Calderón problem for Yang–Mills connections

TL;DR: In this paper, the authors considered the problem of identifying a unitary Yang-Mills connection on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian over compact Riemannian manifolds with boundary.

Abstract: We consider the problem of identifying a unitary Yang-Mills connection $ abla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $ abla^* abla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection, but at the cost of having more measurements. Also, we prove that the DN map is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

Summary

1. Introduction

• The authors consider the Calderón inverse problem for a special type of connections, called the Yang-Mills connections.
• Let us explain some motivation for considering this problem.
• Moreover, by exploiting this analytic structure they observe that the method of Lassas and Uhlmann [20] who prove the analytic Calderón problem for metrics, may be used to extend this local isometry to the whole of the manifold.
• The authors may use a similar unique continuation property (UCP in short) result to yield the equivalence of connections close to the boundary.
• Finally, in section five the authors apply the Runge-type approximation property and prove Theorem 1.3.

2. Preliminaries

• As mentioned previously, Yang-Mills (YM) connections are very important in physics and geometry.
• They satisfy the so called Yang-Mills equations, which are considered as a generalisation of Maxwell's equations in electromagnetism and which provide a framework to write the latter equations in a coordinate-free way (see e.g. [1] or [10] for a geometric overview and definitions).
• PDE theory and vector bundles over complex projective spaces, or algebraic geometry.
• First recall that connections on bundles modulo gauges are classified by their holonomy representation on the so called loop group modulo conjugation (see Kobayashi and Nomizu [17] ).
• This has an analogous geometric interpretation: the difference to the flat case is that the authors now identify homotopic only if they enclose the same area.

2.2. Local expressions for d *

• For the record, the authors will write down the explicit formula in local coordinates for the inner product on the differential forms with values in E. Example 2.2 (An electromagnetic correspondence).
• There are several gauges that have proved to work well in practise, i.e. that fit well into other mathematical formalism in applications.
• Another slightly related gauge is the temporal gauge, which the authors will also make use of -in this gauge, one of the components of the connection vanishes locally (they usually distinguish this variable as "time").
• Which the authors state for convenience, since it will get used frequently throughout the paper: Lemma 2.3.

3. Boundary determination for a connection and a matrix potential

• Before going into proofs, let us briefly lay out some of the notation that goes into pseudodifferential operators on vector bundles over manifolds (see [21, 28, 29] for more details).
• The local symbol calculus developed for scalar operators carries over to the case of vector bundles, as can be seen from the above references.
• The following claim is analogous to Proposition 1.2 in [22] -the main difference is that now the authors are using matrix valued pseudodifferential operators, so they need to make sure that appropriate generalisations hold.
• Assume without loss of generality that A satisfies condition (3.12) (see the paragraph after this Proposition).
• This completes the proof of the induction and of the theorem, since two formal expansions of the same operator in terms of classical symbols that agree modulo S −∞ , must also be congruent.

Remark 3.6 (Local boundary determination). If we assume that Γ ⊂ ∂M is open and Λ

• Remark 3.7 (The case of E topologically non-trivial).
• The authors main result of the chapter, Theorem 3.4, remains valid in the following form.
• In this section the authors consider the main conjecture in the special case of Yang-Mills connections.
• The authors prove Theorem 1.1 for line bundles in the smooth category.
• Where P is a first order, non-linear operator arising from the equality.

Proof. By using that d

• Without loss of generality, assume that the normal components of connections A and B near the boundary vanish (see Lemma 2.3).
• It can of course happen that the zero of g contains an (n−1)-dimensional submanifold, see Figure 1 below for such an example (more precisely, u in this example gives the real part of such a solution, with the imaginary part equal to zero).
• The boundary determination result applied to quantities A and B defined in (4.5) and (4.6) and the degenerate unique continuation result of Mazzeo now applies to equations (4.7) and (4.9), to uniquely extend from ∂M , as before.
• The idea is that drilling the holes connects path components over the possibly disconnecting set N (g).
• From that point, the authors may apply the earlier argument in the same way.

