Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in  $$q^{\pm 2}$$  <math> <msup> <mi>q</mi> <mrow> <mo>±</mo> <mn>2</mn> </mrow> </msup> </math>   of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

Russia

E. Lanina

A. Morozov

Defect and degree of the Alexander polynomial

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> Chern-Simons Wilson loops, while the limit is <math><mi>N</mi><mo>→</mo><mn>0</mn></math>. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ-functions. This result is an extension of what we did in [1], where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton τ-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton τ-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants.

V. Mishnyakov

A. Sleptsov

Perturbative analysis of the colored Alexander polynomial and KP soliton -functions

We estimate the chirality of the cosmological plasma due to parity violating decays of Standard Model particles, focusing on the example of tau leptons. The nontrivial chirality is however too small to make a significant contribution to the cosmological magnetic field via the chiral-magnetic effect.

United States

Tanmay Vachaspati

Alexander Vilenkin

Cosmological chirality and magnetic fields from parity violating particle decays

We describe predictions for top quark pair differential distributions at hadron colliders, by combining the next-to-next-to-leading order quantum chromodynamics calculations and next-to-leading order electroweak corrections with double resummation at the next-to-next-to-leading logarithmic accuracy of threshold logarithms and small-mass logarithms. To the best of our knowledge, this is the first study to present such a combination, which incorporates all known perturbative information. Numerical results are presented for the invariant-mass distribution, transverse-momentum distribution, and rapidity distributions.

Germany

Michał Czakon

Andrea Ferroglia

United Kingdom

Alexander Mitov

Davide Pagani

Andrew S. Papanastasiou

Benjamin D. Pecjak

Netherlands

Darren J. Scott

Sweden

Ioannis Tsinikos

Xing Wang

China

Li Lin Yang

Italy

Marco Zaro

Top quark pair production at complete NLO accuracy with NNLO+NNLL′ corrections in QCD L. L. Yang and X. Wang are supported in part by the National Natural Science Foundation of China (11975030, 11635001, 11575004). X. Wang is also supported in part by the Cluster of Excellence PRISMA+ (project ID 39083149). D. J. S. is supported under the ERC grant ERC-STG2015-677323. The work of D. P. and I. T. is supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja Award Project "Event Simulation for the Large Hadron Collider at High Precision". D. P. is also supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy – EXC 2121"Quantum Univers" – 390833306. I. T. is also supported by the Swedish Research Council under contract number 2016-05996. The research of A. M. and A. P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (683211) as well as from UK STFC grants ST/L002760/1 and ST/K004883/1. The work of M. C. was supported in part by a grant of the BMBF and by the Deutsche Forschungs-gemeinschaft under grant 396021762 - TRR 257
						

The gravitational production of superheavy dark matter, in the Peebles-Vilenkin quintessential inflation model, is studied in two different scenarios: When the particles, whose decay products reheat the universe after the end of the inflationary period, are created gravitationally, and when are produced via instant preheating. We show that the viability of both scenarios requires that the mass of the superheavy dark matter to be approximately between $$10^{16}$$ <math><msup><mn>10</mn><mn>16</mn></msup></math>  and $$10^{17}~\hbox {GeV}$$ <math><mrow><msup><mn>10</mn><mn>17</mn></msup><mspace width="3.33333pt"></mspace><mtext>GeV</mtext></mrow></math> .

Spain

Jaume Haro

Gravitational production of dark matter in the Peebles–Vilenkin model

Predictions in an eternally inflating multiverse are meaningless unless we specify the probability measure. The scale-factor cutoff is perhaps the simplest and most successful measure which avoids catastrophic problems such as the youngness paradox, runaway problem, and Boltzmann brain problem, but it is not well defined in contracting regions with a negative cosmological constant. In this paper, we propose a new measure with properties similar to the scale-factor cutoff which is well-defined everywhere. The measure is defined by a cutoff in the four volume spanned by infinitesimal comoving neighborhoods in a congruence of timelike geodesics. The probability distributions for the cosmological constant and for the curvature parameter in this measure are similar to those for the scale-factor cutoff and are in a good agreement with observations.

Masaki Yamada

Four-volume cutoff measure of the multiverse

We review the well-known Peebles–Vilenkin (PV) quintessential inflation model and discuss its possible improvements in agreement with the recent observations. The improved PV model depends only on two parameters: the inflaton mass m, and another smaller mass M; where the latter has to be chosen in order to undertake that, at present time, the dark energy density of the universe is approximately about 70% of the total energy budget of the universe. The value of the inflaton mass m is calculated using the observational value of the power spectrum of the scalar perturbations, and the value of mass M, which depends on the reheating temperature, is calculated by solving the corresponding dynamical system whose initial conditions are taken at the matter-radiation equality and are obtained from three observational data: the red shift at the matter-radiation equality, the ratio of the matter energy density to the critical one at the present time and the current value of the Hubble parameter.

Jaume Amorós

India

Supriya Pan

The Peebles–Vilenkin quintessential inflation model revisited

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: <math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>R</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mo>[</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>|</mo><mi>R</mi><mo>|</mo></mrow></msup><mo>)</mo></math> for all 1-hook Young diagrams R. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial.

A. Mironov

S. Mironov

Coloured Alexander polynomials and KP hierarchy

We connect two important conjectures in the theory of knot polynomials. The first one is the property $$Al_R(q) = Al_{[1]}(q^{|R|})$$ <math><mrow><mi>A</mi><msub><mi>l</mi><mi>R</mi></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>=</mo><mi>A</mi><msub><mi>l</mi><mrow><mo>[</mo><mn>1</mn><mo>]</mo></mrow></msub><mrow><mo>(</mo><msup><mi>q</mi><mrow><mo>|</mo><mi>R</mi><mo>|</mo></mrow></msup><mo>)</mo></mrow></mrow></math>  for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices $$U_{i}$$ <math><msub><mi>U</mi><mi>i</mi></msub></math>  in the relation $$\mathcal{R}_i = U_i\mathcal{R}_1U_i^{-1}$$ <math><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>=</mo><msub><mi>U</mi><mi>i</mi></msub><msub><mi>R</mi><mn>1</mn></msub><msubsup><mi>U</mi><mi>i</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup></mrow></math>  between the ith and the first generators $$\mathcal{R}_i$$ <math><msub><mi>R</mi><mi>i</mi></msub></math>  of the braid group are universally expressible through the eigenvalues of $$\mathcal{R}_1$$ <math><msub><mi>R</mi><mn>1</mn></msub></math> . Since the above property of Alexander polynomials is very well tested, this relation provides new support to the eigenvalue conjecture, especially for $$i&gt;2$$ <math><mrow><mi>i</mi><mo>&gt;</mo><mn>2</mn></mrow></math> , when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.

Eigenvalue conjecture and colored Alexander polynomials

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

Hessian eigenvalue distribution in a random Gaussian landscape