to S3, we will interpret r as the radius of the sphere. 14There is some freedom in choosing QiH;C, which corresponds precisely to the choice of great circle on S3. (3.1) (3.2) There are Q1H;2-exact twisted translation and dilatation given by L b Lb0 = = = 1 i i 1 1 4 fQ1H ; Q221_ g = 4 fQ2H ; Q122_ g = P3 + 8 fQ1 ; Q1Hyg = H 1 8 fQ2 ; Q2Hyg = H 8 fQ1H ; 2rQ211_ S121_ g 8 fQ2H ; 2rQ112_ + S222_ g = i 2r The twisted translation generated by Lb can be used to move cohomology classes along the line x1 = x2 = 0. In particular, every cohomology class de ned at the origin can be twistedtranslated to the whole line x1 = x2 = 0. It is those observables on the line which where referred to before as twisted operators. Because Lb is Q1H - and Q2H -exact, this twisted translation is a trivial operation at the level of the cohomology of Q1H or Q2H . Therefore, to characterize the local operators in cohomology completely, it is su cient to consider them inserted at the origin. By the state-operator map, this corresponds to studying the state cohomology. Using (3.4), Lb0 = 1 Hy 8 fQ1H ; Q1 g = 8 fQ2H ; Q2Hyg, the standard 1 Hodge theory argument proves that the cohomologies of Q1H and Q2H are identical and are represented by the kernel of Lb0. R3 As shown in [7, 8], these representatives are given by local operators Oa1 an (~0) transforming in the (n + 1; 1) irrep of the su(2)H su(2)C R-symmetry and of conformal dimension = n=2. When translated with Lb they give the twisted operator: R3 O(s) = Oa1 an ~x=(0;0;s)uaR13 uaRn3 ; uR3 1; s 2r ; which de nes a non-trivial cohomology class on the line x1 = x2 = 0. Local operators in the cohomology form a certain algebraic structure under the OPE of the full theory. In particular, because Lb is zero in cohomology, the OPE of operators in the cohomology does not depend on their positions on the line, but it can depend on their ordering. By moving operators to one point, we then de ne a product of cohomology classes. This way we get an algebra in the cohomology, which is associative but not necessary commutative. As explained in [8], the operators O(s), when inserted at the origin s = 0, are just the Higgs branch chiral ring operators. However, as we move away from the origin, they become mixed with anti-chiral operators, because of the twisting factor uR3 = (1; 2sr ). This twisting factor can be thought of as an s-dependent choice of the Cartan generator of su(2)H given by 11+ss22==((22rr))22 3 + 1+s2=(2r)2 1 s=r . A twisted operator O(s) is in the su(2)H highest weight state with respect to this s-dependent Cartan generator. The fact that the twisted operators are not chiral with respect to a xed Cartan generator is responsible for the fact that the algebraic structure we get is not a chiral ring, but rather its deformation quantization.16 The deformation parameter is 21r , which was denoted by in [8]. In fact, it turns out to be slightly more convenient to study the cohomology of a linear combination Q1H + Q2H with some generic . This operator squares to the bosonic transformation Z, and so it plays a role of the equivariant di erential. The cohomology problem 16More precisely, the Higgs (or Coulomb) branch chiral ring has a natural Poisson structure, since it corresponds to the ring of holomorphic functions on the moduli space, which for N = 4 theories is a hyperkahler cone. The algebraic structure we obtain is the deformation quantization of this Poisson algebra. (3.3) (3.4) (3.5) for this operator therefore involves two steps: one has to restrict to the Z = 0 subspace rst, and then compute the cohomology there. The idea to treat a supercharge as an equivariant di erential (with respect to spacetime symmetries) in SUSY gauge theories is an old one and goes back, e.g., to [45, 46], while the associated quantization was rst discussed in [47]. Recall that Z is a sum of rotation in the (x1; x2) plane and a certain R-symmetry transformation. The condition Z = 0 then implies that geometrically, the con guration of operators should be invariant under this rotation. In particular, local operators, as well as line operators, can only be inserted at the line x1 = x2 = 0, which is the locus of this rotation. Surface operators, on the other hand, can only span the orthogonal (x1; x2) plane and correspond to some xed value of x3. Including line operators would of course change the answer, and the cohomology of local operators located at the line defect at x1 = x2 = 0 would give a di erent protected algebra. Surface operators, on the other hand, are expected to give some modules for the protected algebra to act on. They would describe point defects on the line x1 = x2 = 0, acted on by the local operators. This action simply corresponds to merging local operators and the defect together. Including extended operators gives an interesting direction for further explorations, and it would potentially allow one to extract more dynamical information about the theory. However, in this paper, we do not consider any extended operators and study only the protected algebra of local operators. 