Supersymmetric Research Articles

Scalar, fermionic and supersymmetric field theories with subsystem symmetries in d + 1 dimensions

We study various non-relativistic field theories with exotic symmetries called subsystem symmetries, which have recently attracted much attention in the context of fractons. We start with a scalar theory called ϕ-theory in d + 1 dimensions and discuss its properties studied in literature for d ≤ 3 such as self-duality, vacuum structure, ’t Hooft anomaly, anomaly inflow and lattice regularization. Next we study a theory called chiral ϕ-theory which is an analogue of a chiral boson with subsystem symmetries. Then we discuss theories including fermions with subsystem symmetries. We first construct a supersymmetric version of the ϕ-theory and dropping its bosonic part leads us to a purely fermionic theory with subsystem symmetries called ψ-theory. We argue that lattice regularization of the ψ-theory generically suffers from an analogue of doubling problem as previously pointed out in the d = 3 case. We propose an analogue of Wilson fermion to avoid the “doubling” problem. We also supersymmetrize the chiral ϕ-theory and dropping the bosonic part again gives us a purely fermionic theory. We finally discuss vacuum structures of the theories with fermions and find that they are infinitely degenerate because of spontaneous breaking of subsystem symmetries.

On Bailey pairs for mathcal{N} = 2 supersymmetric gauge theories on {S}_b^3/{mathbb{Z}}_r

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional mathcal{N} = 2 supersymmetric gauge theories on {S}_b^3/{mathbb{Z}}_r . The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation, and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic 2 − 3 Pachner move for triangulated 3-manifolds.

Generalized matter parities from finite modular symmetries

AbstractWe classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries ΓN. Since all the couplings in supersymmetric theories of finite modular symmetries ΓN are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight 1/M, we find $\mathbb {Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, R-parity, $\mathbb {Z}_3$ baryon triality and $\mathbb {Z}_6$ proton hexality. Such $\mathbb {Z}_{2M}$ symmetries are enlarged to $\mathbb {Z}_{2M} \rtimes \mathbb {Z}_2^{\text{CP}}$ symmetries together with the CP transformation.

Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron

The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions for contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. It is postulated by Livia Ferro, Tomasz Łukowski and Robert Moerman (2020) that contracted Grassmannian forests are in bijection with boundary strata of the momentum amplituhedron, an object encoding the tree-level S-matrix of maximally supersymmetric Yang-Mills theory. With this assumption, our results give a rank generating function for the boundary strata of the momentum amplituhedron, and imply that the Euler characteristic of the momentum amplituhedron is \(1\).Mathematics Subject Classifications: 05A05, 05A15, 05C10Keywords: Generating functions, permutations, planar forests

Shelling the \(m=1\) amplituhedron

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally nonnegative part of $\text{Gr}_{k,n}$. Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

Koszul duality in quantum field theory

In this article, we introduce basic aspects of the algebraic notion of Koszul duality for a physics audience. We then review its appearance in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories. Though much of the content of this article is well-known to experts, the presentation and examples have not, to our knowledge, appeared in the literature before. Our aim is to provide an elementary introduction for those interested in the appearance of Koszul duality in supersymmetric gauge theories with line defects and, ultimately, its generalizations to higher-dimensional defects and twisted holography.

Brane brick models for the Sasaki-Einstein 7-manifolds Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2)

The 2d (0, 2) supersymmetric gauge theories corresponding to the classes of Yp,k(ℂℙ1× ℂℙ1) and Yp,k(ℂℙ2) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2d (0, 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics.

Seven-dimensional super Yang-Mills at negative coupling

We consider the partition function for Euclidean SU(N)SU(N) super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six ``fixed" three-spheres on the \mathbb{S}^7𝕊7. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the {\mathbb S}^7𝕊7 is well-defined even when the square of the effective Yang-Mills coupling is negative. We show for an SU(2)SU(2) gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super Yang-Mills and, in fact, is gravitational.

Mathcal{N} = 2 higher spins: superfield equations of motion, the hypermultiplet supercurrents, and the component structure

As a continuation of our previous papers arXiv:2109.07639 and arXiv:2202.08196, we study the linearized structure of the manifestly 4D, mathcal{N} = 2 supersymmetric theory of the cubic couplings of the higher spin gauge superfields to the matter hypermultiplets. We consider in detail the superfield equations of motion, construct the conserved hypermultiplet superfield currents, explore their component structure (basically in the bosonic sector) and compare it with the corresponding currents in the conventional higher-spin bosonic theory. We thoroughly study the mathcal{N} = 2 spin 2 and 3 models as instructive examples.

