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 = ��(� ∧ ⋆ �) 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

� = �� + � ∧ � = �� − �� + �, ��� ∧ �� = �, ��� ∧ ��

The general gauge transformation by a group element �: � → � on the field strength tensor then behaves in the following way:

� → � = �(�)��(�)

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)

� → � (�)�(�)�(�) + � (�)��(�)

- 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

� (�), � (� ) = � �(� − � ) ⇒ � (�), �̂ (� ) = �� �(� − � )

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. We proceed by noting,

() �(�, �) = ⟨�|�(�)⟩ = ⟨�| �(�)⟩ = ⟨�|� |�(�)⟩

Inserting a complete set of position eigenstates into the above expression we find,

() = ⟨�|� |�⟩⟨�|�(�)⟩�� = �(�, �; �, �)�(�, �)��

Where we have defined the Kernel as follows,

() �(�, �; �, �) ≔ ⟨�|� |�⟩

The Kernel (called the propagator in Quantum Field Theory) defines the transition probability for a particle to go from position � to some position � in (� − �) amount of time. In the free particle case, the Hamiltonian is given by

� � = 2�

The above expression can be evaluated exactly and is found to be:

()/ ()/ �(�, �; � − �) = ⟨�|� |�⟩ = ⟨�|� |�⟩⟨�|�⟩��

= � ()/�()��

��(� − �) � �(�, �; � − �) = � exp , � = 2(� − �) 2��(� − �)

For an arbitrary Hamiltonian operator with some spatially dependent potential �(�), evaluating the Kernel is no trivial task since not the potential and kinetic terms do not necessarily commute! (This is where the mathematical equivalence of canonical quantization and the path integral formulation is made manifest) To resolve this, we use the following formula, attributed to Sophus Lie,

� = lim �/�/ →

Applying this to our expression for the Kernel we find,

()() / () ⟨�|� |�⟩ = lim ⟨�|� � |�⟩, � = (� − �)/� →

Proceeding, by inserting � − 1 factors of unit operators between the product of matrix exponentials, we find

- 5 - �(�, �; � − �) / () / () = lim �� … �� ⟨�|� |�⟩� ⟨�|� |�⟩ … ⟨�|� |�⟩� →

() () = lim �� … �� �(�, �; �)� �(�, �; �) … �(�, �; �)� → () / () () / () / () = lim � �� … �� � � � … � � →

Giving us the following result

()() �(�, �; �, �) = lim � �� … ��� →

� � − � �()(�) = − �(� ) � , � = (� − � )/� 2 �

In the limit where � → ∞, we arrive at the following expression:

� �()(�) → �̇ − �(�) �� = ℒ(�, �̇; �)�� = �(�) 2

Hence, for Quantum Mechanics, the Kernel for a single particle in a given potential �(�) reduces to () () �(�, �; �, �) = �� � ()

Where �� is the path integral measure, the sum over “all paths”, defined as

�� = lim ��� →

But in this essay, we are concerned with how quantum fields propagate through a spacetime, not single particles with a certain position function labeled by a ‘time’ variable. However, the above is in fact a quantum field propagator; simply one defined for a bosonic field � over some time (� − �) in a point-like universe. Moving forward, in QFT we are only concerned with on-shell behavior; and thus we take the integral to be over the entire background spacetime such that (� − �) → ∞.

By bootstrapping our above analysis, we arrive at the following expression for the transition probability of a quantum scalar field �(�):

() () ��, �; � , � = �� � , �(�) = ℒ�, ��; � � �

Or more succinctly, and in the notation typically seen in the literature on quantum field theory,

- 6 - ⟨�|�⟩ = �� �()

But there is a problem that we have glossed over: as defined, the infinite product of Lebesgue measures over a complex exponential is ill-defined! To resolve this, we must perform a Wick rotation, and complexify our time coordinate, thereby turning our transition amplitude into a partition function, opening the door between the analysis of quantum fields and the study of statistical physics.

Before we move on, let’s note what has been done in the above derivation of the path integral. We have constructed a quantum version of a classical field theory mathematically equivalent to canonical quantization. This is no small adjustment. For we now have an opportunity to stray away from the abstract domain of algebra and vector spaces, and venture into the realm of analysis and differential geometry. By constructing the path integral formulation of QFT, it should now be readily apparent that not all field configurations are created equal; those which produce local extrema will give rise to greater contributions to the transition amplitude by stationary phase (or by exponential suppression in the case of partition functions).

Wick Rotation

As mentioned, the path integral in Minkowski space as posed is ill-defined. There is no way to guarantee that a given field configuration returns a positive valued action and spacetime fields are singular on the light-cone. Since the metric is not positive definite, our action becomes unbounded from below, and the path integral, in general, cannot be solved analytically. In a theory whose dynamics are predicated on statistical probability, this poses a major problem. The resolution to this setback is to return the background spacetime metric to the world of Euclidean geometry.

To the achieve this, we perform a Wick rotation, by “rotating” our time coordinate into the complex plane, thereby performing the following transformation time variable, and its corresponding 1-form and basis vector:

� → ��, �� → ���, � → −��

Such that our 4-vector norm takes the following form:

�� = (� + � + � − � ) → (� + � + � + � )

The effect on the relevant Lorentz transformations are as follows (notice that Wick rotation changes the relative sign of our frame velocity):

� → � = �(� − ��), � → � = �(� + ��)

� → � = �(� − ��), � → �′ = �(� − ��)

/ / � = (1 − � ) , � = (1 + � )

- 7 - In matrix form, we can see the effect it has on the rapidity of a boost:

1 −� cosh � − sinh � Λ = � = , � = cosh � ∈ (−∞, ∞) −� 1 −sinh � cosh �

1 � cos � sin � � � Λ = � = , � = cos � ∈ − , −� 1 −sin � cos � 4 4

This method raises two important questions. First, is the causal structure of our spacetime preserved under Wick Rotation? As we can see above, a Wick rotation changes the definition of the Lorentz factor, �. This redefinition makes it such that now our � factor is bounded, thereby limiting the regions of Euclidean spacetime to which we are causally connected.

This is critical, for it will preserve all the causal properties we expect of fields in a Minkowski spacetime (like the antipodal identification of the Liénard-Wiechert potential at spatial infinity). But most importantly, our space is now everywhere regular; light- cones no longer exist on our complexified spacetime with Euclidean signature. The singularities that would arise on the light-cone in our instanton solutions will now be well behaved under a Wick rotation [12].

Second, how does this change our analysis of the path integral? What are the relevant quantities and why? Before Wick rotation, the relevant field configurations were those that kept the phase stationary. The non-classical solutions would create violent oscillations in the phase and cancel themselves in our evaluation of the Kernel.

Now, a similar phenomenon will occur. Instead of identifying constant phase (action) configurations, one is now concerned with where the Euclidean action is minimized. Small deviations from the minimum will now be exponentially suppressed. This is exactly the same behavior in thermodynamics when summing up configurations in a canonical ensemble: the lowest energy configurations are the most likely to occur and thus give the largest contributions in evaluation of the 2-point function.

Hence, under a Wick Rotation, our propagator can be written as a correlation function in the form familiar to statistical field theory:

⟨�|�⟩ = �� �() where the path integral measure �� as before indicates the sum over all field configurations in our new metric space with Euclidean signature. It now becomes manifestly clear that classically speaking we are concerned with minima of the Euclidean action (sometimes referred to as a quasi-energy with its resemblance to the Gibb’s free energy functional in statistical physics) as all other contributions will be exponentially suppressed.

With this, we begin our quest for instantons, elusive pseudo-particle beasts that serve as valleys in the infinite ‘action space’ of decoherent quantum fields.

- 8 - II. Yang Mills Instantons

Yang Mills Action As was previously shown in the preliminary material, the Yang Mills gauge invariant Lagrangian for a general symmetry group is given by [2]

1 1 ℒ = ��(� ∧ ⋆ �), ℒ = ℒ 2 4�

Where the curvature (field strength tensor) � is defined as always in terms of Lie Algebra valued connection 1-form � as follows [6]:

� = �� + � ∧ �

With the Hodge-star operator ⋆ defined in the usual way

vol = |det �|��

1 … ⋆ � = � … � �� ∧ … ∧ �� �! …

Integrating this over our spacetime manifold � with standard basis of coordinate 1- forms {��} gives us the following expression for the Yang Mills action in terms of the spacetime coordinates [1] 1 � = ���� � � 4

In what follows, we will work exclusively in form notation. We can rewrite the action as,

1 1 � = ��(� ∧ ⋆ �) = ��[(� ± ⋆ �) ∧ (� ± ⋆ �) ∓ (� ∧ �) ∓ (⋆ � ∧⋆ �)] 2 4

This can be further simplified by noting that,

1 1 ⋆ � ∧⋆ � = (⋆ �) (⋆ �) �vol = � � ϵ ϵ �vol (2!) (2!) 1 1 = � � ϵ �vol = � � �vol = � ∧ � (2!) (2!)

Hence, we arrive at the following equivalent expression

1 1 � ≡ ��[(� ± ⋆ �) ] ∓ ��[� ∧ �] 4 2

It is important to note that we have made no changes to our original definition of the Yang Mills action. Hence, any characteristic properties we find from this expression were always there; but hidden by the more compact form in which it was written.

