Twisted Dolbeault cohomology of nilpotent Lie algebras

Liviu Ornea111Liviu Ornea is partially supported by a grant of Ministry of
Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-
PCE-2016-0065, within PNCDI III., Misha Verbitsky222Misha Verbitsky is
partially supported by the Russian Academic Excellence Project ’5-100”, FAPERJ
E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Keywords: nilpotent Lie algebra, twisted cohomology, locally conformally
Kähler, Vaisman manifold. 2010 Mathematics Subject Classification: 53C55,
18H25, 22E25.

Abstract

It is well known that the cohomology of any non-trivial 1-dimensional local
system on a nilmanifold vanishes (this result, due to J. Dixmier, was also
announced and proved in some particular case by Alaniya). A complex
nilmanifold is a quotient of a nilpotent Lie group equipped with a left-
invariant complex structure by an action of a discrete, co-compact subgroup.
We prove a Dolbeault version of Dixmier’s and Alaniya’s theorem, showing that
the Dolbeault cohomology $H^{0,p}(\mathfrak{g},L)$ of a nilpotent Lie algebra
with coefficients in any non-trivial 1-dimensional local system vanishes. Note
that the Dolbeault cohomology of the corresponding local system on the
manifold is not necessarily zero. This implies that the twisted version of
Console-Fino theorem is false (Console-Fino proved that the Dolbeault
cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of
its Lie algebra, when the complex structure is rational). As an application,
we give a new proof of a theorem due to H. Sawai, who obtained an explicit
description of LCK nilmanifolds. An LCK structure on a manifold $M$ is a
Kähler structure on its cover $\tilde{M}$ such that the deck transform map
acts on $\tilde{M}$ by homotheties. We show that any complex nilmanifold
admitting an LCK structure is Vaisman, and is obtained as a compact quotient
of the product of a Heisenberg group and the real line.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Locally conformally Kähler manifolds
    2. 2.2 Vaisman manifolds
    3. 2.3 LCK manifolds with potential
  3. 3 LCK structures on nilmanifolds
    1. 3.1 Invariant geometric structures on Lie groups
    2. 3.2 LCK nilmanifolds
  4. 4 Twisted Dolbeault cohomology on nilpotent Lie algebras
    1. 4.1 Twisted Dolbeault cohomology
    2. 4.2 Console-Fino theorem with coefficients in a local system fails
  5. 5 Classification of LCK nilmanifolds
    1. 5.1 Ugarte’s conjecture
    2. 5.2 Classification theorem for LCK nilmanifolds

##  1 Introduction

In complex dimension $>2$, an LCK (locally conformally Kähler) manifold is a
complex manifold equipped with a Hermitian metric (LCK metric) which is
locally conformally equivalent to a Kähler manifold. Then the Hermitian form
satisfies $d(\omega)=\theta\wedge\omega$, where $\theta$ is a 1-form. Clearly,
$d^{2}(\omega)=0$ gives $d\theta\wedge\omega=0$, which is equivalent to
$d\theta=0$ in complex dimension $>2$. In complex dimension 2, this condition
has to be added artificially.

Summing it up, an LCK metric (or an LCK structure) on a complex manifold of
complex dimension $\geqslant 2$ is a Hermitian metric with Hermitian form
$\omega$ which satisfies $d(\omega)=\theta\wedge\omega$, where $\theta$ is a
closed 1-form.

Consider the differential $d_{\theta}(\alpha):=d(\alpha)-\theta\wedge\alpha$.
Then the equation $d(\omega)=\theta\wedge\omega$ can be written as
$d_{\theta}(\omega)=0$. The differential $d_{\theta}$ is called the Morse-
Novikov differential. It can be interpreted as de Rham differential for forms
with coefficients in a local system, which is done as follows.

Let $L$ be a trivial real bundle of rank 1, and $\nabla_{0}$ the standard
connection. Demote by $\nabla:=\nabla_{0}-\theta$ the connection
$\nabla(b)=\nabla_{0}(b)-b\otimes\theta$. Since $\nabla^{2}=0$, this
connection is flat, and $(L,\nabla)$ defines a local system, which we denote
by the same letter. Then $d_{\theta}$ is the de Rham differential on
differential forms with values in $L$, and the cohomology of $d_{\theta}$ is
identified with the cohomology of this local system.

In this paper we study the locally conformally Kähler structures on
nilmanifolds.

A nilmanifold is a quotient of a connected, simply connected nilpotent Lie
group by a discrete co-compact lattice. It is not hard to see that any
nilmanifold is the total space of an iterated family of circle bundles. Its
cohomology is expressed through the cohomology of the Chevalley-Eilenberg
differential on the corresponding Lie algebra ${\mathfrak{g}}$, and the Morse-
Novikov cohomology is obtained as cohomology of the twisted version of the
Chevalley-Eilenberg differential (Section 4).

In the literature, there are several different (and incompatible) ways to
define a complex nilmanifold. Before 1980-ies, most people defined complex
nilmanifold as a quotient of a complex nilpotent Lie group $G_{\mathbb{C}}$ by
a co-compact lattice. This quotient is a parallelizable complex manifold (that
is, its tangent bundle is holomorphically trivial). Indeed, the right action
of $G_{\mathbb{C}}$ on itself commutes with the left action, and both are
holomorphic. If we take the quotient $M:=G_{\mathbb{C}}/\Gamma$ by (say) left
action of $\Gamma\subset G_{\mathbb{C}}$, the manifold $M$ remains homogeneous
with respect to the right action of $G_{\mathbb{C}}$, and this action
trivializes $TM$, making $M$ parallelizable.

Now such nilmanifolds are called parallelizable complex nilmanifolds or
Iwasawa type nilmanifolds, after the Iwasawa group (the non-commutative
3-dimensional complex nilpotent Lie group).

Before the paper [CFD], in the published literature “complex nilmanifold”
meant the quotient of a complex Lie group; see for example [Fis] or [Oe]. In
[CFD], Cordero, Fernández and de León introduced the modern notion of a
complex nilmanifold. Since then, a complex nilmanifold is defined as a compact
quotient of a nilpotent Lie group equipped with a left-invariant complex
structure (Subsection 3.2). The main advantage of this notion is that it can
be rephrased in terms of a complex structure operator on the corresponding Lie
algebra. However, such a quotient is not homogeneous, unless $G$ is a complex
Lie group.

The same approach can be used to describe other important geometric structures
on nilmanifolds in terms of linear-algebraic structures on their Lie algebras
(Subsection 3.1). In the same paper [CFD], this approach was used to construct
an LCK structure on a nilmanifold obtained from the Heisenberg group. Later,
H. Sawai proved in [Saw1] that any LCK structure on a complex nilmanifold is
obtained this way. In the present paper we give a proof of Sawai’s theorem
based on cohomology vanishing.

Consider an oriented rank 1 local system $(L,\nabla_{0}+\theta)$ on a
nilmanifold $M=G/\Gamma$, where $(L,\nabla_{0})$ is a trivial bundle with
connection, and $\theta$ is a closed, non-exact 1-form. In [Ala] (see also
[Mi]), it was shown that $H^{*}(M,L)=0$. The proof goes as follows: first, one
identifies $H^{*}(M,L)$ with the cohomology of the Lie algebra
${\mathfrak{g}}=\operatorname{Lie}(G)$ with coefficients in a non-trivial rank
one representation. This is done by using the homogeneous $G$-action. Then one
takes the Chevalley-Eilenberg complex which computes the Lie algebra
cohomology, filters it by the central series, and computes the first page of
the corresponding spectral sequence. This first page is identified with the
twisted cohomology of a commutative Lie algebra, which always vanishes.

However, this approach will not work for Dolbeault cohomology. Console and
Fino ([CF]) proved that the Dolbeault cohomology can, indeed, be computed
using the Hodge decomposition on the Chevalley-Eilenberg complex of the
corresponding Lie algebra, when the complex structure is rational (in
dimension up to 6 the rationality of the complex structure is not needed, see
[FRR], but for higher dimensions the question is still open). However, this
result is highly non-trivial, because complex nilmanifolds are not
homogeneous. Moreover, the extension of Console-Fino theorem to the Dolbeault
cohomology with coefficients in a local system (“twisted Dolbeault
cohomology”) is false (Subsection 4.2).

The main result of the present paper is the following theorem, which computes
the Lie algebra version of the Dolbeault cohomology with coefficients in a
local system.

Theorem 1.1: Let ${\mathfrak{g}}$ be a nilpotent real Lie algebra,
$I:\;{\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{g}}$ a complex structure
operator,
${\mathfrak{g}}_{\mathbb{C}}:={\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$,
and
${\mathfrak{g}}_{\mathbb{C}}={\mathfrak{g}}^{1,0}\oplus{\mathfrak{g}}^{0,1}$
the corresponding eigenvalue decomposition, called the Hodge decomposition in
the sequel. Assume that
${\mathfrak{g}}^{1,0}\subset{\mathfrak{g}}_{\mathbb{C}}$ is a Lie subalgebra
(this is equivalent to the left-invariant complex structure on on the Lie
group $G=\operatorname{Lie}({\mathfrak{g}})$ associated with $I$ being
integrable, see 3.1). Consider the Hodge decomposition
$\Lambda^{*}({\mathfrak{g}}_{\mathbb{C}}^{*})=\bigoplus\Lambda^{p,q}({\mathfrak{g}}^{*})$
on the Grassman algebra of ${\mathfrak{g}}_{\mathbb{C}}$, with

$\Lambda^{p,q}({\mathfrak{g}}^{*})=\Lambda^{p}(({\mathfrak{g}}^{1,0})^{*})\otimes_{\mathbb{C}}\Lambda^{q}(({\mathfrak{g}}^{0,1})^{*})$

and define the twisted Dolbeault differentials $\partial_{\theta}$,
$\overline{\partial}_{\theta}$ as Hodge components of the twisted Chevalley-
Eilenberg differential $d_{\theta}(x):=d(x)-\theta\wedge x$, i.e.
$\partial_{\theta}=\partial-\theta^{1,0}$ and
$\overline{\partial}_{\theta}=\overline{\partial}-\theta^{0,1}$ (Subsection
3.1). Then the cohomology of the complex
$(\Lambda^{0,*}({\mathfrak{g}}_{\mathbb{C}}^{*}),\overline{\partial}_{\theta})$
vanishes.

Proof: See 4.1.

We apply this result to get a new proof of the classification of LCK
structures on complex nilmanifolds.

##  2 Preliminaries

###  2.1 Locally conformally Kähler manifolds

Definition 2.1: Let $(M,I)$ be a complex manifold,
$\dim_{\mathbb{C}}M\geqslant 2$. It is called locally conformally Kähler (LCK)
if it admits a Hermitian metric $g$ whose fundamental 2-form
$\omega(\cdot,\cdot):=g(\cdot,I\cdot)$ satisfies

$d\omega=\theta\wedge\omega,\quad d\theta=0,$ (2.1)

for a certain closed 1-form $\theta$ called the Lee form.

Remark 2.2: Definition (2.1) is equivalent to the existence of a covering
$\tilde{M}$ endowed with a Kähler metric $\Omega$ which is acted on by the
deck group $\operatorname{Aut}_{M}(\tilde{M})$ by homotheties. Let

$\chi:\operatorname{Aut}_{M}(\tilde{M})\longrightarrow{\mathbb{R}}^{>0},\quad\chi(\tau)=\frac{\tau^{*}\Omega}{\Omega},$
(2.2)

be the character which associates to a homothety its scale factor. On this
cover, the pull-back of the Lee form is exact.

