Non-trivial Term Research Articles

The giant graviton expansion in $AdS_5 \times S^5$

The superconformal index of \frac1212-BPS states of \mathcal{N}=4𝒩=4U(N)U(N) super Yang-Mills theory has a known infinite qq-series expression with successive terms suppressed by q^NqN. We derive a holographic bulk interpretation of this series by evaluating the corresponding functional integral in the dual AdS_5 × S^55×S5. The integral localizes to a product of small fluctuations of the vacuum and of the collective modes of an arbitrary number of giant-gravitons wrapping an S^3S3 of maximal size inside the S^5S5. The quantum mechanics of the small fluctuations of one maximal giant is described by a supersymmetric version of the Landau problem. The quadratic fluctuation determinant reduces to a sum over the supersymmetric ground states, and precisely reproduces the first non-trivial term in the infinite series. Further, we show that the terms corresponding to multiple giants are obtained precisely by the matrix versions of the above super-quantum-mechanics.

Central limit theorem of overlap for the mean field Ghatak–Sherrington model

The Ghatak–Sherrington spin glass model is a random probability measure defined on the configuration space {0,±1,±2,…,±S}N with system size N and S⩾1 finite. This generalizes the classical Sherrington–Kirkpatrick (SK) model on the boolean cube {−1, +1}N to capture more complex behaviors, including the spontaneous inverse freezing phenomenon. We give a quantitative joint central limit theorem for the overlap and self-overlap array at sufficiently high temperature under arbitrary crystal and external fields. Our proof uses the moment method combined with the cavity approach. Compared to the SK model, the main challenge comes from the non-trivial self-overlap terms that correlate with the standard overlap terms.

Cancellation of quantum corrections on the soft curvature perturbations

We study the cancellation of quantum corrections on the superhorizon curvature perturbations from subhorizon physics beyond the single-clock inflation from the viewpoint of the cosmological soft theorem. As an example, we focus on the transient ultra-slow-roll inflation scenario and compute the one-loop quantum corrections to the power spectrum of curvature perturbations taking into account nontrivial surface terms in the action. We find that Maldacena’s consistency relation is satisfied and guarantees the cancellation of contributions from the short-scale modes. As a corollary, primordial black hole production in single-field inflation scenarios is not excluded by perturbativity breakdown even for the sharp transition case in contrast to some recent claims in the literature. We also comment on the relation between the tadpole diagram in the in-in formalism and the shift of the elapsed time in the stochastic-δN formalism. We find our argument is not directly generalisable to the tensor perturbations.

Geodetic precession and shadow of quantum extended black holes

We study the circular motion of massive and massless particles in a recently proposed quantum-corrected Schwarzschild black hole in loop quantum gravity. This solution is supposed to introduce small but non-zero quantum corrections in the low curvature limit. In this paper, we confine our attention to the shadow of the black hole and the geodetic precession (GP) of a freely falling gyroscope in a circular orbit. Despite the mathematical complexity of the metric, our results are exact and show that the black hole shadow decreases slightly in this solution while the quantum corrections introduce a non-trivial term in the GP frequency of the gyroscope.

A stochastic maximum principle for partially observed general mean-field control problems with only weak solution

In this paper we focus on a general type of mean-field stochastic control problem with partial observation, in which the coefficients depend in a non-linear way not only on the state process Xt and its control ut but also on the conditional law E[Xt|FtY] of the state process conditioned with respect to the past of observation process Y. We first deduce the well-posedness of the controlled system by showing weak existence and uniqueness in law. Neither supposing convexity of the control state space nor differentiability of the coefficients with respect to the control variable, we study Peng’s stochastic maximum principle for our control problem. The novelty and the difficulty of our work stem from the fact that, given an admissible control u, the solution of the associated control problem is only a weak one. This has as consequence that also the probability measure in the solution Pu=LTuQ depends on u and has a density LTu with respect to a reference measure Q. So characterizing an optimal control leads to the differentiation of non-linear functions f(Pu∘{EPu[Xt|FtY]}−1) with respect to (LTu,Xt). This has as consequence for the study of Peng’s maximum principle that we get a new type of first and second order variational equations and adjoint backward stochastic differential equations, all with new mean-field terms and with coefficients which are not Lipschitz. For their estimates and for those for the Taylor expansion new techniques have had to be introduced and rather technical results have had to be established. The necessary optimality condition we get extends Peng’s one with new, non-trivial terms.

