Wess-Zumino Research Articles

Quasi-Lie schemes for PDEs

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.

Anomalous Supersymmetry.

We show that supersymmetry is anomalous in N=1 superconformal quantum field theories (SCFTs) with an anomalous R symmetry. This anomaly was originally found in holographic SCFTs at strong coupling. Here we show that this anomaly is present, in general, and demonstrate it for the massless superconformal Wess-Zumino model via a one-loop computation. The anomaly appears first in four-point functions of two supercurrents either with two R currents or with an R current and an energy-momentum tensor. In fact, the Wess-Zumino consistency conditions together with the standard R-symmetry anomaly imply the existence of the anomaly. We outline the implications of this anomaly.

Gauged Wess-Zumino terms for a general coset space

The low-energy physics of systems with spontaneously broken continuous symmetry is dominated by the ensuing Nambu-Goldstone bosons. It has been known for half a century how to construct invariant Lagrangian densities for the low-energy effective theory of Nambu-Goldstone bosons. Contributions, invariant only up to a surface term – also known as the Wess-Zumino (WZ) terms – are more subtle, and as a rule are topological in nature. Although WZ terms have been studied intensively in theoretically oriented literature, explicit expressions do not seem to be available in sufficient generality in a form suitable for practical applications. Here we construct the WZ terms in d=1,2,3,4 spacetime dimensions for an arbitrary compact, semisimple and simply connected symmetry group G and its arbitrary connected unbroken subgroup H, provided that the d-th homotopy group of the coset space G/H is trivial. Coupling to gauge fields for the whole group G is included throughout the construction. We list a number of explicit matrix expressions for the WZ terms in four spacetime dimensions, including those for QCD-like theories, that is vector-like gauge theories with fermions in a complex, real or pseudoreal representation of the gauge group.

The renormalization group and the effective potential in the Wess–Zumino model

We consider the effective potential V in the massless Wess–Zumino model. By using the renormalization group equation, we show that the explicit and implicit dependence of V on the renormalization mass scale μ cancels. If V has an extremum at some non-vanishing value of the background field, then it follows that V is “flat”, independent of the background field. This is consistent with the general requirement that V be convex. The consequences for supersymmetric gauge theories are briefly considered.

Renormalization properties of a Galilean Wess-Zumino model

We consider a Galilean mathcal{N}=2 supersymmetric theory with F-term couplings in 2 + 1 dimensions, obtained by null reduction of a relativistic Wess-Zumino model. We compute quantum corrections and we check that, as for the relativistic parent theory, the F-term does not receive quantum corrections. Even more, we find evidence that the causal structure of the non-relativistic dynamics together with particle number conservation constrain the theory to be one-loop exact.

Formula omitted][formula omitted] Lifshitz-Wess-Zumino model: A paradigm of reconciliation between Lifshitz-like operators and supersymmetry

By imposing the weighted renormalization condition and the (super)symmetry requirements, we construct a Lifshitz-like extension of the three-dimensional Wess-Zumino model, with dynamical critical exponent z=2. In this context, the auxiliary field F plays a key role by introducing the appropriate Lifshitz operator in the bosonic sector of the theory, avoiding so undesirable time-space mixing derivatives and inconsistencies concerning the critical z exponent, as reported in the literature. The consistency of the proposed model is verified by building explicitly the susy algebra through the Noether method in the canonical formalism. This component-field Lifshitz-Wess-Zumino model is in addition rephrased in the Lifshitz superspace, a natural modification of the conventional one. The one-loop effective potential is computed to study the possibility of symmetry breaking. It is found that supersymmetry remains intact at one-loop order, while the U(1) phase symmetry suffers a spontaneous breakdown above the critical value of the renormalization point. By renormalizing the one-loop effective potential within the cutoff regularization scheme, it is observed an improvement of the UV behavior of the theory compared with the relativistic Wess-Zumino model.

Marginal deformations of WZW models and the classical Yang–Baxter equation

We show how so-called Yang–Baxter (YB) deformations of sigma models, based on an R-matrix solving the classical Yang–Baxter equation (CYBE), give rise to marginal current–current deformations when applied to the Wess–Zumino–Witten (WZW) model. For non-compact groups these marginal deformations are more general than the ones usually considered, since they can involve a non-Abelian current subalgebra. We classify such deformations of the string.

Effective action of a dynamical Dp-brane with background fluxes

We shall construct the Dirac-Born-Infeld like and the Wess–Zumino like actions for a dynamical Dp-brane with the U(1) gauge potential and the Kalb–Ramond background field. The brane dynamics simultaneously has both tangential and transverse components. Our calculations will be in the context of the type II superstring theory, via the boundary state formalism.