Remark 2.3:  For an LCK manifold, coverings with the above property are not
unique. The covering for which the character $\chi$ is injective is called the
minimal cover.

Remark 2.4: The operator $d_{\theta}:=d-\theta\wedge$ obviously satisfies
$d_{\theta}^{2}=0$ and hence $(\Lambda^{*}M,d_{\theta})$ produces a cohomology
called Morse-Novikov or twisted. It can be interpreted as the cohomology of
the local system $L$ associated to the line bundle endowed with (flat)
connection form $\theta$. See Section 4 for details.

Theorem 2.5: ([Va1]) Let $(M,\omega,\theta)$ be a compact LCK manifold, not
globally conformally Kähler (i.e. with non-exact Lee form). Then $M$ does not
admit a Kähler metric.

###  2.2 Vaisman manifolds

Definition 2.6: An LCK manifold $(M,\omega,\theta)$ is called Vaisman if
$\nabla\theta=0$, where $\nabla$ is the Levi-Civita connection of $g$.

The following characterization, very much used in applications, is available:

Theorem 2.7: ([KO]) Let $(M,\omega,\theta)$ be an LCK manifold equipped with a
holomorphic and conformal ${\mathbb{C}}$-action without fixed points, which
lifts to non-isometric homotheties on the Kähler covering $\widetilde{M}$.
Then $(M,\omega,\theta)$ is conformally equivalent to a Vaisman manifold.

The main example of Vaisman manifold is the diagonal Hopf manifold ([OV1]).
The Vaisman compact complex surfaces are classified in [Bel], see also [OVV].

Remark 2.8: There exist compact LCK manifolds which do not admit Vaisman
metrics. Such are the LCK Inoue surfaces, [Bel], the Oeljeklaus-Toma
manifolds, [OT], [Ot], and the non-diagonal Hopf manifolds, [OV1], [OVV].

Remark 2.9:  On a Vaisman manifold, the Lee field $\theta^{\sharp}$ and the
anti-Lee field $I\theta^{\sharp}$ are real holomorphic
($\operatorname{Lie}_{\theta^{\sharp}}I=\operatorname{Lie}_{I\theta^{\sharp}}I=0$)
and Killing
($\operatorname{Lie}_{\theta^{\sharp}}g=\operatorname{Lie}_{I\theta^{\sharp}}g=0$),
see [DO].

Recall that a Killing vector field $X$ satisfies the equation:

$g(\nabla_{A}X,B)=-g(A,\nabla_{B}X).$

Setting $A=X$, we get $g(\nabla_{X}X,B)=-g(X,\nabla_{B}X)$. The last term is
equal to $-1/2\operatorname{Lie}_{B}(g(X,X))=0$, hence $g(\nabla_{X}X,B)=0$
for all $B$. Then $\nabla_{X}X=0$, and hence the trajectories of a Killing
field of constant length are geodesics. Therefore, the canonical foliation in
a Vaisman manifold is totally geodesic.

Remark 2.10: (i) Note that while the LCK condition is conformally invariant
(changing the metric $g\mapsto e^{f}g$ changes the Lee form into
$\theta\mapsto\theta+df$), the Vaisman condition is not conformally invariant.
Indeed, on a Vaisman manifold, the Lee form is coclosed (being parallel) and
hence a Vaisman metric is a Gauduchon metric; but one knows that on a compact
complex manifold, each conformal class contains a unique Gauduchon metric (up
to constant multipliers), [Ga].

(ii) Since $\theta$ is parallel, it has constant norm and thus we can always
scale the LCK metric such that $|\theta|=1.$ In this assumption, the following
formula holds, [Va2], [DO]:

$d\theta^{c}=\theta\wedge\theta^{c}-\omega,\quad\text{where}\quad\theta^{c}(X)=-\theta(IX).$
(2.3)

Moreover, one can see, [Ve], that the (1,1)-form $\omega_{0}:=-d^{c}\theta$ is
semi-positive definite, having all eigenvalues111The eigenvalues of a
Hermitian form $\eta$ are the eigenvalues of the symmetric operator $L_{\eta}$
defined by the equation $\eta(x,Iy)=g(L_{\eta}x,y)$. positive, except one
which is 0.

###  2.3 LCK manifolds with potential

We now introduce a class of LCK manifolds strictly containing the Vaisman
manifolds.

Definition 2.11: We say that an LCK manifold has LCK potential if it admits a
Kähler covering on which the Kähler metric has a global and positive potential
function which is acted on by holomorphic homotheties by the deck group. In
this case, $M$ is called LCK manifold with potential.

Remark 2.12: Note that in several previous papers of ours we asked the
potential function to be proper. It is not the case for the above definition.
See [OV2] for a comprehensive discussion about proper and improper LCK
potentials.

Remark 2.13:  One can prove that $(M,I,g,\theta)$ is LCK with potential if and
only if equation (2.3) is satisfied.

Definition 2.14: A function $\varphi\in C^{\infty}(M)$ is called
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic if
$\omega=d_{\theta}d^{c}_{\theta}(\varphi)$.

Remark 2.15: Note that $d_{\theta}d^{c}_{\theta}$-plurisubharmonic are not
plurisubharmonic (they exist on compact manifolds). Moreover, if $\varphi$ is
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic, then
$\varphi+\operatorname{\text{\sf const}}$ is not necessarily
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic.

Equation (2.3) (and hence the definition of LCK manifolds with potential) can
be translated on the LCK manifold itself:

Theorem 2.16: ([OV2, Claim 2.8]) $(M,I,\theta,\omega)$ is LCK with potential
if and only if $\omega=d_{\theta}d^{c}_{\theta}(\varphi)$ for a strictly
positive $d_{\theta}d^{c}_{\theta}$-plurisubharmonic function $\varphi$ on
$M$.

Theorem 2.17: ([OV4]) Let $(M,\theta,\omega)$ be an LCK manifold which is not
Kähler, and suppose $\omega=d_{\theta}d^{c}_{\theta}(\varphi_{0})$ for some
smooth function $\varphi_{0}\in C^{\infty}(M)$. Then $\varphi_{0}>0$ at some
point of $M$.

Proof: (Courtesy of Matei Toma.) By absurd, $\varphi_{0}\leqslant 0$
everywhere on $M$. Let $\tilde{M}$ be the minimal Kähler cover of $M$ (see
2.1), $\Gamma$ its deck group, and $\rho$ a function on $\tilde{M}$ such that
$d\rho=\theta$. Then the $d_{\theta}d^{c}_{\theta}$-plurisubharmonicity of
$\varphi_{0}$ is equivalent to the plurisubharmonicity of
$\varphi:=e^{-\rho}\varphi_{0}$, see 2.3.

Since the strict $d_{\theta}d^{c}_{\theta}$-plurisubharmonicity is stable
under $C^{2}$-small deformations of $\varphi$, the function
$\varphi-\varepsilon$ is also strictly
$d_{\theta}d^{c}_{\theta}$-plurisubharmonic. Therefore, we may assume that
$\varphi<0$ everywhere.

Define

$\psi:=-\log(-\varphi).$

Since $x{\>\longrightarrow\>}-\log(-x)$ is strictly monotonous and convex, the
function $\psi$ is strictly $d_{\theta}d^{c}_{\theta}$-plurisubharmonic.
Moreover, for every element $\gamma\in\Gamma$, we have

$\gamma^{*}\psi=-\log(-(\varphi\circ\gamma))=-\log(\chi(\gamma))-\log(-\varphi)=\operatorname{\text{\sf
const}}+\psi.$

Therefore, the Kähler form $dd^{c}\psi$ is $\Gamma$-invariant and descends to
$M$, contradiction with 2.1.

Remark 2.18: All Vaisman manifolds are LCK manifolds with potential. Among the
non-Vaisman examples, we mention the non-diagonal Hopf manifolds, [OV2]. On
the other hand, the Inoue surfaces and their higher dimensional analogues, the
Oeljeklaus-Toma manifolds, are compact LCK manifolds which are not LCK
manifolds with potential ([Ot]).

We can characterize the Vaisman metrics among the LCK metrics with potential:

Proposition 2.19: ([OV3, Proposition 2.3 & Corollary 2.4])
Let $(M,\omega,\theta)$ be a compact LCK manifold with potential. Then the LCK
metric is Gauduchon if and only if $\omega_{0}=-d^{c}\theta$ is semi-positive
definite, and hence it is Vaisman. Equivalently, a compact LCK manifold with
potential and with constant norm of $\theta$ is Vaisman.

##  3 LCK structures on nilmanifolds

###  3.1 Invariant geometric structures on Lie groups

Let $G$ be a Lie group, $\Lambda$ a discrete subgroup, and $I$ a left-
invariant complex structure on $G$. Consider the quotient space $G/\Lambda$,
where $\Lambda$ acts by left translations. Since $I$ is left-invariant, the
manifold $G/\Lambda$ is equipped with a natural complex structure. This
construction is usually applied to solvmanifolds or nilmanifolds, but it makes
sense with any Lie group. When we need to refer to this particular kind of
complex structures on $G/\Lambda$, we call them locally $G$-invariant.

A left-invariant almost complex structure $I$ on $G$ is determined by its
restriction to the Lie algebra ${\mathfrak{g}}=T_{e}G$, giving the Hodge
decomposition
${\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}={\mathfrak{g}}^{1,0}\oplus{\mathfrak{g}}^{0,1}$.
This structure is integrable if and only if the commutator of $(1,0)$-vector
fields is again a $(1,0)$-vector field, which is equivalent to
$[{\mathfrak{g}}^{1,0},{\mathfrak{g}}^{1,0}]\subset{\mathfrak{g}}^{1,0}$. This
allows one to define a complex structure on a Lie algebra.

Definition 3.1:  A complex structure on a Lie algebra ${\mathfrak{g}}$ is a
subalgebra
${\mathfrak{g}}^{1,0}\subset{\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$
which satisfies
${\mathfrak{g}}^{1,0}\oplus\overline{{\mathfrak{g}}^{1,0}}={\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$.

Indeed, an almost complex structure operator $I$ can be reconstructed from the
decomposition
${\mathfrak{g}}^{1,0}\oplus\overline{{\mathfrak{g}}^{1,0}}={\mathfrak{g}}\otimes_{\mathbb{R}}{\mathbb{C}}$
by making it act as $\sqrt{-1}\>$ on ${\mathfrak{g}}^{1,0}$ and $-\sqrt{-1}\>$
on $\overline{{\mathfrak{g}}^{1,0}}$.

In a similar way one could define symplectic structures or LCK structures on a
Lie algebra ${\mathfrak{g}}$. Recall that the Grassmann algebra
$\Lambda^{*}({\mathfrak{g}}^{*})$ is equipped with a natural differential,
called Chevalley differential, which is equal to the de Rham differential on
left-invariant differential forms on the Lie group if we identify the space of
such forms with $\Lambda^{*}({\mathfrak{g}}^{*})$. In dimensions 1 and 2 it
can be written explicitly as follows: if $\lambda\in\mathfrak{g^{*}}$, then
$d\lambda(x,y)=-\lambda[x,y]$; if $\beta\in\Lambda^{2}\mathfrak{g}^{*}$, then
$d\beta(x,y,z)=-\beta([x,y],z)-\beta([y,z],x)-\beta([z,x],y)$. The
corresponding complex is called the Chevalley-Eilenberg complex.

