Harmonic Superspace Formalism Research Articles

On the two-loop divergences in 6D, [formula omitted] SYM theory

We continue studying 6D,N=(1,1) supersymmetric Yang-Mills (SYM) theory in the N=(1,0) harmonic superspace formulation. Using the superfield background field method we explore the two-loop divergences of the effective action in the gauge multiplet sector. It is explicitly demonstrated that among four two-loop background-field dependent supergraphs contributing to the effective action, only one diverges off shell. It is also shown that the divergences are proportional to the superfield classical equations of motion and hence vanish on shell. Besides, we have analyzed a possible structure of the two-loop divergences on general gauge and hypermultiplet background.

The renormalization structure of [formula omitted] supersymmetric higher-derivative gauge theory

We consider the harmonic superspace formulation of higher-derivative 6D,N=(1,0) supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives. The theory is quantized in the framework of the superfield background method ensuring manifest 6D,N=(1,0) supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization by dimensional reduction, we find possible counterterms and show that they can be removed by the coupling constant renormalization for any number of loops, while the divergences in the hypermultiplet sector are absent at all. Assuming that the deviation of the gauge-fixing term from that in the Feynman gauge is small, we explicitly calculate the divergent part of the one-loop effective action in the lowest order in this deviation. In the approximation considered, the result is independent of the gauge-fixing parameter and agrees with the earlier calculation for the theory without a hypermultiplet.

An eight-dimensional Taub-NUT-like hyper-Kähler metric in harmonic superspace formalism

Using the harmonic superspace formalism, we find the metric of a certain eight-dimensional manifold. This manifold is not compact and represents an eight-dimensional generalization of the Taub-NUT manifold. Our conjecture is that the metric that we derived is equivalent to the known metric possessing a discrete Z2 isometry, which may be obtained from the metric describing the dynamics of four Bogomol'nyi-Prasad-Sommerfield monopoles by Hamiltonian reduction.

Hidden Supersymmetry as a Key to Constructing Low-Energy Superfield Effective Actions

In this review paper, we outline and exemplify the general method of constructing the supefield low-energy quantum effective action of supersymmetric Yang-Mills (SYM) theories with extended supersymmetry in the Coulomb phase, grounded upon the requirement of invariance under the non-manifest (hidden) part of the underlying supersymmetry. In this way we restore the ${\cal N}=4$ supersymmetric effective actions in $4D, {\cal N}=4$ SYM, ${\cal N}=2$ supersymmetric effective actions in $5D, {\cal N}=2$ SYM and ${\cal N}=(1,1)$ supersymmetric effective actions in $6D, {\cal N}=(1,1)$ SYM theories. The manifest off-shell fractions of the full supersymmetry are, respectively, $4D, {\cal N}=2$, $5D, {\cal N}=1$ and $6D, {\cal N}=(1,0)$ supersymmetries. In all cases the effective actions depend on the corresponding covariant superfield SYM strengths and the hypermultiplet superfields. The whole construction essentially exploits a power of the harmonic superspace formalism.

On gauge dependence of the one-loop divergences in 6D, [formula omitted] and [formula omitted] SYM theories

We study the gauge dependence of one-loop divergences in a general matter-coupled 6D, N=(1,0) supersymmetric gauge theory in the harmonic superspace formulation. Our analysis is based on the effective action constructed by the background superfield method, with the gauge-fixing term involving one real parameter ξ0. A manifestly gauge invariant and N=(1,0) supersymmetric procedure for calculating the one-loop effective action is developed. It yields the one-loop divergences in an explicit form and allows one to investigate their gauge dependence. As compared to the minimal gauge, ξ0=1, the divergent part of the general-gauge effective action contains a new term depending on ξ0. This term vanishes for the background superfields satisfying the classical equations of motion, so that the S-matrix divergences are gauge-independent. In the case of 6D, N=(1,1) SYM theory we demonstrate that some divergent contributions in the non-minimal gauges do not vanish off shell, as opposed to the minimal gauge.

Leading low-energy effective action in 6D, mathcal{N}=left(1,1right) SYM theory

We elaborate on the low-energy effective action of 6D, mathcal{N}=left(1,1right) supersymmetric Yang-Mills (SYM) theory in the mathcal{N}=left(1,0right) harmonic superspace formulation. The theory is described in terms of analytic mathcal{N}=left(1,0right) gauge superfield V++ and analytic ω-hypermultiplet, both in the adjoint representation of gauge group. The effective action is defined in the framework of the background superfield method ensuring the manifest gauge invariance along with manifest mathcal{N}=left(1,0right) supersymmetry. We calculate leading contribution to the one-loop effective action using the on-shell background superfields corresponding to the option when gauge group SU(N) is broken to SU(N − 1) × ϒ(1) ⊂ SU(N). In the bosonic sector the effective action involves the structure sim frac{F^2}{X^2} , where F4 is a monomial of the fourth degree in an abelian field strength FM N and X stands for the scalar fields from the ω-hypermultiplet. It is manifestly demonstrated that the expectation values of the hypermultiplet scalar fields play the role of a natural infrared cutoff.