- 9 - The first term is manifestly positive definite, since we take the background spacetime metric to be Euclidean, 1 ��[(� ± ⋆ �)] ≥ 0 4

Note, that in the Gravitational action, we will get a term that is linear in the curvature and we cannot so easily assert that it is positive. To resolve this, we will exploit the Positive Action Conjecture [8, 10], once we get there. Now back to the Yang Mills case, when that the bound above is saturated (we will work with the case that the curvature is self-dual, i.e. � = ⋆ �) we can conclude that,

1 � = ��(� ∧ �) 2

In other words, a minimum of the Euclidean Yang Mills action is located at the point where the curvature 2-form is self-dual. Now, at such a minimum, expanding the above expressions in terms of the Lie Algebra valued 1-forms, we find

1 � = ��(�� ∧ �� + 2�� ∧ � ∧ � + � ∧ � ∧ � ∧ �) 2

This can be simplified by manipulating the last term and using cyclicity of the trace:

��(� ∧ � ∧ � ∧ �) = ������� vol = −������� vol = −������� vol = −��(� ∧ � ∧ � ∧ �)

∴ ��(� ∧ � ∧ � ∧ �) = 0

Furthermore, the first two terms are exact. We can see this by using the graded Leibniz rule of the exterior derivative:

�� ∧ �� = �� ∧ �� + � ∧ ��� = �(� ∧ ��)

1 1 �� ∧ � ∧ � = (�� ∧ � ∧ � + � ∧ �� ∧ � + � ∧ � ∧ ��) = �(� ∧ � ∧ �) 3 3

Giving us the final expression for the Yang-Mills action in the case that the curvature 2-form is self-dual. We can see that in this special case where � = ⋆ �, the Yang- Mills Lagrangian is an exact form:

1 2 ℒ = �� � � ∧ �� + � ∧ � ∧ � 2 3

Using Stoke’s Theorem, we can evaluate this on the boundary such that

1 2 1 2 � = �� � � ∧ �� + � ∧ � ∧ � = �� � ∧ �� + � ∧ � ∧ � 2 3 2 3

- 10 - Now remember, we are looking for finite action solutions. Hence, we will make the physical assumption that the curvature 2-form dies away on the boundary sphere at infinity. Thus, working in a Euclidean metric, we expect

�� → ∞, � = �� + � ∧ � → 0 ⇒ � → 0 ~ � ��|→

Or equivalently in coordinate notation [1],

� → 0 ~ �(�)� �(�) →

Which is to say, that as the field strength tensor vanishes, we expect our gauge field to vanish in step, up to some gauge transformation (Appendix A). Using the finite action condition, we arrive at the final expression,

1 � = − ��(� ∧ � ∧ �) 6

Where we have used the condition that �� = −� ∧ � on the boundary of our spacetime hypersphere, and of course � = ⋆ �.

Topological Invariants in ��(2)

Proceeding, since only boundary terms will contribute to the action, we plug in this expression for � and take the spatial dependence of the local group transformation to be implicit.

1 � = − ��(� �� ∧ � �� ∧ � ��) 6 ℝ 1 = − � �� � (��)� ��� (��) �� 6

where above, �� is the measure of our surface integral over the spacetime boundary hypersphere.

Now let us assume that our gauge group transformations are elements of a Lie Group, and thus can be parameterized by coordinates on a group manifold ℳ(�). In which case, a given group element � ∈ � corresponds to a set of coordinates (�, �, … , �) ∈ ℳ(�). With this in mind, the group manifold coordinates themselves are maps from our background spacetime to the internal group space ℳ(�). Hence, we carry out the chain-rule on the above expression and find,

�� 1 �� �� �� �� �� � = − � �� � � � �� 6 �� �� �� �� �� ��

- 11 - Now the piece of the integrand inside the trace is just the invariant group measure [1] and is a characteristic value of the symmetry group with which we choose to work, independent of our integral over �. Hence, we will pull it out as an overall constant of −24� (to be derived later) and arrive at the following result

�� � �� �� = �� �� ∈ Πℳ(�) 4� �� �� ��

This is to say that the value of the Yang Mills action is uniquely determined by the homotopy group to which the mapping from the boundary 3-sphere to the internal symmetry group manifold belongs; it is a topological property of the field!

The first non-trivial mapping of the 3-sphere we find for compact simple Lie Groups (since the group manifold ℳ�(1) = � which has trivial homotopy, Π(� ) = ∅) is in the special case that our internal symmetry group � = ��(2). In this case, the group manifold is itself �, and we get following

() � /4� = |�| ∈ Π ℳ��(2) = Π(� ) = ℤ

Hence, the Yang-Mills action of ��(2) gauge theory, in the case that the field strength tensor is self-dual, has the nice property of being integer valued up to some scaling constant. This is of particular interest, since every compact simple Lie Group can be decomposed into ��(2) sub groups. Thus, for a general Yang Mills theory with internal symmetry group G,

�/4� = |�| ∈ Πℳ(�) = ℤ, � �� �������, ���������, ������ ��� �����

The Unit (BPST) Instanton

To find the � = ±1 instanton solution for ��(2), note that if the field strength is self-dual, the Yang-Mills equations of motion are automatically satisfied by the Bianchi Identity [1]. Hence, we are looking for solutions of the following differential equation:

1 � = ⋆ � ⇒ � = ± � � 2

To find an analytic instanton solution, we will make use of ‘t Hooft symbols (outlined in Appendix B). First, we make the Ansatz that the connection 1-form will take the following form, (� + ��⃗ ⋅ �⃗) � = �(�)���, �(�) = = � � �

Where �: ℝ → ��(2). � will possess spherical symmetry and have only radial dependence, where � = � + �⃗ ⋅ �⃗, with � our complexified time coordinate. Above, we have used the following notation:

� � = , � = (�, ��⃗) �

- 12 - In addition, it should be noted that the inverse group element, with �(�) as defined, will be (� − ��⃗ ⋅ �⃗) �(�) = = � �̅ �

With �̅ = (�, −��⃗). First, we will expand the gauge field into its coordinates

� � � � � � � � = �(�)�� � = �(�)�� = �(�)� − � � � � � � � �̅ � − � � = �(�)� − = �(�) = �(�) �̅ � − � � � � � �(�) = � � �

Where � is the ‘t Hooft symbol (Appendix B). We will absorb the 1/� factor into our ( ) definition of our radial dependence function → �. This gives us the compact form of the Ansatz posited in [18]: � = �(�)��

Now plug this redefined Ansatz into our expression of the field strength, giving us

� = �� + � ∧ � = �� − �� + �, ��� ∧ ��

= ��(�)�� − ��(�)�� + �(�)��, �(�)���� ∧ ��

� � = � � + �� � − � � + �� � + � , � �� � �� ∧ �� � � � = � � � − � � � + 2�� + � � [�, �]�� � �� � ∧ �� � = � � � − � � � + 2�� �

+ 2�� + ���� �� �� ∧ �� = � = � � � − � � � + 2�� + 2� � � + � � � � �� � � ∧ �� = + 2� � � � − � � � + 2�� �� ∧ �� �

From here we can see that only the second term is manifestly self-dual, from the self-duality of the ‘t Hooft symbol � (Appendix B). Hence, we are looking to solutions such that the first term cancels when

- 13 - ��(�) = −2�� ��

Solving this differential equation yields,

1 1 = � + � ⇒ �(�) = �(�) � + �

Plugging this into our original Ansatz [1], we find

� � � (� + ��⃗ ⋅ �⃗) � (�) = = �(�)� �(�), �(�) = � + � � + � �

Sometimes this solution is expressed in terms of coordinates of Lie Algebra,

� � �(�) = � + �

As one can see, this gauge field is manifestly self-dual, and it approaches ‘pure gauge’ as we approach the spacetime boundary at � → ∞ [3]. All that is left to check is the winding number of our solution, i.e. what is its topological order? Proceeding, we plug in our expression for the gauge field evaluated on the spacetime boundary

� � � (�) = � Giving us

1 1 � = ���� � � = − ��(� ∧ � ∧ �) = 4 6 1 = − � �� � (��)� ��� (��) �� = 6 1 = − ������ � ��(� � � )�� 6� 1 = − ������ � ��(� � + �� � � )�� 6� 1 = − ������ � ��� � 6� + �� (� + �� � )��

� = − ������ � ��(� � )�� 6� � = − � � � ��� � ��� 3� � = − � � � ��� � �|�| ��Ω 3�

- 14 - Since the Pauli matrices are traceless Hermitian, we pull out an overall antisymmetric tensor and a factor of 2 from ��(�). Now using the ‘t Hooft symbol identities [15]

�� = �� − �� + �

� �� = ��� − �� − �� + ��

We proceed in our calculation,

1 = � � � ��� − �� − �� + �� � �|�| ��Ω 3� 1 = � � � ��� − �� + � 3� − ��� − �� + � − ��� − �� + � + ��� − �� + � � |�| ��Ω

1 = � � � �� − ��� + � 3� � + ��� − � � |�| ��Ω = 2 �Ω �

Hence, we find that the action of the BPST instanton gives us,

1 � = ��(� ∧ ⋆ �) � � = 2 �Ω = 2(2� ) = 4� 2

Notice, that when we defined the Yang-Mills action we had a second definition that was divided by a factor of 4�. This calculation demonstrates why that was the case. For when we define the physical action as such

1 � = ��(� ∧ ⋆ �) � � = � ∈ ℤ 8�

For self-dual solutions of the field equations, where � = ⋆ �. Hence, we can conclude that the action contribution for the BPST instanton is

1 � = ��(� ∧ �) � � = 1 8�

Therefore, we have demonstrated that the BPST instanton does in fact have a winding number of 1, and thus serves as the fundamental instanton structure in gauge theory. It can also be shown that the BPST instanton represents a tunneling event between topological vacua [18]. However, the value of the action is independent of which two vacua the Instanton excites a tunneling transition; which is to say any two single- instanton transitions are gauge-equivalent.