Further on, we shall always interpret the elements of
$\Lambda^{p}({\mathfrak{g}}^{*})$ as left-invariant differential forms on the
corresponding Lie group, and refer to them as to “$p$-forms”, with all the
usual terminology (“closed forms”, “exact forms”) as used for the elements of
de Rham algebra.

Definition 3.2: A symplectic structure on a Lie algebra ${\mathfrak{g}}$ is a
non-degenerate, closed 2-form $\omega\in\Lambda^{2}({\mathfrak{g}}^{*})$. A
Kähler structure on a Lie algebra ${\mathfrak{g}}$ is a complex structure $I$
and a Hermitian form $h$ on ${\mathfrak{g}}$ such that the fundamental 2-form
$\omega(\cdot,\cdot):=h(\cdot,I(\cdot))$ is closed.

Remark 3.3: In [BG] (see also [Has]) it was shown that any nilpotent Lie
algebra admitting a Kähler structure is actually abelian.

###  3.2 LCK nilmanifolds

Definition 3.4:  An LCK structure on a Lie algebra is a complex structure $I$
and a Hermitian form $h$ on ${\mathfrak{g}}$ such that the fundamental 2-form
$\omega(\cdot,\cdot):=h(\cdot,I(\cdot))$ satisfies
$d(\omega)=\theta\wedge\omega$, where
$\theta\in\Lambda^{1}({\mathfrak{g}}^{*})$ is a closed 1-form.

Definition 3.5: A nilmanifold is a quotient of a simply connected nilpotent
Lie group by a co-compact discrete subgroup. Alternatively ([Ma]), one can
define nilmanifolds as manifolds which admit a homogeneous action by a
nilpotent Lie group.

Definition 3.6:  Let $(G,I)$ be a nilpotent Lie group equipped with a left-
invariant complex structure. For any co-compact discrete subgroup
$\Gamma\subset G$, the (left) quotient $G/\Gamma$ is equipped with a natural
complex structure. This quotient is called a complex nilmanifold. Similarly,
if $G$ is a Lie group with left-invariant LCK structure, the quotient
$G/\Gamma$ is called an LCK nilmanifold. This structure is clearly locally
homogeneous. When we need to refer to this particular kind of structures, we
call them locally $G$-invariant.

Remark 3.7: The existence of a co-compact lattice already imposes strong
restrictions: the group should be unimodular, [Mo, §1.1, Exercise 14b]. All
nilpotent Lie groups are unimodular, but this is not sufficient for the
existence of a cocompact lattice. According to [Ma], a nilpotent Lie group
admits a cocompact lattice if and only if its Lie algebra admits a basis in
which the structural constants are all rational.

In this paper we present results concerning LCK structures on nilmanifolds. A
classification of nilpotent Lie algebras which admit LCK structures is given.

Remark 3.8:  As shown by Malčev, [Ma], any nilmanifold is uniquely determined
by its fundamental group, which is a discrete nilpotent torsion-free group,
and any such group uniquely determines a nilmanifold.

Remark 3.9: Note that right translations are not holomorphic with respect to
left invariant complex structures. This means that for a lattice $\Lambda$,
the complex structure induced on the manifold $G/\Lambda$ is not necessarily
invariant, and hence $G/\Lambda$ is not a homogeneous LCK manifold.

Remark 3.10: Note that it may happen that a Kähler manifold be diffeomorphic
(as real manifold) with a nilmanifold but the Kähler structure need not be
invariant. For example, let $E$ be an elliptic curve which is a 2-fold
ramified covering of ${\mathbb{C}}P^{1}$, and take the fibered product
$E\times_{{\mathbb{C}}P^{1}}\operatorname{Tw}(T^{4})$, where
$\operatorname{Tw}(T^{4})$ is the twistor space of a 4-torus. This fibered
product is actually a torus, but with an inhomogeneous complex structure
(since it has non-trivial canonical bundle, in fact its anticanonical bundle
is semi-positive, with many non-trivial sections); compare with [Ca, §5]. This
example suggests that the same phenomenon could appear for LCK manifolds (to
be diffeomorphic with a nilmanifold without having an invariant LCK
structure), but for the moment we don’t have a concrete example.

Example 3.11:  Let $H_{2n-1}$ be the Heisenberg group of matrices of the form

$\begin{pmatrix}1&A&c\\\ 0&I_{2n-1}&B^{t}\\\ 0&0&1\end{pmatrix},\quad
c\in{\mathbb{R}},\quad A,B\in{\mathbb{R}}^{n-1}.$

Its Lie algebra $\mathfrak{h}_{2n-1}$ can be defined in terms of generators
and relations as follows: it has a basis $\\{X_{i},Y_{i},Z\\}$,
$i=1,\ldots,n-1$. with $[X_{i},Y_{i}]=Z$, and the rest of Lie brackets
trivial. Clearly, it is a nilpotent Lie algebra. After taking the quotient of
the corresponding Lie group by a co-compact discrete subgroup $\Lambda$, one
obtains a nilmanifold called the Heisenberg nilmanifold. It has an invariant
Sasakian structure given by requiring the above basis to be orthonormal,
defining the contact form as $\sum y_{i}dx_{i}+dz$ (where $dx_{i},dy_{i},dz$
are the dual invariant one-forms). Then $Z$ is its Reeb field.

Now, the product $H_{2n-1}/\Lambda\times S^{1}$ is a Vaisman nilmanifold whose
universal cover is the product $H_{2n-1}\times{\mathbb{R}}$, a Lie group with
Lie algebra $\mathfrak{g}:=\mathfrak{h}_{2n-1}\times{\mathbb{R}}$,
${\mathbb{R}}=\langle T\rangle$, in which the brackets of the basis
$\\{X_{i},Y_{i},Z,T\\}$ are the above, and $T$ is central. The linear LCK
structure on $\mathfrak{g}$ is given by asking the basis to be orthonormal,
and by defining the complex structure $IX_{i}=Y_{i}$, $IZ=-T$. The fundamental
form is $\omega=\sum(X_{i}^{*}\wedge Y_{i}^{*})-Z^{*}\wedge T^{*}.$ A direct
computation shows that: $d\omega=T^{*}\wedge\omega,$ and hence the Lee form is
$T^{*}$ which can be seen directly to be $g$-parallel.

##  4 Twisted Dolbeault cohomology on nilpotent Lie algebras

###  4.1 Twisted Dolbeault cohomology

Let $\theta$ be a closed 1-form on a complex manifold. Then $d-\theta$ defines
a connection on the trivial bundle. We denote the corresponding local system
by $L$. The twisted Dolbeault differentials
$\partial_{\theta}:=(d_{\theta})^{1,0}=\partial-\theta^{1,0}$ and
$\overline{\partial}_{\theta}:=(d_{\theta})^{0,1}=\overline{\partial}-\theta^{0,1}$
are Morse-Novikov counterparts to the usual Dolbeault differentials. The
cohomology of these differentials corresponds to the Dolbeault cohomology with
coefficients in the holomorphic line bundle $L_{\mathbb{C}}$ obtained as
complexification of the local system $L$.

Working with LCK nilmanifolds and nilpotent Lie algebras, it is natural to
consider their twisted Dolbeault cohomology. As shown by Console and Fino
([CF]), the usual Dolbeault cohomology of complex nilmanifolds is equal to the
cohomology of the Dolbeault version of the corresponding Chevalley-Eilenberg
complex on the Lie algebra. This result does not hold in the twisted Dolbeault
cohomology, as we shall see in 4.2.

However, as we show in 4.1 below, the corresponding Lie algebra cohomology
vanishes.

Definition 4.1: Let ${\mathfrak{g}}$ be a Lie algebra and
$\theta\in{\mathfrak{g}}^{*}$ a closed 1-form and

$d_{\theta}:=d-\theta\in\operatorname{End}(\Lambda^{*}({\mathfrak{g}}^{*})),$

where $d$ is the Chevalley differential and $\theta$ denotes the operation of
multiplication by $\theta$. The twisted Dolbeault differentials on
$\Lambda^{*}({\mathfrak{g}}^{*})$ are $\partial_{\theta}:=(d_{\theta})^{1,0}$
and $\overline{\partial}_{\theta}:=(d_{\theta})^{0,1}$.

The cohomology of $d_{\theta}$ is always zero for non-zero $\theta$ ([Ala,
Mi]). This theorem is due to J. Dixmier, [Di]. The main result of the present
section is the following theorem, which is the Dolbeault version of Dixmier’s
and Alaniya’s theorem.

Theorem 4.2:  Let ${\mathfrak{g}}$ be a nilpotent Lie algebra with a complex
structure (3.1), and $\theta\in{\mathfrak{g}}^{*}$ a non-zero, closed real
1-form. Then the cohomology of
$(\Lambda^{0,*}({\mathfrak{g}}^{*}),\overline{\partial}_{\theta})$ vanishes.

Proof: Consider the central series of the Lie algebra ${\mathfrak{g}}^{0,1}$,
with
$W_{0}={\mathfrak{g}}^{0,1},W_{1}=[W_{0},W_{0}],...,W_{k}=[W_{0},W_{k-1}]$.
Let $A_{k}\subset({\mathfrak{g}}^{0,1})^{*}$ be the annihilator of $W_{k}$.
For any 1-form $\lambda\in({\mathfrak{g}}^{0,1})^{*}$, one has
$\overline{\partial}(\lambda)(x,y)=\lambda([x,y])$, hence
$\overline{\partial}(A_{k})\subset\Lambda^{2}(A_{k-1})$. Consider the
filtration $\Lambda^{*}(A_{1})\subset\Lambda^{*}(A_{2})\subset...$ on
$\Lambda\in({\mathfrak{g}}^{0,1})^{*}$. Since
$\overline{\partial}(A_{k})\subset\Lambda^{2}(A_{k-1})$, the operator
$\overline{\partial}$ shifts the filtration by 1:
$\overline{\partial}(\Lambda^{*}(A_{k}))\subset\Lambda^{*}(A_{k-1})$.

Consider the spectral sequence of the complex
$(\Lambda^{*}({\mathfrak{g}}^{*}),\overline{\partial}_{\theta})$ filtered by
$V_{0}\subset V_{1}\subset V_{2}\subset...$, where
$V_{k}:=\Lambda^{*}(A_{k})$. Since $\overline{\partial}(V_{k})\subset
V_{k-1}$, the operator $\overline{\partial}_{\theta}$ acts on
$\bigoplus_{k}V_{k}/V_{k-1}$ as multiplication by $\theta^{0,1}$. The
corresponding associated graded complex, which is $E_{0}^{*,*}$ of this
spectral sequence, is identified with
$\left(\bigoplus_{k}V_{k}/V_{k-1},\theta^{0,1}\right)$. We identify
$\bigoplus_{k}V_{k}/V_{k-1}$ with the Grassmann algebra
$\bigoplus_{k}V_{k}/V_{k-1}=\Lambda^{*}({\mathfrak{g}}^{*})$. After this
identification, the multiplication

$\theta^{0,1}:\;\bigoplus_{k}V_{k}/V_{k-1}{\>\longrightarrow\>}\bigoplus_{k}V_{k}/V_{k-1}$

becomes multiplication by a 1-form, obtained from $\theta^{0,1}$. The
cohomology of multiplication by a 1-form always vanishes. Then the
$E_{1}^{*,*}$-page of the spectral sequence vanishes, which implies vanishing
of $H^{*}(\Lambda^{*}({\mathfrak{g}}^{0,*}),\overline{\partial}_{\theta})$.

The following corollary will be used in the classification of LCK structures
on nilpotent Lie algebras.