3.2 SCFT on the sphere Now let us identify the counterpart of the above construction on the sphere. After describing it in some detail, we will be able to see that it generalizes to non-conformal theories in a straightforward fashion. Mills theory to an S2. Using the stereographic map, one can place any conformal theory on S3. Under this map, the line x1 = x2 = 0 maps to a great circle S1 S3, along which the cohomology classes of local operators described in the previous subsection will be inserted. The rotation in Z now becomes a U(1) isometry of the sphere, whose xed point locus is precisely this S1. As mentioned in section 2.3.1, it will be useful to represent S3 as a U(1) bration over the disk D2, with bers shrinking at its boundary @D2 = S1. This boundary S1, parameterized by the angle ' at = 2 , is the great circle mentioned above along which local operators in cohomology can be inserted. The situation here is similar to that in [44], where an analogous representation of S4 was used in the localization of 4d N = 4 Yang3.2.1 Twisted operators on S3 by stereographic map (0; 0; si). Let us map them on S3: In R3, we were interested in correlators of twisted operators Oi(si) inserted at points hO1(s1) Ok(sk)iR3 = hO1('1) Ok('k)iS3 : (3.6) Operators on the right are the sphere counterparts of the at space twisted operators, S3 and are given by contraction of the S3 operators Oa1 an (') , inserted on the great = 2 HJEP03(218) O S3 , with O(') = cosn ' circle at have O R3 = =2, with u = (1; x2r3 ) = (1; tan '2 ). For every operator of dimension being the conformal factor, which evaluates to = cos2 '2 at = =2. The de nition (3.5) then implies S3 2 Oa1 an = 2 uaR13 S3 uaRn3 = Oa1 an = 2 uaS13 uaSn3 ; (3.7) where uS3 = uR3 cos '2 = (cos '2 ; sin '2 ). Note that the twisted operators O do not transform with a conformal factor in going from R3 to S3, and this is why they do not carry an R 3 or S3 superscript and why there is no conformal factor in (3.6). We will now interpret this construction in a more intrinsic way using the theory on S3 only. In section 2 and appendix C, we chose an embedding of the su(2j1)` su(2j1)r superalgebra in osp(4j4), such that su(2)` su(2)r sp(4) corresponds to isometries of the sphere and su(2j1)r by Q be found in appendix C.2. so(4)R = su(2)H su(2)C was a Cartan subalgebra of the R-symmetry algebra. The choice of Cartan subalgebra was parametrized by the matrices h and h. To be more precise, h parameterizes the Cartan generator RH in su(2)H and h parameterizes the Cartan generator RC in su(2)C . The generators R` and Rr of u(1)` and u(1)r are given by (2.35). The supercharges of su(2j1)` were denoted by Q (` ), and the supercharges of (r ). Their expressions in terms of conformal supercharges Q aa_ and S aa_ can Using this embedding, it is easy to identify our supercharges Q1H and Q2H as: Each of these supercharges is of course nilpotent, and Q1H = Q1 (`+) + Q1 (r ); Q2H = Q2 (J3(`) + J3(r)) is the -translation acting as P = i@ on gauge-invariant where P operators. Using the su(2j1)` su(2j1)r algebra, we can check that: n H r Q1 ; 4i Q2 (` ) Q2 (r+) o = n r Q2; 4i Q1 (`+) Q1 (r ) o = P' + RH Pb' : (3.10) Here P' = J (r) 3 3 J (`) is simply the ' translation isometry of S3 acting on gauge invariant operators as P' = i@'. The generator Pb' de ned above is a new twisted-translation, which is de ned on the sphere purely in terms of the su(2j1)` su(2j1)r superalgebra. Let Oa1 an be some local operator in the SCFT on the sphere,17 in the spin-n=2 irrep of su(2)H . If O11 1, when inserted at the point = =2, ' = 0 (which corresponds to the origin of R3 upon stereographic projection), is in the cohomology of Q1H and Q2H , (recall 17From now on, we drop the superscript or subscript S3 present in the previous subsection. Integrating out the Q's and X's we get the matrix model [39] Z = d 1d 2 sinh2( ( 1 2)) The U(2) gauge theory with a fundamental and an adjoint hypermultiplet is believed to ow to the same IR xed point as the N = 8 U(2) Yang-Mills theory. The IR SCFT has two N = 8 stress tensor multiplets, one of which corresponds to a free sector and one to an interacting sector. Intuitively, the free sector corresponds to the IR limit of the diagonal U(1) in the Yang-Mills description, while the interacting sector corresponds to the IR limit of SU(2) Yang-Mills theory, as will be made more precise shortly. It was shown in [7] that upon decomposition to N = 4 SCFT notation, the 1d Higgs branch theory has a avor su(2)F symmetry that is a subgroup of the so(8) R-symmetry. Under su(2)F , (Xe ; XT ) form a doublet.33 In order to match the notation in [7], let us introduce polarization variables ya, a = 1; 2, and denote the operators in the 1d theory by O2jF ('; y) = Oa1:::a2jF ya1 yajF ; (7.41) where jF is the spin of the su(2)F representation. We will identify 3 operators in the 1d theory and compute their correlation functions: The twisted Higgs branch representative of the N = 8 free eld multiplet. The N = 8 free eld multiplet consists of 