STRING FIELD THEORY FROM QUANTUM GRAVITY

Quantum field theory is a well defined and well established framework for describing Nature at accessibly short distances and high energies. It will play this role in the foreseeable future, unless gravity/string theory have low fundamental scale (for reviews of the latter possibility see, for example, Refs. [1–4]). A particular quantum field theory—the Standard Model—is extraordinarily successful in describing particles and their interactions probed so far in terrestrial experimens. The only exception is the phenomenon of neutrino oscillations, which can be incorporated by a relatively mild extension of the Standard Model at accessible energies. Yet the Standard Model is incomplete, and it is likely that new physics will be found at the LHC. These lectures give an introduction to quantum field theory and the Standard Model, to set the stage for more specialized discussions of heavy flavours, QCD, physics beyond the Standard Model, cosmology, etc. given in other series of lectures at this School. This overview is certainly incomplete and cannot be used to replace textbooks. The purpose is to expose the basic ideas and emphasize some points which are not always emphasized. There are numerous excellent reviews of the Standard Model, in particular, those published in Proceedings of previous Schools [5, 6]. An important aspect of the Standard Model and all high-energy physics—electroweak symmetry breaking—is reviewed, for example, in Refs. [7–11].

Of the three organizers of the conference N. Hitchin is a mathematician, A. Kapustin and W. Nahm are physicists. Ideas from physics, in particular from string theory and related areas had a profound influence on algebraic and differential geometry. Conversely, many developments in string theory use very recent mathematical results. There were 19 talks at the workshop, 12 by mathematicians and 7 by physicists, but in many cases an outside observer would have had difficulties to sort them out. Quantum field theory often allows continuous interpolations between geometrical structures of topologically different manifolds. A particularly important example is mirror symmetry, where the complexified Kähler cone of one manifold corresponds to the moduli space of complex structures of the mirror manifold. Several talks explored geometrical and number theoretical aspects of this connection. Mirror symmetry can be understood in terms of torus fibrations and an isomorphism between the quantum field theories given by maps to a torus and to its dual (T-duality). The B-field of string theory can be described in terms of non-commutative geometry on the tori, and its effect on T-duality has been investigated. The quantum field theories which provide these dualities are supersymmetric conformally invariant quantum field theories in two dimensions. In mathematics, large parts of their structures have been axiomatised in the language of vertex operator algebras, but much remains to be done. A promising mathematical approach is provided by the chiral de Rham complex. This cohomology of this complex seems to be invariant under mirror symmetry, but its physical meaning is not yet clear. This led to many discussions between mathematicians and physicists. Indeed, the free time for discussions was at least as important as the lectures for learning the language of researchers with a different background and for making use of their ideas. An area in which the common understanding is well advance is topological field theory. Here one has good axioms and many of the analytic problems of conformally invariant quantum field theories can be ignored. One can solve rather complex problems and one can make contact with new geometrical ideas, in particular generalised complex structures. The latter allow to interpolate between complex and symplectic geometry, which should become important for both physics and mathematics. To understand all of conformal field theory at a similar mathematical depth will take more time, but already now relations to mixed Hodge structures have given much insight. Eventually many so far intractable problems should be solvable by the use of quantum field theoretical ideas, for example the construction of the non-singular Ricci flat metrics on K3. Perturbative string theory can be described in terms of conformal field theory, but the complete theory has more excitations than strings. Among them the best understood are the D-branes, many features of which can be understood in terms of boundary states in conformal field theory. Some of these branes are related to complex submanifolds, others to certain Lagrangian submanifolds, and their description involves twisted K-theory and derived categories. The development of a common language between mathematicians and physicists is well under way, and the workshop has made a contribution which all participants found very stimulating. In Europe such meetings between mathematicians and string theorists are rare, so the workshop was particularly useful. Much of the time was reserved for informal discussions. On one evening there was a discussion on the relation between classical and quantum geometry. Weather predictions for the free afternoon had been bad, but finally the weather was splendid and the supply of Black Forest cake matched the demand.

