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Research area 02 - PHYSICS AND SPACE SCIENCES, 02-602 - Quantum field theory, quantum mechanics

KeywordsSigma model, integrable system, self-duality equation, Skyrmion, TTbar-deformation, higher spin theory, supersymmetry, spin chain, Gross-Neveu model

Annotation
One of the key unsolved problems in theoretical physics is the construction “from first principles” of quantum field theories beyond perturbation theory. Related to it is one of the “Millennium problems” that has to do with describing the mass generation mechanism in Yang-Mills theory. A simpler problem, though qualitatively very similar one, is the construction of the theory of 2D sigma models. An additional virtue of these theories is their potential integrability that gives hope of finding an exact solution (i.e. calculating the spectrum, computing correlation functions etc.). In recent years the project leader has found a conceptually new formulation of such models based on a proven equivalence with generalized Gross-Neveu models. This formulation has a number of substantial benefits as compared to the standard, geometric, one. In particular, in the new variables the interactions become polynomial, dependence on the gauge fields turns out to be topological, the known quantum anomalies in the “higher charges” are consequences of simple chiral anomalies, worldsheet supersymmetry turns out to be related to target space supersymmetry etc. This formulation opens up a whole program of research, whose goal is to construct an exact solution for models of this class, which includes, for instance, the well-known CP^{n-1} model. Integrability plays an essential role in string theory, mathematical physics, and condensed matter theory. A considerable number of results occur at the intersection of these fields due to the study of new methods of holography and integrable deformations. Much attention is attracted here by the new integrable TbarT-deformation of two-dimensional field theories, an exceptional example of an irrelevant operator acting in a controlled manner in the high-energy domain. In particular, the flow of the energy spectrum of the theory on the cylinder is governed by the well-studied Burgers differential equation. On the other hand, the deformed S-matrix is accompanied by a CDD-like factor. Within the framework of the holographic description, the TbarT-CFT with one sign of the deformation parameter corresponds to a QFT that is holographically dual to finite radial cutoff AdS. With another sign, the perturbed theory becomes non-local and exhibits the Hagedorn behavior in a density of states, sharing these properties with string theory, yet it is still solvable. In this context, a connection between integrability and conformal field theories turns out to be particularly interesting. A great example of this relation is the correspondence between the Calogero-Sutherland model and conformal blocks. Moreover, there is a non-trivial link between the phase space of integrable systems of particles and topological gauge degrees of freedom. The two-dimensional Yang-Mills theory is the simplest example of this connection. The direct implementation of integrable models from gauge theories helps understand the very nature of integrability. The generalization of these results to TbarT-deformed theories seems a promising step. Exploring and classifying the space of integrable sigma models, beyond the typical examples of the ones with group manifolds or Riemann symmetric homogeneous spaces as target space, is an ongoing problem. Two ways to approach this problem in recent times have been the program of integrable deformations and the generalization to non-symmetric homogeneous spaces. This project hopes to contribute to both. Whereas the former is well developed — including geometric meaning of the deformations, studies of behaviour as a quantum theory, applications to AdS/CFT and string theory — the latter is, as of yet, fairly unexplored. Moreover, the construction of some of the deformations (i.e. the Yang-Baxter deformations) is understood easiest in a geometric formulation of sigma models in terms of so-called ‘generalized geometry’ (a generalization of Riemannian geometry that treats and metric and other gauge fields on the target space on the same footing). Formulations of sigma models based on generalized geometry are not restricted to two dimensions and will be used to construct sigma models in higher dimensions and to study their properties. Another task within the project is the study of topological collective excitations, i. e., topological solitons, in sigma models. Topological solitons appear in various branches of physics and play a key role in understanding non-perturbative phenomena. For example, in condensed matter physics, they are used to characterize ordered phases. In this project, we study topological solitons in SU(N>2) magnets, which have been experimentally realized using ultracold atoms and are mathematically described by the SU(N>2) Heisenberg model with or without symmetry breaking terms. In the low-energy regime, the model reduces to an SU(N) symmetric nonlinear sigma model, i.e., the CP^{N-1}-model or the flag F_{N-1} sigma model, possibly with potential terms. We numerically construct soliton solutions in the nonlinear sigma models and discuss some properties of the phase structure of the SU(N) magnets utilizing these soliton solutions. Last but not least, in the context of the construction of quantum field theories from first principles, it is important to understand all potential general structures behind such theories. For that, the key information may come from theories which have unusual properties: they are known to exist, but we have no standard formulation for them. An interesting example of this kind is the “non-abelian (2,0) CFT” in six dimensions. Existence of a classical action for this theory has been shown to conflict with the standard assumptions in field theory. It is important to understand how to generalize these assumptions to accommodate such theories: they have many lessons to teach in the context of basic principles. Another example is higher-spin gravity, constructing a classical action for which has not been successful so far. We will attempt to tackle these problems with the new approaches starting from the simplest setups.