8 scalar operators of scaling dimension 1=2 and 8 spin-1=2 operators of scaling dimension 1. Under the decomposition to N = 4 supersymmetry, 4 of the scalar operators are interpreted as Higgs branch operators (transforming under su(2)H su(2)F as (2; 2), while the other 4 are Coulomb branch operators. From the 4 Higgs branch operators one can construct the twisted Higgs branch operator O1;free('; y). The twisted Higgs branch representatives of the free and of the interacting N = 8 stress tensor multiplets. Any N = 8 stress tensor multiplet contains 35 scalar operators of scaling dimension 1, 9 of which being Higgs branch operators from an N = 4 point of view. From them, one can construct twisted Higgs branch operators O2('; y). We will denote the operator corresponding to the free stress tensor multiplet by O2;free('; y) and the one corresponding to the interacting stress tensor multiplet by O2;int('; y). 7.3.1 Free N = 8 multiplet The free multiplet operator O1;free('; y) is O1;free('; y) = y1 tr Xe (') + y2 tr X(') : (7.42) by 33Because of the D-term relations, we may construct operators only from X and Xe. Indeed, the equations of motion for the auxiliary eld Dab imply QejQi+XejkXik XekiXkj = 0, so every pair QejQi can be replaced XejkXik + XekiXkj. Since gauge-invariant operators can only contain an equal number of Q's and Qe's such replacements yield expressions depending only on X and Xe. From (7.38), we see that tr Xe and tr X only appear in the kinetic term, so computing correlation functions of these operators can be performed using the propagator htr X('1) tr Xe ('2)i = are necessary to establish (7.43).) Using this expression and (7.42), one obtains hO1;free('1; y1)O1;free('2; y2)i = hy1; y2i sgn('1 '2) : 1 ` where the angle bracket notation is de ned by Higher point functions of O1;free('; y) can be computed using Wick contractions using (7.44). Free N = 8 stress tensor multiplet There are two su(2)F triplets of linearly independent operators that are quadratic in X corresponding to the two stress tensor multiplets of the theory. It is easy to identify the one corresponding to the free N = 8 multiplet because this is the only one appearing in the OPE of O1;free O1;free: it is simply the square of the free N = 8 operator O1;free('; y), We can then easily see that (7.43) (7.44) (7.45) (7.46) (7.47) (7.48) (7.49) (7.50) O2;free('; y) = (y1)2(tr Xe )2 + 2y1y2(tr Xe )(tr X) + (y2)2(tr X)2 : Again using (7.43) gives hO2;free('1; y1)O2;free('2; y2)i = 2 `2 hy1; y2i2 : 7.3.3 Interacting N = 8 stress tensor multiplet The interacting stress tensor multiplet must be orthogonal to the free one. To obtain O2;int, we rst compute the matrix of 2-point functions h(tr X)2(') (tr Xe )2(0)i h(tr X)2(') (tr Xe 2)(0)i! h(tr X2)(') (tr Xe )2(0)i h(tr X2)(') (tr Xe 2)(0)i = 1 1 + (y2)2 (tr X2) (tr X)2 : 1 2 1 2 1 2 (tr Xe 2)(') (tr Xe )2(') (tr X)2(0) = 0 ; which implies that the (y1)2 component of O2;int('; y) is (tr Xe 2)(') 12 (tr Xe )2(') up to an overall normalization factor of our choice. The su(2)F symmetry then implies O2;int('; y) = (y1)2 (tr Xe 2) (tr Xe )2 + 2y1y2 (tr XXe T ) (tr X)(tr Xe ) 1 2 Computing the two-point function of O2;int is more challenging, as one now has to use the non-trivial propagators coming from (7.39). A careful calculation shows that the two-point function is hO2;int('1; y1)O2;int('2; y2)i = hy1; y2i2 1 Z sinh2( d 1d 2 16 cosh4( 12) cosh( 1) cosh( 2) 12) [5 + cosh(2 4`2 Z = 2 We can use the formalism we have developed to calculate the 4-point functions of O2;free('; y) and O2;int('; y) and compare with [7]. In [7] it was found that the 4-point function of an operator O2('; y) corresponding to an N = 8 stress tensor multiplet (which could be any linear combination of O2;free('; y) and O2;int('; y)) is hO2('1; y1)O2('2; y2)O2('3; y3)O2('4; y4)i = C2hy1; y2i2hy3; y4i2 1+ Here, w is de ned as the constant C is given by the normalization of the operator, w hy1; y2ihy3; y4i ; hy1; y3ihy2; y4i hO2('1; y1)O2('2; y2)i = Chy1; y2i2 ; and (2B;2), (2B;+), and s2tress are the squares of the various OPE coe cients of N = 8 superconformal multiplets appearing in the OPE of the N = 8 stress tensor multiplet with itself. The four-point function of O2;free('; y) does not require any integrals, as it again only uses (7.43). When '1 < '2 < '3 < '4, we obtain hO2;free('1; y1)O2;free('2; y2)O2;free('3; y3)O2;free('4; y4)i = `4 hy1; y2i2hy3; y4i2 6+2w 2w2 4 w2 Obtaining O2;int('; y) is slightly more complicated. The nal result is hO2;int('1; y1)O2;int('2; y2)O2;int('3; y3)O2;int('4; y4)i = Comparing these expression with (7.52), we nd 15`4 hy1; y2i2hy3; y4i2 4 + w 8 w2 Free stress tensor: Interacting stress tensor: 2 stress = 16 ; 2 stress = 12 ; 2 (B;+) = 16 ; 2 (B;+) = 64 5 ; 2 (B;2) = 0 ; 2 (B;2) = 0 : (7.51) (7.52) (7.53) (7.54) ; : (7.55) w2 (7.56) (7.57) HJEP03(218) The expressions for the OPE coe cients of the free N = 8 stress tensor multiplet match the result of [7] in the free N = 8 theory (of 8 free massless scalars and 8 free massless Majorana fermions), while the corresponding expressions obtained for the interacting stress tensor match those obtained in [7] for the U(2)2 U(1) 2 ABJ theory. The former theory is the infrared limit of N = 8 super Yang-Mills theory with gauge group U(1), while the latter theory is the infrared limit of N = 8 Yang-Mills theory with gauge group SU(2). These results show quite explicitly how, at the level of the N = 4 Higgs branch theory, the IR limit of N = 8 U(2) Yang-Mills theory (or the U(2) gauge theory with one fundamental and one adjoint hypermultiplet) is a product between a free theory and the IR limit of = 4 QFTs on S3 with non-vanishing mass and FI SU(2) Yang-Mills theory. 