Supersymmetric Gauge Theories Research Articles

Large Order Behavior Near the AD Point: The Case of 𝒩 = 2, su(2), Nf = 2

Abstract A non-perturbative effect in κ (renormalized string coupling) obtained from the large order behavior in the vicinity of the prototypical Argyres–Douglas critical point of su(2), Nf = 2, $\mathcal {N} =2$ supersymmetric gauge theory can be studied in the Gross–Witten–Wadia unitary matrix model with the log term: one as the work done against the barrier of the effective potential by a single eigenvalue lifted from the sea and the other as a non-perturbative function contained in the solutions of the nonlinear differential Painlevé II equation that goes beyond the asymptotic series. The leading behaviors are of the form $\exp \{[-({4}/{3})({1}/{\kappa })] \, [1, \left({s}/{K}\right)^{{3}/{2}}]\}$. We make comments on their agreement.

Whittaker vectors for W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-algebras from topological recursion

We identify Whittaker vectors for Wk(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}^{\ extsf{k}}(\\mathfrak {g})$$\\end{document}-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^2$$\\end{document} for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N} = 2$$\\end{document} four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

Aspects of 4d supersymmetric dynamics and geometry

In this set of five Lectures we present a basic toolbox to discuss the dynamics of four dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional supersymmetric models as compactifications of six dimensional SCFTs. We discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be naturally imbedded in the geometric constructions. The Lectures mostly review results which previously appeared in the literature but also contain some unpublished derivations.

Discovering T-dualities of little string theories

We describe a general method for deducing T-dualities of little string theories, which are dualities between these theories that arise when they are compactified on circle. The method works for both untwisted and twisted circle compactifications of little string theories and is based on surface geometries associated to these circle compactifications. The surface geometries describe information about Calabi-Yau threefolds on which M-theory can be compactified to construct the corresponding circle compactified little string theories. Using this method, we deduce at least one T-dual, and in some cases multiple T-duals, for untwisted and twisted circle compactifications of most of the little string theories that can be described on their tensor branches in terms of a 6d supersymmetric gauge theory with a simple non-abelian gauge group, which are also known as rank-0 little string theories. This includes little string theories carrying 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} = (1, 1) and 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} = (1, 0) supersymmetries. For many, but not all, circle compactifications of 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} = (1, 1) little string theories, we have T-dualities that realize Langlands dualities between affine Lie algebras. Along the way, we find another discrete theta angle distinct from 0 and π for an E-string node.

Massive ambitwistor-strings; twistorial models

Ambitwistor-strings are chiral strings whose targets are spaces of complex massless particles, and whose correlation functions directly lead to simple, compact formulae for scattering amplitudes and loop integrands for massless gauge and gravity theories. This article extends the framework to massive particles in 4d, obtained via a symmetry reduction of higher dimensional massless models based on twistors. The target space of the resulting models turns out to be the phase space of 4d massive particles in a twistorial representation, and the worldsheet theory agrees with the two-twistor string previously introduced by the authors. The paper has been written so as to be largely self-contained. We discuss two interesting classes of massive theories in detail. For gauge theories, the reduction procedure is explicitly adapted to supersymmetric gauge theories on the Coulomb branch. For supergravity theories, the reduction is adapted to give theories obtained via Cremmer, Scherk & Schwartz (CSS) reduction, with broken supersymmetry and massive multiplets. The reduction procedure gives explicit and systematic rules to obtain amplitudes for all these theories and their amplitudes from two compact master formulae that have their origins in 6d based on the polarized scattering equations; in the CSS case the formulae are new, and in both cases their derivation is systematic. The freedom to include mass allows the definition of a loop insertion operator, thereby extending the formulae to 1-loop. Unlike the massless 4d twistorial models, these all display a perfect double copy structure, here incorporating massive particles in the relationship between gauge theory and CSS supergravity amplitudes.

Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model

We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact “ghost instantons”, which are extrema of the one-eigenvalue effective potential on the “unphysical sheet” of the spectral curve and have been discussed in detail recently by Mariño, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the ’t Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.