5. Recovering a Yang-Mills connection for m > 1 via geometric analysis

• Due to the recent work of the author [5] the authors have strong evidence and some counterexamples to even the weak unique continuation principle.
• These counterexample seem not to be generic, so the authors hope that this method can still be pursued.
• Analytic functions satisfy the SUCP by definition and in addition, the zero set is given by a countable union of analytic submanifolds of codimension one.
• The Yang-Mills equations become elliptic and therefore, A is analytic.
• This leaves us in the setting (5.1) from the previous paragraph, suitable for drilling the holes -inductively, the authors perturb γ such that it intersects the M i in the drilled holes.

Figures

Figure 1. In blue – the zero set of the harmonic function with boundary value equal to f(θ) = θ · sin 100 θ on the unit disc, where θ ∈ (−π, π) is the angular coordinate. In orange – the unit circle. The accumulation point is (1, 0).

Full text

CALDER
´
ON PROBLEM FOR YANG-MILLS CONNECTIONS
MIHAJLO CEKI
´
C
Abstract. We consider the problem of identifying a unitary Yang-Mills connection
∇ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the
connection Laplacian ∇
∗
∇ over compact Riemannian manifolds with boundary.
We establish uniqueness of the connection up to a gauge equivalence in the case
of trivial line bundles in the smooth category and for the higher rank case in the
analytic category, by using geometric analysis methods and essentially only one
measurement.
Moreover, by using a Runge-type approximation argument along curves to re-
cover holonomy, we are able to uniquely determine both the bundle structure and
the connection, but at the cost of having more measurements. Also, we prove that
the DN map is an elliptic pseudodifferential operator of order one on the restriction
of the vector bundle to the boundary, whose full symbol determines the complete
Taylor series of an arbitrary connection, metric and an associated potential at the
boundary.
1. Introduction
In this paper, we consider the Calder´on inverse problem for a special type of con-
nections, called the Yang-Mills connections. Given a Hermitian vector bundle E of
rank m over a compact Riemannian manifold (M, g) with non-empty boundary and a
unitary connection A (∇) on E, one may consider the connection Laplacian denoted
by d
∗
A
d
A
(∇
∗
∇), where d
∗
A
(∇
∗
) denotes the formal adjoint of d
A
(∇) with respect
to the Hermitian and Riemannian structures. Sometimes this operator is called the
magnetic Laplacian because it is used to represent the magnetic Schr¨odinger equation,
where A corresponds to the magnetic potential.
Given this, we may define the associated Dirichlet-to-Neumann (DN map in short)
Λ
A
: C
∞
(∂M; E|
∂M
) → C
∞
(∂M; E|
∂M
)
1
by solving the Dirichlet problem:
d
∗
A
d
A
(u) = 0, u|
∂M
= f (1.1)
and setting Λ
A
(f) = d
A
(u)(ν), where ν is the outwards pointing normal at the bound-
ary. The problem can then be posed as asking whether the map A 7→ Λ
A
is injective
modulo the natural obstruction, or in other words whether Λ
A
= Λ
B
implies the
existence of a gauge automorphism F : E → E with F
∗
(A) = B and F |
∂M
= Id.
This problem was considered in [6, 11–13]; for a survey of the Calder´on problem
for metrics, see [31]. In this paper, we take two approaches to uniqueness: one is
via geometric analysis and the other by constructing special gauges along curves
via the Runge approximation property of elliptic equations. As far as we know,
this paper is the first one that considers the connection problem and does not rely
1
By C
∞
(M; E) we denote the space of smooth sections of E over M.
1
arXiv:1704.01362v2 [math.AP] 12 Jun 2018

