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Gravitational Instantons Gravitational Instantons N. H. Pavao 2019 Preface In this essay we will lay out the motivation for understanding finite action topological invariants in a classical field theory, less formally known as instantons, and their importance for quantization. We will begin by investigating how these mathematical structures emerge in Yang-Mills theory, particularly in ��(�) gauge theories with the BPST instanton [1]; and how they might give rise to physically significant corrections in an effective QFT under renormalization [15]. In the quantum version of field theory, instantons, being local minimum of the action functional, play a critical role in introducing non-perturbative corrections to our path integral [10, 15, 18]. Hence, when performing path integral quantization on any classical field theory, one must first be able to identify and classify the instanton configurations that are present. This line of thinking leads us to the goal of what we wish to develop in writing this essay: a classification of spacetime instantons that could prove crucial when performing path integral quantization on a theory of gravity. Though significantly more difficult than quantization of a gauge theory over a Minkowski spacetime, it was shown that the gravitational path integral is positive definite [8, 10] (under suitable physical assumptions) and thus identifying finite action spacetime configurations could prove fruitful in a future perturbative theory of quantum gravity. In our investigation, we will illuminate some of the physical characteristics and constraints on gravitational instantons: the property of self-dual curvature [1, 5, 10], their characteristic topological invariant quantities, their boundary characteristics over polyhedral isometry groups, and their asymptotic behavior; why they are asymptotically locally Euclidean (ALE) [5, 10], instead of globally. Once we have identified the desirable constraints and have fully classified the necessary conditions on a spacetime configuration to be a local minimum of our gravitational path integral, we will proceed by identifying some of the standard such solutions: Taub-NUT, [12] Eguchi-Hanson [5] and Gibbons-Hawking multi-instanton [9]. To make the similarities these gravitational instantons share with gauge theory manifest, we will demonstrate how one might recover the canonical BPST instanton from the Eguchi-Hanson metric [5]. To conclude our review of these mathematical structures (and how they arise naturally in a path integral formulation of gravity), we will demonstrate that gravitational instantons are in fact of a subclass of Kähler manifolds [13]. Kähler geometries are of particular interest because they permit a complex structure (mimicking that of the ��(2, ℂ) universal cover of Minkowski space), they have a nondegenerate metric (critical for any macroscopic theory of gravity), and additionally they have a well-defined symplectic 2-form (which provides the geometry underlying classical mechanics). Moreover, it can be shown that K3 surfaces and Calabi-Yau manifolds, believed to be structure of the compactified extra-dimensions in superstring theory, too satisfy all the conditions that classify an instanton [16]. Could our universe be constructed from a web of instantons? Maybe. Before we can investigate this claim, we must reveal the historical motivation for identifying these nontrivial vacua and formulate the physical theory that underlies them. To L. M. Pavao - 1 - Contents Preface ..…………………………………………………………………………………………..… 1 I. Preliminaries Gauge Theory …………………………………………………………………………………..…. 3 Path Integral Quantization ..………………………………………………………………… 4 Wick Rotation ..…………………………………………………………………………………… 7 II. Yang Mills Instantons Yang Mills Action ...………………………………………………………………………….….. 9 Topological Invariants in SU(2) …………………………………………………………... 11 The Unit (BPST) Instanton …………………………………...…………………………….. 12 The New Quantum Vacuum ……………………………………………………………….. 16 Yang Mills SO(4) Instanton ……….……………………………………………………….. 18 III. Gravitational Instantons Positive Action Conjecture ………..……………………………………………………..….. 20 Schwarzschild and Taub-NUT ……….…………………………………………………..… 20 Eguchi-Hanson Instanton ………………………………………………………………..….. 24 General Instanton Solutions ……...………………………………………………………... 29 Boundary Characteristics of Self-Dual Metrics …...……………………………….... 32 Kähler Geometry and Compact Self-Dual Manifolds …………………...……….... 32 IV. Conclusion Acknowledgements ………………………………………….………………………………... 36 References ……………………………………………………….………………………….…….. 37 Appendix A: Vector Bundles, Gauge Transformation and Connections …… 38 Appendix B: ‘t Hooft Symbols …………………………………………………………..….. 43 Appendix C: Vierbeins and Spin Connection …………………………………………. 