Exact Off Shell Sudakov Form Factor in N=4 Supersymmetric Yang-Mills Theory.

We consider the Sudakov form factor in planar N=4 supersymmetric Yang-Mills theory in the off shell kinematical regime, which can be achieved by considering the theory on its Coulomb branch. We demonstrate that for up to three loops both the infrared-divergent as well as the finite terms do exponentiate, with the coefficient accompanying log^{2}(m^{2}) determined by the octagon anomalous dimension Γ_{oct}. This behavior is in stark contrast to previous conjectural accounts in the literature. Together with the finite terms we observe that for up to three loops the logarithm of the Sudakov form factor is identical to twice the logarithm of the null octagon O_{0}, which was recently introduced within the context of integrability-based approaches to four point correlation functions with infinitely large R charges. The null octagon O_{0} is known in a closed form for all values of the 't Hooft coupling constant and kinematical parameters. We conjecture that the relation between O_{0} and the off shell Sudakov form factor will hold to all loop orders.

Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be N=4 supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with a lower level of supersymmetry, or topological Chern–Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found using the Cauchy integral formula. The solution restricts the form of the unintegrated gluon distribution as a function of momentum transfer and of Bjorken x. Then, we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behavior in the vicinity of the point x=0 and a smooth behavior in the vicinity of the point x=1.

Symmetry TFTs for 3d QFTs from M-theory

We derive the Symmetry Topological Field Theories (SymTFTs) for 3d supersymmetric quantum field theories (QFTs) constructed in M-theory either via geometric engineering or holography. These 4d SymTFTs encode the symmetry structures of the 3d QFTs, for instance the generalized global symmetries and their ’t Hooft anomalies. Using differential cohomology, we derive the SymTFT by reducing M-theory on a 7-manifold Y7, which either is the link of a conical Calabi-Yau four-fold or part of an AdS4× Y7 holographic solution. In the holographic setting we first consider the 3d mathcal{N} = 6 ABJ(M) theories and derive the BF-couplings, which allow the identification of the global form of the gauge group, as well as 1-form symmetry anomalies. Secondly, we compute the SymTFT for 3d mathcal{N} = 2 quiver gauge theories whose holographic duals are based on Sasaki-Einstein 7-manifolds of type Y7 = Yp,k( mathbbm{CP} 2). The SymTFT encodes 0- and 1-form symmetries, as well as potential ’t Hooft anomalies between these. Furthermore, by studying the gapped boundary conditions of the SymTFT we constrain the allowed choices for U(1) Chern-Simons terms in the dual field theory.

Type II double field theory in superspace

We explore type II supersymmetric double field theory in superspace. The double supervielbein is an element of the orthosymplectic group OSp(10, 10|64), which also governs the structure of generalized superdiffeomorphisms. Unlike bosonic double field theory, the local tangent space must be enhanced from the double Lorentz group in order to eliminate unphysical components of the supervielbein and to define covariant torsion and curvature tensors. This leads to an infinite hierarchy of local tangent space symmetries, which are connected to the super-Maxwell∞ algebra. A novel feature of type II is the Ramond-Ramond sector, which can be encoded as an orthosymplectic spinor (encoding the complex of super p-forms in conventional superspace). Its covariant field strength bispinor itself appears as a piece of the supervielbein. We provide a concise discussion of the superspace Bianchi identities through dimension two and show how to recover the component supersymmetry transformations of type II DFT. In addition, we show how the democratic formulation of type II superspace may be recovered by gauge-fixing.

Linear response, Hamiltonian, and radiative spinning two-body dynamics

Using the spinning, supersymmetric worldline quantum field theory formalism we compute the momentum impulse and spin kick from a scattering of two spinning black holes or neutron stars up to quadratic order in spin at third post-Minkowskian (PM) order, including radiation-reaction effects and with arbitrarily misaligned spin directions. Parts of these observables, both conservative and radiative, are also inferred from lower-PM scattering data by extending Bini and Damour's linear response formula to include misaligned spins. By solving Hamilton's equations of motion we also use a conservative scattering angle to infer a complete 3PM two-body Hamiltonian including finite-size corrections and misaligned spin-spin interactions. Finally, we describe mappings to the bound two-body dynamics for aligned spin vectors: including a numerical plot of the binding energy for circular orbits compared with numerical relativity, analytic confirmation of the NNLO PN binding energy, and the energy loss over successive orbits.