- 15 - The New Quantum Vacuum

What we have demonstrated is that the vacuum of our quantum theory is no longer unique. It does not simply make sense to perturb around the trivial vacuum of Yang Mills theory where the global minimum the action lies. The space of vacuum states is now countably infinite in extent, becoming increasingly more populated the larger the gauge group becomes.

��

� ����������

Since we impose the physical condition that the Euclidean action is finite, these vacuum configurations (plotted roughly against the action above) do not decouple; tunneling events can still occur between the topological vacua. We must come up with a way to account for the richer vacuum structure.

The dynamics of quantum particles (if for a moment we forget the universal coupling of gravity) are determined by hierarchies of relative energy. Thus, a fundamental physical requirement of any quantum theory is that the vacuum state must be an eigenstate of the Hamiltonian. Any such theory without a well-defined, time independent reference state, is doomed.

To proceed, let |�⟩ be a vacuum configuration of ��(2) corresponding to a gauge field with instanton number �. These vacua are related by large gauge transformations (Appendix A). The true vacuum of Yang-Mills theory in the literature is called the ‘� − ������’. To construct it, we define a ‘vacuum transition’ operator [18]

�: |�⟩ ↦ |� + �⟩ which serves as a generator of a large gauge transformation with associated winding number �.

As noted, any two topological vacuum states are (large) gauge equivalent. Since our theory is gauge invariant, our transition operator is time independent and will commute with the Hamiltonian. Hence, since we require that the ‘� − ������’ is an eigenstate of the Hamiltonian, it too must be one of the vacuum transition operator.

The natural choice for such a vacuum is,

- 16 - |�⟩ = �|�⟩ where |�⟩ is the ′� − ���������′ vacuum of our theory. Under action of our vacuum transition operator, �, we find,

() �|�⟩ = � �|�⟩ = � |� + �⟩ = � � |� + �⟩ = �|�⟩

We can verify that |�⟩ as defined is in fact the true vacuum eigenstate by constructing an analogous vacuum and measuring the transition probability under time evolution: ⟨�|�|�⟩ = ��⟨�|�|�⟩ = �()�()⟨�|�|�⟩ , ,

But my Hamiltonian by definition is gauge invariant and therefore only the winding number difference will affect the transition probability; we can do away with the additional states since they are gauge equivalent. Thus, setting � = � − �, gives us

⟨�|�|�⟩ = �() � ��0 = 2��(� − �) � ��0

From this we can see that the transition probability vanishes when � ≠ �, which is to say that a vacuum spectrum with a different characteristic mixing angle completely decouples from the true vacuum state of the theory. Therefore, the fundamental vacuum of an ��(2) gauge theory is |�⟩ = �|�⟩ with � serving as a fundamental constant. Remember, � is a generator of gauge transformations, and thus does not generate a symmetry, but more so a redundancy in how we’ve defined our vacuum.

The analysis carried out above can be expanded to larger gauge groups by classifying the full vacuum structure as a sum over a tensor product of topological vacua, depending on the number of ��(2) subgroups within the internal symmetry group.

So, what have we shown? As one can see, it is no longer valid to perturb around the trivial vacuum when constructing a quantum theory of gauge fields. The instantons serve as non-perturbative corrections to our theory and will play a critical role when analyzing dynamics. One such example of how these corrections to the vacuum structure can affect our understanding of physical theory manifested itself in the spin 1/2 axial current anomaly of QCD [15], commonly known as the �(1) problem.

- 17 - Yang Mills ��(4) Instanton

How might we construct the ��(4) analog? The ��(4) curvature differs from ��(2) in that it is characterized by 2 internal group indices. Since the self duality condition of the instanton identifies local minima of the action, we are looking for solutions of the form: 1 � = � � 2

Note, self duality of the internal gauge components, (�, �), would not guarantee us an extremal solution of the action. We will proceed in the same was as when we found the ��(2) instanton, by constructing a self dual vector potential, � using the decomposition of ��(4) discussed in Appendix B.

� = � ⊕ � = ℎ(�)(� ) ⊕ (� ) � � = ℎ(�)� ⋅ � ⊕ � ⋅ �

Note that the repeated indices on the ‘t Hooft symbols do not indicate a sum over those indices. The above connection is to be read as an ��(2) Lie Algebra valued doublet at each point on our manifold, with spacetime component �.

We now plug this into the ��(4) curvature 2-form in order to identify the differential equation that will fix the value of our radial function ℎ(�):

� = �� + � ∧ � = � �� ⊕ � � − � �� ⊕ � � + (�� ⊕ � � ) , �� ⊕ �� �� ∧ ��

Before differentiating and expanding the commutator in terms of the ℒ��(4) structure constants, note that the two ��(2) components completely decouple from one another. Hence, the above expression just becomes a double copy of the result found when calculating the BPST instanton. This gives us

ℎ � = + 2ℎ ��� − � �� + 2√ℎ� �� ∧ �� 2�√ℎ

Since, again, only the second term is self-dual, we solve the differential equation:

�ℎ ℎ = −4�ℎ/ ⇒ � + � = − 2ℎ/

Giving us resulting radial dependence:

1 ℎ(�) = (� + �)

Thus, the lowest order instanton structure for ��(4) is as follows:

- 18 -

� ⋅ � ⊕ � ⋅ � � = (� + �)

Notice that the radial dependence goes as 1/� instead of 1/� as in the BPST case. This � dependence will pop up again. This is really a ����(4) instanton; to get the desired ��(4) structure we must make the following identifications:

� ⋅ � ⊕ � ⋅ � ~ � ⋅ �̅ ⊕ � ⋅ �̅

� ⋅ � ⊕ � ⋅ �̅ ~ � ⋅ �̅ ⊕ � ⋅ �

Analogous to the identification we made between (�, �) ~ (−�, −�) in Appendix B.

The above construction is very closely linked to the gravitational case. However, we’ll find that our final result is significantly less abstract, not requiring a direct sum of instantons. For though the ��(4) Yang-Mills theory has the same internal asymptotic symmetry as a gravitational theory of spacetime, the internal group indices can be transformed to spacetime indices using tetrad fields! (Appendix C) Moreover, in the above case, the condition below

1 ��� ∧ �� = � = � � 2 was argued to be an invalid condition for self-duality. However, in the gravitational instanton, the condition 1 � = � � 2 is sufficient to guarantee a minimum of the gravitational action (here, � is the curvature for the trivialization of the gravitational tangent bundle)

Only now are we fully equipped to answer the pressing question: what is an instanton in Yang Mills theory?

Definition: an Instanton is an everywhere regular (non-singular), self-dual (local minimum), asymptotically pure gauge (finite action) solution of the Euclidean Yang Mills action.

We will carry this definition forward when constructing analogous solutions in a theory of gravity.

- 19 - III. Gravitational Instantons

Positive Action Conjecture

Before we can begin our investigation into identifying Gravitational Instanton solutions, we are required to take a closer look at the Gravitational Action [10]

1 1 � = − ℛ�� � − [�]√�� � 16� 8� where [�] = � − � is the difference between the second fundamental form and the trace [10].

Notice that this action, unlike the Yang-Mills action, has an explicit boundary term contribution [8]. The second fundamental form, �, is related to the extrinsic curvature of our boundary manifold, with √��� the invariant volume form on the surface ��.

But most importantly, we notice that the volume integral for the gravitational action is linear in the Ricci curvature, ℛ, and not quadratic as in the Yang-Mills case. This creates some problems. Since our 4-dimensional field theory of gravity exhibits conformal symmetry, any number of such conformal transformations can be applied to our field, making the action unbounded from below [10].

To resolve this, it was proposed that when evaluating the path integral, one only integrates over conformally inequivalent metrics [8]. In addition, to guarantee we have a specified lower bound, it was suggested we only concern ourselves with field configurations that approach Euclidean space on the boundary, at least locally. In this case, we become only concerned with the boundary term [10].

Under these physical constraints we state the Generalized Positive Action Conjecture [8, 10]:

� ≥ 0, for any complete non-singular positive definite asymptotically locally Euclidean (ALE) metric with ℛ = 0; � = 0 if and only if the curvature is self-dual.

One can see that this conjecture implies minimum action solutions at � = 0 will satisfy precisely the conditions we laid out for instanton structures in Yang Mills: (1) they are non-singular, (2) the are asymptotically Euclidean (in analogy with pure gauge), and (3) their curvature is self-dual.

With this proposition in hand, we set out in search of such spacetimes.