2 Mihajlo Ceki
´
c
on the Complex Geometric Optics solutions (see any of [6, 11–13]), but on unique
continuation principles and geometric analysis of the zero set of a solution to an
elliptic equation.
The Yang-Mills connections generalise flat connections and are important in physics
and geometry. They satisfy the following equation:
D
∗
A
F
A
= 0
where D
A
= d
End
A
is the induced connection on the endomorphism bundle EndE and
F
A
is the curvature of A (see the preliminaries for more details).
Firstly, we prove that the DN map Λ
A
is an elliptic pseudodifferential operator of
order 1 on the restriction of the vector bundle to the boundary and deduce that its
full symbol determines the full Taylor series of the connection, metric and a potential
at the boundary. This was first proved in the case of a Riemannian metric by Lee and
Uhlmann [22] and later considered in the m = 1 case with a connection in [11]. In
this paper, we generalise this approach to the case of systems and prove the analogous
result.
1.1. Motivation. Let us explain some motivation for considering this problem. Partly,
the idea came from the analogy between Einstein metrics in Riemannian geometry
and Yang-Mills connections on Hermitian vector bundles. Also, Guillarmou and S´a
Barreto in [14] prove the recovery of two Einstein manifolds from the DN map for
metrics. The method of their proof relies on a reconstruction near the boundary,
where in special harmonic coordinates Einstein equations become quasi-linear ellip-
tic (the metric is thus also analytic in such coordinates). Hence, by combining the
boundary determination result and a unique continuation result for elliptic systems
they prove one can identify the two metrics in a neighbourhood of the boundary.
Moreover, by exploiting this analytic structure they observe that the method of Las-
sas and Uhlmann [20] who prove the analytic Calder´on problem for metrics, may be
used to extend this local isometry to the whole of the manifold.
2
In our case, the conventionally analogous concept to harmonic coordinates to con-
sider would be the Coulomb gauge [30] which transforms the connection to a form
where d
∗
(A) = 0, so that the Yang-Mills equations become an elliptic system with
principal diagonal part. However, this gauge does not tie well with the DN map, so
in Lemma 4.1 we construct an analogue of the harmonic gauge for connections. In
this gauge, we may use a similar unique continuation property (UCP in short) result
to yield the equivalence of connections close to the boundary. However, for going
further into the interior we designed new methods.
1.2. Uniqueness via geometric analysis. We believe this approach to be entirely
new. Here is one of the main theorems of the paper.
Theorem 1.1 (Global result). Assume dim M ≥ 2, let E = M × C be a Hermitian
line bundle with standard metric and ∅ 6= Γ ⊂ ∂M an open, non-empty subset of the
boundary. Let A and B be two unitary Yang-Mills connections on E. If Λ
A
(f)|
Γ
=
2
This works by embedding the two manifolds in a suitable Sobolev space using Green’s functions
of the metric Laplacians and showing the appropriate composition is an isometry.