Corollary 4.3:  Let $(\mathfrak{g},I)$ be a $2n$-dimensional nilpotent Lie
algebra with complex structure, $\theta\in\Lambda^{1}({\mathfrak{g}}^{*})$ a
non-zero, closed real 1-form, and $\omega\in\Lambda^{1,1}({\mathfrak{g}}^{*})$
a $d_{\theta}$-closed (1,1)-form. Then there exists a 1-form
$\tau\in\Lambda^{1}({\mathfrak{g}}^{*})$ such that $\omega=d_{\theta}(\tau)$,
where $d_{\theta}(I\tau)=0$.

Proof: Denote by $H^{1,1}_{d_{\theta}d^{c}_{\theta}}({\mathfrak{g}}^{*})$ the
twisted Bott-Chern cohomology, that is, all $d_{\theta}$-closed (1,1)-forms up
to the image of $d_{\theta}d^{c}_{\theta}$. The standard exact sequence

$H^{0,1}_{\overline{\partial}_{\theta}}({\mathfrak{g}}^{*})\oplus
H^{1,0}_{\partial_{\theta}}({\mathfrak{g}}^{*})\stackrel{{\scriptstyle
d}}{{{\>\longrightarrow\>}}}H^{1,1}_{d_{\theta}d^{c}_{\theta}}({\mathfrak{g}}^{*})\stackrel{{\scriptstyle\mu}}{{{\>\longrightarrow\>}}}H^{2}(M)$

([OV0], equation (4.7)) implies that the kernel of $\mu$ vanishes when
$H^{0,1}_{\overline{\partial}_{\theta}}({\mathfrak{g}}^{*})=H^{1,0}_{\partial_{\theta}}({\mathfrak{g}}^{*})=0$.
The last equation follows from 4.1, hence $\mu$ is injective. The LCK form
$\omega$ belongs to the kernel of $\mu$, because the $d_{\theta}$-cohomology
of $\Lambda^{*}({\mathfrak{g}}^{*})$ vanishes by Diximier and Alanya theorem
([Ala, Mi]). Therefore, $\omega$ is twisted Bott-Chern exact, i.e.
$\omega=d_{\theta}d_{\theta}^{c}f$ for some
$f\in\Lambda^{0}({\mathfrak{g}}^{*})={\mathbb{R}}$. Set
$\tau:=d^{c}_{\theta}f$. Then $I\tau=d_{\theta}f$ and $d_{\theta}(I\tau)=0$ as
stated.

###  4.2 Console-Fino theorem with coefficients in a local system fails

Example 4.4:  Let $E_{1},E_{2}$ be elliptic curves. Consider a Kodaira surface
$M$, which is a non-Kähler, locally conformally Kähler complex surface,
obtained as the total space of a principal holomorphic $E_{1}$-bundle
$\pi:\;M{\>\longrightarrow\>}E_{2}$. Such bundles are classified by the first
Chern class of the fibration $c_{1}(\pi)$. This class can be identified with
the $d_{2}$-differential of the corresponding Leray spectral sequence, mapping
from $H^{1}(E_{1})$ to $H^{2}(E_{2})$. By Blanchard’s theorem ([Blan]), $M$ is
non-Kähler if and only if $c_{1}(\pi)$ is non-zero. The Kodaira surface is an
example of a nilmanifold obtained from the Heisenberg group (3.2).

Let $\omega_{E_{2}}$ be a Kähler form on $E_{2}$. Then
$\pi^{*}(\omega_{E_{2}})\in\operatorname{im}d_{2}$ is exact, giving
$\pi^{*}(\omega_{E_{2}})=d\xi$. We chose $\xi$ in such a way that
$d(\theta)=0$, where $\theta:=I(\xi)$ (4.1). Consider a trivial complex line
bundle $(L,\nabla_{0})$ with connection defined by $\nabla_{0}-\theta$; this
line bundle is flat, hence equipped with a holomorphic structure operator
$\overline{\partial}:=\nabla_{0}^{0,1}-\theta^{0,1}$ Using the standard
Hermitian metric, we express its Chern connection as
$\nabla_{0}+\theta^{1,0}-\theta^{0,1}$, hence the curvature of this bundle
satisfies $\Theta_{L}=-\sqrt{-1}\>\pi^{*}(\omega_{E_{2}})$. Rescaling $\theta$
and $\omega_{E_{2}}$ if necessarily, we can assume that the cohomology class
of $\omega_{E_{2}}$ is integer. Then $L$ is a pullback of an ample bundle $B$
on $E_{2}$. Now, $H^{0}(M,L)=H^{0}(E_{2},B)\neq 0$, but this space can be
interpreted as $H^{0}(\Lambda^{0,*}(M),\overline{\partial}_{\theta})$. In this
example the twisted Dolbeault cohomology of $M$ is non-zero, but the
corresponding twisted Dolbeault cohomology of the Lie algebra vanishes (4.1).
This gives a counterexample to Console-Fino theorem for twisted cohomology.

##  5 Classification of LCK nilmanifolds

###  5.1 Ugarte’s conjecture

The main question concerning nilmanifolds with LCK structure is to decide if
the following conjecture is true or not.

Conjecture 5.1: ([U]) Let $M$ be a differentiable nilmanifold admitting an LCK
structure (not necessarily locally $G$-invariant, see 3.2). Then $M$ is
conformally biholomorphic to a quotient $S^{1}\times H_{2n-1}/\Lambda$ with
the Vaisman structure described above.

###  5.2 Classification theorem for LCK nilmanifolds

For LCK nilmanifolds, Ugarte’s conjecture was proven by H. Sawai in the
following form:

Theorem 5.2: ([Saw1]) Let $(M,I)$ be a complex nilmanifold, $M=G/\Lambda$. If
$(M,I)$ admits an LCK structure, then it is conformally equivalent to Vaisman.
Moreover, it is biholomorphic to a quotient of the product
$(H_{2n-1}\times{\mathbb{R}},I)$.

We give a new proof of this result, different from the one in [Saw1].

Proof. Step 1: We replace the LCK metric by a locally $G$-invariant LCK
metric. To this purpose, we apply the averaging trick that Belgun used for the
Inoue surface $S^{+}$ ([Bel, Proof of Theorem 7]); later, it was generalized
by Fino and Grantcharov ([FinGra]). This approach does not work, in general,
for solvmanifolds, but it works well for all nilmanifolds. For LCK structures
on nilmanifolds, this construction is due to L. Ugarte:

Theorem 5.3:  ([U, Proposition 34])
Let $M=G/\Lambda$ be a complex nilmanifold admitting an LCK structure
$(\omega,\theta)$. Then it also admits a structure of LCK nilmanifold. In
other words, there exists a left-invariant LCK structure on $G$ which induces
an LCK structure on $M$.

Proof: As $G$ admits a co-compact lattice, it is unimodular, hence it admits a
bi-invariant measure. Let $d\mu$ be a bi-invariant volume element on $M$, and
suppose $\operatorname{vol}(M)=1$. Consider a left-invariant 1-form form
$\check{\theta}$ on ${\mathfrak{g}}=\operatorname{Lie}(G)$ such that the
corresponding 1-form $\theta_{0}$ on $G/\Lambda$ is cohomologous to $\theta$.
Such a 1-form exists because
$H^{1}(G/\Lambda,{\mathbb{R}})=H^{1}(\Lambda^{*}({\mathfrak{g}}^{*}),d)$
([No]). Replacing $\omega$ by a conformally equivalent LCK form $\omega_{0}$,
we can assume that $d\omega_{0}=\omega_{0}\wedge\theta_{0}$. Let $D\subset G$
be the fundamental domain of the action of $\Lambda$. Given left-invariant
vector fields $X,Y$ on $G$, the 2-form

$\check{\omega}(X,Y):=\int_{D}\omega_{0}(X,Y)d\mu$

defines an Hermitian structure on the Lie algebra ${\mathfrak{g}}$ of $G$.
Moreover, as

$d(\check{\omega})(X,Y,Z)\\\
=-\int_{D}\omega_{0}([X,Y],Z)d\mu-\int_{D}\omega_{0}(X,[Y,Z])d\mu+\int_{D}\omega_{0}(Y,[X,Z])d\mu\\\
=-\int_{D}d(\omega_{0})(X,Y,Z)d\mu=-\int_{D}\theta_{0}\wedge\omega_{0}(X,Y,Z)d\mu=-\check{\omega}\wedge\check{\theta}(X,Y,Z),$

it follows that $\check{\omega}$ is an LCK form on ${\mathfrak{g}}$.

Proof of 5.2, Step 2: Now we prove that any LCK nilmanifold is Vaisman. Using
4.1, we obtain that $\omega=d_{\theta}(\tau)$, where $d_{\theta}(I\tau)=0$.

Applying Dixmier and Alaniya’s theorem again, we obtain that
$d_{\theta}(I\tau)=0$ implies $I\tau=d_{\theta}(v)$, where
$v\in\Lambda^{0}({\mathfrak{g}}^{*})$ is a constant. Therefore,
$\omega=d_{\theta}d^{c}_{\theta}(\operatorname{\text{\sf const}})$, which is
the equation for the LCK manifold with potential.111Note that the constant
$\operatorname{\text{\sf const}}$ should be positive by 2.3. However, as shown
in 2.3, an LCK manifold with potential and constant $|\theta|$ is Vaisman.

Step 3: We show that the Lee field $\theta^{\sharp}\in{\mathfrak{g}}$ and the
anti-Lee field $I(\theta^{\sharp})\in{\mathfrak{g}}$ generate an ideal in
${\mathfrak{g}}$. This observation also follows from [FGV, Theorem 3.12].

Let $\Sigma=\langle\theta^{\sharp},I(\theta^{\sharp})\rangle$ be the canonical
foliation on $M$, and ${\mathfrak{s}}\subset{\mathfrak{g}}$ the corresponding
subspace of the Lie algebra of $G$. Since $\theta^{\sharp}$ and
$I(\theta^{\sharp})$ are Killing and have constant length, their trajectories
are geodesics (2.2). Let $S\subset\operatorname{Iso}(M)$ be the group of
isometries of $M$ generated by exponentials of $s\in\mathfrak{s}$.

By construction, $S$ is a subgroup of $G$ which acts on $M=G/\Lambda$.
Therefore, it is invariant under conjugation with $\Lambda$, and the Lie
algebra $\mathfrak{s}\subset{\mathfrak{g}}$ of $S$ is invariant under the
adjoint action of $\Lambda$ on ${\mathfrak{g}}$. Since $G$ is the Malčev
completion of $\Lambda$, its image in $\operatorname{End}({\mathfrak{g}})$
coincides with the Zariski closure of the image of $G$ under the adjoint
action (see Property 1.5 for the Malčev completion functor in [GH]).
Therefore, $\mathfrak{s}$ is $G$-invariant.

Step 4: Now we prove that for any Vaisman nilmanifold $M=G/\Lambda$, the group
$G$ is the product of ${\mathbb{R}}$ with the Heisenberg group. Consider the
ideal $\mathfrak{s}\subset{\mathfrak{g}}$ associated with the canonical
foliation $\Sigma$ (Step 3). The leaf space of $\Sigma$ is Kähler because
$M=G/\Lambda$ is Vaisman. However, this leaf space is a nilmanifold associated
with the Lie algebra ${\mathfrak{g}}/\mathfrak{s}$. A Kähler nilpotent Lie
algebra is abelian ([BG]), hence ${\mathfrak{g}}$ is a central extension of an
abelian algebra. Since $M$ is Vaisman, it is locally isometric to a product of
${\mathbb{R}}$ and a Sasakian manifold, hence the corresponding Lie algebra is
a product of ${\mathbb{R}}$ and a 1-dimensional central extension of
${\mathbb{R}}^{2n-2}$. This finishes the proof.

