Homological projective duality and quantum gauge theory pdf

I will survey a program that seeks to translate the aldaygaiottotachikawa correspondence between gauge theory on r4 and conformal field theory into the language of algebraic geometry. In quantum field theory one says supersymmetric localization to denote. The very formulation of quantum mechanics is based on linear algebra. Geometric langlands duality is then naturally formulated as an. A third major effort in theoretical particle physics is string theory. We will argue that the exact results on partition functions and loop operator expectation values in certain susy gauge theories obtained by pestun, aldaygaiottotachikawa agt and others are best understood in terms of a reduction to an effective quantum mechanics that describes the gauge theory at low energies. Teleman berkeley gauges, mirrors and langlands columbia, 4.

Conference on algebraic geometry, representation theory and. We observe supersymmetry breaking in superyangmills theory and in theories with small numbers of flavors. We construct a structure of transversal 1category on the space of chains of maps from a suspension space sy, with. Abelian glsms, gerbes, and homological projective duality eric sharpe virginia tech t pantev, es, hepth0502027, 0502044, 0502053 s hellerman, a henriques, t pantev, es, m ando, hepth0606034. Quantum field theory ii lecture 31 renormalization of gauge. In this third volume of his modern introduction to quantum field theory, eberhard zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe. This behavior is very different from four dimensions. Langlands as a consequence of homological mirror symmetry for the hitchin. The states in a physical system are described by \rays in a projective hilbert space, and physical observables are identi ed with hermitian linear operators on hilbert space. Donaldsons theory as a topological quantum field theory, which can be thought of as. Classical geometry of quantum integrability and gauge theory.

Gauge theory, mirror symmetry, and langlands duality constantin teleman uc berkeley columbia university, april 2012 revised some refs 427 c. One of our main goals will be to understand nick sheridans 2011 proof of homological. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by hori. Quantum gauge field theory cohesive homotopy type theory. In quantum mechanics, the position and velocity of a particle are noncommuting operators acting.

Since he is unable to make it, i will pretend to be him. These examples violate old unproven lore on glsms e. Apr 15, 2017 this is a course offered by ashoke sen, called quantum field theory ii. Abelian glsms, gerbes, and homological projective duality. Talk at homological projective duality and quantum gauge theory ipmu, tokyo. Introduction to localization in quantum field theory iopscience. Workshop report homological projective duality and. Analogies between number theory and quantum field theory are widespread, resting on such observations as michael atiyah.

Any opinions, findings and conclusions or recommendations expressed on the nlab are those of the authors and do not necessarily reflect the views of the afosr. The name quantum spectrum in the title refers for a compact symplectic manifold to the spectrum of the operator of quantum multiplication by the first chern class of the tangent bundle, considered as a multivalued function on. Topological gauge theory, and gravity derek keith wise. Homological quantum field theory edmundo castillo and rafael daz abstract we show that the space of chains of smooth maps from spheres into a. In general, quantum field theory qft is a puzzling subject for mathemati.

Fabian haiden u vienna measured foliations, tstructures, and wallcrossing. On quantum hydrodynamic and quantum energy transport models degond, pierre, gallego, samy, and mehats, florian, communications in mathematical sciences, 2007. Gauge theory, mirror symmetry, and langlands duality. Kuznetsovs theory of homological projective duality is a very useful tool for proving relations between the derived categories of coherent sheaves of different varieties. Duality in twodimensional 2,2 supersymmetric nonabelian. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge. Duality as a categorytheoretic concept sciencedirect. Gauge theoretic aspects of the geometric langlands. This is a course offered by ashoke sen, called quantum field theory ii. If true, this duality would have tremendous consequences for quantum gauge theory, because it would relate a theory at small values of the coupling constant weak coupling to a theory with large. Curriculum vitae artan sheshmani harvard university. Factorization algebras in quantum field theory by kevin costello.

The physics of gauge theory since the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics. The nlab runs on a server at carnegie mellon university that is supported by muri grant fa95501510053 from the air force office of scientific research. Poincare duality on hochschild cohomology and framed little disk algebra. Oct 21, 20 we observe supersymmetry breaking in superyangmills theory and in theories with small numbers of flavors. Qingyuan jiang categorical plucker formula and homological projective duality. This is done in the brst and batalinvilkovisky formulation. They described the lowenergy limit as a model in the space of vacua, which they identi ed with the atiyahhitchin moduli space of charge 2 monopoles. The branes occurring in topological string theory are naturally organized into linear infinity categories. Without relying on quantum field theory arguments, can one directly deduce a known construction of the jones polynomial or khovanov homology from the gauge theory. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to absence of quantum coulomb branch. Threedimensional quantum supergravity and supersymmetric spin foam models livine, etera r.

