Heat kernel expansion: user's manual

Superstrings in the Early Universe

The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Variational Principles in Mechanics

The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local boundary conditions, in the presence of various types of singularities (e.g., domain walls). In each case the heat kernel coefficients are given in terms of several geometric invariants. These invariants are derived for scalar and spinor theories with various interactions, Yang-Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding anomalous actions, and covariant perturbation expansions of the effective action (both "low-" and "high-energy" ones).

Liouville theory is shown to describe the asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. This is because (i) Chern - Simons theory with a gauge group on a spacetime with a cylindrical boundary is equivalent to the non-chiral SL(2,R) WZW model; and (ii) the anti-de Sitter boundary conditions implement the constraints that reduce the WZW model to the Liouville theory.

/pdf/the-asymptotic-dynamics-of-three-dimensional-einstein-4ss2j8eod8.pdf

The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant

We consider a free (2 k)-form gauge-field on a Euclidean (4 k + 2)-manifold. The parameters needed to specify the action and the gauge-invariant observables take their values in spaces with natural complex structures. We show that the correlation functions can be written as a finite sum of terms, each of which is a product of a holomorphic and an anti-holomorphic factor. The holomorphic factors are naturally interpreted as correlation functions for a chiral (2 k)-form, i.e. a (2 k)-form with a self-dual (2 k + 1)-form field strength, after Wick rotation to a Minkowski signature.

Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory

Chairmen's PrefaceIn 1984 we obtained a grant from the Nobel Foundation to organize a Nobel Symposium on "Unification of the Fundamental Interactions". In our proposal which we submitted in the fall of 1983 we stated that we wanted to cover the various attempts to unification such as GUT'S, supergravity, Kaluza–Klein theories and superstrings. What has happened in particle physics since then is already history. With the realization that certain superstring theories could be anomaly free, it became clear that these models could encompass earlier attempts to unification as well as solving the fundamental problem of quantum gravity. The excitement that some of us had felt for some time now spread through most of the particle physics community and this excitement certainly was evident during the Symposium. With the international advisory committee we originally chose a list of around 30 invitees which could best represent the various subjects listed above. When it came to the final planning of the programme essentially all talks dealt with superstrings! We were very fortunate that almost all of the invitees managed to come to the Symposium. From the western world only three were unable to participate, Andre Neveu, Steven Weinberg and Bruno Zumino. We certainly missed them during the meeting. We were particularly happy that Stephen Hawking managed to take part actively. Our real problem was to get participants from the Soviet Union. Out of eight invitations only one came through. We were very happy to have Renata Kallosh, who really did her utmost to enlighten us about not only her own work but also about recent progress in the USSR, However, we were very sorry that in spite of all our letters, telegrammes and endless attempts to get telephone calls through and despite the good relations between the Swedish and Soviet Academies of Sciences we had to miss Ludwig Faddeev, Valodja Gribov, Andrej Linde, Victor Ogievetsky, Sasha Polyakov, Misha Shifman and Arkadij Vainshtein. We had hoped that the Nobel Symposium should be a signal for improved relations between East and West, but in this respect we failed. The Symposium was held in the small town of Marstrand, a summer resort on an island outside Goteborg. The idea was to have it relatively close to home and to show the participants perhaps the best part of the local nature. Another motive was to keep the participants semi-isolated to provide ample time for discussions. With the somewhat heavy programme we found that even so, the days were not long enough. This Symposium was the second Nobel Symposium on elementary particle physics. The first one, also organized by our group, was held in 1968. We hope and believe that the next one need not be awaited another 18 years! The Symposium was made possible by a generous grant from the Nobel Foundation. An additional grant from the Royal Swedish Academy of Sciences was also essential. A grant from Nordita made it possible to invite some ten Nordic observers. We also gratefully acknowledge help with the organization of the Symposium from the city of Goteborg, the city of Kungalv, Volvo, Skandinaviska Enskilda banken, Ericsson AB and IBM Sweden. We thank all the participants for all their efforts, to come to our remote part of the world, to give excellent talks, to write up their talks astonishingly quickly and to share with us so much of their knowledge and expertise. Last but not least we are indebted to the members of the organizing committee, members of the group and the secretarial staff. Goteborg, August 1986 Editors' NoteIn these Proceedings the lectures are given in the order of their oral presentation at the Symposium. Regrettably, a few contributors could not participate (V. Gribov, A. Linde, V. Ogievetsky, M. Shifman and A. M. Polyakov). Their contributions have been placed close to the end of the Proceedings. As these authors did not have the opportunity of correcting the proofs, responsibility for printing errors must rest with the Editors. It is a pleasure to thank all the authors for their enthusiastic and rapid response in our efforts to bring the contributions to the stage of printing. Without the efficient and fruitful collaboration with Physica Scripta the publication of these Proceedings would have been substantially delayed.

Unification of Fundamental Interactions

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} \lambda_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons.

On Functional Determinants of Laplacians in Polygons and Simplices

We consider type IIB string theory on a seven dimensional orbifold with holonomy in G2. The motivation is to use D1-branes as probes of the geometry. The low energy theory on the D1-brane is a sigma-model with two real supercharges (N = (1,1) in two dimensional language). We study in detail the closed and open string sectors and propose a coupling of the twisted fields to the brane that modifies the vacuum moduli space so that the singularity at the origin is removed. Instead of coming from D-terms, which are not present here, the modification comes from a ``twisted'' mass term for the seven scalar multiplets on the brane. The proposed mechanism involves a generalization of the moment map.

D-brane probes on G2 Orbifolds

Standard-model-like string theories from covariant lattices

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