Expected results
– Applying the recently found equivalence between sigma models and Gross-Neveu models to the solution of long-standing problems in the theory of sigma models and in quantum field theory in general; developing approaches for the construction of their exact quantum solutions, starting from first principles. – Constructing sigma models with worldsheet SUSY, starting from models with target space SUSY (most importantly, in the case of models with Hermitian symmetric target spaces with symmetry groups SO and SP). The analysis of one-loop beta functions. – Proving that the anomalies in the so-called “higher charges”, which are related to the integrability of the models, are canceled automatically whenever chiral anomalies are canceled. The analysis will be based on CFT methods, such as OPE’s of chiral currents. – Investigating whether integrability properties of the theories are preserved, if one places them on higher-genus Riemann surfaces. Up to now mostly the case of the plane and cylinder have been studied, but apparently the Gross-Neveu models may be placed on higher genus surfaces in a canonical way. We expect that the relevant formalism should be somewhat similar to the CFT formalism, employed in string sigma models. – Studying the most general Gross-Neveu models, which correspond to quiver phase spaces; building their trigonometric deformations with the use of classical r-matrices; constructing explicit solutions of the Ricci flow equations and clarifying the geometric consequences (existence of generalized Einstein metrics, etc.). – Proving the conjecture that the dimensional reduction of 4D gravity with matter naturally leads to 2D Gross-Neveu models. It is expected that the pre-images of the fundamental fields of the Gross-Neveu model are the Ashtekar variables of gravity. – Studying the effects of topological (theta) angles in sigma models. According to the Haldane conjectures, the phase structure of bosonic sigma models should crucially depend on the values of theta angles. In particular, for certain values of the angles there can be a nontrivial IR fixed point described by WZNW theory. One can expect that the analysis of discrete ‘t Hooft anomalies in these models will help to clarify these issues. Results in this direction could find applications in the physics of spin chains with extended symmetries. – Studying properties of the newly discovered first-order phase transition in the TbarT-perturbed Yang-Mills theory – Clarifying a connection between conformal blocks and eigenfunctions of the TbarT Calogero-Sutherland Hamiltonian – Formulating the correspondence between the partition function of the deformed YM and topological strings – Finding a geometric interpretation of the new TbarT-like deformation of 1d non-relativistic many-body systems – Exploring the space of sigma-models with Z_N-symmetric homogeneous target spaces, yielding potentially new examples of AdS-compactifications and AdS/CFT – Seeking a unified formulation of sigma models of objects in string and M-theory, using generalized geometry and exceptional field theory. This would help understanding the dualities and connections between different sigma models (of same or different dimensionality). Also, new integrable sigma-models could be found like this. – Constructing various topological solitons in sigma models with SU(N>2)-symmetry. The construction of the solitons and understanding of their properties are of great importance to understanding SU(N) magnets. The phase structure of SU(N>2) magnets is very rich due to their large number of degrees of freedom. It implies that exotic magnets host various kinds of topological solitons, which we hope to construct analytically and numerically in the context of sigma models. The significance of this is rapidly growing due to recent proposals of potential industrial applications of topological solitons in condensed matter. – Constructing S-duality symmetric interacting field theories based on the democratic formulation developed recently and applied successfully to non-linear electrodynamics. – Generalizing the democratic formulation of electrodynamics to the non-abelian case, to gravity and to higher spins. – Constructing (first examples of) Lagrangian interacting theories of Higher-Spin (Coloured) Gravity with matter in three dimensions, holographically dual to 2d CFTs.