On stochastic quantisation of supersymmetric theories

We explain how stochastic TQFT supersymmetry can be made compatible with space supersymmetry. Taking the case of N=2 supersymmetric quantum mechanics, (the proof would be the same for the Wess–Zumino model), we determine the kernels that ensure the convergence of the stochastic process toward the standard path integral, under the condition that they are covariant under supersymmetry. They depend on a massive parameter M that can be chosen at will and modifies the course of the stochastic evolution, but the infinite stochastic time limit of the correlation functions is in fact independent on the choice of M.

Schramm–Loewner-evolution-type growth processes corresponding to Wess–Zumino–Witten theories

A group theoretical formulation of Schramm--Loewner-evolution-type growth processes corresponding to Wess--Zumino--Witten theories is developed that makes it possible to construct stochastic differential equations associated with more general null vectors than the ones considered in the most fundamental example in [Alekseev et al., Lett. Math. Phys. 97, 243-261 (2011)]. Also given are examples of Schramm--Loewner-evolution-type growth processes associated with null vectors of conformal weight $4$ in the basic representations of $\widehat{\mathfrak{sl}}_{2}$ and $\widehat{\mathfrak{sl}}_{3}$.

Amplitudes from anomalous superconformal symmetry

We initiate a systematic study of the consequences of (super)conformal symmetry of massless scattering amplitudes. The classical symmetry is potentially broken at the quantum level by infrared and ultraviolet effects. We study its manifestations on the finite hard part of the scattering process. The conformal Ward identities in momentum space are second-order differential equations, difficult to analyze. We prefer to study superconformal symmetry whose generators are first-order in the momenta. Working in a massless mathcal{N} = 1 supersymmetric Wess-Zumino model, we derive on-shell superconformal Ward identities. They contain an anomaly due to collinear regions of loop momenta. It is given by an integral with one loop less than the original graph, with an extra integral over a collinear splitting parameter. We discuss the relation to the holomorphic anomaly that was previously studied in tree-level amplitudes and at the level of unitarity cuts. We derive and solve Ward identities for various scattering processes in the model. We classify the on-shell superamplitudes according to their Grassmann degree, in close analogy with the helicity classification of gluon amplitudes. We focus on MHV-like and NMHV-like amplitudes with up to six external particles, at one and two loops. Interestingly, the superconformal generator acting on the bosonic part of the amplitudes is Witten’s twistor collinearity operator. We find that the first-order differential equations, together with physically motivated boundary conditions, uniquely fix the answer. All the cases considered give rise to uniform weight functions. Our most interesting example is a five-point non-planar hexa-box integral with an off-shell leg. It gives first indications on the function space needed for Higgs plus two jets production at next-to-next-to leading order.

Brane Wess–Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid

We reinterpret the generalised Lie derivative of M-theory $E_6$ generalised geometry as hamiltonian flow on a graded symplectic supermanifold. The hamiltonian acts as the nilpotent derivative of the tensor hierarchy of exceptional field theory. This construction is an M-theory analogue of the Courant algebroid and reveals the $L_\infty$-algebra underlying the tensor hierarchy. The AKSZ construction identifies that same hamiltonian with the lagrangian of a 7-dimensional generalisation of Chern-Simons theory that reduces to the M5-brane Wess-Zumino term on 5-brane boundaries. The exercise repeats for the type IIB $E_5$ generalised geometry and we discuss the relation to the D3-brane.

Quantum critical duality in two-dimensional Dirac semimetals**Project supported by the National Natural Science Foundation of China (Grant Nos. 11647111, 11504285, and 11674026) and the Research Start-up Funds of Guizhou University, China (Grant No. 201538).

Quantum criticality is closely related to the existence of two phases with unrelated symmetry breaking. In this paper, we study Néel and Kekulé valence bond state (VBS) quantum criticality in Dirac semimetals with four-fermion interactions. Our results show that all possible dynamical masses yield the same critical coupling, which exhibits the phenomenon that all possible phases meet at a multicritical point (e.g., a tricritical point for the Néel, Kekulé-VBS and semimetallic phases). In terms of the well-established Wess–Zumino–Witten field theory, we investigate the typical criticality for the transition between Néel and Kekulé-VBS phases, and the compatible Néel–Kekulé-VBS mass matrices imply the existence of a non-Landau transition between the Néel and Kekulé-VBS phases. We show the existence of mutual duality in the defect-driven Néel–Kekulé-VBS transition near the non-Landau critical point and find that this mutual duality results from the presence of a mutual Chern–Simons term. We also study the mutual duality based on dual topological excitations.