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.

Kinetic axion dark matter in string corrected f(R) gravity

Under the main assumption that the axion scalar field mainly composes the dark matter in the Universe, in this paper we shall extend the formalism of kinetic axion $R^2$ gravity to include Gauss-Bonnet terms non-minimally coupled to the axion field. As we demonstrate, this non-trivial Gauss-Bonnet term has dramatic effects on the inflationary phenomenology and on the kinetic axion scenario. Specifically, in the context of our formalism, the kinetic axion ceases to be kinetically dominated at the end of the inflationary era, since the condition $\dot{\phi}\simeq 0$ naturally emerges in the theory. Thus, unlike the case of kinetic axion $R^2$ gravity, the Gauss-Bonnet corrected kinetic axion $R^2$ gravity leads to an inflationary era which is not further extended and the reheating era commences right after the inflationary era, driven by the $R^2$ fluctuations.

Wald-Zoupas prescription with soft anomalies

We show that the Wald-Zoupas prescription for gravitational charges is valid in the presence of anomalies and field-dependent diffeomorphism, but only if these are related to one another in a specific way. The geometric interpretation of the allowed anomalies is exposed looking at the example of BMS symmetries: They correspond to soft terms in the charges. We determine if the Wald-Zoupas prescription coincides with an improved Noether charge. The necessary condition is a certain differential equation, and when it is satisfied, the boundary Lagrangian of the resulting improved Noether charge contains in general a non-trivial corner term that can be identified a priori from a condition of anomaly-freeness. Our results explain why the Wald-Zoupas prescription works in spite of the anomalous behaviour of BMS transformations, and should be helpful to relate different branches of the literature on surface charges.

Macroscopic Behaviour in a Two-Species Exclusion Process Via the Method of Matched Asymptotics

We consider a two-species simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a cross-diffusion system of partial differential equations for the two species’ densities. First, our result captures non-trivial interaction terms neglected in the mean-field approach, including a non-diagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel (Commun Pure Appl Math 45(6):623–679, 1992), valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, to the case of unequal rates in the dilute regime. In the equal-rates case, by combining matched asymptotic approximations in the low- and high-density limits, we obtain a cubic polynomial approximation of the self-diffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the second-order in the density of the self-diffusion coefficient obtained using a rigorous recursive method.

Asymptotic Evaluation of a Lattice Sum Associated with the Laplacian Matrix

The Laplacian matrix is of fundamental importance in the study of graphs, networks, random walks on lattices, and arithmetic of curves. In certain cases, the trace of its pseudoinverse appears as the only non-trivial term in computing some of the intrinsic graph invariants. Here we study a double sum Fn which is associated with the trace of the pseudo inverse of the Laplacian matrix for certain graphs. We investigate the asymptotic behavior of this sum as n → ∞. Our approach is based on classical analysis combined with asymptotic and numerical analysis, and utilizes special functions. We determine the leading-order term, which is of size n2 log n, and develop general methods to obtain the secondary main terms in the asymptotic expansion of Fn up to errors of $$\mathcal {O}({\rm{log}} \; n)$$ and $$\mathcal {O}(1)$$ as n → ∞. We provide some examples to demonstrate our methods.

Constructing chiral effective field theory around unnatural leading-order interactions

A momentum-dependent formulation based on a stationary spin-0 and isospin-1 dibaryon field is proposed to improve convergence of chiral effective field theory in the ${}^{1}{S}_{0}$ channel of $NN$ scattering. Although the two-parameter leading-order interaction appears to be unnatural, it nevertheless has the necessary features of an effective field theory. A rapid order-by-order convergence is found in ${}^{1}{S}_{0}$. As an application beyond the two-body level, the triton binding energy is studied and compared to standard chiral effective field theory with partly perturbative pions. The consistency of the chiral Lagrangian for the new formulation is examined by working out the pionic radiative corrections, and consequences of nontrivial chiral-connection terms are discussed.

