In this talk, we discuss the relationship between the integral Bailey lemma and mirror symmetry for three-dimensional supersymmetric gauge theories. Three-dimensional mirror symmetry relates infrared fixed points of a certain class of quiver gauge theories. The simplest example of such a duality is the equivalence of 3d N=2 supersymmetric quantum electrodynamics and the theory of three chiral multiplets X, Y, Z with a superpotential W=XYZ. One can check these dualities by computing supersymmetric partition functions. It happens that in some cases, starting with the partition function identity for a certain mirror duality one can construct a family of duality via the integral analog of the Bailey lemma. 

Integrable deformations of sigma-models provide new examples of integrable QFTs which are interesting on their own and also due to their applications to string theory. I will review a class of such models known as Yang-Baxter (YB) deformed sigma-models on symmetric spaces G/H. The undeformed versions of these models are renormalizable and integrable at quantum level provided that H is simple. It is known that YB deformation survives both these properties. Remarkably it changes the UV behavior in a nice manner allowing to study the model by conformal perturbation theory.  In my talk I will advertise a challenging problem of identifying the UV fixed point CFTs and the set of relevant operators that describe the vicinity of the fixed point.

We will discuss applications of knots and quantum groups to the topological quantum computer. We will show which problems in knot theory and quantum groups arise from this connection. Also we will discuss how our knowledge in knot theory modifies our perception of the models of topological quantum computer.

I will make an overview of higher-spin theories starting from earlier no-go theories and ending with recent results and open directions.

In my talk I will review the Stokes phenomenon of perturbative expansions of oscillatory integrals (from the perspective of Picard-Lefschetz theory) and the notion of analytically continued Chern-Simons theory. I will then tell how one can explicitly calculate Stokes constants in Chern-Simons theory on 3-manifolds of certain types. If time permits, I will also comment on categorification of the Stokes constants.

I will talk about what interesting directions there are for research related to the Chern-Simons theory. First, Chern-Simons theory is one of the simplest three-dimensional quantum field theories that we can solve exactly using quantum algebra and the R-matrix. The question of searching for integrable structures in other gauge theories seems promising. Second, vacuum expectation values in Chern-Simons theory are invariants of knots (or links) in a given 3-dimensional manifold. Since the theory of knots and their invariants is far from its complete description, this question is of great interest. Third, there are two dualities with the Chern-Simons theory. One holographic duality between 3D Chern-Simons theory and 2D conformal WZW theory. The second duality arises from string theory and is a mirror symmetry between the A-model (Chern-Simons theory) and the B-model (Gromov-Witten theory). Due to the  progress in the Chern-Simons theory, it is possible to investigate this duality in explicit terms.

The report will focus on the Zamolodchikov tetrahedron equation, which is the next n-simplex equation after the Yang-Baxter one. I will talk about the most striking embodiments of this equation in my opinion: namely, its role in the theory of cluster varieties, the theory of 2-knot invariants, that is, classes of isotopes of embeddings of a two-dimensional surface into a 4-dimensional space, exactly solvable models of statistical physics in dimension 3 and integrable quantum mechanical models on two-dimensional lattices.

We consider general theory of k-inlation and find out, that it may be in strong coupling regime. We derive accurate conditions of classical description validity using unitarity bounds for this model. Next, we choose simple toy model of k-inflation and obtain the explicit condition, which guarantees that the generation of perturbations is performed in a controllable way, i.e the exit from effective horizon occurs in the weak coupling regime. However, for the same toy model the corresponding experimental bounds on a non-linear parameter fequilNL associated with non-Gaussianities of the curvature perturbation provide much stronger constraint than strong coupling absence condition. Nevertheless, for other known models of inflation this may not be the case. Generally, one should always check if classical description is legitimate for chosen models of inflation.

The Catastrophe theory is a powerful tool of mathematical physics to study complicated dynamical systems. It commonly operates with systems of differential equations, but is close in spirit to homological calculus in topology. A kind of hybrid of the two fields is known as a cohomological quantum field theory (CQFT). Such models seem to be very interesting and profound, and they might be useful in various applications as a new tool of the Catastrophe theory. Our goal is to use the knot homology calculus to develop a “Cohomological Catastrophe theory”, which would be associated with a family of constructively defined CQFT models related by a kind of evolution “regular” in the moduli space but special points of “catastrophes” in the knot homology. In the talk we summarise our current knowledge on the Catastrophes for explicitly studied CQFT models associated with the Khovanov--Rozansky homology of several knot families. 

The Euclidean path integral is compared to the thermal (canonical) partition function in curved static space-times. It is shown that if spatial sections are non-compact and there is no Killing horizon, the logarithms of these two quantities differ only by a term proportional to the inverse temperature, that arises from the vacuum energy. When spatial sections are bordered by Killing horizons the Euclidean path integral is not equal to the thermal partition function. It is shown that the expression for the Euclidean path integral depends on which integral is taken first: over coordinates or over momenta. In the first case the Euclidean path integral depends on the scattering phase shift of the mode and it is UV diverge. In the second case it is the total derivative and diverge on the horizon. Furthermore we demonstrate that there are three different definitions of the energy, and the derivative with respect to the inverse temperature of the Euclidean path integral does not give the value of any of these three types of energy. We also propose the new method of computation of the Euclidean path integral that gives the correct equality between the Euclidean path integral and thermal partition function for non-compact spaces with and without Killing horizon.

The celebrated AKSZ construction provides a nice way to encode all the data about a gauge theory (namely it's Batalin–Vilkovisky lagrangian) in geometrical objects. There is a generalization of this construction which allows one to obtain AKSZ-like description of non-topological field theories by relaxing the invertibility of the symplectic form. In my talk I will explain how the common physical notion of background fields and symmetries related to them fit nicely into this geometrical picture by enriching the source manifold.