Schwarzschild and Taub-NUT

The first solution to the Einstein equations considered when identifying a finite action solution was, naturally, the Schwarzschild metric [12]. The Euclidean metric describing this spacetime is,

- 20 - �� = �(�)�� + �(�)�� + ��Ω

With potential term given as, � �(�) = 1 − �

In Euclidean signature, the point singularity of this spacetime is removable, and thus can be made to be everywhere regular. In addition, this metric is asymptotically Euclidean. By the Positive Action Conjecture, it has zero action, and thus serves as a minimum of the gravitational action. However, the Schwarzschild metric is not self-dual in its Ricci curvature [12]. So what does this tell us? Well for one, it certainly can’t be an instanton analog, since self-duality is one of the necessary conditions. But it has zero action, so surely it serves as a minimum action configuration in the path integral.

However, since the Schwarzschild metric is asymptotically Euclidean, it is isometric to the flat Euclidean vacuum outside some compact region [10]. In the evaluation of the Path Integral, we are only concerned with conformally inequivalent space-times [8], and thus the Schwarzschild solution loses its standing among potential instanton solution.

Another metric of interest in the early days of gravitational instantons, was the Taub-NUT metric [6]:

1 � − � 1 � − � �� = �� + (� − �)(�� + sin � ��) + � (�� + cos � ��) 4 � + � 4 � + �

We can show that this metric is self-dual in its spin-connection � and thus serves as a self dual curvature solution to the Einstein equations (Appendix C). First, we will transform to coordinate basis of 1-forms using the Cartan-Maurer forms (defined in the following section):

1 1 � = (�� + cos � ��), � + � = (�� + sin � ��) 4 4

An explicit definition for which is given by

1 1 � = (��� − ��� + ��� − ���) = (sin � �� − sin � cos � ��) � 2

1 1 � = (��� − ��� + ��� − ���) = (− cos � �� − sin � sin � ��) � 2

1 1 � = (��� − ��� + ��� − ���) = (�� + cos � ��) � 2

Where the Euler angles parameterizing the 3-sphere (�, �, �) are related to the cartesian spacetime coordinates by

� � � � � + �� = � cos exp (� + �) , � + �� = � sin exp (� − �) 2 2 2 2 With ranges

- 21 - � ∈ [0, �), � ∈ [0,2�), � ∈ [0,4�)

Note the complex structure of this coordinate transformation. This will pop up again when we formulate the Eguchi-Hanson solution as a Kähler manifold.

We will not explicitly calculate it, but it is useful to know that the structure equation for the Cartan-Maurer 1-forms is as follows [6]:

�� − 2�� ∧ � = 0

Returning to the Taub-NUT metric, this gives us,

1 � + � � − � �� = �� + (� − �)� + � + 4� � 4 � − � � + �

We can now decompose the metric into a Vierbein basis (Appendix C)

1 � + � � − � � = ��, � − �� , � − �� , 2� � 2 � − � � + �

From here we apply the structure equation (Appendix C)

�� + � ∧ � = 0 and find the following expressions for the spin connection using

� �� = �� ∧ � − 2� − � � ∧ � √� − � 2� 1 � + � = � ∧ � − � ∧ � (� + �)√� − � � � − �

� �� = �� ∧ � − 2� − � � ∧ � √� − � 2� 1 � + � = � ∧ � − � ∧ � (� + �)√� − � � � − �

2� � − � �� = �� ∧ � − 2� � ∧ � (� + �)√� − � � + � 2� 2� = � ∧ � − � ∧ � (� + �)√� − � (� + �)√� − � Hence,

2� 2� � = � = �, � = � = � (� + �)√� − � (� + �)√� − �

- 22 - 2� � = � = � (� + �)√� − �

As one can see, the spin connections calculated above are manifestly self-dual by satisfying, 1 � = � � 2

and thus, the Taub-NUT solution serves as a spacetime with self-dual Ricci curvature. Now we must ask ourselves, is this metric everywhere regular? For it looks like there is a singularity at � = �. To investigate whether this singularity is removable, we will change the ‘proper distance coordinate’ [6]:

1 � + � �� = �� 4 � − �

So that now our metric reads:

� − � �� = �� + (� − �)� + � + 4� � � + �

We are concerned with how our new coordinate � behaves near the singular point at � = � (which in our new system is at � = 0), hence, in the region � = � + �, we find

1 � + � / 2� + � � = �� ≈ � ≈ (2��)/ 2 � − � �

Thus, some � away from the singularity at � = � + � we find that the metric behaves as

�� = �� + 2��� + � + � + �(� ) ≈ �� + � � + � + �

Which is just the metric for ℝ and thus the singularity is a removable nut singularity (to be discussed in the sections to follow). Therefore, we can make the metric everywhere regular by transforming to Cartesian coordinates in the neighborhood of the singular point.

This is great, the Taub-NUT solution is both everywhere regular and self-dual in the curvature. All that’s left to check is that it’s asymptotically Euclidean and we have identified our first gravitational instanton!

However, here we run into a problem. Returning back to the original definition of Taub- NUT away from the singularity we find,

1 � + � � − � �� = �� + (� − �)� + � + 4� � 4 � − � � + �

In the large � → ∞ limit, the spacetime metric behaves as

- 23 - 1 �� = �� + 4�� + �� + � 4

This is not the ℝ × � of Euclidean 4-space; it is clear that the 3-sphere at the boundary is distorted by the coefficient of the � Cartan-Maurer form. Thus, Taub-NUT does not qualify as a gravitational instanton solution, though its motivation is clear in [12].

Eguchi-Hanson Instanton

The first true analog of the BPST instanton solution identified in Euclidean gravity was the metric derived by Eguchi and Hanson [5]. The solution has both finite action, as it is asymptotically (locally) Euclidean, and possesses a self-dual Ricci curvature, which thus made it a minimum of the Euclidean action.

To derive such a solution, an algorithm similar to that which was used to discover the BPST instanton is performed. The steps are threefold [5]:

(1) Identify a spin connection � that is itself self dual, thereby guaranteeing self duality of the curvature �. Note that the Einstein equations in the trivialization of our tangent bundle can be expressed as (Appendix B),

� ∧ � = 0

Thus, if the curvature of the trivial bundle is self-dual, such that � = �, then from the cyclic identity (Appendix B),

� ∧ � = 0 ⇒ � ∧ � = 0

and hence the vacuum Einstein equations are immediately satisfied. Thus, by identifying a self-dual connection 1-form, we get a solution to the Einstein equations for free!

Notice that we are not guaranteed a solution in the analogous case for Yang Mills since the internal group indices differ from the spacetime indices, and thus the cyclic identity does not apply in the same way.

(2) Choose an Ansatz that possesses spherical symmetry in the 4 spacetime coordinates (with complex time), such that the solution differs from the trivial vacuum configuration only by a function of the radius � = � + � ⋅ �.

(3) Solve the differential equation of the metric ansatz that makes the connection � self dual.

First, we will write our metric in terms of Cartan-Maurer forms of the previous section, that will make the spherical symmetry of our ansatz manifest. Written in terms of these forms, the flat metric becomes

- 24 - �� = �� + � � + � + � = �� + � �Ω

Now that we have written the flat spacetime metric in polar coordinates so that the ℝ × � boundary behavior of Euclidean space is manifest, we can proceed to step two of our procedure and make a metric Ansatz that varies from the flat metric by some function with solely radial dependence. Our Ansatz is then [3],

�� = � (�)�� + � � + � + � (�)�

First, we decompose the metric into a tetrad basis on the condition that the new trivial metric is locally Euclidean (Appendix C):

� = �(�)��, ��, ��, ��(�)�

The structure equation (Appendix C) gives us

1 2 �� = �� ∧ � − � � ∧ � = � ∧ � − � ∧ � �� ��

1 2 �� = �� ∧ � − � � ∧ � = � ∧ � − � ∧ � �� ��

1 1 � 2� �� = (� + ��′)�� ∧ � − �� � ∧ � = + � ∧ � − � ∧ � � � � �

Thus, the components of the spin connection are

1 � � = − �, � = � �� �

1 � � = − �, � = � �� �

1 1 � 2 � = + �, � = � � � � ��

From the first two pairs of dual components, we find the condition that

� 1 = − = 0 ⇒ �� = −1 � ��

From third pair, we find that

2 1 1 � = + = 0 ⇒ 2� = � + ��′ = 0 �� � � �

- 25 - Hence, to identify a self-dual metric solution we need only solve the following system of first order differential equations:

2 1 − � � = , � ⋅ � = −1 � �

The resulting expression is

� �� � �� = 2 ⇒ 4 log �/� = − log(1 − �) ⇒ 1 − � = , � ∈ ℝ 1 − � � �

Thus, the � dependent functions for our Ansatz are as follows:

−1 �(�) = 1 − (�/�), �(�) = 1 − (�/�)

The anti-self-dual expressions only differ by an overall negative sign in �(�). Hence, we arrive at the Eguchi-Hanson metric, which has characteristic self-dual curvature of an instanton like solution:

�� = [1 − (�/�) ] �� + � � + � + � [1 − (�/�) ]�

This solution has a particularly interesting property – the curvature is either self- dual or anti-self-dual, depending on how we choose to define our orthonormal tetrad basis! This is no coincidence; this is deeply tied to the ��(2) chiral subgroups that form the ��(4) symmetry group of the internal space, as are the antipodal boundary conditions of the manifold (which we will derive momentarily). The analogous solution for an ��(4) Yang-Mill instanton would be as derived in the previous section:

� ⋅ � ⊕ � ⋅ �̅ � = (� + �)

The above ��(4) solution has winding number doublet of (1, 1) and thus would take the tensor product topological vacuum of the theory from

|�, �⟩ → |� + 1, � + 1⟩

Just as the above lowest order instanton has Euler characteristic and signature [9]

|�(�) = 2, �(�) = 1⟩

While flat Euclidean space carries corresponding topological numbers of [9]

|�(�) = 1, �(�) = 0⟩

This correspondence between the two solutions derived is linked to the shared symmetry group ��(4) that acts on the vector bundles of each theory. In addition, we can see that both solutions have a characteristic � fall off as we approach the spacetime boundary.