Finiteness of conformal blocks over the projective line by T. Abe and K. Nagatomo Permutation orbifolds and their applications by P. Bantay Category theory for conformal boundary conditions by J. Fuchs and C. Schweigert GNAVOA, I. Studies in groups, nonassociative and vertex operator by R. L. Griess, Jr. Genera of vertex operator and three-dimensional topological quantum field theories by G. Hohn Riemann surfaces with boundaries and the theory of vertex operator by Y.-Z. Huang Vertex (operator) are algebras of vertex operators by H. Li Correlation functions, differential operators and vertex operator by A. Milas Relations for annihilating fields of standard modules for affine Lie by M. Primc From branes to boundary conformal field theory: Draft of a dictionary by A. Recknagel Open strings and non-commutative geometry by V. Schomerus The world sheet revisited by C. Schweigert and J. Fuchs.

Stochastic quantization

The Standard Model of particle physics describes electromagnetic, weak, and strong interactions, which are three of the four known fundamental forces of nature. The unification of the fourth interaction, gravity, with the Standard Model has been challenging due to incompatibilities of the underlying theories-general relativity and quantum field theory. While quantum field theory utilizes compact, finite-dimensional symmetries associated with the internal degrees of freedom of quantum fields, general relativity is based on noncompact, infinite-dimensional external space-time symmetries. The present work aims at deriving the gauge theory of gravity using compact, finite-dimensional symmetries in a way that resembles the formulation of the fundamental interactions of the Standard Model. For our eight-spinor representation of the Lagrangian, we define a quantity, called the space-time dimension field, which enables extracting four-dimensional space-time quantities from the eight-dimensional spinors. Four U(1) symmetries of the components of the space-time dimension field are used to derive a gauge theory, called unified gravity. The stress-energy-momentum tensor source term of gravity follows directly from these symmetries. The metric tensor enters in unified gravity through geometric conditions. We show how the teleparallel equivalent of general relativity in the Weitzenböck gauge is obtained from unified gravity by a gravity-gauge-field-dependent geometric condition. Unified gravity also enables a gravity-gauge-field-independent geometric condition that leads to an exact description of gravity in the Minkowski metric. This differs from the use of metric in general relativity, where the metric depends on the gravitational field by definition. Based on the Minkowski metric, unified gravity allows us to describe gravity within a single coherent mathematical framework together with the quantum fields of all fundamental interactions of the Standard Model. We present the Feynman rules for unified gravity and study the renormalizability and radiative corrections of the theory at one-loop order. The equivalence principle is formulated by requiring that the renormalized values of the inertial and gravitational masses are equal. In contrast to previous gauge theories of gravity, all infinities that are encountered in the calculations of loop diagrams can be absorbed by the redefinition of the small number of parameters of the theory in the same way as in the gauge theories of the Standard Model. This result and our observation that unified gravity fulfills the Becchi-Rouet-Stora-Tyutin (BRST) symmetry and its coupling constant is dimensionless suggest that unified gravity can provide the basis for a complete, renormalizable theory of quantum gravity.

The theory of fundamental particles and their non-gravitational interactions is encapsulated by an extremely successful quantum field theory known as the Standard Model. This quantum field theory is determined by a particular set of quantum fields and a particular Hamiltonian, which we will outline in this chapter. It is an interacting quantum field theory, not solvable by the methods we have seen so far. In the non-interacting approximation, it includes just the sorts of free quantum field theories that we have studied in earlier chapters. This chapter gives only a very brief sketch of the definition of the theory; for details, one needs to consult a conventional particle physics textbook.

The existence of magnetic monopoles is predicted by many theories of particle physics beyond the standard model. However, in spite of extensive searches, there is no experimental or observational sign of them. I review the role of magnetic monopoles in quantum field theory and discuss their implications for particle physics and cosmology. I also highlight their differences and similarities with monopoles found in frustrated magnetic systems.