Remark 5.4: The general case of 5.1 was proven only for nilmanifolds of
Vaisman type, by G. Bazzoni, [Baz].

Acknowledgment: Liviu Ornea thanks IMPA for financial support and excellent
research environment during the preparation of this work. Both authors thank
Anna Fino for carefully reading a first draft of the paper, Sergiu Moroianu
for observing an error in a first proof of 4.1, and the referees for their
very useful suggestions concerning both the content and the references of the
paper.

## References

  * [Ala] L.A. Alaniya, Cohomology with local coefficients of some nilmanifolds, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 149–150; translation in Russian Math. Surveys 54 (1999), no. 5, 1019-1020.
  * [Baz] G. Bazzoni, Vaisman nilmanifolds, Bull. Lond. Math. Soc. 49 (2017), no. 5, 824–830.
  * [Bel] F.A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.
  * [BG] C. Benson, C. Gordon, Kähler and symplectic structures on compact nilmanifolds, Topology 27 (1988), 513–518.
  * [Blan] A. Blanchard, Sur les variétés analitiques complexes, Ann. Sci. École Norm. Sup. 73, no. 3 (1956), 157–202.
  * [Ca] E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math.Soc. 87 (1958), 407–438.
  * [CFD] L. A. Cordero, M. Fernández, M. De León, Compact locally conformal Kähler nilmanifolds, Geom. Dedicata, 21 (2) (1986), 187–192.
  * [CF] S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transformation Groups 6 (2) (2001), 111–124.
  * [Di] J. Dixmier, Cohomologie des algèbres de Lie nilpotentes, Acta Sci. Math. Szeged. 16 (1955), 246–250.
  * [DO] S. Dragomir, L. Ornea, Locally conformally Kähler manifolds, Progress in Math. 55, Birkhäuser, 1998.
  * [FinGra] A. Fino, G. Grantcharov, On some properties of the manifolds with skew-symmetric torsion and holonomy SU(n) and Sp(n), math.DG/0302358, Adv. Math. 189 (2004), no. 2, 439–450.
  * [FGV] A. Fino, G. Grantcharov, M. Verbitsky, Algebraic dimension of complex nilmanifolds, J. Math. Pures Appl. (9) 118 (2018), 204–218.
  * [FRR] A. Fino, S. Rollenske, J. Ruppenthal, Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups, arXiv:1808.08090, to appear in Quart. J. Math.
  * [Ga] P. Gauduchon, La 1-forme de torsion d’une variete hermitienne compacte, Math. Ann., 267 (1984), 495–518.
  * [Fis] R. J. Fisher, Jr., On the Picard group of a compact complex nilmanifold, Rocky Mountain J. Math. 13 (1983), no. 4, 631-638.
  * [GH] F. Grunewald, J. O’Halloran, Nilpotent groups and unipotent algebraic groups, J. Pure Appl. Algebra 37 (1985), no. 3, 299–313.
  * [Has] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), 65–71.
  * [KO] Y. Kamishima, L. Ornea, Geometric flow on compact locally conformally Kähler manifolds, Tohoku Math. J., 57 (2) (2005), 201–221.
  * [Ma] A.I. Mal’cev, On a class of homogeneous spaces, AMS Translation No. 39 (1951).
  * [Mi] D.V. Millionshchikov, Cohomology of solvmanifolds with local coefficients and problems in the Morse-Novikov theory (Russian) Uspekhi Mat. Nauk 57 (2002), no. 4(346), 183–184; translation in Russian Math. Surveys 57 (2002), no. 4, 813–814.
  * [Mo] D.W. Morris, Ratner’s Theorems on Unipotent Flows, Univ. of Chicago Press, Chicago, 2005.
  * [No] K. Nomizu, On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups, Ann. Math. 59 (1954), No. 3, 531–538.
  * [Oe] K. Oeljeklaus, Hyperflächen und Geradenbündel auf homogenen komplexen Mannigfaltigkeiten, [Hypersurfaces and line bundles on homogeneous complex manifolds] Schriftenreihe des Mathematischen Instituts der Universität Münster, 2, 1985.
  * [OT] K. Oeljeklaus, M. Toma, Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier 55, no. 1 (2005), 1291–1300.
  * [OV0] L. Ornea and M. Verbitsky, Morse-Novikov cohomology of locally conformally Kähler manifolds, J. Geom. Phys. 59 (2009), 295–305. arXiv:0712.0107
  * [OV1] L. Ornea, M. Verbitsky, Locally conformally Kähler metrics obtained from pseudoconvex shells, Proc. Amer. Math. Soc. 144 (2016), 325–335. arXiv:1210.2080.
  * [OV2] L. Ornea, M. Verbitsky, LCK rank of locally conformally Kähler manifolds with potential, J. Geom. Phys. 107 (2016), 92–98.
  * [OV3] L. Ornea, M. Verbitsky, Hopf surfaces in locally conformally Kahler manifolds with potential, Geom. Dedicata, https://doi.org/10.1007/s10711-019-00495-5. arxiv:1601.07421, v2.
  * [OV4] L. Ornea, M. Verbitsky, Positivity of LCK potential, J. Geom. Anal. 29 (2019), 1479–1489.
  * [OVV] L. Ornea, M. Verbitsky, V. Vuletescu, Classification of non-Kähler surfaces and locally conformally Kähler geometry, arxiv:1810.05768.
  * [Ot] A. Otiman, Morse-Novikov cohomology of locally conformally Kähler surfaces, Math. Z. 289 (2018), no. 1-2, 605–628. arXiv:1609.07675.
  * [Saw1] H. Sawai, Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures, Geom. Dedicata 125 (2007), 93–101.
  * [U] L. Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175–202.
  * [Va1] I. Vaisman, On locally and globally conformal Kähler manifolds, Trans. Amer. Math. Soc., 262 (1980), 533-542.
  * [Va2] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231–255.
  * [Ve] M. Verbitsky, Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds, Proc. Steklov Inst. Math. no. 3 (246), 54–78 (2004).

Liviu Ornea
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania, and:
Institute of Mathematics ”Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Str. 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, National Research University Higher School of
Economics, Department of Mathematics, 9 Usacheva str., Moscow, Russia.
<EMAIL_ADDRESS><EMAIL_ADDRESS>


The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# Search for Neutrino-Induced Neutral Current $\Delta$ Radiative Decay in
MicroBooNE and a First Test of the MiniBooNE Low Energy Excess Under a Single-
Photon Hypothesis