- 26 - All of these characteristic properties can trace their origins to the BPST instanton. It is reassuring that the ��(2) decomposition of all simple Lie Group becomes manifest in the construction of more complex instantons in a variety of different theories.

Now, before we proceed, let us illuminate some of the topological properties and why the boundary geometry of the Eguchi-Hanson Gravitational instanton has this aforementioned antipodal association.

The asymptotic symmetry group of Euclidean gravity (under the assumptions of the Positive Action Conjecture) is the Lie Group ����(4). The homotopy group of this internal symmetry can be shown as follows:

Π����(4) ~ Π��(2) × ��(2) = Π��(2) ⊕ Π��(2) = ℤ ⊕ ℤ

Hence, we expect there to be two topologically invariant quantities in a gravitational theory of instantons. These happen to be the Euler characteristic, �, and the Hirzebruch signature, �, studied extensively in [4]. For the given metric, these happen to be

�(�) = 2, �(�) = 1

Furthermore, it is important to understand the geometry of the Eguchi-Hanson instanton when we approach the boundary as � → ∞. First, note that as written, the metric has an apparent singularity at � = �. To make the boundary properties manifest, we will perform the coordinate transformation

� = �[1 − (�/�)]

[1 + (�/�)] 2��� = 2�[1 + (�/�)]�� ⇒ �� = �� 1 − (�/�) Plugging this into our metric gives us

�� = [1 + (�/�) ] �� + � � + � + � � and in terms of our Euler angles,

� � �� = [1 + (�/�)]�� + (�� + sin � ��) + (�� + cos � ��) 4 4

Now let us investigate how our metric behaves near the singularity at � = � (which in our new coordinate system, is located at � = 0). By inspection we find,

1 �� ≈ [�� + �(�� + cos � ��) + �(�� + sin � ��)] 4

We can see that near the singular point we recover the metric for a 2-sphere in the last term, with coordinate angles � and �. For a fixed � and �,

- 27 - 1 ��| ≈ [�� + ���] ,. 4

Which is just the flat metric in polar coordinates. Near the coordinate singularity, the Eguchi-Hanson metric behaves in the following way

�|→ = ℝ × �

Hence, by analogy with the removable polar coordinate singularity at the origin of a given ℝ plane, the singularity at � = 0 is removable provided that we restrict the domain of the Euler angle �. Hence, when

� ∈ [0,2�) then Eguchi-Hanson metric is everywhere regular, and thus under these boundary conditions, it serves as a true instanton-like spacetime solution. But this comes at a cost. When � was free to take values over its full range from 0 to 4�, the metric approached Euclidean space on the boundary,

�� = [1 − (�/�) ] �� + � � + � + � [1 − (�/�) ]�

� → ∞, �� → �� + � � + � + �

However, by removing the singularity at � = �, we have identified antipodal points and contracted the boundary surface from � → � /� = �(ℝ). Thus, we deduce that the Eguchi Hanson instanton has the following boundary

�� = � /� = �(ℝ)

What we have done here is removed something called a ‘bolt’ singularity in our metric (to be discussed further), which in turn constrained the boundary, taking it from a space that was asymptotically globally Euclidean to one that is locally asymptotically Euclidean. This is in agreement with the Positive Action Conjecture which states that [6]

An everywhere regular self-dual metric will have action � = 0, and if it is asymptotically Euclidean, it must be the trivial flat vacuum.

Hence, if we have an asymptotically Euclidean space with self-dual curvature, that is not the flat vacuum, there must exist a singular point. If removable, we are free to remove this singularity, but at the cost of redefinition of our boundary manifold, such that it is only locally Euclidean. In following sections, we will show that to remove such a singularity, we must exchange it with an additional factor of boundary surface identifications.

Notice that we have careful about what we call the Eguchi-Hanson metric vs. instanton. The asymptotically globally Euclidean manifold described by the EH metric, is not an instanton, since it is not everywhere regular! Going forward, when we are referring to the Eguchi-Hanson instanton, we mean specifically the spacetime manifold endowed with EH metric that has the necessary boundary conditions in place.

- 28 - General Instanton Metrics

Nuts and Bolts

Before proceeding to talk about a more general class of instanton metrics, a nice formalism exists that will allow us to identify removable singularities that pop-up when constructing instanton spacetimes. Gibbons and Hawking demonstrated a way to classify spacetime singularities into groups of ‘nut’ and ‘bolt’ singularities [11].

Essentially, a ‘nut’ singularity is a removable point singularity that might arise in a spacetime metric; the canonical case being that in the Taub-NUT metric. A ‘bolt’ singularity, named such for obvious reasons, is a 2-sphere singularity, that is removable by proper point identification on the boundary manifold of the spacetime [11].

To display the concepts underlying ‘nut’ and ‘bolt’ singularities, consider the following metric: �� = �� + � (�)� + � (�)� + � (�)�

Now we will consider the limiting case where � → 0 and our radial functions take on the form �(�) = �(�) = �(�) = �

The given metric becomes the flat metric of ℝ:

�� = �� + � � + � + � = (�� + � �٠)

As one can see, the metric is singular at � = 0. This is the standard polar coordinate singularity at the origin of our coordinate system. To remove it, we note that the manifold, �, approaches ℝ near the singular point. Thus, we simply transform to Cartesian coordinates in a region outside the neighborhood of the singularity, and then replace the point, thereby removing the singularity. Hence, we define a removable ‘nut’ singularity as follows:

Nut Singularity: a point singularity where the metric approaches the standard ℝEuclidean metric in the neighborhood of singularity

Alternatively, consider the case where in the limit of � → 0

�(�) = ��, �(�) = �(�) < ∞

The metric in this limit becomes

�� �(�) �� = �� + (�� + cos � ��) + (�� + sin � ��) 4 4

The 2-sphere metric is clearly visible, with radius �(� → 0)/2. For fixed � and �, we find that the metric behaves in the following way:

- 29 - �� �� ≈ �� + �� 4

This is exactly the metric of the ℝ plane, with a subtle difference. Now proper distances measured in the patch created when � sweeps out an angle of 4�/� behave like distances in a complete ℝ plane. Hence, the 2-sphere singularity at � = 0 now is of degree �. To remove it, we must

(1) Restrict the domain of � from [0,4�) to [0,4�/�)

(2) Remove the polar coordinate singularity by transforming to cartesian coordinates

Step 2 is trivial. Step 1 is critical in constructing gravitational instantons with Euclidean boundary manifold modulo a cyclic group. Hence, the working definition of a ‘bolt’ singularity will be:

Bolt Singularity: a 2-sphere singularity where the metric locally approaches that of ℝ × � in the neighborhood of singularity, where globally one might find

ℝ /� × �

where ‘k’ is the degree of the singularity of interest.

With this handy formalism in mind, we will proceed to handling more general metrics that contain multi-instanton solutions of higher topological order.

Multi-Instanton Metric

Above we have laid out the first example of a gravitational instanton; one that is self-dual and everywhere regular, but whose topological order differs from that of flat Euclidean space. Now, expanding on the Eguchi-Hanson Instanton, we state a general solution for multi-center instanton solutions attributed to Gibbons and Hawking. This general class of solutions possesses the same characteristic self-duality, but unlike the self-dual Taub-NUT metric, they approach Euclidean space (locally) on the boundary. It can be expressed as [9]

�� = �(�⃗)(�� + �⃗ ⋅ ��⃗) + �(�⃗)��⃗ ⋅ ��⃗

+, ���� − ���� ∇⃗� = ±∇⃗ × �⃗ −, ���� − ���� − ����

In this metric, the spacetime is asymptotically Euclidean when we let � be periodic with the range � ∈ [0,4�)

just as we had for the azimuthal angle of the Eguchi-Hanson metric when the singularity was left alone. Notice, that �(�) is harmonic,

- 30 - ∇�(�) = ∇⃗ ⋅ ∇⃗ �(�) = ±∇⃗ ⋅ ∇⃗ × �⃗ = 0

And hence we can construct a general solution as

1 �(�) = Λ + |�⃗ − �⃗|

Where Λ is an arbitrary constant and �⃗ serve as locations of singular points in our metric, or the center of a given instanton. Hawking and Gibbons found something interesting in the case that Λ = 0. For that spacetime metric, all the singularities are ‘bolt’ singularities (mentioned in the previous section). They can be removed by restricting the range of � such that

� ∈ [0,4�/�)

Thus, the multi-instanton solution constructed by Hawking and Gibbons has the property that for � centers, we have � boundary conditions for the self-dual metric. Therefore, in the case that � = 1, the boundary surface is equivalent to � /� = � ; which is to say that it is asymptotically Euclidean out of some compact region and hence is isometrically equivalent to the trivially flat Euclidean vacuum [10].