46 - 2 - I. Preliminaries Gauge Theory The great triumph of 20th century physics was formulating dynamical theories in terms of symmetries. It gave rise to the unification of electroweak theory and paved the way in QCD for the construction of quarks governed by an exact ��(3) symmetry. Global transformations that leave the dynamics of physical observables invariant provide the structure of physical principles that we understand today. However, one could argue that more important than symmetry in physics, is an apparent redundancy of physical theory: the property of gauge symmetry. It was posited that if a theory possessed an internal symmetry, local transformations belonging to that symmetry group could be made on a field that would leave Lagrangian, and thus equations of motion, invariant. But formulation of such a theory is difficult; how ought one construct a Lagrangian that is preserved when transformations across the entire base manifold are wildly varying? The solution was constructed by Yang and Mills, and takes the following form [2]: 1 ℒ = ��(� ∧ ⋆ �) qr 2 where � is the field strength tensor, or curvature 2-form, for a given G-valued vector bundle (Appendix A). In terms of the Lie algebra valued connection 1-forms, the tensor � can be written as ~ • ~ • � = �� + � ∧ � = |�~�• − �•�~ + •�~, �•‚ƒ�� ∧ �� = •�~, �•‚�� ∧ �� The general gauge transformation by a group element �: � → � on the field strength tensor then behaves in the following way: � → �Š = �(�)‹Œ��(�) One can think of this as analogous to a coordinate transformation on some manifold with metric �~•. Here, the gauge transformation serves as a transformation from one section of the vector bundle to another, such that �(�): Γ(�) → Γ(�) Here, Γ(�) is the space of sections on the total space, �, of our vector bundle. At • the level of the vector potential, �~, for a general internal symmetry group �, the transformation acts in the following way (Appendix A) • ‹Œ • ‹Œ �~ → � (�)�~(�)�(�) + � (�)�~�(�) - 3 - As we proceed, the takeaway is that when identifying instanton solutions in Yang- Mills theory, we will be looking for them modulo some gauge transformation. For each solution we identify, there will exist an infinitely large class of equivalent solutions. Hence, we will often conduct our search in a gauge that makes the physical symmetries of our solution manifest. A second key motivation for instanton solutions in gauge theory arises from the path integral formulation of quantum field theory. A brief discussion on this topic is in the section to follow. Path Integral Quantization The inception of Yang Mills theory was a breakthrough in the classical field theory formulation of particle physics. It extended the class of gauge invariant theories beyond the trivial case of Electromagnetism with �(1) phase symmetry, to an arbitrarily large internal symmetry group. The next step towards constructing a theory of microscopic particle dynamics was quantizing this new theory. There are two standard methods for transitioning from the classical to quantum world. The first being canonical quantization. In standard Quantum Mechanics, this takes the form of •� (�), � (� )• = � �(� − � ) ⇒ •�š (�), �̂ (� )‚ = �� �(� − � ) ‘ “ ” –— ‘“ ” ‘ “ ” ‘“ ” By promoting the position and conjugate momentum functions to operators and replacing the bilinear antisymmetric Poisson bracket with the Lie bracket, we thereby are able to encode all the wave like properties of single particle quantum mechanics. The way one would carry out this scheme on a classical field is as follows: Define a scalar field �(�⃗, �) = �(�~) over some � + 1 dimensional spacetime, and conjugate momentum as �ℒ �(�⃗, �) = �(�~) = ��̇ Above we have taken the freedom to let the argument of our field, �~, to be the components of the spacetime � + 1-vector with standard coordinate basis elements •�~•. We impose the following commutation relations on the field operators as such: �� , (� − �)¦ > 0 •� (�~), � (�~)‚ = ¥ ¢£ ¢ £ 0, ��ℎ������ This is to say that when the field � and its conjugate � are causally disconnected, they commute. From these second quantization relations, all dynamics of quantum field theory will follow. As one can see, the commutation relations for standard single particle QM are the same as those for QFT, with spatial dimensions � = 0. With this in mind, we will investigate the second method of quantization only in the world of 0 + 1 dimensional quantum field theory to motivate what it might look like in a richer theory of arbitrary spatial dimensions. - 4 - Suppose we wish to find the value of a spatially dependent wave function �(�, �) at some time �, given a reference state |�(�”)⟩ at time � = �”. Working in the Schrodinger picture, the time dependence is encoded in the state. Our Hamiltonian, �°, as always serves as our time evolution operator.
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