Asymptotic Bethe Ansatz from String Sigma Model on S^3 x R

We derive the asymptotic Bethe ansatz (AFS equations) for the string on S^3 x R sector of AdS_5 x S^5 from the integrable nonhomogeneous dynamical spin chain for the string sigma model proposed in GKSV. It is clear from the derivation that AFS equations can be viewed only as an effective model describing a certain regime of a more fundamental inhomogeneous spin chain.Comment: 16 page

QCD Pomeron from AdS/CFT Quantum Spectral Curve

Using the methods of the recently proposed Quantum Spectral Curve (QSC) originating from integrability of ${\cal N}=4$ Super--Yang-Mills theory we analytically continue the scaling dimensions of twist-2 operators and reproduce the so-called pomeron eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation. Furthermore, we recovered the Faddeev-Korchemsky Baxter equation for Lipatov's spin chain and also found its generalization for the next-to-leading order in the BFKL scaling. Our results provide a non-trivial test of QSC describing the exact spectrum in planar ${\cal N}=4$ SYM at infinitely many loops for a highly nontrivial non-BPS quantity and also opens a way for a systematic expansion in the BFKL regime.Comment: 22 pages, 2 figures, minor corrections, references adde

Quantum spectral curve for arbitrary state/operator in AdS5_5/CFT4_4

We give a derivation of quantum spectral curve (QSC) - a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed in our recent paper Phys.Rev.Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -- a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.Comment: 96 pages, 15 figures; Some mathematica examples added in v2; Published version is v

Wronskian Solution for AdS/CFT Y-system

Using the discrete Hirota integrability we find the general solution of the full quantum Y-system for the spectrum of anomalous dimensions of operators in the planar AdS5/CFT4 correspondence in terms of Wronskian-like determinants parameterized by a finite number of Baxter's Q-functions. We consider it as a useful step towards the construction of a finite system of non-linear integral equations (FiNLIE) for the full spectrum. The explicit asymptotic form of all the Q-functions for the large size operators is presented. We establish the symmetries and the analyticity properties of the asymptotic Q-functions and discuss their possible generalization to any finite size operators.Comment: 31 pages, 4 figures, 1 attached mathematica fil

The universal order one invariant of framed knots in most S^1-bundles over orientable surfaces

It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant I of framed knots in orientable total spaces of S^1-bundles over an orientable not necessarily compact surface F not S^2. We show that if F is not S^2 or S^1 X S^1, then I is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-3.abs.htm

Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers

In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operatorsComment: LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two Figures; v2: minor improvements and corrections; v3: a section on raising/lowering operators is added; v4: typos are fixed; v5: Introduction is added, open problems are expanded.v6: Russian translation is added, references areupdated, NoWeb and C++ source codes are added as ancillary files. arXiv admin note: substantial text overlap with arXiv:0707.402

N=4 SYM Quantum Spectral Curve in BFKL regime

We review the applications of the Quantum Spectral Curve (QSC) method to the Regge (BFKL) limit in N=4 supersymmetric Yang-Mills theory. QSC, based on quantum integrability of the AdS$_5$/CFT$_4$ duality, was initially developed as a tool for the study of the spectrum of anomalous dimensions of local operators in the N=4 SYM in the planar, $N_c\to\infty$ limit. We explain how to apply the QSC for the BFKL limit, which requires non-trivial analytic continuation in spin $S$ and extends the initial construction to non-local light-ray operators. We give a brief review of high precision non-perturbative numerical solutions and analytic perturbative data resulting from this approach. We also describe as a simple example of the QSC construction the leading order in the BFKL limit. We show that the QSC substantially simplifies in this limit and reduces to the Faddeev-Korchemsky Baxter equation for Q-functions. Finally, we review recent results for the Fishnet CFT, which carries a number of similarities with the Lipatov's integrable spin chain for interacting reggeized gluons.Comment: Contribution to the memorial volume "From the past to the future - the legacy of Lev Lipatov", 28 pages, 7 figures; v2: references added, typos fixe

Assessing the Complementarity of Preferential Tax Regimes in the Sakhalin Region

State regulation of the anthropogenic impact on nature, including the policy of limiting greenhouse gas emissions, makes it necessary to support those enterprises that are willing to invest in climate projects. The relevance of incentives in this field is confirmed by the task set by the Arctic Council chaired by Russia at the Research and Training Conference on Climate Change and Permafrost Thawing 2023, namely to find reasonable, practical solutions for adapting the global economy to such changes. In the area of taxation, there is a solution prepared by the Russian Ministry of Economic Development willing to provide a package of tax benefits for investors. According to the draft, they should be exempted from paying VAT and income tax. At the same time, the variety of previously introduced preferential ‘investment’ tax regimes calls into question the rationality of new tax incentives due to the assumption of their redundancy. The fact is that they have similarities in the basic idea of support aimed at the growth of regional investment activity. For this reason, the purpose of the article is to answer the following questions: Is it really worthwhile to expand the number of tax regimes once again? Is it possible to use existing incentives instead of introducing new ones? To achieve the purpose, we rely on the results of comprehensive, comparative analysis of the developed incentives and the current tax preferential regimes on the example of the Sakhalin region. It stands out among other regions of Russia by its carbon-neutral pilot project and the greater number of preferences already enjoyed by investors, so we can provide representative and practically significant results. The hypothesis of the study is based on the concept of complementarity, meaning that the newly developed tax incentives will contribute to the complexity of the Russian tax system but will not cross the scope of the other mentioned tax regimes. The results show that climate projects have their own specifics in terms of objectives as well as areas of investment that require special approaches in tax policy, since investors create a useful economic effect in the form of reducing greenhouse gas emissions rather than producing goods, and no taxable profit is generated until the investor profitably sells carbon units in the market