Annotation of the results obtained in 2022
We have continued developing the theory related to the observed equivalence between 2D sigma models of a certain type (those with complex homogeneous target spaces) and chiral Gross-Neveu models. In particular, we investigated the so-called level-zero Gross-Neveu model, which was shown to be equivalent to the CP^{n-1} sigma model with N=(2,2) supersymmetry. The beta function of the latter model (when expressed in superspace) can be proven to be one-loop exact. On the other hand, the same statement is rather non-trivial for Gross-Neveu models. There is a conjecture on the structure of the exact beta function in Gross-Neveu models, however it requires a more refined specification of regularization/renormalization scheme. We constructed a well-defined perturbative setup for the calculation of the beta function, based on the analysis of the 4-point function [1-1]. It was shown that, in the level zero model, only crossed ladder diagrams contribute to the 4-point function. Using a non-zero momentum as an IR cutoff and the so-called momentum subtraction scheme (MOM), we found a four-loop contribution to the beta function proportional to zeta(3) (at two and three loops there are no corrections). Finding a natural renormalization scheme for GN models that would yield a one loop-exact beta function remains a challenge for the future. Our second result is related to the classification of sigma models admitting a GN-formulation. In the past only models with SU(n) symmetry were considered (Su(n) flag manifolds), and in the last year we have extended this construction to the case of O- and Sp- Grassmannians. Just like unitary Grassmannians of m-planes in C^n, these can be treated for all `m’ uniformly, although there are important distinctions when the manifold in question is a symmetric space (this happens for special values of `m’ only). Only in the latter case is the resulting metric Kähler, and the B-field topological. In general, the target space metric is the so-called normal metric, which is a special point in the parameter space of invariant metrics. The fact that this particular metric emerges is related to the integrability of the models and may already be understood at the level of 1D (mechanical) reductions. We have been able to construct GN formulations for all orthogonal and symplectic Grassmannian sigma models. Unlike their geometric counterparts, these are simple quartic models, akin to the conventional phi^4 theories. We have been able to calculate one-loop beta functions of all these models showing that they are proportional to the dual Coxeter numbers of the respective symmetry groups. This agrees with the known result for the case of symmetric spaces, generalizing it to the non-symmetric setup. Again, for the case of symmetric spaces, we have related the values of the beta function to the first Chern class of the respective target space, performing a corresponding geometric calculation. As a tool for checking our methods in a very robust setup, we have considered the 1D quantum-mechanical CP^{n-1} sigma model, with N=2 supersymmetry, expressed using GN-type variables [1-2]. In fact, in this case the GN formulation is equivalent to the first-order (i.e. Hamiltonian) formulation of the system, if one uses homogeneous coordinates and their superpartners as the basic variables. The 1D setup also allows adding an SU(n)-symmetric monopole configuration without breaking N=2 supersymmetry. The supercharges and the Hamiltonian can be expressed in a manifestly SU(n)-invariant way, and a natural ansatz for the wave functions leads to a complete description of the spectrum. In particular, we have been able to identify the Young diagrams of the respective SU(n) representations corresponding to the spectrum degeneracies. By a well-known relation between the Dolbeault and Dirac complexes for Kähler manifolds, essentially the same result yields the spectrum of the Dirac operator on CP^{n-1}, in a monopole field. These N=2 quantum-mechanical models may be thought of as dimensional reductions of sigma models with N=(0,2) SUSY, which so far have not been constructed in the GN formalism, but we believe that similar methods may be used to accomplish this shortly. In [2], certain three-dimensional N=2 supersymmetric dualities were studied on a three-dimensional lens space. These dualities have been studied via the gauge/YBE correspondence, which connects dualities with integrable models in statistical mechanics. We consider three-dimensional SU(2) gauge theory with six flavors. The corresponding star-triangle relation for this theory was known. The reduction of the gauge symmetry to U(1) via the gauge symmetry breaking gives another solution to the star-triangle relation. It is shown that this leads to a generalized Faddeev-Volkov model. It was noted that the integral identity for the partition functions, obtained using