[Paper Review] Reminiscences of Collaborations with Joel Scherk

John H. Schwarz|ArXiv.org|Jul 14, 2000

Particle physics theoretical and experimental studies35 references3 citations

TL;DR

This paper recounts John H. Schwarz's collaborative work with Joël Scherk across three pivotal periods (1970, 1974, 1978–79), highlighting their contributions to early string theory, including the identification of critical spacetime dimensions (26 for bosonic, 10 for superstring), the resolution of unitarity issues in loop amplitudes via Virasoro constraints, and the development of a field-theoretic mechanism for spontaneous supersymmetry breaking through twisted compactification. The key contribution is a consistent, classically formulated mechanism for breaking supersymmetry in higher-dimensional theories using R-symmetries and nontrivial monodromies on compactified dimensions.

ABSTRACT

I had the privilege of collaborating with Joel Scherk on three separate occasions: in 1970 at Princeton, in 1974 at Caltech, and in 1978-79 at the Ecole Normale Superieure. In this talk I give some reminiscences of these collaborations.

Motivation & Objective

• To document and reflect on the collaborative research journey with Joël Scherk across three distinct periods in theoretical physics.
• To elucidate the development of early string theory, particularly the emergence of critical dimensions (26 and 10) and the resolution of unitarity issues in loop amplitudes.
• To present and analyze a field-theoretic mechanism for spontaneous supersymmetry breaking via twisted compactification on a circle with R-symmetry monodromy.
• To explore the implications of this mechanism for constructing nonsupersymmetric four-dimensional effective field theories from higher-dimensional supersymmetric theories.
• To highlight the lasting influence of Scherk’s ideas, especially in the context of modern brane-world scenarios and string compactifications.

Proposed method

• Analyzing one-loop amplitudes in dual resonance models using Jacobi transformations of theta functions to isolate divergences and propose counterterms.
• Applying Virasoro subsidiary constraints in higher-dimensional spacetime (26D for bosonic, 10D for superstring) to convert unitarity-violating branch points into physical poles.
• Studying nonplanar loop amplitudes to identify unexpected singularities and linking them to Regge trajectories, particularly the Pomeron.
• Formulating a mechanism for supersymmetry breaking via compactification on a circle where fields acquire a phase transformation under monodromy, induced by an R-symmetry group element.
• Constructing a four-dimensional effective theory from five-dimensional maximal supergravity (N=8) compactified on a circle, with the R-symmetry element specifying four real mass parameters for the gravitino multiplet.
• Using group-theoretic analysis of USp(8) R-symmetry to classify the resulting four-parameter deformation of N=8 supergravity in four dimensions.

Experimental results

Research questions

• RQ1 How can divergences in one-loop amplitudes of the dual resonance model be systematically isolated and canceled using modular properties of theta functions?
• RQ2 Why do nonplanar loop amplitudes in four-dimensional string theories exhibit unitarity-violating branch points, and how can this be resolved by extending spacetime to 26 dimensions?
• RQ3 What is the role of the Virasoro constraints in transforming unphysical singularities into physical poles in the nonplanar amplitude?
• RQ4 How can supersymmetry be broken in a consistent, local, and classically well-defined manner in a higher-dimensional field theory with extra compact dimensions?
• RQ5 Can a four-dimensional effective theory with massive gravitinos arise from a compactified higher-dimensional supergravity theory with nontrivial R-symmetry monodromy?

Key findings

• The divergent piece in the planar one-loop amplitude was isolated via a Jacobi transformation of theta functions, leading to a natural counterterm that corresponds to the tachyon exchange in the dual channel.
• The nonplanar amplitude contains unexpected singularities that are unitarity-violating branch points in four dimensions, but become physical poles when the spacetime dimension is raised to 26, due to the Virasoro constraints.
• Lovelace's analysis showed that in 26 dimensions with transverse oscillator dominance, the singularities become poles, proving that unitarity requires the critical dimension and specific intercept values.
• A consistent mechanism for spontaneous supersymmetry breaking was developed in a field-theoretic setting: compactifying on a circle with an R-symmetry monodromy that gives mass to the gravitino without breaking gauge invariance.
• In five-dimensional N=8 supergravity compactified on a circle, the R-symmetry element induces four real mass parameters for the four complex gravitinos, resulting in a four-parameter deformation of the four-dimensional N=8 theory.
• The mechanism remains viable in modern contexts such as brane-world scenarios with large compactification scales, suggesting potential relevance for realistic model building.