In the � = 2 case, we recover the Eguchi-Hanson solution derived in the previous section, with the desired antipodal identification such that

�� = � /� ≃ �(ℝ )

In general, we find that boundary manifold for the � center instanton would have the following characteristic �� = � /� = � /�

Where � is the cyclic group of order �.

Polygons and Gravitons

All instanton solutions that are ALE with boundary � modulo a cyclic permutation are of the form described by the Gibbons-Hawking multicenter metric [9]. What it seems we have discovered is that there runs a deep connection between the topological quantities of a spacetime instanton and it’s boundary symmetry group �, that serves as a discrete subgroup of ��(4). But could we generalize this further such that the boundary takes on the more general form:

�� = S/Γ, Γ ⊂ SO(4), Γ � �������� ��������

In fact, it has been conjectured that a spacetime will have a self-dual Ricci curvature if and only if it has a boundary � modulo some discrete symmetry group [8]. So now we finally can go about answering the question: what is a gravitational instanton?

- 31 - Boundary Characteristics of Gravitational Instantons

Definition: a Gravitational Instanton is a 4 dimensional spacetime, M, that is everywhere regular, has self dual Ricci curvature, and is asymptotically locally Euclidean, such that

� = � + � ⋅ � → ∞, � → �/Γ × ℝ where that the boundary �� has the following form [8]:

� : ������ ����� �� ������ � ⎧ ⎪�: ��ℎ����� ����� �� ������ � �� = �/Γ, Γ = � ∶ �����ℎ����� ����� ⎨ ⎪ � ∶ ����ℎ����� ����� ⎩ � ∶ �����ℎ����� �����

Spacetimes with these properties, together with the trivial flat vacuum, thus characterize all zero action configurations permitted by the Positive Action Conjecture. Hence, in the path integral formulation of gravity, the identification of every such self-dual metric would completely classify the full vacuum configuration of spacetime, and functionally solve the problem of a quantum theory of gravity [13].

Easier said than done.

Though the aforementioned surfaces described by Hitchin that fully classify the class of instanton boundaries in 4 dimensional spacetimes certainly have lofty implications, they too have been used for humbler investigations. Because of their unique complex structure, some of the above metrics have been used in constructing compact self-dual manifolds, those without boundary [16].

Albeit of little interest in a path integral formulation of quantum gravity, compact self- dual manifolds are of particular interest in String Theory, where the small extra dimensions are necessarily without boundary. We will conclude the essay with a brief investigation into this class of mathematical structures.

Kähler Geometry and Compact Self-Dual Manifolds

Alluded to, but not explicitly stated, is the fact that the spacetime metrics described above by Eguchi, Hanson, Gibbons, Hawking and Hitchin are in fact a special class of surfaces; they are all Kähler manifolds and more specifically they are Hyper-Kähler.

A Kähler structure satisfies the following two conditions:

(1) It has a non-degenerate (Riemannian) metric that has a complex structure. In other words, near any point � the coordinates can be written in complex holomorphic coordinates � = � + �� ∈ ℂ

- 32 - Such that the full metric takes the form:

� = ����̅ + �(|�| )

(2) The manifold in question is equipped with a Kähler form, which is a non-degenerate symplectic 2-form that satisfies the following condition

�(��, ��) = �(�, �), ∀�, � ∈ �� where � is an almost complex structure, such that

�: �� ⊗ ℂ → �� ⊗ ℂ, � = −1

The map, �, can be thought of as a projection operator, taking vector fields in the tangent space, ��, to their holomorphic and anti-holomorphic parts in the following way [17]:

1 �,� = � ∈ � � ⊗ ℂ: � = (1 − ��)� 2 1 �,� = � ∈ � � ⊗ ℂ: � = (1 + ��)� 2

If our manifold is equipped with such a complex structure and a Riemannian metric, we automatically get a symplectic 2-form that satisfies condition (2). We do so by defining,

�(�, �) = �(��, �)

We can see that condition (2) then follows by noting the non-degeneracy of the metric and observing

�(��, ��) = �(��, ��) = −�(�, ��) = −�(��, �) = −�(�, �) = �(�, �)

As desired. Below, we show that the Eguchi-Hanson instanton can be written as such a manifold. To do so, we need only show that the Eguchi-Hanson metric can be written in the form of condition (1) and then impose the necessary conditions to make it non- degenerate (everywhere regular). We start with the given metric

�� = [1 − (�/�) ] �� + � � + � + � [1 − (�/�) ]�

First, we make the coordinate transformation [10]

� = � − �,

This will place the singular point in our metric at the origin of the new coordinate system. Some useful relations that follow are:

[1 − (�/�)]�� = (1 + (�/�))/��

�[1 − (�/�)] = �(1 + (�/�))/, � = �(� + (�/�))/

- 33 -

Giving us a new form of the metric,

/ / �� = [1 + (�/�) ] (�� + � � ) + � [1 + (�/�) ] � + �

Now we impose a complex coordinate transformation on the metric:

� = � + ��, � = � + ��

� = ��̅ + ��̅ ⇒ 2��� = ���̅ + ���̅ + ���̅ + ���̅ ⇒ �� = ����̅ + ����̅

Under this coordinate transformation, we find that the Cartan-Maurer forms transform in the following way [10]:

� � + � = (����̅ + ����̅ ) − � (�̅ �� + �̅ ��)(���̅ + ���̅ )

� � = (�̅ �� + �̅ �� − ���̅ − ���̅ )

Plugging this into the above expression, we find

� �� = (�� ��̅ + �� ��̅ ) [� + �]/ 1 + (�̅ �� + �̅ �� − � ��̅ − � ��̅ ) [� + �]/ [� + �]/ + (�� ��̅ + �� ��̅ ) � [� + �]/ − (�̅ �� + �̅ �� )(� ��̅ + � ��̅ ) �

This new definition of the metric looks messy and very far from the desired complex form of the metric that we desire. However, since we are working in a complex space, we can split our exterior derivative up into the following way [17]:

� � � � � � � � � = �� + �� + �� + �� = �� + ��̅ + �� + ��̅ �� �� �� �� �� ��̅ �� ��̅

Since, �� = �� + ���. Using this, we define a new set of operators,

� � ̅ � ≡ �� , � ≡ ��̅ �� ��̅

And then the above metric can be written in the compact form [10]:

� � �� = (�� ��̅ + �� ��̅ ) + � �̅ ln(�) [� + �]/ [� + �]/

- 34 - Thus, we have written the Eguchi-Hanson metric as a complex metric. However, there’s a problem. In order to satisfy condition (1) for a Kähler manifold (and thus condition 2) we require that the metric is non-degenerate, i.e. everywhere regular. But as it stands, �� has a singularity at � = 0.

To resolve this, we make the pointwise identification,

� ~ −�

(�, �) ~ (−�, −�), (�̅ , �̅ ) ~ (−�̅ , −�̅ )

Now when � = 0, it’s no problem; for the singular jump at the origin becomes smooth when we impose these antipodal periodic boundary conditions. Thereby, we restore � to a non-degenerate metric. With this we can conclude that

The Eguchi-Hanson Instanton is a Kähler Manifold with complex metric,

� � �� = (�� ��̅ + �� ��̅ ) + � �̅ ln(�) [� + �]/ [� + �]/

and boundary manifold �(ℝ) = � /�from antipodal identification on the Eguchi- Hanson metric.

All instantons described in [13] will possess such a complex structure. In fact, we can go further. It can be shown that these instantons actually belong to a more symmetric subclass of Kähler manifolds, called Hyper-Kähler manifolds.

A Hyper-Kähler manifold is like a Kähler manifold, but it has 3 symplectic Kähler forms. These can be constructed from three complex structures, �, � and � [14]. In addition, these complex structures must satisfy the following relations:

� = � = � = ��� = −1

That is, the 3 complex structures of Hyper-Kähler manifolds behave like the Quaternions, ℍ [14]. Hence, we can think of spacetime instantons as having a quaternion like structure, such that there is a quaternion coordinate transformation that allows us to write our metric as

� = �ℎ�ℎ + �(|ℎ| ) where

ℎ = � + �� + �� + �� ∈ ℍ

This is no surprise. We have demonstrated throughout the essay a general theme that all instantons derive their structure from the fundamental ��(2) Yang-Mills instanton; and the generators of ��(2) also satisfy the quaternion algebra stated above.

- 35 - Though the insight of Hitchin seemed a promising step forward in the path integral formulation of quantum gravity, as he was able to identify all possible boundary manifolds of spacetime instantons, actually identifying a generalized set of associated self-dual metrics is an open problem. So, if there is no more ground to be gained with instantons when classifying the vacuum structure of quantum gravity, then maybe those without boundary could give us some insight.

These geometries are exactly what string theorists have posited make up the extra dimensions of spacetime in the stringy version of quantum gravity. K3 surfaces and Calabi-Yau manifolds are a class of compact Hyper-Kähler, self-dual manifolds that many suggest could describe the geometry of the 6 extra dimensions in superstring theory [16]. If this were in fact the case, it appears our spacetime is a tight-knit quilt of action quanta on which the dynamics of the universe unfold.