Mathcal{N} = 4 super-Yang-Mills in LHC superspace part I: classical and quantum theory

We present a formulation of the maximally supersymmetric mathcal{N} = 4 gauge theory in Lorentz harmonic chiral (LHC) superspace. It is closely related to the twistor formulation of the theory but employs the simpler notion of Lorentz harmonic variables. They parametrize a two-sphere and allow us to handle efficiently infinite towers of higher-spin auxiliary fields defined on ordinary space-time. In this approach the chiral half of mathcal{N} =4 supersymmetry is manifest. The other half is realized non-linearly and the algebra closes on shell. We give a straightforward derivation of the Feynman rules in coordinate space. We show that the LHC formulation of the mathcal{N} = 4 super-Yang-Mills theory is remarkably similar to the harmonic superspace formulation of the mathcal{N} = 2 gauge and hypermultiplet matter theories. In the twin paper arXiv:1601.06804 we apply the LHC formalism to the study of the non-chiral multipoint correlation functions of the mathcal{N} = 4 stress-tensor supermultiplet.

Harmonic superspaces forN= (1,1), 6DSYM theory

This is a short account of the off-shell N = (1,0) and on-shell N = (1,1), 6D harmonic superspace formalism and its applications for the analysis of higher-dimension invariants in N = (1,1) SYM theory.

SU(4) harmonic superspace and supersymmetric gauge theory

We consider the harmonic-superspace formalism in the $N=4$ supersymmetry using the $SU(4)/SU(2)\times SU(2)\times U(1)$ harmonics which was earlier applied to the abelian gauge theory. The N=4 non-abelian constraints in a standard superspace are reformulated as the harmonic-superspace equations for two basic analytic superfields: the independent superfield strength W of a dimension 1 and the dimensionless harmonic gauge 4-prepotential V having the $U(1)$ charge 2. These constraint equations I manifestly depend on the Grassmann coordinates $\theta$, although they are covariant under the unusual N=4 supersymmetry transformations. We analyze an alternative harmonic formalism of the supergauge theory for two unconstrained nonabelian analytic superfields W and V. The gauge-invariant action A(W,V) in this formalism contains $\theta$ factors in each term, it is invariant under the $SU(4)$ automorphism group. In this model, the interaction of two infinite-dimensional N=4 supermultiplets with the physical and auxiliary fields arises at the level of component fields. The action A(W,V) generate analytic equations of motion II alternative to the harmonic-superspace superfield constraints I. Both sets of equations give us the equivalent equations for the physical component fields of the $N=4$ gauge supermultiplet, they connect auxiliary and physical fields of two superfields. The nonlinear effective interaction of the abelian harmonic superfield W is constructed.

SQM with non-abelian self-dual fields: harmonic superspace description

We present a Lagrangian formulation for N=4 supersymmetric quantum-mechanical systems describing the motion in external non-Abelian self-dual gauge fields. For any such system, one can write a component supersymmetric Lagrangian by introducing extra bosonic variables with topological Chern-Simons type interaction. For a special class of such system when the fields are expressed in the `t Hooft ansatz form, it is possible to give a superfield description using harmonic superspace formalism. As a new explicit example, the N=4 mechanics with Yang monopole is constructed.

Harmonic superspace, minimal unitary representations and quasiconformal groups

We show that there is a remarkable connection between the harmonic superspace (HSS) formulation of N=2, d=4 supersymmetric quaternionic Kaehler sigma models that couple to N=2 supergravity and the minimal unitary representations of their isometry groups. In particular, for N=2 sigma models with quaternionic symmetric target spaces of the form G/HXSU(2) we establish a one-to-one mapping between the Killing potentials that generate the isometry group G under Poisson brackets in the HSS formulation and the generators of the minimal unitary representation of G obtained by quantization of its geometric realization as a quasiconformal group. Quasiconformal extensions of U-duality groups of four dimensional N=2, d=4 Maxwell-Einstein supergravity theories (MESGT) had been proposed as spectrum generating symmetry groups earlier. We discuss some of the implications of our results, in particular, for the BPS black hole spectra of 4d, N=2 MESGTs.

