Using the harmonic superspace approach, we perform a comprehensive study of the structure of divergences in the higher-derivative 6D, N=(1,0) supersymmetric Yang-Mills theory coupled to the hypermultiplet in the adjoint representation. The effective action is constructed in the framework of the superfield background field method with the help of the N=(1,0) supersymmetric higher-derivative regularization scheme which preserves all symmetries of the theory. The one-loop divergences are calculated in a manifestly gauge invariant and 6D, N=(1,0) supersymmetric form, hopefully admitting a generalization to higher loops. The β function in the one-loop approximation is found and analyzed. In particular, it is shown that the one-loop β function for an arbitrary regulator function is specified by integrals of double total derivatives in momentum space, like it happens in 4D, N=1 superfield gauge theories. This points to the potential possibility to derive the Novikov-Shifman-Vainshtein-Zakharov-like exact β function in the considered theory. Published by the American Physical Society 2025

[formula omitted] higher-spin supercurrents

The Batalin-Vilkovisky formalism is applied to quantise the N = 1 supersymmetric generalisation of the Freedman-Townsend (FT) model, which was proposed by Lindström and Roček in 1983 in Minkowski superspace and is lifted to a supergravity background in this paper. This super FT theory describes a non-Abelian tensor multiplet and is known to be classically equivalent to a supersymmetric nonlinear sigma model. Using path integral considerations, we demonstrate that this equivalence holds at the quantum level in the sense that the quantum supercurrents in the two theories coincide. A modified Faddeev-Popov procedure is employed to quantise models for the N = 2 tensor multiplet in harmonic superspace. The obtained results agree with those derived by applying the Batalin-Vilkovisky scheme within the harmonic superspace setting.

Using the harmonic superspace approach, we construct, at the linearized level, N=2 supersymmetric curvatures generalizing scalar curvature, Ricci curvature and Weyl tensor. These supercurvatures are the building blocks of various linearized 4D,N=2 Einstein supergravity invariants. The supercurvatures involving the scalar and Ricci curvatures are analytic harmonic N=2 superfields, while the Weyl supertensor is a chiral N=2 superfield. As the basic distinguished feature of our construction, all these objects are expressed through the fundamental analytic gauge prepotentials h++M,M=(αα˙,α+,α˙+,5). The related characteristic features are the heavy use of harmonic derivatives and harmonic zero-curvature equations. On a number of instructive examples, we describe the component reduction of the superfield invariants constructed. Published by the American Physical Society 2024

We construct an off-shell N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal cubic vertex for the hypermultiplet coupled to an arbitrary integer higher spin s gauge N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supermultiplet in a general N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 conformal supergravity background. We heavily use N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4D harmonic superspace that provides an unconstrained superfield Lagrangian description. We start with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 global superconformal symmetry transformations of the free hypermultiplet model and require invariance of the cubic vertices of general form under these transformations and their gauged version. As a result, we deduce N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4D unconstrained analytic superconformal gauge potentials for an arbitrary integer s. These are the basic ingredients of the approach under consideration. We describe the properties of the gauge potentials, derive the corresponding superconformal and gauge transformation laws, and inspect the off-shell contents of the thus obtained N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal higher-spin s multiplets in the Wess-Zumino gauges. The spin s multiplet involves 8(2s− 1)B + 8(2s− 1)F essential off-shell degrees of freedom. The cubic vertex has the generic structure higher spin gauge superfields × hypermultiplet supercurrents. We present the explicit form of the relevant supercurrents.

We consider six-dimensional higher-derivative N=(1,0) supersymmetric gauge theory coupled with the hypermultiplet. We use the background superfield method in six-dimensional N=(1,0) harmonic superspace to study the effective action in the theory. Using the dimensional regularization scheme we analyze the one-loop divergent contributions to the effective action. We demonstrate that UV behavior is determined by the higher-derivative term for gauge multiplet sector.

We study the off-shell structure of the two-loop effective action in 6D, mathcal{N} = (1, 1) supersymmetric gauge theories formulated in mathcal{N} = (1, 0) harmonic superspace. The off-shell effective action involving all fields of 6D, mathcal{N} = (1, 1) supermultiplet is constructed by the harmonic superfield background field method, which ensures both manifest gauge covariance and manifest mathcal{N} = (1, 0) supersymmetry. We analyze the off-shell divergences dependent on both gauge and hypermultiplet superfields and argue that the gauge invariance of the divergences is consistent with the non-locality in harmonics. The two-loop contributions to the effective action are given by harmonic supergraphs with the background gauge and hypermultiplet superfields. The procedure is developed to operate with the harmonic-dependent superpropagators in the two-loop supergraphs within the superfield dimensional regularization. We explicitly calculate the gauge and the hypermultiplet-mixed divergences as the coefficients of frac{1}{varepsilon^2} and demonstrate that the corresponding expressions are non-local in harmonics.

In this paper the harmonic superspace action of the tensor multiplet of N = (1, 0), d = 6 supersymmetry is constructed which in the bosonic limit reduces to the known Pasti-Sorokin-Tonin action for the self-dual tensor field. The action involves, besides the potential containing the dynamical fields, also an auxiliary tensor multiplet and a set of analytic superfields with gauge PST scalar among them. For each of gauge symmetries of the PST action, a superfield analog is found. The equations of motion are calculated and it is shown that no extra degrees of freedom appear.