Calder
´
on problem for Yang-Mills connections 3
Λ
B
(f)|
Γ
for all f ∈ C
∞
0
(Γ; E|
Γ
), then there exists a gauge automorphism (unitary) h
with h|
Γ
= Id such that h
∗
(A) = B on the whole of M.
We now explain this geometric analysis type method in more detail. Our gauge F
from Lemma 4.1 (m × m matrix function on M) satisfies the equation d
∗
A
d
A
F = 0
and so we cannot guarantee that it is non-singular globally. We show that the zero
set of the determinant of F is suitably small in the smooth case when m = 1 and in
the analytic case for arbitrary m – it is covered by countably many submanifolds of
codimension one, or in the language of geometric analysis it is (n −1)-C
∞
-rectifiable.
Since (the complement of) this singular set can be topologically non-trivial (see Figure
1), we end up with barriers consisting of singular points of F that prevent us to use
the UCP and go inside the manifold. This is addressed by looking at the sufficiently
nice points of the barriers and locally near these points, using a degenerate form of
UCP (in the smooth case) or a suitable form of analytic continuation (in the analytic
case) to extend an appropriate gauge equivalence between the two given connections
beyond the barriers; we name this procedure as “drilling”. Since we show there is
a dense set of such nice points, we may perform the drilling to extend our gauges
globally.
Here is what we prove in the analytic case, for arbitrary m:
Theorem 1.2. Let (M, g) be an analytic Riemannian manifold with dim M ≥ 2 and
let Γ be as in Theorem 1.1.
3
If E = M × C
m
is a Hermitian vector bundle with
the standard structure and if A and B are two unitary Yang-Mills connections on E,
then Λ
A
(f)|
Γ
= Λ
B
(f)
Γ
for all f ∈ C
∞
0
(Γ; E|
Γ
) if and only if there exists a gauge
automorphism H of E, with H|
Γ
= Id, such that H
∗
(A) = B.
We briefly remark that the proof the above theorem also relies on using the Coulomb
gauge locally, since in this gauge the connection is analytic, so det F satisfies the
SUCP (see Lemma 5.1); we additionally apply this gauge in the drilling procedure.
The main difficulty in proving uniqueness via the geometric analysis method for the
smooth, higher rank (m > 1) case is that the strong unique continuation property
(SUCP) for the determinant det F of a solution to d
∗
A
d
A
F = 0 might not hold –
see Remark 5.2 for more details. Indeed, in the subsequent work [5] we treat this
question in more detail and prove a positive answer for n = 2 and also provide some
counterexamples.
1.3. Uniqueness via Runge approximation. Next, we outline our second ap-
proach to uniqueness by using Runge-type approximation for elliptic equations to
recover holonomy, which we use to prove the stronger statement of uniqueness for
arbitrary bundles. In general, Runge approximation is known to be applicable to
inverse problems (see e.g. [2, 19, 27]).
Theorem 1.3. Let (M, g) be a smooth compact Riemannian manifold with boundary
of dimension dim M ≥ 2, Γ ⊂ ∂M a non-empty open set, E and E
0
Hermitian vector
bundles over M such that we have the identification E|
Γ
= E
0
|
Γ
. Let A and B be two
smooth unitary Yang-Mills connections on E and E
0
respectively, such that Λ
A
(f)|
Γ
=
3
The metric g is only assumed to be analytic in the interior of M and smooth up to the boundary.

4 Mihajlo Ceki
´
c
Λ
B
(f)|
Γ
for all f ∈ C
∞
0
(Γ; E). Then there exists a unitary bundle isomorphism
H : E
0
→ E with H|
Γ
= Id, such that H
∗
A = B.
This theorem clearly generalises Theorems 1.1 and 1.2. Note in particular that
we are able to uniquely determine the topology of the bundles and the Hermitian
structures.
Let us point out the main differences between the two approaches. Although Theo-
rem 1.3 is stronger than the first two theorems, the advantage of the former approach
is in its method of proof. More precisely, the geometric analysis technique is minimal
with respect to the necessary data – essentially, we only need one arbitrary measure-
ment to uniquely identify the connections – see Remark 4.6 for more details. Also,
this method is entirely new, so it gives hope that it can be generalised to different
settings, such as the metric Calder´on problem.
On the other hand, in the Runge-type density approach we need many measure-
ments that concentrate in a suitable sense on closed loops (see Lemma 6.1). In the
context of the Calder´on problem for connections, this method is also new and gives
hope to be generalised to other settings.
1.4. Organisation of the paper. The paper is organised as follows: in the next
section, we recall some formulas from differential geometry and a make a few obser-
vations about choosing appropriate gauges. In the third section we prove that Λ
A
is a pseudodifferential operator of order 1 for systems and prove that its full symbol
determines the full jet of A at the boundary. Furthermore, in section four we consider
the smooth case and prove the global result for m = 1, Theorem 1.1. In the same
section, we construct the new gauge and deduce the UCP result we need. In section
five we consider the m > 1 case for analytic metrics, Theorem 1.2, by adapting the
proof of the line bundle case and exploiting real-analyticity. Finally, in section five
we apply the Runge-type approximation property and prove Theorem 1.3.
This paper also has two appendices: in Appendix A we recall some well-posedness
condition for the heat equation and prove a few elementary statements about extend-
ing functions smoothly over small sets. Next, in Appendix B we lay out the technical
results needed to prove the Runge type approximation result we need – this requires
some well-posedness for Dirichlet problem in negative Sobolev spaces and a duality
argument; the aim is to prove that for L
A
= d
∗
A
d
A
, we can build m smooth solutions
that span the bundle over a given curve.
Acknowledgements. The author would like to express his gratitude for the support
of his supervisor, Gabriel Paternain. Furthermore, he would also like to acknowledge
the help of Mikko Salo, Rafe Mazzeo, Colin Guillarmou and Gunther Uhlmann, es-
pecially in clarifying the unique continuation results used in the paper. The author
thanks Guillaume Bal and Francois Monard for telling him about Runge type ap-
proximation results which inspired the last section of the paper.
He is grateful to the Trinity College for financial support, where the most of this
research took place and to Max-Planck Institute for Mathematics in Bonn, where a
part of this research took place.