IV. Conclusion

There is much left undiscussed throughout this essay. In particular, we did not expand upon how the Yang Mills instanton gives rise to a spin 1/2 axial anomaly, demonstrated by ‘t Hooft [15]. Furthermore, analogous to the gravitational multi- instanton solution of Gibbons and Hawking, there exists a similar solution in the Yang Mills construction by Jackiw, Nohl and Rebbi. One can then show that gravitational instantons would induce a spin 3/2 axial anomaly in a similar way to that of the spin 1/2 anomaly [6]. More in depth discussion is required to describe the Hyper-Kähler structure of instantons and the emergence of their compact analogs in other theories of quantum gravity.

Acknowledgements

The author would like to thank Professor Nick Dorey for setting the essay on Instantons and providing the impetus to expand upon the topic in the gravitational regime. The author also gives thanks to Professor Csaba Csaki for first introducing the him to the field while an undergraduate at Cornell. Finally, appreciation is due to the University of Cambridge for providing the resources to pursue research on this subject, a platform for intellectual discourse, and the venue for lectures of graduate level mathematical physics, without which the authors intellectual development required to engage in the essay material would not have been possible.

- 36 - References

1. A. A. Belavin, A. M. Polyakov, A. S. Schwarz, Yu. S. Tyupkin, “Pseudoparticle Solutions of the Yang-Mills Equations”Phys. Lett. B 59 (1975), 85. 2. J. Baez and J. P. Munian, “Gauge Fields, Knots and Gravity”, Series on Knots and Everything, (1994) 3. S. Coleman “Aspects of Symmetry” CUP (1985) 4. T. Eguchi and P. G. O. Freund, “Quantum Gravity and World Topology” Phys. Rev. Lett. B 37 (1978) 5. T. Eguchi and A. J. Hanson, “Asymptotically Flat Self-Dual Solutions to Euclidean Gravity” Phys. Lett. B 74 (1978), 249. 6. T. Eguchi and A. J. Hanson “Self-Dual Solution to Euclidean Gravity” Ann. Phys. 120 (1979) 82-106 7. T. Eguchi, P. B. Gilkey and A. J. Hanson, “Gravitation, Gauge Theories and Differential Geometry” Phys. Lett. Review Section 66 (1980) 213-393 8. G. W. Gibbons and S. W. Hawking “Action Integrals and Partition Functions in Quantum Gravity” Phys. Rev. D 15 (1977) 9. G. W. Gibbons and S. W. Hawking, “Gravitational Multi-Instantons” Phys. Lett. B 78 (1978) 430 10. G. W. Gibbons and C. N. Pope, “Positive Action Conjecture and Asymptotically Euclidean Metrics in Quantum Gravity” Comm. Math. Phys. 66 (1979) 11. G. W. Gibbons and S. W Hawking, “Classification of Gravitational Instanton Symmetries” Comm. Math. Phys. 66 (1979) 12. S. W. Hawking, “Gravitational Instantons” Phys. Lett. A 60 (1977), 81. 13. N. J. Hitchin, “Polygons and Gravitons”, Math. Proc. Camb. Phil. Soc. 85 (1979) 465 14. N. J. Hitchin “Hyperkähler Manifolds”, Séminaire Bourbaki, (1991) 15. G. ‘t Hooft “Computation of Quantum Effects Due to a Four-Dimensional Pseudoparticle” Phys. Lett. D 14 (1976) 3432 16. D. N. Page, “A Physical Picture of the K3 Gravitational Instanton” Phys. Lett. B 80 (1978) 55 17. D. Skinner, “Lecture Notes on Supersymmetry: Nonlinear Sigma Models” (2019) 18. E. Weinberg “Classical Solutions in Quantum Field Theory” CUP (2012)

- 37 - Appendix A: Vector Bundles, Gauge Transformations and Connections

Much of the physics in this essay is heavily weighted on the mathematical construction of vector bundles. This appendix will serve as a brief introduction to some of the key concepts used throughout the text. To begin, take a manifold �, and for each point � ∈ �, associate a fiber �. Generally, this fiber will be some type of vector space (a tangent space, a representation space of some Lie group, etc.). In the case that the fibers of our bundle are vector spaces, we call it a ������ ������. Now, we define the total space of our fiber bundle to be the union of all such fibers over the manifold � [6]:

� = �

In the case that all fibers of our bundle are the same field, �, the total space is just

� = � × �

We call this the ������� ������, and it will be particularly useful when constructing gravitational instantons using Vierbein (or tetrad) fields.

Proceeding, we construct a projection map � that maps the total space ���� the manifold �, such that �: � → �

A fiber bundle is then just a mathematical structure consisting of a total space, a manifold, and a well-defined onto map; but sometimes we’ll be lazy and simply refer to the total space � as the fiber bundle in question. From here on, we will assume that our total space is composed of vector spaces, and hence we will be working with vector bundles.

Now that we have defined the projection map, we can define a related map, �, such that

�: � → �

This map is called a section of the vector bundle and it maps a point � ∈ � to an element of the fiber over that point:

�(�) ∈ �

We will call the space of all sections on � to be Γ(�).

So, we ask ourselves, what is a vector field? A vector field over a given manifold � is just section of the associated vector bundle! What then is a gauge transformation? If the fibers of our bundle are representation spaces of some group, we can imagine that performing a global group transformation (one that is uniform across the manifold) will shift the a given section:

- 38 -

But equivalently, we can imagine performing a local group transformation, such that every vector in a section is shifted by a different amount. Thus, the effect of performing a gauge transformation is just smooth map from a given section � ∈ Γ(�) to a different section � ∈ Γ(�).

A brief aside: when gauge transformation can be constructed from infinitesimal gauge transformations, we call it a small gauge transformation. When a gauge transformation cannot be constructed from infinitesimals, we call it large. Thus, we find that small gauge transformation when acting on a toroidal total space are broken up into gauge equivalent classes of characteristic winding number.

�(�): � ⟼ �, �, � ∈ Γ(�)

In other words, maps from the base space manifold to the internal symmetry group with different winding number are small gauge inequivalent. This is the basic idea that underlies instanton physics. But if the gauge transformation is large, the section winding number, �, can be shifted

- 39 -

�(�): Π(� ) → Π(� ), �(�): �(�) = 0 ⟼ �(�) = 2

With this little bit of formalism in hand, we can start to understand how we might define a dynamical theory of fields over a vector bundle. As in any such theory, we need to construct a means for differentiating between any two given points. In the case that the vector bundle is a trivial bundle, and our base space is flat, differentiation behaves the same way as in standard vector calculus.

But what are we to do if the base manifold has some associated curvature, like the 2-sphere? [2]

Moreover, what if the fibers over a given point vary as we move along the manifold, such that the bundle is no longer trivial? It is not explicitly obvious how one would compare two vectors at different points, even in the neighborhood of each other, because the spaces to which they belong are completely different. What we need is a derivative operator that can act on any smooth section � ∈ Γ(�) and return its ‘derivative’, a different section � ∈ Γ(�). Such an operator would serve as a universal connection between fibers in our bundle.

Hence, we define a connection, �, to be the operator that maps every vector field � on the manifold � to a derivative operator � [2]:

�: � ⟼ �

We call the image of the connection the ��������� ���������� and it has the desired behavior such that �: Γ(�) → Γ(�)

- 40 - where it differentiates any section of our bundle in the direction of the specified vector field � = � (�)�. Often, we will use the shorthand notation:

� = � � = � �

This is starting to look a lot like differentiation in the direction of a vector field in Euclidean space. Now, any section can be broken up into a standard basis {�}, and since � is a linear map, we define it in the following way:

� � = � �

where � are the components of the vector potential (sometimes itself called the connection), the same one in physical gauge theories. Thus, given a section � ∈ Γ(�), action of the covariant derivative yields

�(�) = �� � = �� � + � �� = �� + � ��

Hence, the � component of the resultant section from acting the covariant derivative on a section � ∈ Γ(�) will be:

�� = �� + ��

Contracting this with a tensor product of section basis elements, and defining

� = � � ⊗ �

We can express this in the way most familiar to those in the physics community:

� = � + �

As we mentioned above, when working with the trivial bundle, differentiation behaves as expected, and the connection used in this case is the standard flat connection:

� ≡ �

An important property of the connection, in fact the property that underlies the entire foundation of vector bundle differentiation, is that it can act on any section in Γ(�), whether or not it is defined over locally trivial subspaces of �. Hence, we require that if � is a connection then so is � = � + � And so is � = � + �

Now how might we find ourselves needing to differentiate sections in nontrivial sectors of our total space? Gauge transformations, of course! Suppose we use a gauge transformation to shift a section we wish to differentiate,

- 41 - � → �(�)� = � ∈ Γ(�), �(�) ∈ �

Then the new connection for this section � would be

�(�) = �(� �) = � �(�) ⇒ �(�) = � �(��)

Using this, we find that our vector potential under a gauge transformation will change in the following way:

�(�) = � �(��) = � � + �(��) = � �� + � ��� + (� �)�� = � �� + � ��� + �� and since �(�) = � + �� we are given the following expression for a gauge transformation on our vector potential: � = � �� + � ��

As a quick check, suppose that � ∈ �(1) such that

�(�) = �(), �(�) ∈ ℒ�(1) = �ℝ

Then we find that the Lie-Algebra valued connection 1-form of a �(1) gauge theory will transform in the following way:

� = � + ��

This is exactly the gauge transformation for the vector potential in Electromagnetism.