The workshop was attended by nearly 40 mathematicians and physicists, including many of the persons mentioned above, as well as others who work on different and important aspects of homological algebra, algebraic geometry and supersymmetric gauge theories. In particular, it has been used to prove derived equivalences between nonbirational calabiyau 3folds. Topological gauge theories represent a sort of compromise. Homological projective duality and quantum gauge theory. Glsms, gerbes, and kuznetsovs homological projective duality eric sharpe virginia tech t pantev, es, hepth0502027, 0502044, 0502053 s hellerman, a henriques, t pantev, es, m ando, hepth0606034. The realization of symmetry in quantum mechanics is through representation theory. We obtain a new perspective on noncommutative gauge theory on a torus. The basic starting point is that the axioms of special relativity on the one hand and those of quantum mechanics on the other, should be combined into one theory. A dual pair can be seen as an adjoint pair in the sense of definition 1. We cannot have a section on derived categories without sasha kuznetsov. Modern moduli theory september 2529, 2017 abstracts of talks.

In quantum physics, in order to quantize a gauge theory, for example the yangmills theory, chernsimons theory or the bf model, one method is to perform gauge fixing. Institute for the physics and mathematics of the universe, tokyo. For larger numbers of flavors, we discover duality between regular theories with different gauge groups and matter contents, where regularity refers to. Gauge theory and langlands duality edward frenkel arxiv. Homological projective duality have a signiycant overlap but there are also some differences. Gravity, on the other hand, is a kind of gauge theory that determines the spacetime geometry itself. A major feature underlying our analysis is the emphasis on free quantum field theory and locality expressed through hochschild co homology.

Once the local net is given, the space of states is obtained as a subset of the dual of am as a. The duality provides a unifying scheme to understand many of the known derived equivalences and also produces more examples. Modern moduli theory september 2529, 2017 abstracts of. Coulomb branches of 3dimensional gauge theories and related. The quantum phases of matter and gauge gravity duality harvard university of florida, gainesville, february 21, 20 subir sachdev thursday, february 21. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to. In many respects duality in field theory and string theory can be used as an. Of the four dimensional gauge theory, and conclude that the quantum hitchin. I am interested in understanding string theory, and some of its relationships to.

Monopoles and 4d gauge theory 3d reduction of yangmills theory seiberg and witten studied the 3d reduction of 4d quantum yangmills theory for su2 a cousin of the donaldson invariants. Kuznetsov has conjectured that pfaffian varieties should admit noncommutative crepant resolutions which satisfy his homological projective duality. With emphasis on special varieties ill discuss what information about homological projective duality one can see from quantum cohomology, and other way around. We prove cases of rietsch mirror conjecture that the amodel of projective homogeneous varieties is isomorphic to the bmodel of its mirror, which is a partially compactified landauginzburg model constructed from lie theory and geometric crystals. Shingtung yau, geometric structures in field and string theory. We obtain a new perspective on noncommutative gauge theory on. Glsms, gerbes, and kuznetsovs homological projective duality. This is a brief survey of work by the authors developed in detail elsewhere 47, 44. The quantum phases of matter and gaugegravity duality harvard university of michigan, ann arbor, march, 20 subir sachdev wednesday, march. Motives, quantum field theory, and pseudodifferential operators. Les houches lectures on fields, strings and duality.

Yang institute for theoretical physics state university of new york at stony brook stony brook, new york 117943840 usa. Gauge theory, mirror symmetry, and langlands duality constantin teleman uc berkeley simons center, stony brook, 21 february 2012 c. This seminar will trace mirror symmetry from its origins to some modern developments. Pdf an introduction to algebraic quantum field theory. Khovanov homology is a knot invariant that is a categorification of the. Relativistic quantum field theory is a mathematical scheme to describe the subatomic particles and forces. Apr, 2017 this is a course offered by ashoke sen, called quantum field theory ii. Then we can define the homology groups, which are given by cycles of. Apr 29, 2010 glsms, gerbes, and kuznetsovs homological projective duality. Louis crane, igor frenkel, fourdimensional topological quantum field theory, hopf categories. The quantum phases of matter and gaugegravity duality harvard university of florida, gainesville, february 21, 20 subir sachdev thursday, february 21.

Quantum yangmills theory the physics of gauge theory. The affine grassmannian grg gkgo is an indprojective scheme, the union. Douglas simons center foundations of qft stringmath 2011 1 38. The path integral in quantum field theory is supposed to be the integral over. Workshop report homological projective duality and quantum. If the theory is successful, it may be considered a. A natural choice of orientation leads to the holomorphic projection of.

The conjecture relates to deep objects in algebraic combinatorics. I will describe some examples of homological projective duality projective bundles, double veronese embeddings and grassmannians of lines. Institute for the physics and mathematics of the universe conference, homological projective duality and quantum gauge theory, march fractional grade restriction rules and autoequivalences of derived categories uc berkeley workshop, tensors and their geometry in high dimensions, sept. Bachtold, on the finite dimensional bv formalism, 2005. Type ii factors have projection classes which can be labeled by real. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from abelian chernsimons theory. This summer in moscow i discussed with kuznetsov, and we found that the examples from gauge theory duality and homological projective duality have a signiycant. Projective duality, quantum cohomology and exceptional approximations. On solution of selfduality equation in quantumgroup gauge. These are gauge theories in two dimensions with n 2. The quantum phases of matter and gaugegravity duality.