P. Abratenko  Tufts University, Medford, MA, 02155, USA  R. An  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  J. Anthony  University
of Cambridge, Cambridge CB3 0HE, United Kingdom  L. Arellano  The University
of Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi  University of
Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv University, Tel Aviv,
Israel, 69978  S. Balasubramanian  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  B. Baller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  C. Barnes  University of Michigan,
Ann Arbor, MI, 48109, USA  G. Barr  University of Oxford, Oxford OX1 3RH,
United Kingdom  V. Basque  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Bathe-Peters  Harvard University, Cambridge, MA 02138, USA
O. Benevides Rodrigues  Syracuse University, Syracuse, NY, 13244, USA  S.
Berkman  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
A. Bhanderi  The University of Manchester, Manchester M13 9PL, United Kingdom
A. Bhat  Syracuse University, Syracuse, NY, 13244, USA  M. Bishai  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster
University, Lancaster LA1 4YW, United Kingdom  T. Bolton  Kansas State
University (KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University,
Cambridge, MA 02138, USA  L. Camilleri  Columbia University, New York, NY,
10027, USA  D. Caratelli  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  I. Caro Terrazas  Colorado State University, Fort
Collins, CO, 80523, USA  R. Castillo Fernandez  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  F. Cavanna  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Cerati  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y. Chen
Universität Bern, Bern CH-3012, Switzerland  D. Cianci  Columbia University,
New York, NY, 10027, USA  J. M. Conrad  Massachusetts Institute of Technology
(MIT), Cambridge, MA, 02139, USA  M. Convery  SLAC National Accelerator
Laboratory, Menlo Park, CA, 94025, USA  L. Cooper-Troendle  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  J. I.
Crespo-Anadón  Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Madrid E-28040, Spain  M. Del Tutto  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  S. R. Dennis
University of Cambridge, Cambridge CB3 0HE, United Kingdom  P. Detje
University of Cambridge, Cambridge CB3 0HE, United Kingdom  A. Devitt
Lancaster University, Lancaster LA1 4YW, United Kingdom  R. Diurba  University
of Minnesota, Minneapolis, MN, 55455, USA  R. Dorrill  Illinois Institute of
Technology (IIT), Chicago, IL 60616, USA  K. Duffy  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  S. Dytman  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  B. Eberly  University of Southern
Maine, Portland, ME, 04104, USA  A. Ereditato  Universität Bern, Bern CH-3012,
Switzerland  J. J. Evans  The University of Manchester, Manchester M13 9PL,
United Kingdom  R. Fine  Los Alamos National Laboratory (LANL), Los Alamos,
NM, 87545, USA  G. A. Fiorentini Aguirre  South Dakota School of Mines and
Technology (SDSMT), Rapid City, SD, 57701, USA  R. S. Fitzpatrick  University
of Michigan, Ann Arbor, MI, 48109, USA  B. T. Fleming  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  N. Foppiani
Harvard University, Cambridge, MA 02138, USA  D. Franco  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  A. P.
Furmanski  University of Minnesota, Minneapolis, MN, 55455, USA  D. Garcia-
Gamez  Universidad de Granada, Granada E-18071, Spain  S. Gardiner  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Ge
Columbia University, New York, NY, 10027, USA  S. Gollapinni  University of
Tennessee, Knoxville, TN, 37996, USA  Los Alamos National Laboratory (LANL),
Los Alamos, NM, 87545, USA  O. Goodwin  The University of Manchester,
Manchester M13 9PL, United Kingdom  E. Gramellini  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  P. Green  The University of
Manchester, Manchester M13 9PL, United Kingdom  H. Greenlee  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W. Gu  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  R. Guenette  Harvard
University, Cambridge, MA 02138, USA  P. Guzowski  The University of
Manchester, Manchester M13 9PL, United Kingdom  L. Hagaman  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  O. Hen
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  C.
Hilgenberg  University of Minnesota, Minneapolis, MN, 55455, USA  G. A.
Horton-Smith  Kansas State University (KSU), Manhattan, KS, 66506, USA  A.
Hourlier  Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139,
USA  R. Itay  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA
C. James  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
X. Ji  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  L. Jiang
Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA  J. H.
Jo  Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  R. A. Johnson  University of Cincinnati, Cincinnati, OH, 45221,
USA  Y.-J. Jwa  Columbia University, New York, NY, 10027, USA  D. Kalra
Columbia University, New York, NY, 10027, USA  N. Kamp  Massachusetts
Institute of Technology (MIT), Cambridge, MA, 02139, USA  N. Kaneshige
University of California, Santa Barbara, CA, 93106, USA  G. Karagiorgi
Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Kobilarcik
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  I.
Kreslo  Universität Bern, Bern CH-3012, Switzerland  R. LaZur  Colorado State
University, Fort Collins, CO, 80523, USA  I. Lepetic  Rutgers University,
Piscataway, NJ, 08854, USA  K. Li  Wright Laboratory, Department of Physics,
Yale University, New Haven, CT, 06520, USA  Y. Li  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  K. Lin  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  B. R. Littlejohn  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  W. C. Louis  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  X. Luo  University of
California, Santa Barbara, CA, 93106, USA  K. Manivannan  Syracuse University,
Syracuse, NY, 13244, USA  C. Mariani  Center for Neutrino Physics, Virginia
Tech, Blacksburg, VA, 24061, USA  D. Marsden  The University of Manchester,
Manchester M13 9PL, United Kingdom  J. Marshall  University of Warwick,
Coventry CV4 7AL, United Kingdom  D. A. Martinez Caicedo  South Dakota School
of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  K. Mason  Tufts
University, Medford, MA, 02155, USA  A. Mastbaum  Rutgers University,
Piscataway, NJ, 08854, USA  N. McConkey  The University of Manchester,
Manchester M13 9PL, United Kingdom  V. Meddage  Kansas State University (KSU),
Manhattan, KS, 66506, USA  T. Mettler  Universität Bern, Bern CH-3012,
Switzerland  K. Miller  University of Chicago, Chicago, IL, 60637, USA  J.
Mills  Tufts University, Medford, MA, 02155, USA  K. Mistry  The University of
Manchester, Manchester M13 9PL, United Kingdom  A. Mogan  University of
Tennessee, Knoxville, TN, 37996, USA  T. Mohayai  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. Moon  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  M. Mooney  Colorado State
University, Fort Collins, CO, 80523, USA  A. F. Moor  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Mousseau  University of
Michigan, Ann Arbor, MI, 48109, USA  M. Murphy  Center for Neutrino Physics,
Virginia Tech, Blacksburg, VA, 24061, USA  R. Murrells  The University of
Manchester, Manchester M13 9PL, United Kingdom  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  R. K. Neely  Kansas State
University (KSU), Manhattan, KS, 66506, USA  D. A. Newmark  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  J. Nowak  Lancaster
University, Lancaster LA1 4YW, United Kingdom  M. Nunes  Syracuse University,
Syracuse, NY, 13244, USA  O. Palamara  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  V. Paolone  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  A. Papadopoulou  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  V. Papavassiliou  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  S. F. Pate  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  N. Patel  Lancaster
University, Lancaster LA1 4YW, United Kingdom  A. Paudel  Kansas State
University (KSU), Manhattan, KS, 66506, USA  Z. Pavlovic  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv
University, Tel Aviv, Israel, 69978  I. D. Ponce-Pinto  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  S. Prince
Harvard University, Cambridge, MA 02138, USA  X. Qian  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V. Radeka  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Rafique  Kansas State
University (KSU), Manhattan, KS, 66506, USA  M. Reggiani-Guzzo  The University
of Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New Mexico State
University (NMSU), Las Cruces, NM, 88003, USA  L. C. J. Rice  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  L. Rochester  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Rodriguez Rondon  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  M.
Rosenberg  University of Pittsburgh, Pittsburgh, PA, 15260, USA  M. Ross-
Lonergan  Columbia University, New York, NY, 10027, USA  G. Scanavini  Wright
Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA
D. W. Schmitz  University of Chicago, Chicago, IL, 60637, USA  A. Schukraft
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  W.
Seligman  Columbia University, New York, NY, 10027, USA  M. H. Shaevitz
Columbia University, New York, NY, 10027, USA  R. Sharankova  Tufts
University, Medford, MA, 02155, USA  J. Shi  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  J. Sinclair  Universität Bern, Bern
CH-3012, Switzerland  A. Smith  University of Cambridge, Cambridge CB3 0HE,
United Kingdom  E. L. Snider  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  M. Soderberg  Syracuse University, Syracuse, NY,
13244, USA  S. Söldner-Rembold  The University of Manchester, Manchester M13
9PL, United Kingdom  P. Spentzouris  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  J. Spitz  University of Michigan, Ann Arbor,
MI, 48109, USA  M. Stancari  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  J. St. John  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  T. Strauss  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  K. Sutton  Columbia University, New
York, NY, 10027, USA  S. Sword-Fehlberg  New Mexico State University (NMSU),
Las Cruces, NM, 88003, USA  A. M. Szelc  University of Edinburgh, Edinburgh
EH9 3FD, United Kingdom  W. Tang  University of Tennessee, Knoxville, TN,
37996, USA  K. Terao  SLAC National Accelerator Laboratory, Menlo Park, CA,
94025, USA  C. Thorpe  Lancaster University, Lancaster LA1 4YW, United Kingdom
D. Totani  University of California, Santa Barbara, CA, 93106, USA  M. Toups
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  Y.-T.
Tsai  SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  M. A.
Uchida  University of Cambridge, Cambridge CB3 0HE, United Kingdom  T. Usher
SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA  W. Van De
Pontseele  University of Oxford, Oxford OX1 3RH, United Kingdom  Harvard
University, Cambridge, MA 02138, USA  B. Viren  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  M. Weber  Universität Bern, Bern CH-3012,
Switzerland  H. Wei  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  Z. Williams  University of Texas, Arlington, TX, 76019, USA  S. Wolbers
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T.
Wongjirad  Tufts University, Medford, MA, 02155, USA  M. Wospakrik  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  K. Wresilo
University of Cambridge, Cambridge CB3 0HE, United Kingdom  N. Wright
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  W. Wu
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E.
Yandel  University of California, Santa Barbara, CA, 93106, USA  T. Yang
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G.
Yarbrough  University of Tennessee, Knoxville, TN, 37996, USA  L. E. Yates
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  H. W.
Yu  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA  G. P. Zeller
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J.
Zennamo  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
C. Zhang  Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We report results from a search for neutrino-induced neutral current (NC)
resonant $\Delta$(1232) baryon production followed by $\Delta$ radiative
decay, with a $\langle 0.8\rangle$ GeV neutrino beam. Data corresponding to
MicroBooNE’s first three years of operations (6.80$\times$1020 protons on
target) are used to select single-photon events with one or zero protons and
without charged leptons in the final state ($1\gamma 1p$ and $1\gamma 0p$,
respectively). The background is constrained via an in-situ high-purity
measurement of NC $\pi^{0}$ events, made possible via dedicated $2\gamma 1p$
and $2\gamma 0p$ selections. A total of 16 and 153 events are observed for the
$1\gamma 1p$ and $1\gamma 0p$ selections, respectively, compared to a
constrained background prediction of $20.5\pm 3.65\text{(sys.)}$ and $145.1\pm
13.8\text{(sys.)}$ events. The data lead to a bound on an anomalous
enhancement of the normalization of NC $\Delta$ radiative decay of less than
$2.3$ times the predicted nominal rate for this process at the 90% confidence
level (CL). The measurement disfavors a candidate photon interpretation of the
MiniBooNE low-energy excess as a factor of $3.18$ times the nominal NC
$\Delta$ radiative decay rate at the 94.8% CL, in favor of the nominal
prediction, and represents a greater than $50$-fold improvement over the
world’s best limit on single-photon production in NC interactions in the sub-
GeV neutrino energy range.

††preprint: APS/123-QED

For over two decades, the anomalous signals consisting of MiniBooNE’s low-
energy excess (LEE) [1, 2, 3] and the prior LSND [4] $\overline{\nu}_{e}$
appearance results have been at the forefront of neutrino physics. Each has
been interpreted as evidence for new types of neutrinos or other physics
beyond the Standard Model (SM). The existence of new particles would be the
first evidence for a new paradigm of physics associated with the neutrino
sector since the discovery of neutrinos mass via their observed oscillations,
and would have profound ramifications for all particle physics, astrophysics,
and cosmology. At the heart of this puzzle of anomalies in need of
interpretation is the fact MiniBooNE could not differentiate neutrino
interactions producing an electron (such as from $\nu_{e}$ appearance due to
light sterile neutrinos) from those with a single photon in the final state.
Thus, both types of interactions must be examined independently as a source of
the LEE.

Neutrino-induced neutral current (NC) production of the $\Delta$(1232) baryon
resonance with subsequent $\Delta$ radiative decay is predicted to be the
dominant source of single photons in neutrino-argon scattering below 1 GeV
[5]. Although $\Delta$ radiative decay is predicted in the SM, and
measurements of photoproduction [6] and virtual compton scattering [7] are
well described by theory, this process has never been directly observed in
neutrino scattering. Previous searches have been performed by the T2K [8] and
NOMAD [9] experiments with average incident neutrino energies, $E_{\nu}$, of
0.85 and 25 GeV, respectively, resulting in leading limits on this process.
Although on a different target, T2K’s result is closest in $E_{\nu}$ to that
of the MiniBooNE beam. However, the 90% confidence level (CL) limit is $\sim
100$ times the theoretically predicted rate of NC $\Delta$ radiative decay.

In this letter, we present the world’s most sensitive search for NC
$\Delta\rightarrow N\gamma$, where $N=p,n$, using neutrino-argon scattering
data collected by the MicroBooNE detector [10]. MicroBooNE is an 85 metric ton
active volume liquid argon time projection chamber (LArTPC) situated at a
similar baseline in the same muon neutrino dominated Booster Neutrino Beam
(BNB) at Fermilab [11] as MiniBooNE, with $\langle E_{\nu}\rangle=0.8$ GeV.
The measurement makes use of data corresponding to a BNB exposure of
$6.80\times 10^{20}$ protons on target (POT), collected during 2016-2018.
LArTPC technology allows MicroBooNE to distinguish electromagnetic showers
originating from electrons or photons based on ionization energy deposition
($dE/dx$) at the start of the shower, and the non-zero conversion distance of
the photon relative to the interaction vertex.

This search represents a first for this process with argon as the neutrino
target, and also constitutes the first test of the MiniBooNE LEE under a
single-photon interpretation. In a fit to the radial distribution of the
MiniBooNE data with statistical errors only, an enhancement of NC
$\Delta\rightarrow N\gamma$ (as predicted by the NUANCE [12] neutrino event
generator on CH2) by a normalization factor of $x_{\text{MB}}=3.18$ (quoted
with no uncertainty) was found to provide the best fit for the observed LEE
[3]. We perform an explicitly model-dependent test of this interpretation,
cast as a factor of 3.18 enhancement to the predicted NC $\Delta\rightarrow
N\gamma$ rate in MicroBooNE, under a two-hypothesis $\Delta\chi^{2}$ test
between the enhanced rate and the nominal NC $\Delta\rightarrow N\gamma$
prediction.

MicroBooNE uses a custom tune [13] of the genie neutrino event generator
v3.0.6 [14, 15] to simulate neutrino-argon interactions. At BNB energies, the
dominant source of single-photon production with no charged leptons or pions
in the final state is NC $\Delta(1232)\rightarrow N\gamma$. This process is
included in the MicroBooNE nominal prediction exactly as modeled in genie.
Heavier resonances and non-resonant processes, including coherent single-
photon production [16], are not currently included in the simulation, but are
each estimated to contribute at the 10% level or less. Both these processes
would produce slightly higher-energy photons than the $\Delta$(1232)
resonance, and a more forward (in the direction of the neutrino beam) photon
in the case of coherent production. Although such events may be selected by
this analysis, we do not explicitly quantify their selection efficiency and in
this letter we focus on the dominant NC $\Delta(1232)\rightarrow N\gamma$
process.