The relativistic Dirac equation in four-dimensional spacetime reveals a coherent relation between the dimensions of spacetime and the degrees of freedom of fermionic spinors. A massless Dirac fermion generates new symmetries corresponding to chirality spin and charge spin as well as conformal scaling transformations. With the introduction of intrinsic W-parity, a massless Dirac fermion can be treated as a Majorana-type or Weyl-type spinor in a six-dimensional spacetime that reflects the intrinsic quantum numbers of chirality spin. A generalized Dirac equation is obtained in the six-dimensional spacetime with a maximal symmetry. Based on the framework of gravitational quantum field theory proposed in Ref. [] with the postulate of gauge invariance and coordinate independence, we arrive at a maximally symmetric gravitational gauge field theory for the massless Dirac fermion in six-dimensional spacetime. Such a theory is governed by the local spin gauge symmetry SP(1,5) and the global Poincaré symmetry P(1,5)=SO(1,5)⋉P1,5 as well as the charge spin gauge symmetry SU(2). The theory leads to the prediction of doubly electrically charged bosons. A scalar field and conformal scaling gauge field are introduced to maintain both global and local conformal scaling symmetries. A generalized gravitational Dirac equation for the massless Dirac fermion is derived in the six-dimensional spacetime. The equations of motion for gauge fields are obtained with conserved currents in the presence of gravitational effects. The dynamics of the gauge-type gravifield as a Goldstone-like boson is shown to be governed by a conserved energy-momentum tensor, and its symmetric part provides a generalized Einstein equation of gravity. An alternative geometrical symmetry breaking mechanism for the mass generation of Dirac fermions is demonstrated.

The 2nd International Conference on Particle Physics and Astrophysics (ICPPA-2016) was held in Moscow, Russia, from October 10 to 14, 2016. The conference is organized by the National Research Nuclear University “MEPhI”. The aim of the Conference is to promote contacts between scientists and to develop new ideas in fundamental research. We bring together experts and young scientists working on experimental and theoretical aspects of nuclear, particle and astroparticle physics and cosmology.The conference covers a wide range of topics such as accelerator physics, (astro) particle physics, cosmic rays, cosmology and methods of experimental physics: detectors and instruments. These directions are unified by development of the Standard Model which is evidently not complete. There are deviations from the Standard Model: neutrino oscillations, the dark matter existence. Together with strong interactions they are main subjects of the Conference.New results from leading high energy physics collaborations are discussed. Main LHC experiments (ATLAS, CMS, ALICE) presented their results and detector upgrade prospects on the conference. Various aspects of Standard Model testing and search for new phenomena are main subjects of the conference. Among them: flavor physics at B factories, precision multi-boson production measurements, dark matter searches.Electroweak interaction was discussed in the talks given by participants of neutrino physics experiments (Borexino, Neutrino-4, SOX, T2K and others). Enigmatic properties of neutrinos such as their tiny masses, oscillations between different neutrino types, cannot be explained in frame of the modern theory and require new approaches. Properties of neutrinos influenced the formation of the large-scale structure of the Universe and may be neutrinos are partially responsible for the excess of matter over anti-matter. Also our current challenge is the nature of the Dark matter. Many opportunities arise with the development of neutrino detection technologies comprising: measurements of neutrinos from the Sun, measurements of geo-neutrino from the deep interior of the Earth, detecting neutrino fluxes from supernovae and different astrophysical sources. They were the subject of discussion which have been held at the conference. Neutrino physics experiments make huge demands to technologies like ultra-high precision measurements, huge detector size, excellent reliability and ultra-low radioactive background.AcknowledgmentsWe are thankful to Anotonova M M, Buzhan P Zh, Gurov Yu B, Vdovkina S S, Zherebtsova E S, Esipova E A, Kovylyaeva A A, Kozlov E S, Koshelenko D D, Matveev N A, Mayorov A G, Nigmatkulov G A, Novikov A S, Svadkovsky I V, Smirnov Yu S, Voronov S A and Yurin K O for the assistance in organizing the conference and heavy work on proceedings.Conference organizers would like to express their gratitude to the TFT company (supply and maintenance of CAEN equipment, delivery of equipment and components for scientific research) for their support.

