Wess-Zumino Research Articles

Yangian symmetry of string theory on AdS3 × S3 × S3 × S1 with mixed 3-form flux

We find the Yangian symmetry underlying the integrability of type IIB superstrings on AdS3×S3×S3×S1 with mixed Ramond–Ramond and Neveu–Schwarz–Neveu–Schwarz flux. The abstract commutation relations of the Yangian are formulated via RTT realisation, while its matrix realisation is in an evaluation representation depending on the quantised coefficient of the Wess–Zumino term. The construction naturally encodes a secret symmetry of the worldsheet scattering matrix whose generators map different Yangian levels to each other. We show that in the large effective string tension limit the Yangian becomes a deformation of a unitary loop algebra and we derive its universal classical r-matrix.

Curious aspects of three-dimensional mathcal{N}=1 SCFTs

We study the dynamics of certain 3d mathcal{N}=1 time reversal invariant theories. Such theories often have exact moduli spaces of supersymmetric vacua. We propose several dualities and we test these proposals by comparing the deformations and supersymmetric ground states. First, we consider a theory where time reversal symmetry is only emergent in the infrared and there exists (nonetheless) an exact moduli space of vacua. This theory has a dual description with manifest time reversal symmetry. Second, we consider some surprising facts about mathcal{N}=2 U(1) gauge theory coupled to two chiral superfields of charge 1. This theory is claimed to have emergent SU(3) global symmetry in the infrared. We propose a dual Wess-Zumino description (i.e. a theory of scalars and fermions but no gauge fields) with manifest SU(3) symmetry but only mathcal{N}=1 supersymmetry. We argue that this Wess-Zumino model must have enhanced supersymmetry in the infrared. Finally, we make some brief comments about the dynamics of mathcal{N}=1 SU(N) gauge theory coupled to Nf quarks in a time reversal invariant fashion. We argue that for Nf< N there is a moduli space of vacua to all orders in perturbation theory but it is non-perturbatively lifted.

3D Current Algebra and Twisted K Theory

Equivariant twisted K theory classes on compact Lie groups [Formula: see text] can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra [Formula: see text] using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

Coadjoint Orbits, Cocycles and Gravitational Wess–Zumino

About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group [Formula: see text]. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group [Formula: see text]. In the case of [Formula: see text] being a central extension, we construct Wess–Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov–Wiegmann (PW) formula with boundary term given by the 2-cocycle which defines the central extension. In particular, we obtain a PW type formula for Polyakov’s gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess–Zumino terms obtained from geometric actions, and that in this case, the path integral is expressed in terms of the 2-cocycle which defines the central extension. In memory of our teacher Ludwig Faddeev

Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS3 characters

We analyze the partition function of three-dimensional quantum gravity on the twisted solid torus and the ensuing dual field theory. The setting is that of a non-perturbative model of three-dimensional quantum gravity, the Ponzano–Regge model, which is here reviewed in a self-contained manner and then used to compute quasi-local amplitudes for its boundary states. In this second paper of the series, we choose a particular class of boundary spin network states which impose Gibbons–Hawking–York boundary conditions to the partition function. The peculiarity of these states is to encode a two-dimensional quantum geometry peaked around a classical quadrangulation of the finite toroidal boundary. Thanks to the topological properties of three-dimensional gravity, the theory easily projects onto the boundary while crucially still keeping track of the topological properties of the bulk. This produces, at the non-perturbative level, a specific non-linear sigma-model on the boundary, akin to a Wess–Zumino–Novikov–Witten model, whose classical equations of motion can be used to reconstruct different bulk geometries: the expected classical one is accompanied by other “quantum” solutions. The classical regime of the sigma-model becomes reliable in the limit of large boundary spins, which coincides with the semi-classical limit of the boundary geometry. In a 1-loop approximation around the solutions to the classical equations of motion, we recover (with corrections due to the non-classical bulk geometries) results obtained in the past via perturbative quantum General Relativity and through the study of characters of the BMS3 group. The exposition is meant to be completely self-contained.

Dark matter as ultralight axion-like particle in E6 × U(1)X GUT with QCD axion

Axion-like fields are naturally generated by a mechanism of anomaly cancellation of one or more anomalous gauge abelian symmetries at the Planck scale, emerging as duals of a two-form from the massless bosonic sector of string theory. This suggests an analogy of the Green–Schwarz mechanism of anomaly cancellation, at field theory level, which results in one or more Stueckelberg pseudoscalars. In the case of a single Stueckelberg pseudoscalar b, vacuum misalignments at phase transitions in the early Universe at the GUT scale provide a small mass – due to instanton suppression of the periodic potential – for a component of b, denoted as χ and termed the “axi-Higgs”, which is a physical axion-like particle. The coupling of the axi-Higgs to the gauge sector via Wess–Zumino terms is suppressed by the Planck mass, which guarantees its decoupling, while its angle of misalignment is related to MGUT. We build a gauged E6×U(1) model with anomalous U(1). It contains both an automatic invisible QCD axion and an ultra-light axi-Higgs. The invisible axion present in the model solves the strong CP problem and has mass in the conventional range while the axi-Higgs, which can act as dark matter, is sufficiently light (10−22 eV<mχ<10−20 eV) to solve short-distance problems which confront other cold dark matter candidates.

Comment about the vanishing of the vacuum energy in the Wess–Zumino model

We check the cancellation of the vacuum energy in the Wess–Zumino model at the two-loop order in the component field formalisms with and without auxiliary fields. We show that in both cases the vacuum energy is equal to zero. However, in the formalism where the auxiliary fields are excluded, the vanishing of the vacuum energy arises due to the cancellation between the potential and kinetic energies, while in the formalism with the auxiliary fields, both terms vanish separately.

Interaction of supersymmetric nonlinear sigma models with external higher spin superfields via higher spin supercurrents

We consider a four dimensional generalized Wess-Zumino model formulated in terms of an arbitrary Kähler potential mathcal{K}left(varPhi, overline{varPhi}right) and an arbitrary chiral superpotential mathcal{W}left(varPhi right) . A general analysis is given to describe the possible interactions of this theory with external higher spin gauge superfields of the (s + 1, s + 1/2) supermultiplet via higher spin supercurrents. It is shown that such interactions do not exist beyond supergravity (s ≥ 2) for any mathcal{K} and mathcal{W} . However, we find three exceptions, the theory of a free massless chiral, the theory of a free massive chiral and the theory of a free chiral with linear superpotential. For the first two, the higher spin supercurrents are known and for the third one we provide the explicit expressions. We also discuss the lower spin supercurrents. As expected, a coupling to (non-minimal) supergravity (s = 1) can always be found and we give the generating supercurrent and supertrace for arbitrary mathcal{K} and mathcal{W} . On the other hand, coupling to the vector supermultiplet (s = 0) is possible only if mathcal{K} = mathcal{K}left(overline{varPhi}varPhi right) and mathcal{W}=0 .

Tensor network formulation for two-dimensional lattice mathcal{N} = 1 Wess-Zumino model

Supersymmetric models with spontaneous supersymmetry breaking suffer from the notorious sign problem in stochastic approaches. By contrast, the tensor network approaches do not have such a problem since they are based on deterministic procedures. In this work, we present a tensor network formulation of the two-dimensional lattice mathcal{N} = 1 Wess-Zumino model while showing that numerical results agree with the exact solutions for the free case.

Classical and quantum aspects of Yang-Baxter Wess-Zumino models

We investigate the integrable Yang-Baxter deformation of the 2d Principal Chiral Model with a Wess-Zumino term. For arbitrary groups, the one-loop β-functions are calculated and display a surprising connection between classical and quantum physics: the classical integrability condition is necessary to prevent new couplings being generated by renormalisation. We show these theories admit an elegant realisation of Poisson-Lie T-duality acting as a simple inversion of coupling constants. The self-dual point corresponds to the Wess-Zumino-Witten model and is the IR fixed point under RG. We address the possibility of having supersymmetric extensions of these models showing that extended supersymmetry is not possible in general.

Deriving all p-brane superalgebras via integrability

In previous work we demonstrated that the enlarged super-Poincare algebras which underlie p-brane and D-brane actions in superstring theory can be directly determined based on the integrability of supersymmetry transformations assigned to fields appearing in Wess–Zumino terms. In that work we derived p-brane superalgebras for p=2 and 3. Here we extend our previous results and give a compact expression for superalgebras for all valid p.

Decoding a three-dimensional conformal manifold

We study the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional mathcal{N} = 2 supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the well-known XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group isomorphic to S4 and can be thought of as an orbifold of CP1. We use the 4 − ε expansion and the numerical conformal bootstrap to calculate the spectrum of conformal dimensions of low-lying operators and their OPE coefficients, and find a very good quantitative agreement between the two approaches.

Application of tensor network method to two dimensional lattice N = 1 Wess–Zumino model

We study a tensor network formulation of the two dimensional lattice N = 1 Wess–Zumino model with Wilson derivatives for both fermions and bosons. The tensor renormalization group allows us to compute the partition function without the sign problem, and basic ideas to obtain a tensor network for both fermion and scalar boson systems were already given in previous works. In addition to improving the methods, we have constructed a tensor network representation of the model including the Yukawatype interaction of Majorana fermions and real scalar bosons. We present some numerical results.

4D N = 1 SYM supercurrent on the lattice in terms of the gradient flow

The gradient flow [1–5] gives rise to a versatile method to construct renor-malized composite operators in a regularization-independent manner. By adopting this method, the authors of Refs. [6–9] obtained the expression of Noether currents on the lattice in the cases where the associated symmetries are broken by lattice regularization. We apply the same method to the Noether current associated with supersymmetry, i.e., the supercurrent. We consider the 4D N = 1 super Yang–Mills theory and calculate the renormalized supercurrent in the one-loop level in the Wess–Zumino gauge. We then re-express this supercurrent in terms of the flowed gauge and flowed gaugino fields [10].

