Topological Quantum Field Theory and Four Manifolds

Labastida, J.M.F.; Mariño, Marcos

doi:10.1023/b103664

Cited by 3 publications

References 19 publications

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An Introduction to Donaldson-Witten Theory

Mariño

2003

Geometric and Topological Methods for Quantum Field Theory

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In these lecture notes a pedagogical introduction to Donaldson-Witten theory is given. After a survey of four-manifold topology, some basic aspects of Donaldson theory are presented in detail. The physical approach to Donaldson theory is based on topological quantum field theory (TQFT), and some general properties of TQFT's are explained. Finally, the TQFT underlying Donaldson theory (which is usually called Donaldson-Witten theory) is constructed in detail by twisting N = 2 super Yang-Mills theory. ContentsGeometric and Topological Methods for Quantum Field Theory Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM LIBRARY -INFORMATION SERVICES on 03/22/15. For personal use only. 266 3 N = 1 supersymmetry 3.1 The supersymmetry algebra 3.2 N = 1 superspace and superfields 3.3 Construction of N = 1 Lagrangians 4 n/ = 2 super Yang-Mills theory 5 Topological field theories from twisted supersymmetry 5.1 Topological field theories: basic properties 5.2 Twist of N = 2 supersymmetry 6 Donaldson-Witten theory 6.1 The topological action 6.2 The observables 6.3 Evaluation of the path integral 7 Conclusions and further developments A Appendix: Conventions for spinors References

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An Introduction to Donaldson-Witten Theory

Mariño

2003

Geometric and Topological Methods for Quantum Field Theory

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Introduction to Chern-Simons Theory

Zanelli¹,

Huerta²

2011

Proceedings of 4th International Conference on Fundamental Interactions — PoS(ICFI 2010)

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Chern-Simons theories occur in Physics more often than one may think. In particular, the Hamiltonian action for any mechanical system with finite number of degrees of freedom is a CS Lagrangian. CS forms can be seen as generalizations of the coupling between the electromagnetic field and a point charge. It should come as no surprise, therefore, that the CS forms are perfectly suited to describe the consistent coupling of a (nonabelian) gauge connection to extended charged objects (branes). An application of this is provided by a spacetime region M where the electromagnetic field is described by the Lagrangian 1 4 (F ∧ * F + θ F ∧ F). Since the Maxwell equations are unaffected by the presence of the θ-term, this represents a slight modification of the vacuum in M (θ-vacuum). This modification is shown to correspond to the addition of a form of matter that couples through a CS form on the surface of M. The connection to superconductivity is also briefly mentioned.

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Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

Tan¹

2011

Advances in Theoretical and Mathematical Physics

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We continue our program initiated in [1] to consider supersymmetric surface operators in a topologically twisted N = 2 pure SU (2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsváth and Szabó [2,3] and Taubes [4], will be furnished. In particular, we will show that Taubes' groundbreaking and difficult result -that the ordinary SW invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold X -nonetheless lends itself to a simple and concrete physical derivation in the presence of "ordinary" surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and "ramified" SW invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the "ramified" Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important e-print archive: http://lanl.arXiv.org/abs/1006.3313 MENG-CHWAN TANmathematical results concerning (i) knot homology groups from "ramified" instantons by Kronheimer and Mrowka [5]; and (ii) monopole Floer homology and SW theory on symplectic four-manifolds by KutluhanTaubes [4,6]. Supersymmetry, as well as other physical concepts such as R-invariance, electric-magnetic duality, spontaneous gauge symmetrybreaking and localization onto supersymmetric configurations in topologically twisted quantum field theories, play a pivotal role in our story.

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Topological Quantum Field Theory and Four Manifolds

Cited by 3 publications

References 19 publications

An Introduction to Donaldson-Witten Theory

An Introduction to Donaldson-Witten Theory

Introduction to Chern-Simons Theory

Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

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