8 Applications to N parameters either m or non-vanishing. The S3 partition function is Let us now present a few examples of correlation functions in non-conformal theories with Deformation by FI parameters SQED with non-zero FI parameter Turning on a non-zero FI parameter in SQED is easily implemented by replacing Z in (7.1) and subsequent formulas in section 7.1 by Z = e2 i ` N 2 i ` 2 (N N2 + i ` 1)! : The two-point function of J (p) is then still given by (7.11). The integrals evaluate to (8.1) (8.2) (8.3) : (8.4) Comparing with eq. (1.4) of [58] we see that (8.4) agrees with the bilinear form of the generalized higher spin algebra hs (sl(N )) with parameter = 2i `=N . (p)('1; y1; y1)J (p)('2; y2; y2)i ( 1)p (N ) N 2 = i ` + p N 2 i ` N2 + i ` + p N2 + i ` (y1 y2)p(y1 y2)p : These two-point functions can be combined into a single formula upon using the de nition (7.14). We have hJ ('1; y1; y1)J ('2; y2; y2)i = 3F2 N 2 N 2 i `; + i `; 1; N N + 1 2 2 ; (y1 y2)(y1 y2) 4 = N N X j=1 j = 0 : Because the j sum to zero, it is possible to write them as Z = Y N j=1 e2 i` j j j = !j 1 !j ; for some !j , and then using summation by parts one can write One can also compute 3-point functions. We have, for example, hJ (1)('1; y1; y1)J (1)('2; y2; y2)J (1)('3; y3; y3)i 1 2i ` N + 2 (y1 y2)(y2 y3)(y3 y1) ; (y1 y3)(y3 y2)(y2 y1) (8.5) (8.6) (8.7) (8.8) (8.9) JI J ? JK L = JIK JL + K JI L + ILJK J J ( IJ JK L + KL JI J ) 2 N 2 N (N + 1) L J I K 2` K JI L + N 4`2(N + 1) L J I K In the limit ` ! 1, (8.6) reduces to the commutative product on the deformed Higgs branch chiral ring. Indeed, using (7.6) and (7.8), one can check that the multiplication of J JI JK L yields the ` ! 1 limit of (8.6) provided that the relation QeI QI = i is satis ed, as appropriate for the deformed Higgs branch chiral ring. 8.1.2 N -node quiver with non-zero FI parameters The N node quiver has gauge group U(1)N =U(1) containing N 1 Abelian factors. Conse quently, there are N 1 linearly independent FI parameters that can be introduced. Let us introduce an FI parameter j for each one of the N gauge group factors with the constraint The deformation to non-zero j 's is realized by modifying the expression of Z in (7.19) to which matches eq. (1.4) of [58] upon making the identi cation = 2i `=N . (See footnotes 31 and 32.) The star product of the generators of the chiral ring becomes N X j=1 N X j=1 N j=1 X !j ( j j j = j (!j 1 !j ) = j+1) : (8.10) we obtain condition This expression can be substituted into (8.8). Upon performing the Fourier transform to the j coordinates using now e 2 i`!j( j j+1) = Z e2 i`( j j+1) j d j 2 cosh( ( j + `!j)) ; Z Z = QjN=1 [2 cosh( ( + `!j))] : The S3 partition function (8.12) agrees with that of SQED with N hypermultiplets with real masses !j, as required by mirror symmetry. (See [59] where this equivalence was rst shown at the level of the S3 partition function.) Note that an overall shift in !j can be \gauged away" by shifting the integration variable . We will thus impose a gauge xing In the presence of the FI terms, we can use a modi ed de nition of the operators (7.22): X = Q1Q2 QN ; Y = Qe1Qe2 QeN ; Z = Qe1Q1 i!1 = Qe2Q2 i!2 = : : : = QeN QN i!N : They obey the classical relation X Y = (Z + i!1)(Z + i!2) the deformation of the Kleinian singularity X Y = Z N with parameters !j. (Z + i!N ), corresponding to With the de nition in (8.14), we have hZ('1)Z('2)i = `2 Z d 1 1 Z (i )2 QjN=1 [2 cosh( ( + `!j))] : Then the second equation in (7.27) still holds, and we have hX ('1)Y('2)i = More generally, de ning 1 1 Z `N Z d QjN=1 i( + `!j) 1 2 QjN=1 [2 cosh( ( + `!j))] p j=1 Zcp = Y(QejQj i!j) we nd that for '1 < '2 < '3 we have hZcp('1)Zcq('2)i = `p+q Z hZcp('1)X ('2)Y('3)i = 1 1 Z 1 1 Z `N+p Z d d (i )p+q QjN=1 [2 cosh( ( + `!j))] (i )p QjN=1 i( + `!j) QjN=1 [2 cosh( ( + `!j))] ; 1 2 : From these expressions, it is straightforward to extract the corresponding star product deformed by the parameters !j. N j=1 X !j = 0 : (8.11) (8.12) (8.13) (8.14) (8.15) (8.17) (8.18) ; for '1 < '2 : (8.16) The N -node quiver has a U(1) avor symmetry under which the Qi carry charge +1=N while Qei carry charge operators X and Y carry charges +1 and 1, respectively. 