Geometric structure of multi-form-field isotropic inflation and primordial fluctuations

An inflationary scenario is expected to be embedded into an ultraviolet (UV) complete theory such as string theory. The effect of UV complete theories may appear as nontrivial kinetic terms in the low energy effective field theory, which provides a nontrivial geometry in field space. In this paper, we study the effect of the geometry of multi-form-field space on an inflationary scenario. In particular, we focus on the geometric destabilization mechanism which induces the phase transition from the conventional slow-roll inflation to a novel inflationary scenario. Anisotropic inflation is a typical example of the new phase. To conform to observations, we restrict us to isotropic configuration of form fields. We clarify the conditions for the onset of the destabilization and reveal the geometric structure of attractors after the destabilization. We classify the viable models from the observational point of view. We also investigate the features of the primordial fluctuations and find the similarity to hyperbolic inflation. By calculating the power spectrum, we make several phenomenological predictions which are useful to discriminate our models from others inflation models. We found the scalar-to-tensor r will be suppressed by large one-form gauge fields, while it has the same order as the slow roll parameter r ∼ \U0001d4aa(1)ϵ for large two-from gauge fields.

Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation.

Simulation and experimental studies have demonstrated non-equilibrium ordering in driven colloidal suspensions: with increasing driving force, a uniform colloidal mixture transforms into a locally demixed state characterized by the lane formation or the emergence of strongly anisotropic stripe-like domains. Theoretically, we have found that a linear stability analysis of density dynamics can explain the non-equilibrium ordering by adding a non-trivial advection term. This advection arises from fluctuating flows due to non-Coulombic interactions associated with oppositely driven migrations. Recent studies based on the dynamical density functional theory (DFT) without multiplicative noise have introduced the flow kernel for providing a general description of the fluctuating velocity. Here, we assess and extend the above deterministic DFT by treating electric-field-driven binary ionic mixtures as the primitive model. First, we develop the stochastic DFT with multiplicative noise for the laning phenomena. The stochastic DFT considering the fluctuating flows allows us to determine correlation functions in a steady state. In particular, asymptotic analysis on the stationary charge-charge correlation function reveals that the above dispersion relation for linear stability analysis is equivalent to the pole equation for determining the oscillatory wavelength of charge–charge correlations. Next, the appearance of stripe-like domains is demonstrated not only by using the pole equation but also by performing the 2D inverse Fourier transform of the charge–charge correlation function without the premise of anisotropic homogeneity in the electric field direction.

Asymptotics and zeta functions on compact nilmanifolds

In this paper, we obtain asymptotic formulae on nilmanifolds Γ﹨G, where G is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup Γ. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group G (and more generally any positive Rockland operators when G is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions.

The massive supermembrane on a knot

We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice punctured torus times a 9D Minkowski space-time. It corresponds to a M2-brane formulated in 11D space with ten non-compact dimensions. The critical points like the poles and the zeros of the fields describing the embedding of the Supermembrane in the target space are treated rigorously. The non-trivial compactification generates non-trivial mass terms appearing in the bosonic potential, which dominate the full supersymmetric potential and should render the spectrum of the (regularized) Supermembrane discrete with finite multiplicity. The behaviour of the fields around the punctures generates a cosmological term in the Hamiltonian of the theory.The massive supermembrane can also be seen as a nontrivial uplift of a supermembrane torus bundle with parabolic monodromy in M9 × T2. The moduli of the theory is the one associated with the punctured torus, hence it keeps all the nontriviality of the torus moduli even after the decompactification process to ten noncompact dimensions. The formulation of the theory on a punctured torus bundle is characterized by the (1, 1) − knots associated with the monodromies.

