Peter van Nieuwenhuizen

Consistency of the AdS7×S4 reduction and the origin of self-duality in odd dimensions

We discuss the full nonlinear Kaluza–Klein (KK) reduction of the original formulation of d=11 supergravity on AdS7×S4 to gauged maximal (N=4) supergravity in 7 dimensions. We derive the full nonlinear embedding of the d=7 fields in the d=11 fields (“the ansatz”) and check the consistency of the ansatz by deriving the d=7 supersymmetry laws from the d=11 transformation laws in the various sectors. The ansatz itself is nonpolynomial but the final d=7 results are polynomial. The correct d=7 scalar potential is obtained. For most of our results the explicit form of the matrix U connecting the d=7 gravitino to the Killing spinor is not needed, but we derive the equation which U has to satisfy and present the general solution. Requiring that the expressionδF=dδA in d=11 can be written as δd(fields in d=7), we find the ansatz for the 4-form F. It satisfies the Bianchi identities. The corresponding ansatz for the 3-form A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for A in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field Sαβγ,A. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.

On Euclidean spinors and Wick rotations

Abstract We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosons. The result is a recipe to construct a supersymmetric Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a five-dimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder-Schrader approach in which spinors are “doubled” but the action is not hermitian. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields.

Topological boundary conditions, the BPS bound, and elimination of ambiguities in the quantum mass of solitons

Abstract We fix the long-standing ambiguity in the one-loop contribution to the mass of a 1 + 1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N = 1 supersymmetric case we find a result which for the supersymmetric sine-Gordon mdoel agrees with the known exact solution of the S -matrix but seems to violate the BPS bound. We analyze the non-trivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N = 2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond one-loop there are no ambiguities in any theory, which we explicitly check by a two-loop calculation in the sine-Gordon model.

Loop calculations in quantum-mechanical non-linear sigma models☆

Abstract By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved spacetime. Although the prescription how to deal with the products of distributions that appear in the computation of Feynman diagrams in configuration space is surprising, this prescription follows unambiguously from the discretized path integral. We check our results by an explicit two-loop calculation.

Consistent boundary conditions for open strings

Abstract We study boundary conditions for the bosonic, spinning (NSR) and Green–Schwarz open string, as well as for (1+1)-dimensional supergravity. We consider boundary conditions that arise from (1) extremizing the action, (2) BRST, rigid or local supersymmetry, or κ(Siegel)-symmetry of the action, (3) closure of the set of boundary conditions under the symmetry transformations, and (4) the boundary limits of bulk Euler–Lagrange equations that are “conjugate” to other boundary conditions. We find corrections to Neumann boundary conditions in the presence of a bulk tachyon field. We discuss a boundary superspace formalism. We also find that path integral quantization of the open string requires an infinite tower of boundary conditions that can be interpreted as a smoothness condition on the doubled interval; we interpret this to mean that for a path-integral formulation of open strings with only Neuman boundary conditions, the description in terms of orientifolds is not just natural, but is actually fundamental.

The Massless Spectrum of Covariant Superstrings

We obtain the correct cohomology at any ghost number for the open and closed covariant superstring, quantized by an approach which we recently developed. We define physical states by the usual condition of BRST invariance and a new condition involving a new current which is related to a grading of the underlying affine Lie algebra.

Rigid supersymmetry with boundaries

We construct rigidly supersymmetric bulk-plus-boundary actions, both in x-space and in superspace. For each standard supersymmetric bulk action a minimal supersymmetric bulk-plus-boundary action follows from an extended F- or D-term formula. Additional separately supersymmetric boundary actions can be systematically constructed using co-dimension one multiplets (boundary superfields). We also discuss the orbit of boundary conditions which follow from the Euler-Lagrange variational principle.

LOOP CALCULATIONS IN QUANTUM MECHANICAL NON-LINEAR SIGMA MODELS WITH FERMIONS AND APPLICATIONS TO ANOMALIES

Abstract We construct the path integral for one-dimensional non-linear sigma models, starting from a given Hamiltonian operator and states in a Hilbert space. By explicit evaluation of the discretized propagators and vertices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems, are now extended to fermionic systems. We then generalize the work of Alvarez-Gaume and Witten by developing a framework to compute anomalies of an n-dimensional quantum field theory by evaluating perturbatively a corresponding quantum mechanical path integral. Finally, we apply this formalism to various chiral and trace anomalies, and solve a series of technical problems: (i) the correct treatment of Majorana fermions in path integrals with coherent states (the methods of fermion doubling and fermion halving yield equivalent results when used in applications to anomalies), (ii) a complete path integral treatment of the ghost sector of chiral Yang-Mills anomalies, (iii) a complete path integral treatment of trace anomalies, (iv) the supersymmetric extension of the Van Vleck determinant, and (v) a derivation of the spin - 3 2 Jacobian of Alvarez-Gaume and Witten for Lorentz anomalies.

On field theory for massive and massless spin52 particles

Abstract Wave equations and actions are derived for free, massive and massless, spin - 5 2 fields, which lead to ghost-free propagators with the correct number of propagating states. The massive theory is constructed from the root method, whereas the massless theory is derived from a direct analysis of the propagator. A discontinuity exists in the description of massive and massless spin - 5 2 fields in the sense that the zero-mass limit of the massive theory does not lead to the massless theory, but contains too many propagating modes.

Quantum corrections to mass and central charge of supersymmetric solitons

Abstract We review some recent developments in the subject of quantum corrections to soliton mass and central charge. We consider in particular approaches which use local densities for these corrections, as first discussed by Hidenaga Yamagishi. We then consider dimensional regularization of the supersymmetric kink in 1+1 dimensions and an extension of this method to a (2+1)-dimensional gauge theory with supersymmetric abelian Higgs vortices as the solitons. In the case of the supersymmetric N =1 kink, the anomalous contribution to the central charge which is required for BPS saturation can be understood as the remnant of parity violation of the (2+1)-dimensional theory where a minimally supersymmetric kink can be embedded keeping the same field content. In the case of the (2+1)-dimensional N =2 vortex, BPS saturation at the quantum level follows from explicit calculations of mass and central charge at the one-loop level. The number of fermionic zero modes in the vortex background is such as to enforce BPS saturation to all orders by multiplet shortening.