1=N . This normalization of the U(1) charge is such that the We can introduce a real mass term associated with this avor symmetry by adding to the exponent of (7.18). This amounts to replacing Z in (7.19) by The partition function is given by the equation ` Z d' mr N QeiQi Z = Y N j=1 1 2 cosh ( ( j j+1 + mr=N )) : Z = N 2 imr 2 (N N2 + imr 1)! ; which, upon the replacement mr ! `, can be seen to agree with eq. (8.2) of the partition function of SQED with N charged hypers and FI parameter . Indeed, under mirror symmetry the real masses and FI parameters are interchanged. Eqs. (7.27) and (7.28) still hold, with the only change that j(j+1) is replaced by j(j+1) + mr=N . We obtain, for instance, that 1 1 Z `2 Z d e2 imr More generally, we can de ne the operators Zcp with p functions is given by N , whose matrix of two point hZcp('1)Zcq('2)i = 1 1 Z `p+q Z d e2 imr )]N (i )p+q : where m ! 1: The mixing of these operators can be removed by performing a Gram-Schmidt procedure as was the case for SCFTs. For example, we can remove the mixing with the identity operator by subtracting the expectation values of the operators. Explicitly, (8.22){(8.23) imply that the connected correlation function of Z is hZ('1)Z('2)i hZ('1)ihZ('2)i = (1) N 2 imr + (1) N2 + imr 4 2`2 ; (8.24) (n)(z) is the polygamma function. One can see that this function vanishes as (8.19) (8.20) (8.21) (8.22) (8.23) The SQED theory with N charged hypermultiplets has an SU(N ) avor symmetry. One can introduce N by adding 1 real mass parameters corresponding to the U(1)N 1 Cartan of SU(N ) N I=1 ` Z d' X mI r QeI Q ; I N I=1 X mI = 0 to the exponent of the second equation in (7.1). The condition PN that the mI are real masses for the Cartan of SU(N ). The expression for the S3 partition function in (7.1) gets replaced by Z Z = d Z ; Z = 1 QIN=1 2 cosh( ( + mI r)) : The S3 partition function agrees with that of the N -node quiver (8.12) upon the replacement mI r ! `!I , in agreement with mirror symmetry. While it is possible to perform computations for arbitrary N , for simplicity let us give an example in the case N = 2 where we take m1 = m2 = m. The partition function in (8.26) evaluates to Z = mr csch(2 mr) in this case. Let us de ne the quadratic operators J3 = 1 2 Qe1Q 1 Qe2Q 2 ; 2 J+ = Qe1Q ; 1 = Qe2Q : The operator J3 is neutral under the U(1) Cartan of avor SU(2) symmetry, so it's correlation functions are independent of position. We obtain, for instance, hJ3('1)J3('2)i hJ3('1)ihJ3('2)i = hJ3(')i = 1 2 mr coth(2 mr) 4 mr` 1+8 2m2r2 cosh(4 mr) csch2(2 mr) 32 2m2r2`2 : (8.29) On the other hand, the operators J carry charges 2 under the Cartan of the avor SU(2). Their expectation values must vanish because they cannot mix with the identity operator. Their correlation functions, however, do depend on position as in (6.16) with FJ (mr) = 2mr. We obtain hJ+('1)J ('2)i = e2mr('1 '2) [1 2mr coth(2mr )] [coth(2mr ) One can see that both (8.29) and (8.30) interpolate between a non-trivial topological expression at mr = 0 and they both tend to zero as mr ! 1. Indeed, if we interpret mr as the RG scale, then at small mr we are probing the UV SCFT, while at large mr we are probing the infrared. (8.25) (8.26) (8.27) (8.28) : (8.30) In this paper we used supersymmetric localization to derive a 1d theory coupled to a matrix model, given in (6.1), that can be used to calculate correlation functions of twisted Higgs branch operators of N = 4 QFTs on S4. In the case of N = 4 SCFTs, this theory provides a Lagrangian realization of the protected Higgs branch topological sector discussed in [7, 8]. The immediate practical application of (6.1) is to the computation of 2- and 3-point functions of Higgs branch operators. Our results can be used to perform more detailed tests of mirror symmetry. We have seen, for instance, that in the N -node necklace quiver, the twisted operator Z has the 1 1 Z `2 Z d 1 This theory is mirror dual to SQED with N avors. One expects the twisted Higgs branch operator Z in the N -node quiver to be mirror dual to the twisted Coulomb branch operator constructed from the vectormultiplet scalars in SQED. In section 5.2.2 we explained that the 2-point function of can be computed by replacing each insertion of by 2 =r in the KWY matrix model, thus obtaining (9.1) (9.2) h ('1) ('2)i = 64 2 1 Z `2 Z d 1 Comparing (9.1) and (9.2), we can thus identify Z in the N -node quiver with i =(8 ) in SQED. A similar exercise shows that, at least for p p in the N -node quiver can be identi ed with [ i =(8 )]p in SQED. These are, of course, rather simple tests of mirror symmetry. It should be possible to perform more non-trivial tests in non-Abelian gauge theories. There are a few generalizations of our results that we have left for the future. One such generalization is to N = 4 gauge theories that include twisted vectormultiplets and twisted hypermultiplets, which would then open the possibility of including Chern-Simons interactions. Another such generalization would be to complete the Coulomb branch localization computation by allowing for insertions of monopole operators. Yet another such generalization would be to Higgs branch operators in theories with 8 supercharges de ned in a di erent number of spacetime