We study the quantum structure of four-dimensional {{mathcal {N}}}=2 superfield sigma-model formulated in harmonic superspace in terms of the omega-hypermultiplet superfield omega . The model is described by harmonic superfield sigma-model metric g_{ab}(omega ) and two potential-like superfields L^{++}_{a}(omega ) and L^{(+4)}(omega ). In bosonic component sector this model describes some hyper-Kähler manifold. The manifestly {{mathcal {N}}}=2 supersymmetric covariant background-quantum splitting is constructed and the superfield proper-time technique is developed to calculate the one-loop effective action. The one-loop divergences of the superfield effective action are found for arbitrary g_{ab}(omega ), L^{++}_{a}(omega ), L^{(+4)}(omega ), where some specific analogy between the algebra of covariant derivatives in the sigma-model and the corresponding algebra in the {{mathcal {N}}}=2 SYM theory is used. The component structure of divergences in the bosonic sector is discussed.

We present, for the first time, the complete off-shell 4D, mathcal{N} = 2 superfield actions for any free massless integer spin s ≥ 2 fields, using the mathcal{N} = 2 harmonic super-space approach. The relevant gauge supermultiplet is accommodated by two real analytic bosonic superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{++} , {h}_{alpha left(s-2right)dot{alpha}left(s-2right)}^{++} and two conjugated complex analytic spinor superfields {h}_{alpha left(s-1right)dot{alpha}left(s-1right)}^{+3} , {h}_{alpha left(s-2right)dot{alpha}left(s-1right)}^{+3} , where α(s) := (α1. . . αs), dot{alpha} (s) := ( dot{alpha} 1. . . dot{alpha} s). Like in the harmonic superspace formulations of mathcal{N} = 2 Maxwell and supergravity theories, an infinite number of original off-shell degrees of freedom is reduced to the finite set (in WZ-type gauge) due to an infinite number of the component gauge parameters in the analytic superfield parameters. On shell, the standard spin content (s,s−1/2,s−1/2,s−1) is restored. For s = 2 the action describes the linearized version of “minimal” mathcal{N} = 2 Einstein supergravity.

We study the six-dimensional $\mathcal{N}=(1,0)$ supersymmetric model of the interacting gauge multiplet and hypermultiplet with arbitrary self-coupling. Using the background-field method in the harmonic superspace, we calculate the divergent part of the one-loop effective action and discuss the possible finite contribution to the low-energy effective action. We demonstrate that the one-loop divergences do not vanish even in the case of the on-shell background superfields. An application of the developed technique to the hypermultiplet model in four dimensions is briefly discussed.

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.

We exploit the 6D, mathcal{N} = (1, 0) and mathcal{N} = (1, 1) harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension d = 12 in 6D, mathcal{N} = (1, 1) supersymmetric Yang-Mills (SYM) theory. Both single- and double-trace invariants are derived. Only four single-trace and two double-trace invariants prove to be independent. The invariants constructed can provide the possible counterterms of mathcal{N} = (1, 1) SYM theory at four-loop order, where the first double-trace divergences are expected to appear. We explicitly exhibit the gauge sector of all invariants in terms of mathcal{N} = (1, 0) gauge superfields and find the absence of mathcal{N} = (1, 1) supercompletion of the F6 term in the abelian limit.

We develop a novel bi-harmonic mathcal{N} = 4 superspace formulation of the mathcal{N} = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions. In this approach, the mathcal{N} = 4 SYM superfield constraints are solved in terms of on-shell mathcal{N} = 2 harmonic superfields. Such an approach provides a convenient tool of constructing the manifestly mathcal{N} = 4 supersymmetric invariants and further rewriting them in mathcal{N} = 2 harmonic superspace. In particular, we present mathcal{N} = 4 superfield form of the leading term in the mathcal{N} = 4 SYM effective action which was known previously in mathcal{N} = 2 superspace formulation.

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.

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.

We quantize super Yang-Mills action in $\mathcal{N}=3$ harmonic superspace using "Fermi-Feynman" gauge and also develop the background field formalism. This leads to simpler propagators and Feynman rules that are useful in performing explicit calculations. The superspace rules are used to show that divergences do not appear at 1-loop and beyond. We also compute a finite contribution to the effective action from a 4-point diagram at 1-loop, which matches the expected covariant result.

We construct three-pronged junctions of mass-deformed nonlinear sigma models on SO(2N)/U(N) and Sp(N )/U(N ) for generic N. We study the nonlinear sigma models on the Grassmann manifold or on the complex projective space. We discuss the relation between the nonlinear sigma model constructed in the harmonic superspace for- malism and the nonlinear sigma model constructed in the projective superspace formalism by comparing each model with the mathcal{N} = 2 nonlinear sigma model constructed in the mathcal{N} = 1 superspace formalism.

We apply the harmonic superspace approach for calculating the divergent part of the one-loop effective action of renormalizable 6D, mathcal{N} = (1, 0) supersymmetric higher-derivative gauge theory with a dimensionless coupling constant. Our consideration uses the background superfield method allowing to carry out the analysis of the effective action in a manifestly gauge covariant and mathcal{N} = (1, 0) supersymmetric way. We exploit the regularization by dimensional reduction, in which the divergences are absorbed into a renormalization of the coupling constant. Having the expression for the one-loop divergences, we calculate the relevant β-function. Its sign is specified by the overall sign of the classical action which in higher-derivative theories is not fixed a priori. The result agrees with the earlier calculations in the component approach. The superfield calculation is simpler and provides possibilities for various generalizations.

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.