Neveu–Schwarz model

Neveu–Schwarz model
Name	Neveu–Schwarz model
Inventors	André Neveu, John H. Schwarz
Introduced	1971
Field	Theoretical physics
Related	Ramond sector, Superstring theory, Dual models

Neveu–Schwarz model

Introduction

The Neveu–Schwarz model is an early dual resonance model developed by André Neveu and John H. Schwarz that introduced worldsheet fermions into the framework of dual models and influenced the formulation of string theory; its origins connect to work by Gabriele Veneziano, Miguel Virasoro, Pierre Ramond and contributions at CERN and Princeton. The model provided a route from the dual resonance program to modern superstring constructions such as those pursued by Edward Witten, Michael Green, David Gross, and Philip Candelas, and it played a role in the development of heterotic constructions by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm.

Historical Development and Motivation

Neveu and Schwarz proposed the model in the context of attempts to reconcile the Veneziano amplitude and the Virasoro algebra with fermionic degrees of freedom; their work interacted with contemporaneous efforts by Pierre Ramond, Joël Scherk, and John Schwarz at institutions including CERN, Princeton University, and Caltech. Motivations drew on spectra studied by Stanley Mandelstam, Leonard Susskind, and Yoichiro Nambu and on algebraic structures analyzed by Miguel Virasoro and Sergio Fubini, while later impact was seen in research by Michael Green and John Schwarz that led to anomaly cancellation results with contributions from Michael Duff and Paul Townsend.

Mathematical Formulation

The model is formulated on a two-dimensional worldsheet with bosonic coordinates and anticommuting fermionic fields satisfying mode expansions that realize an extension of the Virasoro algebra, incorporating generators akin to those studied by Virasoro, Pierre Ramond, and Claudio Rebbi; the algebraic structure informed subsequent operator formalisms by Stanley Mandelstam and Joel Scherk. Mode operators obey anticommutation relations analogous to those used by Pascual Jordan, Wolfgang Pauli, and Freeman Dyson in early quantum field theory, and the construction employs techniques refined by Paul Dirac, Julian Schwinger, and Richard Feynman. The Neveu–Schwarz algebra couples to central charge considerations encountered in the work of Alexander Polyakov, Belavin, Alexander Zamolodchikov, and Ludwig Faddeev.

Physical Spectrum and States

The spectrum of the model contains bosonic excitations including a tachyonic ground state and higher excitations comparable to those found in the Veneziano model and studied by Gabriele Veneziano, Miguel Virasoro, and Leonard Susskind; fermionic excitations emerge through Neveu–Schwarz boundary conditions paralleling analyses by Pierre Ramond and Michael Green. Physical state conditions involve constraints analogous to those examined by Claudio Rebbi, Stanley Mandelstam, and Joël Scherk and informed massless sectors relevant to later work by Edward Witten, David Gross, and Paul Townsend on anomaly cancellation and compactification issues explored by Philip Candelas and Andrew Strominger.

Worldsheet Supersymmetry and Boundary Conditions

Worldsheet supersymmetry in the model links to supersymmetry concepts developed by Julius Wess, Bruno Zumino, and Peter West and interfaces with boundary conditions distinguishing Neveu–Schwarz and Ramond sectors, which were analyzed alongside work by Pierre Ramond and Ferdinando Gliozzi. Implementations of periodic and antiperiodic conditions relate to spectral-flow ideas studied by Alexander Belavin, Alexander Polyakov, and Alexander Zamolodchikov, and these choices impacted modular invariance considerations examined by Edward Witten, Michael Green, and John Schwarz in later formulations.

Vertex Operators and Amplitudes

Vertex operators in the Neveu–Schwarz model generalize constructions introduced by Gabriele Veneziano and Miguel Virasoro and make use of operator-state correspondence techniques developed by Belavin, Polyakov, and Zamolodchikov; amplitudes computed via these operators were evaluated using methods refined by Richard Feynman, Julian Schwinger, and Stanley Mandelstam. Correlators and sewing prescriptions informed later path integral treatments by Alexander Polyakov and the BRST quantization techniques of Carlo Becchi, Alain Rouet, and Raymond Stora, with anomaly analyses that connect to results by Michael Green and John Schwarz.

Relation to Ramond Sector and Superstring Models

The Neveu–Schwarz model is paired with the Ramond sector introduced by Pierre Ramond to produce full superstring models such as the type II and heterotic strings developed by Michael Green, John Schwarz, David Gross, and Jeffrey Harvey; these combined constructions were central to the first superstring revolution led by Edward Witten and Michael Green. The interplay between sectors informed GSO projection methods proposed by Ferdinando Gliozzi, Joël Scherk, and David Olive and influenced compactification scenarios analyzed by Philip Candelas, Andrew Strominger, and Edward Witten, ultimately impacting phenomenological explorations by John Schwarz, Michael Green, David Gross, and others.

Category:String theory