We develop Schwinger-Keldysh in-in formalism for generic nonequilibrium dynamical systems with mixed initial states. We construct the generating functional of in-in Green's functions and expectation values for a generic density matrix of the Gaussian type and show that the requirement of particle interpretation selects a distinguished set of positive/negative frequency basis functions of the wave operator of the theory, which is determined by the density matrix parameters. Then we consider a special case of the density matrix determined by the Euclidean path integral of the theory, which in the cosmological context can be considered as a generalization of the no-boundary pure state to the case of the microcanonical ensemble, and show that in view of a special reflection symmetry its Wightman Green's functions satisfy Kubo-Martin-Schwinger periodicity conditions which hold despite the nonequilibrium nature of the physical setup.

I will tell about Wilson networks in AdS₂ constructed from Wilson lines and cap states. Cap states regulate the boundary behaviour of Wilson line operators and by imposing right restrictions on cap states one can obtain the global conformal block at the boundary of the AdS₂. I will show how to find cap states and obtain dictionary relation between Wilson networks and global conformal blocks. Then I will tell about calculation of the n-point global conformal block in the comb channel.

In this talk we will develop a precise analytic description of oscillons — long-lived quasiperiodic field lumps — in scalar field theories with nearly quadratic potentials. These oscillons are essentially nonperturbative due to large amplitudes, and they are known to achieve extreme longevities. The proposed method is based on a consistent expansion in both the the inverse width of the oscillon and the anharmonicity of the potential at strong fields, which is made accurate by introducing a field-dependent “running mass". 

We study the means to geometrically engineer a top-down holographic model for the composite Higgs scenario. Our construction modifies the Sakai-Sugimoto model for the holographic AdS/QCD, namely its variation based on the D4 brane stack wrapped over the orientifold with the orthogonal stack of D8 branes. The difficult question is how to geometrically engineer elementary fermion fields in the specific representations that allow the composite fermionic operators to emerge, which can be coupled appropriately to the Standard model fermions. We present the possible solutions to this problem and review the properties of the resulting composite Higgs models.

With the use of the non-stationary Keldysh-Schwinger diagrammatic technique, we find the one-loop effective action for graviton that interacts with the scalar field on the background of de Sitter space. We show that the graviton does not acquire mass for the most symmetric Bunch-Davies state in one-loop effective theory. However, we have shown that even in this case, there is a nontrivial modification of the theory at one loop in the scalar sector of gravity. 

We study the vacuum zero point energy associated to an interacting scalar field with an arbitrary mass and conformal coupling in a dS background. Employing dimensional regularization scheme, we calculate the regularized zero point energy density, pressure and the trace of the energy momentum tensor. It is shown that the classical relation for the vacuum stress energy tensor receives anomalous quantum correction which depends on the mass and the conformal coupling does not hold. We calculate the density contrast associated to the vacuum zero point energy indicating an inhomogeneous and nonperturbative distribution of the zero point energy. Finally, we calculate the skewness associated to the distribution of the zero point energy and pressure and show that they are highly non-Gaussian.

We consider coupling constant and beta-function in holographic models supported by Einstein-dilaton-Maxwell action for heavy and light quarks. The significant dependence of the coupling constant and beta function on chemical potential and temperature is obtained. Near the first-order phase transitions, the beta function undergoes junctions depending on temperature and chemical potential.

I will discuss how a family of Schur-Jack type polynomials naturally emerges as a representation space for infinite-dimensional quiver Yangian algebras. An initial example would be the affine Yangian of gl_1, in which the representation space is Jack polynomials (enumerated by Young diagrams) in time variables, and the generators of the algebra itself are generated by the cut-and-join operator. By analogy with the Cartan classification of Lie algebras, quiver Yangians are defined using a quiver and a superpotential. I will show how free-field representations of Fock-like representations are modified using the simplest examples, that is, how the set of “Young diagrams/time variables/Jack polynomials/cut-and-glue operator” generalizes for quivers beyond the gl_1 case.

I'll review the known results about BPS indices, which encode in particular the entropy of BPS black holes, appearing in string compactifications down to four dimensions with various number of supersymmetries. First, I'll recall the well-known results about BPS states in N=8 and N=4 compactifications, and then present what is known about them in the N=2 case. Depending on time, I hope to cover some recent advances where an important role was played by (mock) modular symmetry.

The AdS3/CFT2 correspondence is an intriguing instance of holography. Despite its long and rich history, only recently we have developed the techniques to obtain exact results for generic (non-protected) observables at generic points of the parameter space. In this talk I will overview this progress, describe the major stumbling blocks and how they have been overcome, and outline the remaining challenges for the field.

I will review the correspondence between configurations of branes of Type IIB string theory and representation theory of quantum toroidal algebras. In particular I will show that the R-matrices of these complicated algebras arise from brane crossings similarly to how R-matrices of finite quantum groups arise in Chern-Simons theory from crossings of the Wilson lines. Time permitting, I will also present some applications to supersymmetric gauge theories in various dimensions.

Solutions of Zamolodchikov's tetrahedron equation define integrable 3D lattice models in statistical mechanics, just as solutions of the Yang-Baxter equation define integrable 2D lattice models. I will explain how we can construct solutions of the tetrahedron equation using quantum cluster algebras and how they are related to 3D supersymmetric gauge theories. This is based on joint work with Xiao-yue Sun, Rei Inoue, Atsuo Kuniba and Yuji Terashima.