We present a wormhole solution in four dimensions. It is a solution of an Einstein Maxwell theory plus charged massless fermions. The fermions give rise to a negative Casimir-like energy, which makes the wormhole possible. It is a long wormhole that does not lead to causality violations in the ambient space. It can be viewed as a pair of entangled near extremal black holes with an interaction term generated by the exchange of fermion fields. The solution can be embedded in the Standard Model by making its overall size small compared to the electroweak scale.

The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large N limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large N limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N), O(N) 2 , or O(N) 3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large N limits. In the latter example we review how the dominance of melonic diagrams follows when a special "tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N) 3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N) 2 × O(N) × U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.

We study the OðN 1 Þ × OðN 2 Þ × OðN 3 Þ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks N i are all equal, this model has a large N limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of group invariant states for any set of N i . It is non-vanishing only when each N i is even. For equal ranks the number of singlets exhibits rapid growth with N: it jumps from 36 in the Oð4Þ 3 model to 595 354 780 in the Oð6Þ 3 model. We derive bounds on the values of energy, which show that they scale at most as N 3 in the large N limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order 1=N. For N 3 ¼ 1 the tensor model reduces to OðN 1 Þ × OðN 2 Þ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with SUðN 1 Þ × SUðN 2 Þ × Uð1Þ symmetry. Finally, we study the N 3 ¼ 2 case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only OðN 1 Þ × OðN 2 Þ × Uð1Þ. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large N limits where the ground state energies are of order N 2 , while the energy gaps are of order 1.

We study the O(N ) 3 symmetric quantum field theory of a bosonic tensor φ abc with sextic interactions. Its large N limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large N solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for 2.81 < d < 3 and for d < 1.68 the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in 3 − dimensions including eight O(N ) 3 invariant operators necessary for the renormalizability. For sufficiently large N , we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large N limit of the resulting expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the expansion allows us to calculate the 1/N corrections to operator dimensions. The prismatic fixed point in 3 − dimensions survives down to N ≈ 53.65, where it merges with another fixed point and becomes complex. We also discuss the d = 1 model where our approach gives a slightly negative scaling dimension for φ, while the spectrum of bilinear operators is free of complex dimensions.

We study the expectation value of the energy momentum tensor during thin shell collapse for a massive, real, scalar field theory. At tree-level, we find thermal, Hawking-type, behaviour for the energy flux. Using the Schwinger-Keldysh technique, we calculate two-loop corrections to the tree-level correlation functions and show that they exhibit secular growth, suggesting the breakdown of the perturbation theory.Comment: 28 pages, but general physical explanation is given in the introduction; Final version that is going to appear in PR

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We calculate one-loop corrections to the vertexes and propagators of photons and charged particles in the strong electric field backgrounds. We use the SchwingerKeldysh diagrammatic technique. We observe that photon's Keldysh propagator receives growing with time infrared contribution. As the result, loop corrections are not suppressed in comparison with tree-level contribution. This effect substantially changes the standard picture of the pair production. To sum up leading IR corrections from all loops we consider the infrared limit of the Dyson-Schwinger equations and reduce them to a single kinetic equation.

We continue a previous study about the infrared loop effects in the D-dimensional de Sitter space for a real scalar φ 4 theory from the complementary series whose bare mass belongs to the interval √ 3 4 (D − 1) < m ≤ D−1 2 , in units of the Hubble scale. The lower bound comes from the appearance of discrete states in the mass spectrum of the theory when that bound is violated, causing large IR loop effects in the vertices. We derive an equation which allows to perform a selfconsistent resummation of the leading IR contributions from all loops to the two-point correlation functions in an expanding Poincaré patch of the de Sitter manifold. The resummation can be done for density perturbations of the Bunch-Davies state which violate the de Sitter isometry. There exist solutions having a singular (exploding) behavior and therefore the backreaction can change the de Sitter geometry.