The MicroBooNE NC $\Delta\rightarrow N\gamma$ search exclusively targets
events with a single, photon-like electromagnetic shower and either no other
visible activity or one visible final-state proton. These are referred to as
$1\gamma 0p$ and $1\gamma 1p$ events and primarily probe $\Delta\rightarrow
n\gamma$ and $\Delta\rightarrow p\gamma$ decays, respectively. The analysis
selects and simultaneously fits $1\gamma 1p$ and $1\gamma 0p$ data-to-Monte
Carlo (MC) simulated distributions together with two additional, mutually
exclusive but highly correlated event samples: $2\gamma 1p$, and $2\gamma 0p$.
The signal, defined as all true NC $\Delta\rightarrow N\gamma$ events whose
true interaction vertex is inside the active TPC, contributes predominantly to
the $1\gamma$ event samples. The high-statistics $2\gamma$ samples are
enhanced in NC $\Delta\rightarrow N\pi^{0}$ production, which is the dominant
source of mis-identified background to the $1\gamma 1p$ and $1\gamma 0p$ event
samples.

Reconstruction of all four event samples makes use of the Pandora framework
[17]. Reconstructed ionization charge hits are clustered and matched across
three 2D projected views of the MicroBooNE active TPC volume into 3D
reconstructed objects. These are then classified as tracks or showers based on
a multivariate classifier score and aggregated into candidate neutrino
interactions. The topological selection of interactions with exactly one
shower and zero or one tracks represents the basis of the $1\gamma$
selections. Subsequently, pre-selection requires that the reconstructed
vertex, shower-start point and track (as applicable) are all fully contained
within the detector fiducial volume. A minimum energy requirement is imposed
on the shower, ensuring good reconstruction performance, and a maximum track
length requirement is imposed on the track, rejecting obvious muon
backgrounds. Tracks are also required to have a high $dE/dx$ consistent with
that of a proton. Finally, an opening angle requirement between the track and
shower directions is applied to eliminate co-linear events where the start of
a shower can be mis-reconstructed as a track.

The pre-selected events are fed into a set of boosted decision trees (BDTs),
each designed to reject a distinct background and select NC $\Delta\rightarrow
N\gamma$ events. The gradient boosting algorithm XGBoost [18] is used to train
the BDTs. A cosmic BDT rejects cosmogenic backgrounds and is trained on cosmic
ray data events collected when no neutrino beam was present. Track calorimetry
is used to reject cosmic muons, with track and shower directionality-based
variables proving powerful discriminators. A NC $\pi^{0}$ BDT compares the
relationship of the reconstructed shower and track to those expected from
$\pi^{0}$ decay kinematics to separate true single-photon events from those
containing a $\pi^{0}$ decay where a second photon is not reconstructed. A
charged current (CC) $\nu_{e}$ BDT targets the intrinsic $\nu_{e}$ background
events. Here, the photon conversion distance and shower calorimetry play
important roles. A fourth BDT is designed to veto events in which a second
shower from a $\pi^{0}$ decay deposits some charge, but fails 3D shower
reconstruction. Such events can result in 2D charge hits near the neutrino
interaction that are not associated with a 3D object. A plane-by-plane
clustering algorithm, DBSCAN [19], is used to group these unassociated hits,
and properties including direction, shape and energy of the cluster are used
to determine consistency with a second shower from a $\pi^{0}$ decay. A final
CC $\nu_{\mu}$-focused BDT removes any remaining backgrounds, primarily
targeting the muon track through track calorimetry variables.

The $1\gamma 1p$ selection uses all five BDTs. The absence of a track in the
$1\gamma 0p$ sample means that the $1\gamma 0p$ selection cannot use these
BDTs identically, as it is limited to only shower variables. As such, it uses
variations of the cosmic and NC$\pi^{0}$ BDTs, and a third BDT merging the
functionality of the CC $\nu_{e}$ and CC $\nu_{\mu}$-focused BDTs, targeting
all remaining backgrounds. All BDTs are trained explicitly to select well-
reconstructed NC $\Delta\rightarrow N\gamma$ events. While model-dependent,
this leverages the kinematics and correlations between the track and shower
associated with $\Delta(1232)$ resonance decay, particularly for the $1\gamma
1p$ selection. The BDTs for the $1\gamma 1p$ and $1\gamma 0p$ selections are
trained and optimized for each selection independently, through a scan over
grids of their BDT classifier scores. The optimized BDT classifier score cuts
correspond to the highest statistical significance of the NC
$\Delta\rightarrow N\gamma$ signal over background in each sample. The
topological, pre-selection, BDT selection, and combined signal efficiencies
are summarized in Table 1.

Selection Stage |  $1\gamma 1p$ eff. |  $1\gamma 0p$ eff.
---|---|---
Topological | 19.4% | 13.5%
Pre-selection | 63.9% | 98.4 %
BDT Selection | 32.1% | 39.8%
Combined | 3.99% | 5.29%
Table 1: Signal efficiencies for the $1\gamma 1p$ and $1\gamma 0p$ selections.
The topological and combined efficiencies are evaluated relative to all true
NC $\Delta\rightarrow N\gamma$ events inside the active TPC in the simulation
(124.1 events expected for $6.80\times 10^{20}$ POT). The pre-selection and
BDT selection efficiencies are evaluated relative to their respective
preceding selection stage.

The number of predicted background (both from simulation and cosmic ray data)
events and NC $\Delta\rightarrow N\gamma$ signal events after BDT selection
are summarized in Table 2. A significant background to the search for single-
photon events is NC $\pi^{0}$ events in which one of the decay photons is not
reconstructed. This happens for a variety of reasons: (a) one of the photons
from the $\pi^{0}$ decay may leave the detector active TPC volume before
interacting, (b) the $\pi^{0}$ decay may be highly asymmetric leading to a
secondary photon that is low in energy and not reconstructed, (c) both photons
may be approximately co-linear and overlapping and thus reconstructed as a
single shower, or (d) the secondary photon may fall in a region of
unresponsive wires, leading to poor reconstruction efficiency. Motivated by
the background contribution of NC $\pi^{0}$ events, the $2\gamma 1p$ and
$2\gamma 0p$ event samples serve to constrain the rate of NC $\pi^{0}$
background. The $2\gamma$ samples follow the same topological, pre-selection
and BDT selection scheme as the $1\gamma$ samples (see supplemental
materials). The selected $2\gamma 1p$ and $2\gamma 0p$ events are shown in
Fig. 1 as a function of reconstructed $\pi^{0}$ momentum, with a true NC
$1\pi^{0}$ event purity of 63.4% and 59.6%, respectively. The data-to-MC
simulation ratio in the $2\gamma 1p$ and $2\gamma 0p$ samples is $0.80\pm
0.22$(stat.$\oplus$sys.) and $0.91\pm 0.19$(stat.$\oplus$sys.), respectively,
showing an overall deficit but one that is within $1\sigma$.

Process | $1\gamma 1p$ | $1\gamma 0p$
---|---|---
NC $1\pi^{0}$ Non-Coherent | 24.0 | 68.1
NC $1\pi^{0}$ Coherent | 0.0 | 7.6
CC $\nu_{\mu}$ $1\pi^{0}$ | 0.5 | 14.0
CC $\nu_{e}$ and $\bar{\nu}_{e}$ | 0.4 | 11.1
BNB Other | 2.1 | 18.1
Dirt (outside TPC) | 0.0 | 36.4
Cosmic Ray Data | 0.0 | 10.0
Total Background (Unconstr.) | 27.0 | 165.4
NC $\Delta\rightarrow N\gamma$ | 4.88 | 6.55
Table 2: The expected event rates in the $1\gamma 1p$ and $1\gamma 0p$
samples. “Dirt (outside TPC)” represents any neutrino interaction that
originates outside the active TPC, but scatters inside. Relative to
topological selection stage, the $\nu_{e}$ CC rejection is 99.8% and 87.6% for
$1\gamma 1p$ and $1\gamma 0p$, respectively.

(a)

(b)

Figure 1: Data and MC comparisons of the reconstructed $\pi^{0}$ momentum
distributions for the (a) 2$\gamma$1p and (b) 2$\gamma$0p selected events.

The selected data and MC predictions are compared in a fit with a single free
parameter corresponding to the normalization ($x_{\Delta}$) of the nominal
rate of NC $\Delta\rightarrow N\gamma$. A single bin is used for each of the
$1\gamma 1p$ and $1\gamma 0p$ event samples, with reconstructed shower energy
bin boundaries of 0-600 MeV and 100-700 MeV, respectively. The one-bin
$1\gamma 1p$ and $1\gamma 0p$ event rates are fit simultaneously with the
$2\gamma 1p$ and $2\gamma 0p$ distributions shown in Fig. 1. The fit makes use
of a covariance matrix that encapsulates statistical and systematic
uncertainties and bin-to-bin correlations, allowing for both the expected rate
and uncertainties of the NC $\pi^{0}$ backgrounds in the $1\gamma$ samples to
be effectively constrained by the high-statistics data observed in the
$2\gamma$ samples.

The normalization $x_{\Delta}$ can also be reinterpreted in various ways.
First, it can be reinterpreted as a scaling of an effective branching fraction
$\mathcal{B}_{\text{eff}}(\Delta\rightarrow N\gamma)$, where the nominal
prediction ($x_{\Delta}=1$) corresponds to an effective branching fraction of
0.6%. This effective branching fraction can be thought of as a metric to
account for any uncertain nuclear effects that might modify the $\Delta$
behavior inside the nuclear medium, as we cannot observe the true
$\Delta\rightarrow N\gamma$ branching fraction directly. In addition, any BSM
effect that can contribute as an NC $\Delta$-like process (with a single
photon-like shower in the final state) could lead to an effective modification
to the observed branching fraction. Although genie prescribes a normalization
uncertainty for $\mathcal{B}_{\text{eff}}(\Delta\rightarrow N\gamma)$, this
uncertainty is not included in the fit. In addition, with the knowledge that
genie predicts a cross-section for NC $\Delta\rightarrow N\gamma$ production
to be $\sigma^{\text{GENIE,Ar}}_{NC\Delta\rightarrow N\gamma}=8.61\times
10^{-42}$cm-2/nucleon, we can also reinterpret $x_{\Delta}$ as scaling on this
production cross-section. The Feldman-Cousins [20] approach is followed to
construct the confidence intervals for $x_{\Delta}$ given the best fit to the
observed data, with a metric of $\Delta\chi^{2}$ defined using the Combined-
Neyman-Pearson $\chi^{2}$ [21] as an approximation of the log-likelihood
ratio.

Systematic uncertainties include contributions from flux, cross-section
modeling, hadron re-interactions, detector effects, and finite statistics used
in the background predictions (both MC and cosmic ray data). The flux
uncertainties incorporate hadron production uncertainties, uncertainties on
pion and nucleon scattering in the beryllium target and surrounding aluminum
magnetic horn, and mis-modeling of the horn current. Following [22], these are
implemented by reweighting the flux prediction and studying the propagated
effects on event distributions. The cross-section uncertainties incorporate
modeling uncertainties on the genie prediction [13, 23, 15], evaluated also by
reweighting tools. The hadron-argon re-interaction uncertainties are
associated with the propagation of hadrons through the detector, as modeled in
geant4 [24]. The detector modeling and response uncertainties are evaluated
using a novel data-driven technique. This uses in-situ measurements of
distortions in the TPC wire readout signals due to various detector effects,
such as diffusion, electron drift lifetime, electric field, and electronics
response, to parametrize these effects at the TPC wire level, and provides a
detector model-agnostic way to study and evaluate their effects on event
distributions [25]. Additional systematics varying the charge recombination
model, the scintillation light yield, and space charge effects [26, 27] are
separately included. The uncertainty on photo-nuclear absorption of photons on
argon was evaluated to be at the sub-percent level, and is therefore omitted.
There is also no assigned uncertainty for heavier resonances or coherent
single-photon production, which are not simulated in genie. Finally, an
inconsistency was identified in the genie v3.0.6 reweighing code used to
evaluate a small subset of systematic uncertainties, but was found to have
negligible impact on the analysis sensitivity and thus has been ignored.