Scaling corrections to the ground state energy of the spin-½ isotropic anti-ferromagnetic Heisenberg chain

Solutions to the Bethe Ansatz equations for the ground state of spin-½ isotropic anti-ferromagnetic periodic Heisenberg chains to length L = 221 are obtained by combining Lagrange interpolation with Newton–Raphson iteration. The long chain lengths allow many powers of a renormalization group running coupling constant to be included in fits to the ground state energy and make possible the confirmation of the convergence of the leading logarithmic term. The amplitude of this term is consistent with that expected on the basis of conformal field theory and the connection of the discrete spin-½ system to the continuum Wess–Zumino–Witten model. This resolves a decades old discrepancy based on analysis of shorter chains. An analytical improvement to the Hulthén wave-vector distribution is also provided.

P-Brane superalgebras via integrability

It has long been appreciated that superalgebras with bosonic and fermionic generators additional to those in the super-Poincare algebra underlie p-brane and D-brane actions in superstring theory. These algebras have been revealed via “bottom up” approaches, involving consideration of Noether charges, and by “top down” approaches, involving the construction of manifestly supersymmetry invariant Wess–Zumino actions. In this paper, we give an alternative derivation of these algebras based on integrability of supersymmetry transformations assigned to fields in order to solve a cohomology problem related to the construction of Wess–Zumino terms for p-brane and D-brane actions.

Effective potential at three loops

I present the effective potential at three-loop order for a general renormalizable theory, using the \MSbar renormalization scheme and Landau gauge fixing. As applications and illustrative points of reference, the results are specialized to the supersymmetric Wess-Zumino model and to the Standard Model. In each case, renormalization group scale invariance provides a consistency check. In the Wess-Zumino model, the required vanishing of the minimum vacuum energy yields an additional check. For the Standard Model, I carry out the resummation of Goldstone boson contributions, which provides yet more opportunities for non-trivial checks, and obtain the minimization condition for the Higgs vacuum expectation value at full three-loop order. An infrared divergence due to doubled photon propagators appears in the three-loop Standard Model effective potential, but it does not affect the minimization condition or physical observables and can be eliminated by resummation.

Weakness of gravity as illusion which hides true path to unification of gravity with particle physics

Well known weakness of gravity in particle physics is an illusion caused by underestimation of the role of spin in gravity. Relativistic rotation is inseparable from spin, which for elementary particles is extremely high and exceeds mass on 20–22 orders (in units [Formula: see text]). Such a huge spin generates frame-dragging that distorts space much stronger than mass, and effective scale of gravitational interaction is shifted from Planck to Compton distances. We show that compatibility between gravity and quantum theory can be achieved without modifications of Einstein–Maxwell equations, by coupling to a supersymmetric Higgs model of symmetry breaking and forming a nonperturbative super-bag solution, which generates a gravity-free Compton zone necessary for consistent work of quantum theory. Super-bag is naturally upgraded to Wess–Zumino supersymmetric QED model, forming a bridge to perturbative formalism of conventional QED.

(1 + 1)-dimensional gauge symmetric gravity model and related exact black hole and cosmological solutions in string theory

We introduce a four-dimensional extension of the Poincaré algebra (N) in (1+1)-dimensional space-time and obtain a (1+1)-dimensional gauge symmetric gravity model using the algebra N. We show that the obtained gravity model is dual (canonically transformed) to the (1+1)-dimensional anti de Sitter (AdS) gravity. We also obtain some black hole and Friedmann–Robertson–Walker (FRW) solutions by solving its classical equations of motion. Then, we study A4,8A1⊗A1 gauged Wess–Zumino–Witten (WZW) model and obtain some exact black hole and cosmological solutions in string theory. We show that some obtained black hole and cosmological metrics in string theory are same as the metrics obtained in solutions of our gauge symmetric gravity model.

Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g. in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as $p= \langle F, F\rangle$ where $F$ is the curvature 2-form and $\langle \cdot, \cdot\rangle$ is an invariant scalar product on the corresponding Lie algebra $\mathfrak{g}$. The descent for $p$ gives rise to an element $\omega=\omega_3 + \omega_2 + \omega_1 + \omega_0$ of mixed degree. The 3-form part $\omega_3$ is the Chern-Simons form. The 2-form part $\omega_2$ is known as the Wess-Zumino action in physics. The 1-form component $\omega_1$ is related to the canonical central extension of the loop group $LG$. In this paper, we give a new interpretation of the low degree components $\omega_1$ and $\omega_0$. Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class $p$. In more detail, we define a 1-cocycle $C$ which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation $F$ is mapped to $\omega_1=C(F)$. Furthermore, the component $\omega_0$ is related to the associator corresponding to $F$. It is surprising that while $F$ and $\Phi$ satisfy the highly non-linear twist and pentagon equations, the elements $\omega_1$ and $\omega_0$ solve the linear descent equation.