- 42 - Appendix B: ‘t Hooft Symbols

There is a nice homomorphism between the 4-dimensional rotation group, ��(4), and the special unitary group of degree 2, ��(2). To see this, suppose we have some 4- component 1-form (like our connection ��� ) in a Euclidean space with components �. We can pack these components into a complex 2 × 2 matrix in the following way:

� − �� −� − �� � = ��� ⟼ ��̅ = � − �� � + ��

Where we define:

��, � = 1,2,3 −��, � = 1,2,3 � = , �̅ = �, � = 4 �, � = 4

There {�} are the Pauli Matrices. Notice that the way we have written the components � in the complex matrix � has the following properties

� det � = ��, ∈ ��(2) √det �

In other words, we have constructed a complex matrix out of a 4-component 1-form, such that our matrix is unitary up to some constant. Now, we define the following transformation on our newly defined complex matrix �:

� ⟼ � = � ��, �, � ∈ ��(2)

Under action of this transformation, we find the following two properties

det � = det(� ��) = det � ⋅ det � ⋅ det � = det �

⇒ � ⋅ � = � ⋅ � which is to say that the transformation as defined preserves the norm of the components of our original 1-form, �. In addition,

� � � = � �� = √det � � ⋅ ⋅ � , � ⋅ ⋅ � ∈ ��(2) √det � √det �

Thus, our transformed matrix is a new complex matrix with the same scaling constant as before our transformation; all that has changed is the components of the unitary matrix.

Therefore, this transformation we have defined rotates the original 1-form in the same way a group element of ��(4) would. One further note, since both � = � = � and � = � = −� return the identity element in ��(4), there is two-fold degeneracy in our group homomorphism as defined. Hence, we identify these two distinct transformations, such that

- 43 - ��(4) ≃ ��(2) × ��(2)/�

This relationship is critical in constructing ��(2) instantons in 4-dimensional space and recovering them from spacetimes instantons with Euclidean signature in the gravitational case. We will find this congruence important in the derivation of the ��(4) solution.

For now, we will demonstrate how our newly constructed matrix � is subject to change under the above defined transformation. We will then use our result to exploit symmetries when constructing Yang-Mills instanton structures. Proceeding, we can define our unitary transformations by exponentiating the ��(2) group generators:

� = � , � = � , � = 1, 2, 3 where � is the internal group index over which we sum, and {�} the Pauli matrices. Hence,

� + �� = (� + �� � + ⋯ )�(� − ��� + ⋯ ) = � + �� � � − ���� + �(� )

⇒ �� = �(� � � − ��� )

Great, but we are concerned with how the components of our 1-form will transform under a combination of rotations and unitary transformations. To do this we will exploit the following result: 1 � = ��� ⋅ � 2

We derive this by expanding our definition of �

1 1 � � ��(� ⋅ � ) = ��(� ⋅ �) = ��(�) − � ��(� ) = � 2 2 2 2

1 � � � � ��(� ⋅ � ) = � ��(� ) + ��(� � ) = � ��(� ) + ��(� + �� � ) = � 2 2 2 2 2 since the Pauli matrices are traceless. Proceeding we find that varying the components of our original form gives us

� � �� = ��(��� − ���) ⋅ � = �����̅ � − �̅ ��� � 2 2 � = �����̅ � − ����� �̅ � 2

Where we have exploited linearity and cyclicity of the trace. Now we define the following quantities [15],

� � � = − ���� �̅ , �̅ = ����̅ � 2 2

- 44 - These are ‘t Hooft symbols and they behave in the following way for given values of spacetime indices � and � [15]:

� , �, � = 1,2,3 � , �, � = 1,2,3 ⎧ ⎧ �, � = 4 −�, � = 4 � = , �̅ = ⎨−�, � = 4 ⎨�, � = 4 ⎩0, �, � = 4 ⎩0, �, � = 4

They are sometimes more generally expressed in the following closed form:

� = � + �� − ��, � = � + �� − ��

The above symbols have a very nice property that is exploited in the construction of gauge fields with self dual curvature: they themselves are self dual and anti self dual. 1 1 1 � � = � � + � � − � � = 2! � � + � − � = � 2 2 2 [ ]

And similarly, 1 �̅ = − � �̅ 2

- 45 -

Appendix C: Vierbeins and Spin Connections

As alluded to in the opening section on gauge fields, one can think of general relativity as a theory described by a vector bundle of tangent spaces �� over our spacetime manifold �, rather than a �-valued vector bundle in gauge theory. In this case, the total space of our tangent bundle is ��, defined in the following way:

�� = ��

Generally, the connection is then the torsion free Christoffel connection, Γ[] = 0.

However, there is an equivalent formulation of GR that uses a trivial bundle over the manifold, where sections of the bundle can be written in terms of an orthonormal basis {�}. In this case, the total space of Euclidean Gravity becomes � × ℝ . This is sometimes called the Palatini formalism [6]. The way we relate these two total spaces is by use of vierbein (or tetrad) fields

�: � × ℝ → �� Defined in the following way:

��(�) = �(�) = � (�)�, ��� (�) = � (�) = � (�)��

� ∈ �, �(�) = � ∈ ℝ

where {�� } would be an orthonormal basis of 1-form dual to {�}. To emphasize the difference between the trivial bundle with an internal space ℝ and our standard tangent bundle, we use Latin characters for the internal space and Greek characters for indices of coordinates on our spacetime manifold.

By constructing this redefinition of the vector bundle that describes our theory, we are given the nice property that the metric of our internal space is now Euclidean; and thus our understanding of geometry when using these Vierbein fields becomes remarkably intuitive.

Here are some of the following properties:

�� = �(�)�� �� = � (�)� (�)��� �� = �� �

� � = � , � � = �

Where � is used to raise and lower spacetime indices, and (trivially so) � raises and lowers the indices of the internal space (if we were working in Minkowski space, this would be more important)

- 46 - In addition, we are now endowed with a new connection between the standard fibers in our bundle, � , called the spin connection. This new connection 1-form is constructed with the following constraints:

� �� ≡ � = −� , �� = �� + � ∧ � = 0

The curvature 2-form in this total space, analogous to the Ricci curvature of our tangent bundle, now becomes defined in the following way:

� = �� + � ∧ � where

1 1 1 � = � �� ∧ �� = � �� �� ∧ �� = � � ∧ � 2 2 2

Notice the remarkable similarity this now shares with our definition of the field strength tensor in Yang-Mills theory! In fact, the field strength tensor of an ��(4) gauge theory, � , behaves in exactly the same way as our Euclidean spacetime analog of the Riemann curvature, �.

By using the construction of ��(4) with a double copy of ��(2) × ��(2)/�, this fact will be exploited when recovering ��(2) instantons from the Eguchi-Hanson metric in the main body of the test.

Under action of the exterior derivative on our structure equation, we can also recover the Bianchi identity: �(�� ) = 0

⇒ �(�� ) = �(�� + � ∧ � ) = �� ∧ � − � ∧ �� = �� ∧ � − � ∧ (−� ∧ � ) 1 = �� ∧ � + � ∧ � ∧ � = � ∧ � = � � ∧ � ∧ � = 0 2

The Bianchi Identity then reads

� ∧ � = 0 ⇒ �[] = 0

We can also write the Euclidean Einstein equation in a similar form

1 � − � � ∧ � 2 1 = � � � + � � � + � � � − � � � − � � � 4 1 − � � � � � ∧ � ∧ � = � � ∧ � = � ∧ � = 0 2 ⇒ � ∧ � = 0

Once again, this indicates to us that self-dual field configurations automatically satisfy the field equations for our theory. One final note, suppose we have a spin connection that is self-dual such that,

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1 � = � � = � 2 Then we find that,

1 1 � = �� + � ∧ � � = � �� + �� � � ∧ � 2 4 1 1 1 1 = � �� − � � � ∧ � = � �� − � � ∧ � 2 4 2 2 1 1 = � (�� + � ∧ � ) = � � = � 2 2

To further demonstrate the similarities between Yang-Mills and the Pallatini formulation of GR, we note how our connection will changed when the Vierbein fields are transformed by a local ��(4) transformation, Λ(�):

� → � = (Λ )(�)�

Using the structure equation, we find:

�� + � ∧ � → �� + � ∧ � = �((Λ )� ) + � ∧ (Λ ) � = (Λ )�� + �((Λ ) ) ∧ � + � Λ ∧ � = (Λ )�� + � (Λ ) + �(Λ ) ∧ �

For this to vanish, we require that,

� (Λ ) = (Λ )� − �(Λ ) ⇒ � = (Λ )� Λ − �(Λ ) Λ

Then applying the chain rule,

�(Λ ) Λ = �((Λ ) Λ) − (Λ ) �Λ = �(� ) − (Λ ) �Λ = −(Λ ) �Λ

This gives the final expression for how the connection transforms under a ��(4) gauge transformation:

� = Λ�Λ + Λ�Λ

In direct analogy to the law we derived for how a connection changes under a gauge transformation in Appendix A. Hence, by employing the use of tetrad fields we have functionally turned GR into a standard ����(4) gauge theory.

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