Calder
´
on problem for Yang-Mills connections 5
2. Preliminaries
2.1. Yang-Mills connections. As mentioned previously, Yang-Mills (YM) connec-
tions are very important in physics and geometry. They satisfy the so called Yang-
Mills equations, which are considered as a generalisation of Maxwell’s equations in
electromagnetism and which provide a framework to write the latter equations in a
coordinate-free way (see e.g. [1] or [10] for a geometric overview and definitions). The
Yang-Mills connections are critical points of the functional:
F
Y M
(A) =
Z
M
|F
A
|
2
dω
g
Here F
A
= dA + A ∧ A is the curvature 2-form with values in the endomorphism
bundle of E determined by the map d
2
A
s = F
A
∧s on sections s ∈ C
∞
(M; E) and ω
g
is the volume form. It can then be shown by considering variations of this functional,
that the equivalent conditions for A being its critical point are (the Euler-Lagrange
equations):
(D
A
)
∗
F
A
= 0 and D
A
F
A
= 0 (2.1)
where D
A
= d
End
A
is the induced connection on the endomorphism bundle, given
locally by D
A
S = dS + [A, S] or equivalently by D
A
S = [d
A
, S], where [·, ·] denotes
the commutator. The second equation in (2.1) is actually redundant, since it is the
Bianchi identity.
Yang-Mills connections clearly generalise flat connections, for which the curvature
vanishes, i.e. F
A
= 0.
They have been a point of unification between pure mathematics and theoretical
physics, but moreover have brought a few areas of pure mathematics together, such
as e.g. PDE theory and vector bundles over complex projective spaces, or algebraic
geometry.
Example 2.1 (Yang-Mills connections over Riemann surfaces). We give an idea of
the size of the set of YM connections in the simplest non-trivial example of Riemann
surfaces. First recall that connections on bundles modulo gauges are classified by
their holonomy representation on the so called loop group modulo conjugation (see
Kobayashi and Nomizu [17]). In the setting of flat connections, this correspondence
simplifies significantly for a Riemann surface Σ:

ρ : π
1
(Σ) → U(m)

/conj. ←→ {unitary flat bundles of rank m}
since homotopic loops have the same holonomy. The direct map (going left to right)
here is the one taking a representation ρ and defining an associated flat bundle via
e
Σ ×
ρ
C
m
, where
e
Σ is the universal cover of Σ and ×
ρ
means we identified the two by
the diagonal action. Somewhat surprisingly, we may still obtain a correspondence in
the case of YM connections, where π
1
(Σ) is replaced by a certain central extension
bπ
1
(Σ) (see [1] for more details). This has an analogous geometric interpretation: the
difference to the flat case is that we now identify homotopic only if they enclose the
same area. In particular, for the sphere S
2
this simplifies, so that we have bπ
1
(S
2
) = S
1
.