Оценка бюджетных последствий введения необлагаемого минимума в налогообложение личных доходов в России

The social problem of poverty can be mitigated by introduction of a personal tax-free allowance. In this paper the likely effects that a personal tax-free allowance will have on the Russian budget is investigated. It has been assumed that a tax-free allowance will hit regional budgets because they depend greatly on income tax revenue. The indicated effects were estimated by applying a personal tax-free allowance to the data on economic conditions in 2019. Rosstat data on population, poverty, wages and gross regional product and Federal Tax Service data on the number of taxpayers and personal income tax revenues were used. For the purpose of the paper, two scenarios were calculated. In the first scenario, a zero personal income tax rate is applied to wages below the minimum cost of living. We found that under this scenario the consolidated budget of Russia loses over 1 trillion rubles while regional tax revenues reduce by more than 10%. In the second scenario, citizens whose income is below the minimum cost of living are exempt from personal income tax. We found that under this scenario regional tax revenues would be reduced by 1-5%. In both cases the introduction of the personal tax-free allowance puts greater pressure on regions that critically depend on the personal income tax receipts. It was concluded that the negative effect of an introduction of a personal tax-free allowance would be greater, the greater the prevalence of low-income taxpayers in a region. Also considerable regional disparities create a risk that such tax reform will deepen regional inequality and be disruptive for the Russian budgetary system. For citationGromov V.V. Introduction of the personal tax-free allowance in Russia and its budget implications. Journal of Tax Reform. 2021;7(1):20–38. DOI: 10.15826/jtr.2021.7.1.088 Article infoReceived October 7, 2020; Revised January 10, 2021; Accepted January 29, 2021Социальную проблему бедности может смягчить введение необлагаемого подоходным налогом минимума доходов граждан. В статье исследуются вероятные бюджетные потери от введения в России необлагаемого подоходным налогом минимума доходов граждан. Выдвигается гипотеза, что введение необлагаемого минимума негативно скажется на регионах, поскольку зависимость их бюджетов от поступления подоходного налога велика. Расчет последствий введения необлагаемого минимума проводился для экономических условий 2019 г. Источником данных являлись данные Росстата о населении, бедности, заработной плате и валовом региональном продукте, а также данные Федеральной налоговой службы о количестве налогоплательщиков и налоговых поступлениях. Для целей исследования были рассчитаны два сценария. По первому сценарию к заработной плате ниже прожиточного минимума применяется нулевая ставка налога на доходы физических лиц. Мы обнаружили, что при этом сценарии консолидированный бюджет России теряет более 1 трлн р., в то время как налоговые поступления в региональные бюджеты сокращаются более чем на 10%. По второму сценарию не облагаются налогом на доходы физических лиц граждане, чей доход ниже прожиточного минимума. Данные, полученные по второму сценарию, показывают, что региональные налоговые поступления сократятся на 1–5%. В любом случае, введение необлагаемого налогом минимума оказывает более сильное давление на те регионы, которые критически зависят от поступлений подоходного налога с физических лиц. Можно сделать вывод, что введение льготы при преобладании налогоплательщиков с низкими доходами приведет к существенным потерям для региональных бюджетов. Сделан вывод, что негативное влияние введения необлагаемого минимума будет тем больше, чем больше в регионе преобладают малообеспеченные налогоплательщики. Кроме того, значительные региональные диспропорции создают риск того, что такая налоговая реформа усугубит региональное неравенство и подорвет бюджетную систему России.Для цитированияGromov V.V. Introduction of the personal tax-free allowance in Russia and its budget implications. Journal of Tax Reform. 2021;7(1):20–384. DOI: 10.15826/jtr.2021.7.1.088 Информация о статьеДата поступления 7 октября 2020 г.; дата поступления после рецензирования 10 января 2021 г.; дата принятия к печати 29 января 2021 г

Compatibility of Gauss maps with metrics

We give necessary and sufficient conditions on a smooth local map of a Riemannian manifold $M^m$ into the sphere $S^m$ to be the Gauss map of an isometric immersion $u:M^m \to R^n$, $n=m+1$. We briefly discuss the case of general $n$ as wellComment: 14 pages, no figure