arXiv:hep-th/9304011 v1 Apr 5 1993

Exercise. Missing lessons
Determine which of the lessons below are covered quite elegantly in portions of text
that have been omitted from these lecture notes [0] but will be restored for the book
version [19].
15.1. Lessons
From the quantum gravity point of view, the main lessons we have learned from the matrix
model are:
• Euclidean Quantum Gravity makes sense, at least in two dimensions.
• The nature of quantum states in Euclidean quantum gravity, and their interpretation
within the quantum mechanical framework is surprising, and requires the introduction
of non-normalizable wavefunctions as well as normalizable wavefunctions.
• The Wheeler–DeWitt constraint is violated in topology-changing processes.
• The contributions of singular geometries to the path integral of quantum gravity are
important.
• There is a phase of topological gravity which can be connected to phases of nontopological
gravity.
From the string theory point of view, the main lessons we have learned from the matrix
model are:
• Nonperturbative definitions of string physics, at least in some target spaces, exist.
• There are backgrounds with large unbroken symmetries, e.g., w 1+∞ and volume preserving
diffeomorphism algebras.
• The large order behavior of perturbation theory at order g has the typically “stringy”
(2g)! growth.
• In solvable string theories, there is a beautiful mathematical framework (KP flow,
W -constraints, etc.) that relates string physics in different backgrounds.
• With current understanding, it is fundamentally impossible to achieve complete background
independence: There is always dependence on boundary and initial conditions
associated with non-normalizable states.
• There is a phase of string theory which is topological, and can be connected to nontopological
phases with local physics (such as string scattering in two dimensions).
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