dimensions. We hope to report on these questions in )]N N , Z 2 : future publications. Acknowledgments We thank Chris Beem, Cyril Closset, Thomas Dumitrescu, Jaume Gomis, Bruno Le Floch, Wolfger Peelaers, Herman Verlinde, and Edward Witten for useful discussions. The work of SSP was supported in part by the US NSF under Grant No. PHY-1418069, and that of RY by NSF Grant No. PHY-1314198. Work of MD was supported in part by Walter Burke Institute for Theoretical Physics and the U.S. Department of Energy, O ce of Science, O ce of High Energy Physics, under Award Number DE-SC0011632, as well as Sherman Fairchild foundation. { 69 { Conventions Curved space vector indices are denoted by ; ; : : :, while frame indices are denoted by i; j; : : : = 1; 2; 3. We label the doublet (spinor) representation of the SU(2)rot. frame rotation group by ; ; : : : = 1; 2, of SU(2)H by a; b; : : : = 1; 2, and of SU(2)C by a_; b_; : : : = 1; 2. Spinor indices are raised and lowered from the left with the antisymmetric tensors " and " , where "12 = SU(2)C SU(2)H indices (e.g., aa_ "12 = their contraction is de ned with the convention: 1. The same conventions are used for raising and lowering "ab"a_ b_ bb_ ). When SU(2) spinor indices are suppressed ( ) : (A.1) HJEP03(218) In particular, for any three spinors x, y and z (either commuting or anti-commuting) we have the Fierz identity: x (yz) + (xy)z + x y z = 0 : We will always take variation spinors , as in (2.4){(2.9), to be commuting, while the symbol itself to be anti-commuting. The at space gamma matrices are the usual Pauli matrices, ( i) i, which satisfy i j = ij + i"ijk k ; ( i) ( i) = 2 : ("123 = 1) ; Given a Euclidean metric g an orthonormal frame is de ned by and R = R , respectively. (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) A spin connection ! ij = ! ji is then xed from the conditions The Riemann tensor is while the Ricci tensor and scalar are de ned by R With this de nition R = 6 for a round unit 3-sphere. The space covariant derivative of spinors is de ned as g = ei ej ij ; ij = g ei ej : dei + !ij ^ ej = 0 ; e ei dx : ) ; = R while the Lie derivative L^v along v acting on scalars , spinors , and vector elds A , is given by r ! ij ijk k) ; v r r v i We will hereby summarize various details on di erential geometry on S3 that are used in the main text. Let S3 be the radius r 3-sphere embedded into C2 as Each point on S3 can be represented by an SU(2) element jz1j2 + jz2j2 = 1 ; r~z 2 C2 : g = z2 iz1! iz1 z2 : dei(`) + dei(r) 1r ijke(`) 1r ijke(r) k ^ e(j`) = 0 ; k ^ e(jr) = 0 ; The su(2)-valued left/right invariant 1-forms !(`=r), and the frame 1-forms e(`=r) associated with them are de ned as i e(`) i r i dgg 1 = i e(r) i r i : They satisfy the Maurer-Cartan equations (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) i ds2 = e(`)ie(`) = e(r)ie(r) = r2(d 2 + cos2( )d 2 + sin2( )d'2) : 34Li generates the left SU(2) action Lig = 12 ig, while Ri generates the right action Rig = 12 g i. from which the spin-connections can be directly read-o by using (A.6). The su(2)` and Rj , which are dual to the 1-forms ei(r) and ei(`) up to proportionality constants that su(2)r isometries of S3 are generated, respectively, by the vector elds Li ei(`)(Rj ) = ir j 2 i ; ei(r)(Lj ) = ir j 2 i : [Li; Lj ] = i"ijk k L ; [Ri; Rj ] = i"ijk k R : They satisfy the su(2) algebra the metric on S3 is given by In the round coordinates z1 = cos( )ei ; z2 = sin( )ei' ; and the vectors Li = Li @ Ri = R tan( ) sin( + ')@ + cot( ) sin( + ')@') ; tan( ) cos( + ')@ + cot( ) cos( + ')@') ; L L L R R R 1 = 2 = 3 = 1 = 2 = 3 = 2 2 2 2 2 2 ( sin( ')@ + tan( ) cos( It will also be useful to introduce stereographic coordinates. Let rz1 = X1 + iX2 ; rz2 = X3 + iX4 : The stereographic coordinates xi (i = 1; 2; 3) are de ned as X1;2 = x1;2 = 1 + 4xr22 ; 2X1;2 1 + X3=r X4 = x3 = 2X4 1 + X3=r X3 = r x2 1 + 44xrr222 ; ; x 2 x21 + x22 + x32 : In our de nition the origin ~x = (0; 0; 0) is mapped to X~ = (0; 0; r; 0). The induced metric on S3 is conformally at and we de ne the stereographic frame as g = e2 e = 1 1 + 4xr22 ; e i = e i : Let us summarize how Killing spinors on S3 look in the di erent frames that we introduced. The spinor covariant derivatives in the left and right invariant frames are given, respectively, by r left inv. i 2r right inv. 2r : Let (`) and (r) be spinors satisfying (`) ; (r) = (r) : Then in the left invariant frame (`) is some constant spinor (`), while (r) is some constant spinor (r) in the right invariant frame. In the stereographic frame one can check that (`) = e =2 1 (`) ; (r) = e =2 1 + (r) : (A.34) r (`) = 2r 2r i i x i r r 2r 2r i i x i (A.21) (A.22) (A.23) (A.24) (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (A.31) (A.32) (A.33) Closure of superconformal algebra For any two spinors aa_ and ~aa_ satisfying (2.3), the anti-commutator of supreconformal transformations (2.4){(2.9) acting on any eld closes up to equations of motion into The operator K^ ;~ is de ned as where f ; ~g K^ ;~ + G^ + e.o.m. : K^ ;~ L^v + R^C + R^H + ^ ; Lv is the Lie derivative along v i ~aa_ aa_ . R^C=H is an su(2)C=H transformation, acting on doublets with the matrices Ra_ b_ Rab i( ~c(a_ j0cjb_) + c(a_ ~j0cjb_)) ; i( ~(ac_ 0 b)c_ + (ac_ ~b0)c_) ; for triplets: R^H Dab = RacDcb + RbcDac, etc. such that, e.g., R^C a_ = Ra_ b_ b_ , R^H qa = Rabqb, and with the obvious generalization is the dilation transformation parameter = i( ~ab_ a0b_ + ab_ ~a0b_ ) : = ( ~c a_ cb_ ) a_ b_ v A ; and those of the hypermultiplet (2.2) have ^ [H] = ( 12 ; 1). The components of the vectormultiplet (2.1) appear with dimensions ^ [V] = (0; 32 ;1; 2), G^ is a gauge transformation with parameter such that, e.g., G^ A = D , G^ a_ b_ = i[ ; a_ b_ ], G^ qa = i qa, G^ qea = i qea, etc. C C.1 N = 4 algebras Superconformal algebra The 3d N = 4 superconformal algebra is osp(4j4), and its bosonic sub-algebra so(3; 2) su(2)H consists of conformal and R-symmetry transformations. In at space, the conformal symmetry generators can be divided into translations P , rotations M , dilatations D and special conformal transformations K . The su(2)C and su(2)H R-symmetry generators will be denoted by Rab and Ra_ b_ , respectively. The corresponding sub-algebra is P K M K M P ; K ; [D; P ] = P ; [D; K K ; [P ; K ] = 4 ( ( M ) ) + 4 ( ) D ; [Rab; Rcd] = adRcb + cbRad ; [Ra_ b_ ; Rc_d_] = a_ dRc_b + c_bRa_ d_ ; _ _ _ (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (C.1) (C.2) (C.3) (C.4) (C.5) HJEP03(218) P ( ) P ; K ( ) K ; M The algebra (C.1){(C.5) is represented on a dimension in the (2; 2) irrep of su(2)C su(2)H as M : (C.6) scalar primary operator Oaa_ (x) [P ; Oaa_ (x)] = i@ Oaa_ (x) ; 2x (x @) 2 x )Oaa_ (x) ; [M [Rab; Occ_(x)] = cbOac_ 2 abOcc_ ; [D; Oaa_ (x)] = (x @ + )Oaa_ (x) ; [R a_b_ ; Occ_(x)] = c_bOca_ _ 1 _ 2 a_bOcc_ ; (C.7) (2.9) as follows. The solution of the conformal Killing spinor equation (2.3) on R3 is The transformations of the odd generators in osp(4j4) can be read from the variations (2.4){ We then de ne the action of the Poincare supercharges Q aa_ and conformal supercharges S aa_ by algebra is and also aa_ = aa_ + xi i aa_ ; a0a_ = aa_ : 2 O [ aa_ Q aa_ + aa_ S aa_ ; O] : The commutators of the odd generators Q aa_ and S aa_ then follow by matching the action of K^ ;~ de ned in appendix B with (C.7). The resulting odd-odd and even-odd part of the fQ aa_ ; Q bb_ g = 4"ab"a_ b_ P fS aa_ ; S bb_ g = 4"ab"a_ b_ K ; ; Q aa_ ] = i S aa_ + Q aa_ Q aa_ ; [M ; S aa_ ] = S a a_+ S aa_ ; 1 2 1 2 abQ cc_ ; 1 _ 2 a_ bQ cc_ ; (C.8) (C.9) (C.10) (C.12) (C.13) (C.14) (C.15) (C.16) (C.17) 1 2 [D; S aa_ ] = S aa_ ; [Rab; S cc_] = cbS ac_ [Ra_ b_ ; S cc_] = c_b_ S ca_ 1 2 1 2 abS cc_ ; 1 _ b 2 a_ S cc_ ; i : fQ aa_ ; S bb_ g = 4i h"ab"a_ b_ M D "a_ b_ Rab + "abRa_ b_ C.2 Non-conformal N = 4 algebra on S3 We will now construct the S3 N = 4 algebra su(2j1)` su(2j1)r explicitly, as a sub-algebra be decomposed into commuting spinors (\twistors") ua and ua_ as of the osp(4j4) superconformal algebra de ned in (C.1){(C.5) and (C.10){(C.16). The matrices hab and ha_ b_ in (2.13) are traceless and square to 1. Therefore, they can always hab = u+aub + u aub+ ; ha_ b_ = ua+_u b_ + ua_ u+b_ ; where (u+u ) = (u+u ) = 1. The decomposition to twistors (C.17) simpli es the construction of the non-conformal sub-algebra, because it eliminates the need to carry the su(2)H indices. The twistors u and u are simply the eigenvectors of hab and ha_ b_ : ua hab = ub ; Let us parameterize the Cartan of su(2)C ha_ b_ u b_ = ua_ : su(2)H as35 3 RH 1 2 habRba = (u+Ru ) ; 2 1 ha_ b_ Rba_ = (u+Ru ) : _ R` = RH3 + RC3 ; 3 Rr = RH RC3 : The generators of the u(1)` su(2)C are then de ned in terms of (C.19) to be su(2)H R-symmetry of the S3 N = 4 algebra HJEP03(218) su(2j1)r are given by Q (` ) 2 1+i ua ua_ 2r Q aa_ + Q (r ) 2 1+i ua ua_ 2r Q aa_ S aa_ : (C.21) stant by demanding that Q commutators are The relative coe cients between Q a a_ and S aa_ in (C.21) can be xed up to one con(` ) anti-commute with Q (r ). The only non-trivial odd-odd where J (`) and J (r) are the su(2)` su(2)r isometry generators of S3 de ned by fQ g = fQ 4i 4i J (`) + J (r) + 1 2 1 2 J (`) = J (r) = 2 2 P P 1 4r2 K 1 4r2 K M 1 1 M R` ; Rr ; In particular, if we denote their components as then the only non-trivial even-even commutators are J (`) J (`) J +(`) ! 3 J (`) 3 (J (`)) ; (J (r)) J (r) J (r) J +(r) ! 