Nonperturbative renormalization of the supercurrent in N=1 supersymmetric Yang-Mills theory

In this work, we study the nonperturbative renormalization of the supercurrent operator in $\mathcal{N}=1$ supersymmetric Yang-Mills theory, using a gauge-invariant renormalization scheme (GIRS). The proposed prescription addresses successfully the unwanted mixing of the supercurrent with other operators of equal or lower dimension, which respect the same global symmetries. This mixing is introduced by the unavoidable breaking of supersymmetry on the lattice. In GIRS all gauge-noninvariant operators, which mix with the supercurrent, are excluded from the renormalization procedure. The one remaining mixing operator is accessible by numerical simulations. We present results for the renormalization of the supercurrent using GIRS. We also compute at one-loop order the conversion matrix which relates the nonperturbative renormalization factors in GIRS to the reference scheme $\overline{\mathrm{MS}}$.

Bootstrapping mathcal{N} = 4 sYM correlators using integrability

How much spectral information is needed to determine the correlation functions of a conformal theory? We study this question in the context of planar supersymmetric Yang-Mills theory, where integrability techniques accurately determine the single-trace spectrum at finite ’t Hooft coupling. Corresponding OPE coefficients are constrained by dispersive sum rules, which implement crossing symmetry. Focusing on correlators of four stress-tensor multiplets, we construct combinations of sum rules which determine one-loop correlators, and we study a numerical bootstrap problem that nonperturbatively bounds planar OPE coefficients. We observe interesting cusps at the location of physical operators, and we obtain a nontrivial upper bound on the OPE coefficient of the Konishi operator outside the perturbative regime.

Addendum to: 4D supersymmetric gauge theories of spacetime translations

Addendum to: 4D supersymmetric gauge theories of spacetime translations

The trilinear Higgs self-couplings at mathcal {O}(alpha _t^2) in the CP-violating NMSSM

In supersymmetric theories the Higgs boson masses are derived quantities where higher-order corrections have to be included in order to match the measured Higgs mass value at the precision of current experiments. Closely related through the Higgs potential are the Higgs self-interactions. In addition, the measurement of the trilinear Higgs self-coupling provides the first step towards the reconstruction of the Higgs potential and the experimental verification of the Higgs mechanism sui generis. In this paper, we advance our prediction of the trilinear Higgs self-couplings in the CP-violating Next-to-Minimal Supersymmetric extension of the SM (NMSSM). We provide the mathcal{O}(alpha _t^2) corrections in the gaugeless limit at vanishing external momenta. The higher-order corrections turn out to be larger than the corresponding mass corrections but show the expected perturbative convergence. The inclusion of the loop-corrected effective trilinear Higgs self-coupling in gluon fusion into Higgs pairs and the estimate of the theoretical uncertainty due to missing higher-order corrections indicate that the missing electroweak higher-order corrections may be significant.

Infrared renormalons in supersymmetric theories

I argue that in large-$N$ supersymmetric QCD infrared renormalons are absent in the conformal window, there is no need in conspiracy, and vacuum expectation values of at least some gluon operators vanish. Basing on this conclusion I conjecture that in supersymmetric gluodynamics (supersymmetric Yang-Mills theory without matter) at least the leading renormalon ambiguity disappears which would be consistent with the fact that the gluon condensate vanishes in this theory, $\langle G_{\mu\nu}^a G^{\mu\nu\,a}\rangle =0$.

Poles at infinity in on-shell diagrams

In this paper we study on-shell diagrams in \U0001d4a9 < 4 supersymmetric Yang-Mills (SYM) theory. These are on-shell gauge invariant objects which appear as cuts of loop integrands in the context of generalized unitarity and serve as building blocks for amplitudes in recursion relations. In the dual formulation, they are associated with cells of the positive Grassmannian G+(k, n) and the on-shell functions can be reproduced as canonical differential forms. While for the case of the \U0001d4a9 = 4 maximally supersymmetric Yang-Mills theory all poles in on-shell diagrams correspond to IR poles when the momentum flows in edges are zero, for \U0001d4a9 < 4 SYM theories there are new UV poles when the loop momenta go to infinity. These poles originate from the prefactor of the canonical dlog form and do not correspond to erasing edges in on-shell diagrams. We show that they can be interpreted as a diagrammatic operation which involves pinching a loop and performing a “non-planar twist” on external legs, which gives rise to a non-planar on-shell diagram. Our result provides an important clue on the role of poles at infinite momenta in on-shell scattering amplitudes, and the relation to non-planar on-shell functions.