Vacua of Massive Hyper-Kahler Sigma Models with Non-Abelian Quotient

The Higgs branch of N=2 supersymmetric gauge theories with non-Abelian gauge groups are described by hyper-Kahler (HK) nonlinear sigma models with potential terms. With the non-Abelian HK quotient by U(M) and SU(M) gauge groups, we give the massive HK sigma models that are not toric in the N=1 superfield formalism and the harmonic superspace formalism. The U(M) quotient gives N!/[M! (N-M)!] discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua.

Non-anticommutative chiral singlet deformation of [formula omitted] gauge theory

We study the SO ( 4 ) × SU ( 2 ) invariant Q-deformation of Euclidean N = ( 1 , 1 ) gauge theories in the harmonic superspace formulation. This deformation preserves chirality and Grassmann harmonic analyticity but breaks N = ( 1 , 1 ) to N = ( 1 , 0 ) supersymmetry. The action of the deformed gauge theory is an integral over the chiral superspace, and only the purely chiral part of the covariant superfield strength contributes to it. We give the component form of the N = ( 1 , 0 ) supersymmetric action for the gauge groups U(1) and U ( n > 1 ) . In the U(1) and U(2) cases, we find the explicit nonlinear field redefinition (Seiberg–Witten map) relating the deformed N = ( 1 , 1 ) gauge multiplet to the undeformed one. This map exists in the general U ( n ) case as well, and we use this fact to argue that the deformed U ( n ) gauge theory can be nonlinearly reduced to a theory with the gauge group SU ( n ) .

Manifest supersymmetry for BPS walls in [formula omitted] nonlinear sigma models

BPS equations and wall solutions are studied keeping (part of) supersymmetry (SUSY) manifest. Using N=1 superfields, massive hyper-Kähler quotient is introduced to obtain massive N=2 (8 SUSY) nonlinear sigma models in four dimensions with T ∗ C P n target manifold, which yield BPS wall solutions for the n=1 case. We also describe massive hyper-Kähler quotient of T ∗ C P n by using the harmonic superspace formalism which preserves all SUSY manifestly, and BPS equations and wall solutions are obtained in the n=1 case.

Quantum superconformal symmetry in N = 2 harmonic superspace

In gauge theories, not all rigid symmetries of the classical action can be maintained manifestly in the quantization procedure, even in the absence of anomalies. If this occurs for an anomaly-free symmetry, the effective action is invariant under a transformation that differs from its classical counterpart by quantum corrections. In this note, we set up a harmonic superspace formalism for computing quantum deformations of superconformal symmetry in the N = 4 supersymmetric Yang–Mills theory.

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n).

Non-holomorphic effective potential in N=4 SU( n) SYM

We compute the one-loop non-holomorphic effective potential for the N=4 SU( n) supersymmetric Yang-Mills theory with the gauge symmetry broken down to the maximal torus U(1) n−1 . Our approach remains powerful for arbitrary gauge groups and is based on the use of N=2 harmonic superspace formulation for general N=2 Yang-Mills theories along with the superfield background field method.

3-point functions in [formula omitted] Yang-Mills

Three-point functions of analytic (chiral primary) operators in N=4 Yang-Mills theory in four dimensions are calculated using the harmonic superspace formulation of this theory. In the case of the energy-momentum tensor multiplet anomaly considerations determine the coefficient. Analyticity in N=2 harmonic superspace is explicitly checked in a two-loop calculation.

Effective action of the N = 2 Maxwell multiplet in harmonic superspace

We present, in the N=2, D=4 harmonic superspace formalism, a general method for constructing the off-shell effective action of an N=2 abelian gauge superfield coupled to matter hypermultiplets. Using manifestly N=2 supersymmetric harmonic supergraph techniques, we calculate the low-energy corrections to the renormalized one-loop effective action in terms of N=2 (anti)chiral superfield strengths. For a harmonic gauge prepotential with vanishing vacuum expectation value, corresponding to massless hypermultiplets, the only non-trivial radiative corrections to appear are non-holomorphic. For a prepotential with non-zero vacuum value, which breaks the U(1)-factor in the N=2 supersymmetry automorphism group and corresponds to massive hypermultiplets, only non-trivial holomorphic corrections arise at leading order. These holomorphic contribution are consistent with Seiberg's quantum correction to the effective action, while the first non-holomorphic contribution in the massless case is the N=2 supersymmetrization of the Heisenberg-Euler effective Lagrangian.

Manifest supersymmetry and the ADHM construction of instantons

We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart.