Non-trivial Background Research Articles

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Vacuum structure of an eight-dimensional SU(3) gauge theory on a magnetized torus

We analyze the vacuum structure of an eight-dimensional non-Abelian gauge theory with a compactified four-dimensional torus as the extra dimensions. As a nontrivial background configuration of the gauge field of an SU(n) gauge group, we suppose a magnetic flux in two extra dimensions, and continuous Wilson line phases are also involved. We introduce matter fields and calculate the mass spectrum of low-energy modes appearing in a four-dimensional effective theory in an SU(3) model as an explicit example. As expected, potentially tachyonic states in four-dimensional modes appear from extra-dimensional gauge fields that couple to the flux background since the gauge group is simply connected. The Wilson line phases give a nonvanishing contribution to their masses, and we have a low-energy mass spectrum without tachyonic states, given that these phases take an appropriate value. To verify the validity of the values of the Wilson line phases, we examine the one-loop effective potential for these phases and explicitly show the contribution from each type of field present in our model. It is clarified that, although there seems to be no local minimum in the potential for the Wilson line phases in the pure Yang-Mills case, by including matter fields, we could find a vacuum configuration where tachyonic states disappear. Published by the American Physical Society 2024

On the anomaly interpretation of amplitudes in self-dual Yang-Mills and gravity

We investigate the integrability anomalies arising in the self-dual sectors of gravity and Yang-Mills theory, focusing on their connection to both the chiral anomaly and the trace anomaly. The anomalies in the self-dual sectors generate the one-loop all-plus amplitudes of gravitons and gluons, and have recently been studied via twistor constructions. On the one hand, we show how they can be interpreted as an anomaly of the chiral U(1) electric-magnetic-type duality in the self-dual sectors. We also note the similarity, for the usual fermionic chiral anomaly, between the 4D setting of self-dual Yang-Mills and the 2D setting of the Schwinger model. On the other hand, the anomalies in the self-dual theories also resemble the trace anomaly, sharing the same type of non-local effective action. We highlight the role of a Weyl-covariant fourth-order differential operator familiar from the trace anomaly literature, which (i) explains the conformal properties of the one-loop amplitudes, and (ii) indicates how this story may be extended to non-trivial spacetime backgrounds, e.g. with a cosmological constant. Moving beyond the self-dual sectors, and focusing on the gravity case, we comment on an intriguing connection to the two-loop ultraviolet divergence of pure gravity, whereby cancelling the anomaly at one-loop eliminates the two-loop divergence for the simplest helicity amplitudes.

Moving mirrors and event horizons in non-flat background geometry

Moving mirrors have been used for a long time as simple models for studying various properties of black hole radiation, such as the thermal spectrum and entanglement entropy. These models are typically constructed to mimic the collapse of a spherically symmetric distribution of matter in the Minkowski background. We generalize this correspondence to the case of non-trivial background geometry and consider two examples, the Schwarzschild—de Sitter black hole and the Bañados–Teitelboim–Zanelli (BTZ) black hole. In the BTZ case we were also able to show that this approach works for the spinning black hole which has only axial symmetry.

Positivity-causality competition: a road to ultimate EFT consistency constraints

Effective field theories (EFT) are strongly constrained by fundamental principles such as unitarity, locality, causality, and Lorentz invariance. In this paper, we consider the EFT of photons (or other U(1) gauge field) and compare different approaches to obtain bounds on its Wilson coefficients. We present an analytic derivation of the implications of unitarity (linear and non-linear positivity bounds) and compare these constraints with the requirement of causal propagation of the photon modes around non-trivial backgrounds generated by external sources. Within this setup, we find that the low energy causality condition can give complementary constraints to our analytic positivity bounds. In other words, simple analytic techniques can give strong constraints on the allowed region of the photon EFT parameters even when the positivity bounds are not numerically optimized.

Electric shocks: bounding Einstein-Maxwell theory with time delays on boosted RN backgrounds

The requirement that particles propagate causally on non-trivial backgrounds implies interesting constraints on higher-derivative operators. This work is part of a systematic study of the positivity bounds derivable from time delays on shockwave backgrounds. First, we discuss shockwaves in field theory, which are infinitely boosted Coulomb-like field configurations. We show how a positive time delay implies positivity of four-derivative operators in scalar field theory and electromagnetism, consistent with the results derived using dispersion relations, and we comment on how additional higher-derivative operators could be included.We then turn to gravitational shockwave backgrounds. We compute the infinite boost limit of Reissner-Nordström black holes to derive charged shockwave backgrounds. We consider photons traveling on these backgrounds and interacting through four-derivative corrections to Einstein-Maxwell theory. The inclusion of gravity introduces a logarithmic term into the time delay that interferes with the straightforward bounds derivable in pure field theory, a fact consistent with CEMZ and with recent results from dispersion relations. We discuss two ways to extract a physically meaningful quantity from the logarithmic time delay — by introducing an IR cutoff, or by considering the derivative of the time delay — and comment on the bounds implied in each case. Finally, we review a number of additional shockwave backgrounds which might be of use in future applications, including spinning shockwaves, those in higher dimensions or with a cosmological constant, and shockwaves from boosted extended objects.

Instantons in sine-Liouville theory

We compute instanton corrections to the partition function of sine-Liouville (SL) theory, which provides a worldsheet description of two-dimensional string theory in a non-trivial tachyon background. We derive these corrections using a matrix model formulation based on a chiral representation of matrix quantum mechanics and using string theory methods. In both cases we restrict to the leading and subleading orders in the string coupling expansion. Then the CFT technique is used to compute two orders of the expansion in the SL perturbation parameter λ, while the matrix model gives results which are non-perturbative in λ. The matrix model results perfectly match those of string theory in the small λ expansion. We also generalize our findings to the case of perturbation by several tachyon vertex operators carrying different momenta, and obtain interesting analytic predictions for the disk two-point and annulus one-point functions with ZZ boundary condition.

Multisoliton interactions approximating the dynamics of breather solutions

AbstractNowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.

Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background

Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background

Propagation of anisotropic gravitational and electromagnetic waves at very high energies

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

On anomalies and gauging of U(1) non-invertible symmetries in 4d QED

In this work we propose a way to promote the anomalous axial U(1)U(1) transformations to exact non-invertible U(1)U(1) symmetries. We discuss the procedure of coupling the non-invertible symmetry to a (dynamical or background) gauge field. We show that as part of the gauging procedure, certain constraints are imposed to make the gauging consistent. The constraints emerge naturally from the form of the non-invertible U(1)U(1) conserved current. In the case of dynamical gauging, this results in new type of gauge theories we call non-invertible gauge theories: These are gauge theories with additional constraints that cancel the would-be gauge anomalies. By coupling to background gauge fields, we can discuss ’t-Hooft anomalies of non-invertible symmetries. We show in an example that the matching conditions hold but they are realized in an unconventional way. Turning on non-trivial background for the non-invertible gauge field changes the vacuum even when the symmetry is not broken and the background is very weak. The anomalies are then matched by the appearance of solitons in the new vacuum.

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin–Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

Irregular universe in the Nieh-Yan modified teleparallel gravity

The Nieh-Yan modified teleparallel gravity is a model which modifies the general relativity equivalent teleparallel gravity by a coupling between the Nieh-Yan density and an axion-like field. This model predicts parity violations in the gravitational waves if the axion-like field has a non-trivial background, and more importantly it is ghost free and avoids the pathologies presented in other parity-violating gravity models. The cosmological dynamics and perturbations of the Nieh-Yan modified teleparallel gravity have been investigated in detail, but all these previous investigations rely on the symmetry requirement that in the background universe both the metric and affine connection are homogeneous and isotropic. In this paper we relax the symmetry constraint on the connection and leave it arbitrary at the beginning, after all the cosmological principle only needs the metric of the background spacetime to meet the symmetry requirement. We find a new flat universe solution for the Nieh-Yan modified teleparallel gravity, for which the background dynamics itself is unchanged but the perturbations around it present a new feature that the scalar and tensor perturbations are coupled together at the linear level. The implications of this peculiar feature in primordial perturbations from inflation are also discussed.

Deforming soft algebras for gauge theory

Symmetry algebras deriving from towers of soft theorems can be deformed by a short list of higher-dimension Wilsonian corrections to the effective action. We study the simplest of these deformations in gauge theory arising from a massless complex scalar coupled to F2. The soft gauge symmetry ‘s-algebra’, compactly realized as a higher-spin current algebra acting on the celestial sphere, is deformed and enlarged to an associative algebra containing soft scalar generators. This deformed soft algebra is found to be non-abelian even in abelian gauge theory. A two-parameter family of central extensions of the s-subalgebra are generated by shifting and decoupling the scalar generators. It is shown that these central extensions can also be generated by expanding around a certain non-trivial but Lorentz invariant shockwave type background for the scalar field.

The propagator matrix reloaded

The standard way to perform calculations for quantum field theories involves the S-matrix and the assumption that the theory is free at past and future infinity. However, this assumption may not hold for field theories in non-trivial backgrounds such as curved spacetimes or finite temperature. In fact, even in the simple case of finite temperature Minkowski spacetime, there are a lot of misconceptions and confusion in the literature surrounding how to correctly take interactions into account when setting up the initial conditions. The objective of this work is to clear up these misconceptions and provide a clean and simple derivation of a formalism which includes interactions in the initial conditions and assesses whether or not it is legitimate to ignore them. The ultimate conclusion is that we cannot ignore them: quantum field theories at finite temperature are not free in the infinite past.