Equivariant localization and holography

We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi–Yau singularities. A key object in our constructions is the equivariant volume, for which we describe two methods of evaluation: the Berline–Vergne fixed point formula and the Molien–Weyl formula, supplemented by the Jeffrey–Kirwan prescription. We present two applications in supersymmetric field theories. Firstly, we describe a method for integrating the anomaly polynomial of SCFTs on compact toric orbifolds. Secondly, we discuss equivariant orbifold indices that are expected to play a key role in the computation of supersymmetric partition functions. In the context of supergravity, we propose that the equivariant volume can be used to characterize universally the geometry of a large class of supersymmetric solutions. As an illustration, we employ equivariant localization to prove the factorization in gravitational blocks of various supergravity free energies, recovering previous results as well as obtaining generalizations.

Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order.

We compute the radiation reacted momentum impulse Δp_{i}^{μ}, spin kick ΔS_{i}^{μ}, and scattering angle θ between two scattered spinning massive bodies (black holes or neutron stars) using the N=1 supersymmetric worldline quantum field theory formalism up to fourth post-Minkowskian (4PM) order. Our calculation confirms the state-of-the-art nonspinning results, and extends them to include spin-orbit effects. Advanced multiloop Feynman integral technology including differential equations and the method of regions are applied and extended to deal with the retarded propagators arising in a causal description of the scattering dynamics. From these results we determine a complete set of radiative fluxes at subleading PM order: the 4PM radiated four-momentum and, via linear response, the 3PM radiated angular momentum, both again including spin-orbit effects.

Burge multipartitions and the AGT correspondence

Burge multipartitions are tuples of partitions that satisfy a cyclic embedding condition. When uncoloured, Burge m-multipartitions give a combinatorial model for characters of the Wm algebras (W 2 is the Virasoro algebra). When n-coloured, they generalise the “cylindric partitions” that provide a combinatorial model (and a crystal graph) for integrable characters of the affine Lie algebra .Here, we show that the n-coloured Burge m-multipartitions yield the characters of the CFT cosets . Having previously shown that the same combinatorial objects give the SU(m) instanton partition functions in 𝒩 = 2 supersymmetric gauge theories on ℂ2/ℤ n , we have thus established a wide-ranging extension of the AGT correspondence.This talk is based on a collaboration with N.Macleod (Melbourne).

Supersymmetric backgrounds in (1 + 1) dimensions and inhomogeneous field theory

We find a (1 + 1)-dimensional metric solution for a background hosting various supersymmetric field theories with a single non-chiral real supercharge. This supersymmetric background is globally hyperbolic even though it contains a naked null singularity. In this regard, we show that scalar wave propagation on the background is well-defined and so the curvature singularity is a mild one. Taking inspiration from our previous work, we relate the field theory on this curved background to some classes of (1 + 1)-dimensional inhomogeneous field theory in the supersymmetric setup. Utilizing our supersymmetric background, we elucidate the limitations of canonical quantization and highlight the conceptual advantages of the algebraic approach to quantization.

3d exceptional gauge theories and boundary confinement

We find boundary confining dualities of 3d 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 supersymmetric gauge theories with exceptional gauge groups. The half-indices which enumerate the boundary BPS local operators in the presence of Neumann and Dirichlet boundary conditions for gauge fields are identified with the Askey-Wilson type q-beta integrals and Macdonald type sums respectively. New conjectural identities of E6 and E7 integrals and sums are derived from the boundary confining dualities. We also consider theories with a vector multiplet and adjoint chiral, which correspond to an 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} = 4 vector multiplet, with appropriate boundary conditions. We argue for the boundary confinement of the 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} = 4 vector multiplet and comment on such theories also with classical gauge groups.