3 J (r) 3 ; [J3(`); J (`)] = [J3(r); J (r)] = J (`) ; [J +(`); J (`)] = 2J3(`) ; J (r) ; [J +(r); J (r)] = 2J3(r) : 35Recall that (u+u ) ua+u a and (u+Ru ) ua+Rabu b, etc. (C.18) (C.19) (C.20) (C.22) (C.23) (C.24) (C.25) (C.26) (C.27) (C.28) The action of the generators J (`=r) on a scalar operator O(x) on S3 is given by [J3(`); O(x)] = [J (`); O(x)] = L3O(x) ; [J3(r); O(x)] = iL2)O(x) ; [J (r); O(x)] = R3O(x) ; iR2)O(x) ; where Li and R i were de ned in (A.21){(A.26). Finally, the non-trivial even-odd commutators are (` ) [Rr; Q(r )] = Q (` ) ; [(J (r)) ; Q(r )] = Q (r ) ; Q (r ) 1 2 For the choice of hab = 2 and ha_ b_ = 3, we can take ua+ = 1 2 1 1 + i u a = 1 1 i ! ua+_ = 1 u a_ = 1 One then nds that (C.29) (C.30) (C.31) Q (r ) : (C.32) (C.33) (C.34) (C.35) (D.1) (D.2) (D.3) (r ) + Q1 (`+) = Q112_ + Q2 (r+) = Q211_ + 1 2r 1 2r S121_ ; are the nilpotent supercharges used to de ne the Higgs branch cohomology in [7, 8]. D 1d Green's function from 3d theory The Green's function (5.22) of the fundamental twisted Higgs branch operators (4.14) inserted on the = 2 circle in S3, can be calculated directly from the 3d Gaussian action (5.10). Without loss of generality let us consider a U(1) gauge theory with one hypermultiplet. The bosonic part of the action is where the operator Dab(x) is de ned by Sfree hyper = Z d3x pg qa(x)Dab(x)qb(x) ; Dab(x) 2 2 + 43r2 + r2 r2 r2 2 2 + 43r2 + r2 ! hqa(x)qeb(x0)i by solving the di erential equation It is a straightforward exercise to determine the two-point function Gab(x; x0) = Dac(x)Gcb(x; x0) = b a 3(x x0) : Gab(x; x0) = hqa(x)qeb(x0)i = 1 8 r cosh( sin( =2) cos( =2) ) sinh( ) cosh( cos( =2) sin( =2) ) ! where is the relative angle between the points x and x0. In the coordinates ( ; ; ') used previously, it is given by cos = cos cos 0 cos( 0) + sin sin 0 cos(' '0) : In particular, when both x and x0 belong to the circle at =2, we have Using the de nition (4.14) of Q(') and Qe(') in terms of the elds qa(x) and qa(x) evaluated = j' HJEP03(218) (D.5) (E.1) H c_b_ ; (E.2) (E.3) (E.4) (E.5) hQ(')Qe(0)i = cos 2 G12('; 0) sin 2 G22('; 0) ; which, when using (D.4), can be seen to agree precisely with (5.22). E Q H BPS equations In this section we will study the full set of BPS equations aa_ = ; ea_ = 0 ; where the transformations were de ned in (2.5), (5.32) and (5.33), the Killing spinor = is de ned in (5.5), and aa_ satis es (5.34).36 Here, we study the consequences of (E.1) before the reality conditions are imposed on the elds. Let us unpack the contents of these equations. The gaugino BPS equations, can be used to solve for the auxiliary elds Dab. This solution can be written as iDab = 1 4 v + ac( aa_ _ bb)D a_ b_ + 2 ac( ca_ b0b_ ) a_ b_ 1 2 ab a_ b_ a_ where the symmetric matrices ab and a_ b_ are given by and v is the Killing vector generating translations along : ab = ( ac_ bc_) ; a_ b_ = ( ca_ cb_ ) ; det( ab) = det( a_ b_ ) = 2 cos2( ) ; v i aa_ aa_ : The remaining gaugino BPS equations imply that the elds are independent of up to a eld dependent gauge transformation. The result is more conveniently expressed in terms of the twisted elds e1_1_ i 1_1_ ; e2_2_ e i 2_2_ ; 36In this section, we will always write for the particular spinor H de ned in (5.5) to avoid clutter. squares to Z. Let us also de ne a modi ed connection D ? as37 which satisfy [Z; e1_1_ ] = [Z; e2_2_ ] = 0, up to a gauge transformation, where Z was de ned in (5.36). The BPS con gurations are naturaly expressed in terms of (E.5) since Q H D ? = D cos( ) ir 1 e1_ 1_ + e2_2_ ; D?;' = D ;' : Using the de nitions (E.5) and (E.6), one can show that ab_ = 0 implies that F ? = F'? = D?Dab = D? e1_1_ = D? e2_2_ = D ? 1_2_ = 0 ; where F ? = i[D? ; D?]. As implied by (E.7), the modi ed connection (E.6) actually satis es D It then follows that all elds in V are similarly -independent. = 0, and so is literally independent of up to a gauge transformation. The analysis of the H0 hypermultiplet BPS equations ; a_ = ; ea_ = 0, is similar. One rst solves for the auxiliary elds: Ga = Gea = 1 (ad) h( da_ 1 (ad) h( da_ ba_ )D q b ( da_ bc_) c_a_ qb + ( da_ 0ba_ )qb ; ba_ )D qeb + ( da_ bc_)qeb c_a_ + ( da_ 0ba_ )qeb ; i where we de ned ( ac_ bc_). The remaining equations then imply that D?Ga = D?Gea = D?qa = D?qea = 0 : Note that the solutions (E.8) and (E.9) for the auxiliary elds depend on the spinors aa_ . Nevertheless, the conditions these spinors satisfy (5.34) can be shown to imply that (E.10) holds for any choice of aa_ . The solutions (E.2), (E.8), (E.9) for the auxiliary elds in terms of the dynamical ones, together with the -independence conditions (E.7) and (E.10), comprise the full set of restrictions that follow from the BPS equations (E.1) without imposing additional reality conditions on the elds. These conditions are su cient in order to show that the action Sh0yper[H0] de ned in (5.31) localizes to the 1d action (5.43). Indeed, after dimensional reduction on , plugging (E.2), (E.8) and (E.9) in Sh0yper[H0], one can show that (E.6) (E.8) (E.10) Sh0yper[H0] Q H BPS Z D2 d2xpgD2 r K ; cb_ )qebD qc + a_ b_ ( ba_ 0cb_ )qebqc + ac( ca_ (E.11) _ bb)qa a_ b_ qb ; (E.12) where K r cos( ) completes the derivation. imaginary. where runs over the coordinates and ' of D2. Using the explicit form of (E.12), one can check that the boundary term left from (E.11) is precisely the 1d action (5.43). This 37Note that in our conventions ( 1_1_)y = 2_2_, so the connection (E.6) is complex unless is pure Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N , Adv. Theor. Math. 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Mykola Dedushenko, Silviu S. Pufu, Ran Yacoby. A one-dimensional theory for Higgs branch operators, Journal of High Energy Physics, 2018, 138, DOI: 10.1007/JHEP03(2018)138