Amplitude’s positivity vs. subluminality: causality and unitarity constraints on dimension 6 & 8 gluonic operators in the SMEFT

We derive the causality and unitarity constraints on dimension 6 and dimension 8 Gluon field strength operators in the Standard Model Effective Field Theory (SMEFT). In the first part of the paper, we use the ‘amplitude analysis’ i.e. dispersion relation for 2 → 2 scattering in the forward limit, to put bounds on the Wilson coefficients. We show that the dimension 6 operators can exist only in the presence of certain dimension 8 operators. It is interesting that the square of the dimension 6 Wilson coefficients can be constrained in this case even at the tree level. In the second part of this work, we successfully rederive all these bounds using the classical causality argument that demands that the speed of fluctuations about any non-trivial background should not exceed the speed of light. We also point out some subtleties in the superluminality analysis regarding whether the low-frequency phase velocity can always be used as the relevant quantity for Causality violation: as an example, we show that, due to these subtleties, if a small pion mass is added in the chiral Lagrangian, it is unclear if any strict positivity bound can be derived on the dimension 8 Wilson coefficient. Finally, we mention an interesting non-relativistic example where the subluminality requirement produces a stronger bound than the ‘amplitude analysis’.

Bilinearization-reduction approach to integrable systems

The paper is a review of the bilinearization-reduction method which provides an approach to obtain solutions to integrable systems. Many integrable coupled systems can be bilinearized and their solutions are presented in terms of double Wronskians (or double Casoratians in discrete case). The bilinearization-reduction method is based on bilinear equations and solutions in double Wronskian/Casoratian form. For those integrable equations that are reduced from coupled systems, one can first solve the unreduced coupled system, obtaining their solutions in double Wronskian/Casoratian form, then, implement suitable reduction techniques, so that solutions of the reduced equation can be obtained as reductions of those of the unreduced coupled system. The method proves effective in solving not only classical integrable equations but also the nonlocal ones. The so-called nonlocal integrable equations were introduced by Ablowitz and Musslimani via reductions with reverse-space (or reverse-time, or reverse-space-time). Note that this method particularly provides a convenient bilinear approach to solve nonlocal integrable systems. In this review, the nonlinear Schrödinger hierarchy and the differential-difference nonlinear Schrödinger equation are employed as demonstrative examples to elaborate this method. These two examples will be pedagogically helpful in understanding the reduction technique. The reduction is implemented by imposing suitable constraints on the basic column vectors of the double Wronskian/Casoratian. Realizations of the constraints are converted to solve a set of matrix equations which varies with the constraints. Special solutions of the matrix equations are provided, which are also helpful in understanding the eigenvalue structure of the involved spectral problems corresponding to the considered equations. Other examples include the Fokas-Lenells equation and the nonlinear Schrödinger equation with nontrivial background. Since many nonlinear equations with physical significance are integrable as reductions of integrable coupled systems, the paper provides a review as well as an introduction about the bilinearization-reduction method that can be used to solve these nonlinear integrable models.

Non-Abelian Gauge Theories with Composite Fields in the Background Field Method

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green’s functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green’s functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion.

Celestial Liouville theory for Yang-Mills amplitudes

We consider Yang-Mills theory with the coupling constant and theta angle determined by the vacuum expectation values of a dynamical (complex, zero mass) dilaton field. We discuss the tree-level N-gluon MHV scattering amplitudes in the presence of a nontrivial background dilaton field and construct the corresponding celestial amplitudes by taking Mellin transforms with respect to the light cone frame energies. In this way, we obtain two-dimensional CFT correlators of primary fields on the celestial sphere. We show that the celestial Yang-Mills amplitudes evaluated in the presence of a spherical dilaton shockwave are given by the correlation functions of primary field operators factorized into the holomorphic current operators times the “light” Liouville operators. They are evaluated in the semiclassical limit of Liouville theory (the limit of infinite central charge) and are determined by the classical Liouville field describing metrics on the celestial sphere.

One small step for an inflaton, one giant leap for inflation: A novel non-Gaussian tail and primordial black holes

We report a novel prediction from single-field inflation that even a tiny step in the inflaton potential can change our perception of primordial non-Gaussianities of the curvature perturbation. Our analysis focuses on the tail of probability distribution generated by an upward step transition between two stages of slow-roll evolution. The nontrivial background dynamics with off-attractor behavior is identified. By using a non-perturbative δN analysis, we explicitly show that a highly non-Gaussian tail can be generated by a tiny upward step, even when the conventional nonlinearity parameters fNLlocal, gNLlocal, etc. remain small. With this example, we demonstrate for the first time the sensitive dependence of non-perturbative effects on the tail of probability distribution. Our scenario has an inconceivable application to primordial black holes by either significantly boosting their abundance or completely forbidding their appearance.