Numerical study of the mathcal{N}=2 Landau-Ginzburg model with two superfields

In the low energy limit, the two-dimensional massless mathcal{N}=2 Wess-Zumino (WZ) model with a quasi-homogeneous superpotential is believed to become a superconformal field theory. This conjecture of the Landau-Ginzburg (LG) description has been studied numerically in the case of the A2, A3, and E6 minimal models. In this paper, by using a supersymmetric-invariant non-perturbative formulation, we simulate the WZ model with two superfields corresponding to the D3, D4, and E7 models. Then, we numerically determine the central charge, and obtain the results that are consistent with the conjectured correspondence. We hope that this numerical approach, when further developed, will be useful to investigate superstring theory via the LG/Calabi-Yau correspondence.

Critical Wess-Zumino models with four supercharges in the functional renormalization group approach

We analyze the $\mathcal{N}=1$ supersymmetric Wess-Zumino model dimensionally reduced to the $\mathcal{N}=2$ supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the $\mathcal{N}=(2,2)$ model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the non-trivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a K\"ahler metric. The K\"ahler metric is not protected by supersymmetry and we calculate its explicit form at the fixed point. In addition we determine the exact quantum dimension of the chiral superfield and several critical exponents of interest, including the correction-to-scaling exponent $\omega$, within the functional renormalization group approach. We compare the results obtained at different levels of truncation, exploring also a momentum-dependent wave function renormalization. Finally we briefly describe a tower of multicritical models in continuous dimensions.

Condensation of non-Abelian anyons in a one-dimensional fermion model

The color excitations of interacting fermions carrying an color and U(Nf) flavor index in one spatial dimension are studied in the framework of a perturbed Wess–Zumino–Novikov–Witten model. Using Bethe ansatz methods the low energy quasi-particles are found to be massive solitons forming quark and antiquark multiplets. In addition to the color index the solitons carry an internal degree of freedom with non-integer quantum dimension. These zero modes are identified as non-Abelian anyons satisfying fusion rules. Controlling the soliton density by external fields allows to drive the condensation of these anyons into various collective states. The latter are described by parafermionic cosets related to the symmetry of the system. Based on the numerical solution of the thermodynamic Bethe ansatz equations we propose a low temperature phase diagram for this model.

Cosets, characters and fusion for admissible-level [formula omitted] minimal models

We study the minimal models associated to osp(1|2), otherwise known as the fractional-level Wess–Zumino–Witten models of osp(1|2). Since these minimal models are extensions of the tensor product of certain Virasoro and sl2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp(1|2). In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles.

Chiral 3d SU(3) SQCD and mathcal{N}=2 mirror duality

Recently a very interesting three-dimensional mathcal{N}=2 supersymmetric theory with SU(3) global symmetry was discussed by several authors. We denote this model by Tx. This was conjectured to have two dual descriptions, one with explicit supersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using Tx as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of mathcal{N}=2 mirror dualities involving such constructions with the dual being either a simple mathcal{N}=2 Wess-Zumino model or a discrete gauging thereof.

Renormalizability of pure 𝒩 = 1 super-Yang–Mills in the Wess–Zumino gauge in the presence of the local composite operators A2 and λ̄λ

The [Formula: see text] super-Yang–Mills theory in the presence of the local composite operator [Formula: see text] is analyzed in the Wess–Zumino gauge by employing the Landau gauge fixing condition. Due to the supersymmetric structure of the theory, two more composite operators, [Formula: see text] and [Formula: see text], related to the SUSY variations of [Formula: see text] are also introduced. A BRST invariant action containing all these operators is obtained. An all-order proof of the multiplicative renormalizability of the resulting theory is then provided by means of the algebraic renormalization setup. Though, due to the nonlinear realization of the supersymmetry in the Wess–Zumino gauge, the renormalization factor of the gauge field turns out to be different from that of the gluino.

Probing the Holomorphic Anomaly of the D = 2, $$\mathcal{N}$$ = 2, Wess–Zumino Model on the Lattice

We study a generalization of the Langevin equation, that describes fluctuations, of commuting degrees of freedom, for scalar field theories with worldvolumes of arbitrary dimension, following Parisi and Sourlas and correspondingly generalizes the Nicolai map. Supersymmetry appears inevitably, as defining the consistent closure of system+fluctuations and it can be probed by the identities satisfied by the correlation functions of the noise fields, sampled by the action of the commuting fields. This can be done effectively, through numerical simulations. We focus on the case where the target space is invariant under global rotations, in Euclidian signature, corresponding to global Lorentz transformations, in Lorentzian signature. This can describe target space supersymmetry. In this case a cross--term, that is a total derivative for abelian isometries, or when the fields are holomorphic functions of their arguments, can lead to obstructions. We study its effects and find that, in two dimensions, it cannot lead to the appearance of the holomorphic anomaly, in any event, when fluctuations are taken into account, because continuous symmetries can't be broken in two dimensions.