Dissipation in holographic superfluids

We study dissipation in holographic superfluids at finite temperature and zero chemical potential. The zero overlap with the heat current allows us to isolate the physics of the conserved current corresponding to the broken global U(1). By using analytic techniques we write constitutive relations including the first non-trivial dissipative terms. The corresponding transport coefficients are determined in terms of thermodynamic quantities and the black hole horizon data. By analysing their behaviour close to the phase transition we show explicitly the breakdown of the hydrodynamic expansion. Finally, we study the pseudo-Goldstone mode that emerges upon introducing a perturbative symmetry breaking source and we determine its resonant frequency and decay rate.

TT¯/JT¯ -deformed WZW models from Chern-Simons AdS3 gravity with mixed boundary conditions

In this work we consider AdS$_3$ gravitational theory with certain mixed boundary conditions at spatial infinity. Using the Chern-Simons formalism of AdS$_3$ gravity, we find that these boundary conditions lead to non-trivial boundary terms, which, in turn, produce exactly the spectrum of the $T\bar{T}/J\bar{T}$-deformed CFTs. We then follow the procedure for constructing asymptotic boundary dynamics of AdS$_3$ to derive the constrained $T\bar{T}$-deformed WZW model from Chern-Simons gravity. The resulting theory turns out to be the $T\bar{T}$-deformed Alekseev-Shatashvili action after disentangling the constraints. Furthermore, by adding a $U(1)$ gauge field associated to the current $J$, we obtain one type of the $J\bar T$-deformed WZW model, and show that its action can be constructed from the gravity side. These results provide a check on the correspondence between the $T\bar{T}/J\bar{T}$-deformed CFTs and the deformations of boundary conditions of AdS$_3$, the latter of which may be regarded as coordinate transformations.

Exact relations for two-photon-exchange effect in elastic ep scattering by dispersion relation and hadronic model * *Supported in part by the National Natural Science Foundation of China (11375044,11975075)

The two-photon-exchange (TPE) effect plays a key role in extracting the form factors (FFs) of the proton. In this work, we discuss several exact properties of the TPE effect in the elastic scattering. By taking four low energy interactions as examples, we analyze kinematical singularities, asymptotic behaviors, and branch cuts of TPE amplitudes. The analytical expressions clearly indicate several exact relations between the dispersion relation (DR) and hadronic model (HM) methods. This suggests that the two methods must be modified to general forms, while novel forms yield the same results. After the modification, new DRs include a non-trivial term with two singularities. Furthermore, new DRs automatically include contributions due to the seagull interaction, meson-exchange effect, contact interactions, and off-shell effect. To analyze the elastic scattering data sets, the new forms must be used.

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity * *Supported by the Polish NCN OPUS Grant 2017/25/B/ST1/00931

In Stróżyna E and Żołądek H (2015 Moscow Math. J. 15 141) a complete formal normal forms for germs of two-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

MATHEMATICAL MODELING AND NUMERICAL ANALYSES OF TRANSPORT PHENOMENA WITH VARIABLE CROSS-DIFFUSION AND NONLINEAR RADIATION

This study proposes a mathematical model and the numerical scheme, with rigorous stability and convergence analyses, for a channel flow problem incorporating a nonstandard variable cross-diffusion (Soret-Dufour effects), an unconventional nonlinear radiative heat flux, and time-dependent wall velocity and wall temperature. This led to a system of highly coupled nonlinear partial differential equations. Backward difference scheme is considered for time derivatives, while conservative-type central scheme is adopted for spatial derivatives. The nonlinear terms are discretized by freezing coefficients in discrete time. This resulted in an implicit-explicit algorithm for each nondimensional equation. To investigate the stability, we adopt the discrete maximum principle and derive two-sided estimates for the numerical solution. To establish convergence, we consider a general scheme that encompasses all three schemes, and prove convergence in the grid L2 norm. A problem with nontrivial source terms is used to confirm the theoretical results. Finally, the flow is investigated and the results showed that the nonlinear radiation parameter increased the fluid temperature.