Spontaneous supersymmetry breaking in inhomogeneous supersymmetric field theories and BPS vacua

We study spontaneous supersymmetry breaking in inhomogeneous extensions of 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} = 1 supersymmetric field theory models in 4-dimensions. The 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} = 1 Abelian Higgs model with the inhomogeneous mass parameter and the FI coefficient that are dependent on spatial coordinates, as well as the O’Raifeartaigh model with all its parameters being dependent on spatial coordinates, are studied in detail. In the presence of inhomogeneous parameters, half supersymmetry can be preserved by adding appropriate inhomogeneous deformations to the original Lagrangians. The inhomogeneous deformations often break the R-symmetry explicitly. In cases where the inhomogeneous deformations do not break the R-symmetry explicitly, we demonstrate that spontaneous breaking of the R-symmetry is infeasible. We argue that those models can not be spontaneous supersymmetry breaking models, according to the Nelson-Seiberg argument. We comment on this issue in the context of a generic 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} = 1 supersymmetric model as well.

Witten index in 4D supersymmetric gauge theories

We present a review of Witten index calculations in different supersymmetric gauge theories in four dimensions: supersymmetric electrodynamics, pure [Formula: see text] supersymmetric Yang–Mills (SYM) theories and also SYM theories including matter multiplets — both with chirally symmetric and asymmetric content.

Supersymmetric conformal field theories from quantum stabilizer codes

Supersymmetric conformal field theories from quantum stabilizer codes

Dualities and loops on squashed $S^3$

We consider \mathcal{N}=4𝒩=4 supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these 2/3-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large NN expression for the ABJM partition function.

Theta -diagram technique for {mathcal {N}}=1, d=4 superfields

We describe a diagrammatic procedure to carry out the Grassmann integration in super-Feynman diagrams of 4d theories expressed in terms of {mathcal {N}}=1 superfields. This method is alternative to the well known D-algebra approach. We develop it in detail for theories containing vector, chiral and anti-chiral superfields, with the type of interactions which occur in {mathcal {N}}=2 SYM theories with massless matter, but it would be possible to extend it to other cases. The main advantage is that this method is algorithmic; we implemented it as a Mathematica program that, given the description of a super Feynman diagram in momentum space, returns directly the polynomial in the momenta produced by the Grassmann integration.

Coulomb branches have symplectic singularities

We show that Coulomb branches for 3-dimensional {mathcal {N}}=4 supersymmetric gauge theories have symplectic singularities. This confirms a conjecture of Braverman–Finkelberg–Nakajima.

Bethe-state counting and Witten index

We count the physical Bethe states of quantum integrable models with twisted boundary conditions using the Witten index of 2d supersymmetric gauge theories. For multi-component models solvable by the nested Bethe ansatz, the result is a novel restricted occupancy problem. For the SU(3) spin chain and the t-J model, we propose formulae for the solution count on singular loci in the space of twist parameters.

Symplectic gauge group on the Lens space

We compute the Lens space index for 4d supersymmetric gauge theories involving symplectic gauge groups. This index can distinguish between different gauge groups from a given algebra and it matches across theories related by supersymmetric dualities. We provide explicit calculations for mathcal{N} = 4 SYM and for classes of mathcal{N} = 2 and mathcal{N} = 1 Lagrangian quivers related by S-duality. In these cases the index matches across the S-dual phases, while models in different S-duality orbits have a different Lens index. We provide analogous computations for a 4d mathcal{N} = 1 toric quiver gauge theory corresponding to a ℤ7 orbifold of ℂ3. This SU(n)7 gauge theory becomes interesting in the case of n = 2 because it is conformally dual to other two models, with symplectic and unitary gauge groups, bifundamentals and antisymmetric tensors. We explicitly check this triality at the level of the Lens space index.

Little string instanton partition functions and scalar propagators

We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the Ω-background with gauge group U(N) and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the Ω-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are also characterised by counting specific holomorphic curves in a Calabi-Yau threefold XN,1 which engineers the LST. Furthermore, using the formulation of the partition function in terms of the Kronecker-Eisenstein series, we argue for new recursive structures which relate higher instanton contributions to products of lower ones.