supersymmetric duality with the gauge group U(1), can also be written as a pentagonal identity. New Bailey pairs were constructed for hyperbolic hypergeometric integral identities. The construction of Bailey pairs for the star-triangle relation led to the derivation of the vertex type of the Yang-Baxter equation through the Coxeter relations. Bailey pairs are also constructed for the generalized Faddeev-Volkov model and the corresponding pentagon identity. The next result includes the development of a technique based on the boost automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces [3]. We provide its implementation for two-dimensional models and resolve the classification problem, which not only confirms the known vertex solution space, but also extends to the new sl(2) deformed sector. In this respect we demonstrate that already for D=2 there exist models with non-trivial deformed Yangians and discuss their relation to other deformed models with Jordan structure (Kulish, ADHR). From the proposed generalisation of the automorphic structure for string integrable AdSn тbackgrounds one finds new integrable deformations (called 6vB and 8vB) and associated R-matrices. We provide the proof for the underlying boost operator and implement it for finite periodic lattice integrable systems. In the framework of the holographic approach, Wilson loops in holographic dual of N = 4 SYM quark-gluon plasma were studied in [4]. To this end, we consider open strings in Schwarzschild-AdS_5 and Kerr-AdS_5 black holes, which are dual to the non-rotating and rotating QGP, respectively. The behavior of the interquark potential and the jet-quenching parameter of a fast parton are studied. Above the critical temperature, the Coulomb behavior of the interquark potentials is observed. It is found that the distance between quarks decreases with increasing rotation, and it is also shown that an increase in the temperature of the medium leads to a similar behavior. It is found that at high temperatures the values of the interquark potential in Kerr-AdS_5 are close to those calculated in the Schwarzschild-AdS_5 black hole. Holographic light-like Wilson loops were also studied, from which the parameters of the jet quenching of a fast parton propagating in the QGP were found. It has been found that rotation increases the value of the jet-quenching parameter. At high temperatures, the jet-quenching parameters have a cubic dependence on temperature, as in the AdS_5 black brane. Considering the integrable T\bar T deformations, we studied the deformation of Hamiltonian systems by the bilinear T\bar T-like operators. These operators are a bilinear combination of the conserved energy and momentum currents of the system. We have considered the deformation of a system of noninteracting particles on a line in an external confining potential. The particles of a system deformed in this way can be considered like dyons with charges corresponding to the currents of the given deformation. We have shown that the deformation leads to a dynamic change of the phase-space coordinates; in the action-angle variables, this coordinate transformation results in a shift in the action variable; a nontrivial geometric phase also appears and can be associated with the presence of a topological defect in the phase space. These new results will be useful in the further study of T\barT-deformed field theories. [1-1] D. Bykov, Beta function of the level-zero Gross-Neveu model, принята к публикации в Scipost Physics https://scipost.org/submissions/scipost_202303_00022v1/ [1-2] D. Bykov, A. Smilga, Monopole harmonics on CP^{n-1}, https://arxiv.org/abs/2302.11691 [2] I. Gahramanov, B. Keskin, D. Kosva, M. Mullahasanoglu “On Bailey pairs for N = 2 supersymmetric gauge theories on S^3_b/Z_r”, J. High Energ. Phys. 2023, 169 (2023). https://link.springer.com/article/10.1007/JHEP03(2023)169 [3] A. Pribytok,"New string integrability from automorphic symmetries", accepted for publication in TMP (2023) [4] A. A. Golubtsova and N. S. Tsegelnik, ``Probing the holographic model of N =4 SYM rotating quark-gluon plasma,'' (статья принята к публикации в Physical Review D) [arXiv:2211.11722 [hep-th]]. https://journals.aps.org/prd/accepted/72072Q36Qa811f3fe18378994e11aa84b6027586c

Publications

1. A. A. Golubtsova, N. S. Tsegelnik Probing the holographic model of N=4 SYM rotating quark-gluon plasma Physical Review D, - (year - 2023)

2. A. Pribytok Новая струнная интегрируемость из автоморфных симметрий Теоретическая и математическая физика, - (year - 2023)

3. D. Bykov Beta-function of the level-zero Gross-Neveu model SciPost Physics, - (year - 2023)

4. Ilmar Gahramanov, Batuhan Keskin, Dilara Kosva & Mustafa Mullahasanoglu On Bailey pairs for N = 2 supersymmetric gauge theories on S^3_b/Z_r Journal of High Energy Physics, J. High Energ. Phys. 2023, 169 (2023) (year - 2023) https://doi.org/10.1007/JHEP03(2023)169