An imaginary-time method was developed for calculating the probability of particle transmission through smooth barriers variable with time. Within the imaginary-time method, the tunneling process is described by using classical equations of motion written in terms of an imaginary time (t → it), while the probability of tunneling is determined by the imaginary part of the action functional, this imaginary part being calculated along the subbarrier particle trajectory. The fundamentals of the imaginary-time method are surveyed, and its applications in the theory of atomic-state ionization under the effect of constant electric and magnetic fields that have various configurations, in the field of intense monochromatic laser radiation and of an ultrashort electromagnetic pulse, in the process of Lorentz ionization of atoms and ions during their motion in a strong magnetic field, etc., are outlined. The applications of the imaginary-time method in relativistic cases—for example, in the theory of ionization of levels of multiply charged ions whose binding energy is commensurate with the electron rest energy—and in quantum field theory (Schwinger effect, which consists in the production of electron-positron pairs from a vacuum by a superstrong external field) are briefly described. Particular attention is given to methodological issues and details of the imaginary-time method that are of importance in solving specific physics problems, but which are usually skipped in original publications.

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book "Factorization algebras in quantum field theory", by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold.

“In recent years the ideas of decay of a metastable state by quantum tunnelling of a field, and also of tunnelling between degenerate vacua, have played a major role in particle physics. Analogous problems occur also in condensed-matter physics, where they tend to be more tractable quantitatively because the role of the field is mainly to provide dissipation for the “primary” tunnelling variable. While the explicitly field-theoretic aspects of the problem are in this sense rather trivial, many of the techniques popular in quantum field theory have been applied successfully to such condensed-state problems, and I review here some of these applications.”

Approaches to Ontology: Candidate General Ontologies for Situating Quantum Field (P Simons) 'Quanta', Tropes, or Processes: Ontologies for QFT Beyond the Myth of Substance (J Seibt) Analytical Ontologists in Action: A Comment on Seibt and Simons (M Kuhlmann) How Do Field Theories Refer to Entities in a Field? (S Y Auyang) Field Ontologies for QFT: A Naive View of the Quantum Field (A Wayne) Comments on Paul Teller's Book, An Interpretive Introduction to Quantum Field Theory (G Fleming) So What Is the Quantum Field? (P Teller) Relativity, Measurement and Renormalization: On the Nature of Measurement Records in Relativistic Quantum Field (J A Barrett) No Place for Particles in Relativistic Quantum Theories? (H Halvorson & R Clifton) Events and Covariance in the Interpretation of Quantum Field (D Dieks) Measurement and Ontology: What Kind of Evidence Can We Have for Quantum Fields? (B Falkenburg) Renormalization and the Disunity of Science (N Huggett) Gauge Symmetries and the Vacuum: The Interpretation of Gauge Symmetry (M Redhead) Comment on Redhead: The Interpretation of Gauge Symmetry (M Drieschner et al.) Is the Zero-Point Energy Real? (S Saunders) Two Comments on the Vacuum in Algebraic Quantum Field (M Redei).

Part I of this book is introductory in nature. Its purpose is to motivate our geometric approach to supersymmetric field theory. We show how geometric structures arise in classical and quantum field theories on quite general grounds. In Chapter 1 we consider the basic geometric structures which hold independently of supersymmetry. In Chapter 2 we specialize to the supersymmetric case (rigid and local) where more elegant structures emerge. Not being part of the technical body of the book, these chapters are rather elementary and sketchy. However, we show how dualities, modularity, and other stringy patterns are universal features of field theory.

Determining the nature of matter’s smallest constituents, and the interactions among them, is the subject of a branch of fundamental physics called “The Physics of Elementary Particles” – the subject of this book. During the last decades this field has gone through a phase transition. It culminated in the formulation of a new theoretical scheme, known as “The Standard Model”, which brought profound changes in our ways of thinking and understanding nature’s fundamental forces. Its agreement with experiment is impressive, to the extent that we should no longer talk about “The Standard Model” but instead “The Standard Theory”. This new vision is based on geometry; the interactions are required to satisfy a certain geometrical principle. In the physicists’ jargon this principle is called “gauge invariance”; in mathematics it is a concept of differential geometry. It is the purpose of this book to present and explain this modern viewpoint to a readership of well motivated undergraduate students. We propose to guide the reader to the more advanced concepts of gauge symmetry, quantum field theory and the phenomenon of spontaneous symmetry breaking through concrete physical examples. The presentation of the techniques required for particle physics is self-contained, and the mathematics is kept at the absolutely necessary level. The reader is invited to join the glorious parade of the theoretical advances and experimental discoveries of the last decades which established our current view. Our ambition is to make this fascinating subject accessible to undergraduate students and, hopefully, to motivate them to study it further.