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On polyvector fields in instantonic theories, geometric formalism  and A-I-B mirror
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
ITEP, Moscow.
(English)In: Journal of High Energy Physics (JHEP), ISSN 1029-8479, E-ISSN 1126-6708Article in journal (Other academic) Submitted
##### Abstract [en]

We develop the geometric formulation for SUSY theories  proposed by Frenkel, Losev and Nekrasov.The formalism is based on localization on instanton moduli spaceand it's deformation and  leads to a rich family of  non-perturbatively well-defined QFTs. Among these theories are the Morse theory, $\beta\gamma-bc$ system and A-model type theories, but with more effort much widerrange of theories may be covered.  The advantage of geometric formalism is it's explicit target space coordinate-independence.  We develop further the geometric formalism and study in detail the relation of this formalism with the conventional ones.  We study the ways to define polyvector field observables in a coordinate-independent wayand compare it to free-field methods.

We investigate the local limit of polyvector fields: for holomorphic polyvector fields we find a nice symmetric regularization prescription,while for non-holomorphic vector fields there arises an interesting dependence on the angleof point-splitting.

Comparing the geometric calculation with free-field approach to first-order non-linear sigma-models based on holomortices, we show the origin of conditionally convergent integrals in the latter approach and prove the way to deal with them.

##### Identifiers
OAI: oai:DiVA.org:uu-123190DiVA: diva2:312943
Available from: 2010-04-26 Created: 2010-04-26 Last updated: 2011-03-08Bibliographically approved
##### In thesis
1. Yang-Mills Theory in Gauge-Invariant Variables and Geometric Formulation of Quantum Field Theories
Open this publication in new window or tab >>Yang-Mills Theory in Gauge-Invariant Variables and Geometric Formulation of Quantum Field Theories
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

In Part I we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables. For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddeev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, used to model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariant field variables called supercurrents.

We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model. Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravity with 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs field as a dilaton (papers I and II).

In Part II of the thesis we study rather simple field theories, called “geometric” or “instantonic”. Their defining property is exact localization on finite-dimensional spaces – the moduli spaces of instantons. These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory.

In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory – the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and in the Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields). We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.

##### Place, publisher, year, edition, pages
Uppsala: Uppsala University, 2010. 93 p.
##### Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 761
##### Keyword
Yang-Mills theory, spin-charge separation, Kaluza-Klein theory, branes, curved beta-gamma systems
##### National Category
Other Physics Topics
##### Research subject
Theoretical Physics
##### Identifiers
urn:nbn:se:uu:diva-129670 (URN)978-91-554-7873-5 (ISBN)
##### Public defence
2010-10-11, Häggsalen, Ångström lab, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Supervisors
Available from: 2010-09-16 Created: 2010-08-22 Last updated: 2011-03-08

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Slizovskiy, Sergey
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Journal of High Energy Physics (JHEP)