The fractional systematic uncertainties on the $1\gamma 1p$ and $1\gamma 0p$
total background events are summarized in Table 3. The genie cross-section
uncertainties dominate. This stems from the uncertainties on NC $\pi^{0}$
production on argon, which forms the largest background and has not been
measured to high precision to date. Both cross-section and flux uncertainties
are strongly correlated between the $1\gamma$ and $2\gamma$ event samples. The
simultaneous fit to the $1\gamma$ and $2\gamma$ samples is equivalent to a
$1\gamma$-only fit where the background and uncertainty are conditionally
constrained [28] by the $2\gamma$ samples. Given the $2\gamma$ samples’
statistics, this constraint effectively reduces the total background
systematic uncertainty of the $1\gamma 1p$ and $1\gamma 0p$ samples by 40% and
50%, and the total background prediction by 24.1% and 12.3%, respectively.

Type of Uncertainty | $1\gamma 1p$ | $1\gamma 0p$
---|---|---
Flux model | 7.4% | 6.6%
genie cross-section model | 24.8% | 16.3%
geant4 re-interactions | 1.1% | 1.3%
Detector effects | 12.2% | 6.4%
Finite background statistics | 8.3% | 4.0%
Total Uncertainty (Unconstr.) | 29.8% | 19.2%
Total Uncertainty (Constr.) | 17.8% | 9.5%
Table 3:  Breakdown of background systematic uncertainties for the $1\gamma
1p$ and $1\gamma 0p$ samples.

The 90% CL sensitivity is quantified for a Feldman-Cousins-corrected limit in
the case of a background-only observation, $x_{\Delta}=0$, to be less than
$x_{\Delta}=2.5$, corresponding to $\mathcal{B}_{\text{eff}}(\Delta\rightarrow
N\gamma)=1.50\%$ and $\sigma^{\text{Ar}}_{NC\Delta\rightarrow
N\gamma}=21.5\times 10^{-42}$cm-2/nucleon . Under a two-hypothesis
$\Delta\chi^{2}$ test, the expected sensitivity of the median experiment
assuming the nominal prediction, to reject the LEE hypothesis
($x_{\text{MB}}=3.18$) in favor of the nominal hypothesis ($x_{\Delta}=1$) is
1.5$\sigma$; in the case of the median experiment assuming the LEE hypothesis,
the sensitivity to reject the nominal hypothesis in favor of the LEE
hypothesis is 1.6$\sigma$.

(a)

(b)

Figure 2: Energy spectra for the (a) $1\gamma 1p$ and (b) $1\gamma 0p$
selected events. The upper section in each figure shows the unconstrained
background predictions and breakdowns as a function of reconstructed shower
energy. The lower section shows the total background prediction with
systematic uncertainty both before and after the $2\gamma$ constraint. The
local significance of the data fluctuation in the 200-250 MeV bin of (b)
corresponds to $1.6\sigma$ ($\chi^{2}/dof=3.66/1$) before the $2\gamma$
constraint, and $2.7\sigma$ ($\chi^{2}/dof=8.54/1$) after. From MC studies,
the probability of any one bin across all 16 $1\gamma$ bins giving rise to a
constrained $\chi^{2}\geq 8.54$ is 4.74%.

The reconstruction, selection, and fitting methods employed in this search
were developed adhering to a signal-blind analysis strategy, whereby the data
was kept blind until the analysis was fully developed, with the exception of a
small subset of the data consisting of $0.51\times 10^{20}$ POT, used for
analysis validation. After $1\gamma 1p$ and $1\gamma 0p$ event samples were
unblinded, 16 data events with an expected constrained background of $20.5\pm
3.6\text{(sys.)}$ events were observed in the $1\gamma 1p$ event sample, and
153 data events with an expected constrained background of $145.1\pm
13.8\text{(sys.)}$ events were observed in the $1\gamma 0p$ event sample. The
reconstructed shower energy distributions of selected $1\gamma 1p$ and
$1\gamma 0p$ events are shown in Fig. 2. Overall, a systematic deficit of data
relative to the unconstrained MC prediction is observed, which is within
systematic and statistical uncertainties, and consistent with a similar
deficit in the $2\gamma$ event samples. The expected signal and background
predictions are summarized in Table 4 and Fig. 3, and compared to the observed
data, both before and after applying the $2\gamma$ conditional constraint. The
$2\gamma$ constraint reduces the total background prediction, consistently
with the data to MC simulation ratio observed in the $2\gamma$ event samples.

| $1\gamma 1p$ |  | $1\gamma 0p$
---|---|---|---
Unconstr. bkgd. | 27.0 $\pm$ 8.1 |  | 165.4 $\pm$ 31.7
Constr. bkgd. | 20.5 $\pm$ 3.6 |  | 145.1 $\pm$ 13.8
NC $\Delta\rightarrow N\gamma$ | 4.88 |  | 6.55
LEE ($x_{\text{MB}}=3.18$) | 15.5 |  | 20.1
Data | 16 |  | 153
Table 4: Number of predicted background, predicted signal, and observed data
events for the $1\gamma 1p$ and $1\gamma 0p$ samples, with background
systematic uncertainties.

(a)

(b)

Figure 3: The observed event rates for the (a) $1\gamma 1p$ and (b) $1\gamma
0p$ event samples, and comparisons to unconstrained (left) and constrained
(right) background and LEE model predictions. The event rates are the sum of
all events with reconstructed shower energy between 0-600 MeV and 100-700 MeV
for (a) and (b), respectively. The one-bin background only conditionally
constrained $\chi^{2}$ is 0.63 and 0.18 for $1\gamma 1p$ and $1\gamma 0p$
respectively.

The best-fit value for $x_{\Delta}$ obtained from the fit is 0, with a
$\chi^{2}_{bf}$ of 5.53 for 15 degrees of freedom ($dof$). This measurement is
in agreement with the nominal NC $\Delta\rightarrow N\gamma$ rate
(corresponding to $x_{\Delta}=1$) within $1\sigma$ (67.8% CL) with a
$\chi^{2}$ of 6.47 for 16 $dof$. The Feldman-Cousins calculated confidence
limit leads to a one-sided bound on the normalization of NC $\Delta\rightarrow
N\gamma$ events of $x_{\Delta}<2.3$, corresponding to
$\mathcal{B}_{\text{eff}}(\Delta\rightarrow N\gamma)<1.38\%$ and
$\sigma^{\text{Ar}}_{NC\Delta\rightarrow N\gamma}<19.8\times
10^{-42}$cm-2/nucleon, at 90% CL. This is summarized in Fig. 4.

This result represents the most stringent limit on neutrino-induced NC
$\Delta\rightarrow N\gamma$ on any nuclear target [9, 8], and a significant
improvement over previous searches, in particular in the neutrino energy range
below 1 GeV. Under a two-hypothesis test, the data rules out the
interpretation of the MiniBooNE anomalous excess [29] as a factor of 3.18
enhancement to the rate of $\Delta\rightarrow N\gamma$, in favor of the
nominal prediction at $94.8$% CL (1.9$\sigma$). While this is a model-
dependent test of the MiniBooNE LEE, and does not apply universally to all
other photon-like interpretations, it provides an important constraint on this
process and a first direct test of the MiniBooNE LEE, and opens the door to
further searches that focus on a broader range of models. Those include
coherent single-photon production [5], anomalous contributions of which could
give rise to additional events and would be expected to leave an imprint in
the $1\gamma 0p$ selection, as well as more exotic beyond-SM processes that
manifest as single-photon events, such as co-linear $e^{+}e^{-}$ pairs from
$Z^{\prime}$ [30, 31] or scalar [32] decays, among others. Follow-up
MicroBooNE analyses will explicitly target and quantify sensitivity to these
alternative hypotheses, as well as model-independent single-photon searches.

Figure 4:  The resulting $\Delta\chi^{2}$ curve after fitting to $x_{\Delta}$
using the Feldman-Cousins procedure, showing extracted confidence intervals.
The best fit is found to be at $x_{\Delta}=0$ with a $\chi^{2}_{bf}=5.53$.
Shown also is the reinterpretation of this scaling factor as both an effective
branching fraction, $\mathcal{B}_{\text{eff}}(\Delta\rightarrow N\gamma)$, and
a cross-section, $\sigma^{\text{Ar}}_{NC\Delta\rightarrow N\gamma}$. The
default genie value corresponds to $x_{\Delta}=1$. The error on the LEE model
is estimated from the MiniBooNE result [3] with statistical and systematic
uncertainty. It should be noted that this uncertainty does not account for
systematic correlations between MiniBooNE and MicroBooNE.

###### Acknowledgements.

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); and The European
Union’s Horizon 2020 Marie Skłodowska-Curie Actions. Additional support for
the laser calibration system and cosmic ray tagger was provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland.

## References

  * Aguilar-Arevalo  _et al._ [2009a] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 102, 101802 (2009a), arXiv:0812.2243 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2018] A. A. Aguilar-Arevalo _et al._ (MiniBooNE Collaboration), Phys. Rev. Lett. 121, 221801 (2018).
  * Aguilar-Arevalo  _et al._ [2021] A. A. Aguilar-Arevalo _et al._ (MiniBooNE Collaboration), Phys. Rev. D 103, 052002 (2021).
  * Athanassopoulos  _et al._ [1996] C. Athanassopoulos _et al._ (LSND), Phys. Rev. C 54, 2685 (1996), arXiv:nucl-ex/9605001 .
  * Wang  _et al._ [2014] E. Wang, L. Alvarez-Ruso, and J. Nieves, Phys. Rev. C 89, 015503 (2014).
  * Blanpied  _et al._ [1997] G. Blanpied _et al._ , Phys. Rev. Lett. 79, 4337 (1997).
  * Sparveris  _et al._ [2008] N. F. Sparveris _et al._ , Phys. Rev. C 78, 018201 (2008), arXiv:0804.1169 [nucl-ex] .
  * Abe  _et al._ [2019] K. Abe _et al._ (T2K), J. Phys. G: Nucl. Part. Phys. 46, 08LT01 (2019).
  * Kullenberg  _et al._ [2012] C. Kullenberg _et al._ (NOMAD), Phys. Lett. B 706, 268–275 (2012).
  * Acciarri  _et al._ [2017] R. Acciarri _et al._ (MicroBooNE), JINST 12, P02017 (2017).
  * Aguilar-Arevalo  _et al._ [2009b] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 79, 072002 (2009b).
  * Casper [2002] D. Casper, Nucl. Phys. B - Proc. Suppl. 112, 161 (2002).
  * Abratenko  _et al._ [2021a] P. Abratenko _et al._ (MicroBooNE), (2021a), arXiv:2110.14028 [hep-ex] .
  * Andreopoulos  _et al._ [2010a] C. Andreopoulos _et al._ , Nucl. Instrum. Meth. A 614, 87 (2010a), arXiv:0905.2517 [hep-ph] .
  * Tena-Vidal  _et al._ [2021] J. Tena-Vidal _et al._ (GENIE Collaboration), arXiv:2104.09179 (2021).
  * Alvarez-Ruso  _et al._ [2018] L. Alvarez-Ruso, J. Nieves, E. Sala, and E. Wang, Journal of Physics: Conference Series 1056, 012001 (2018).
  * Acciarri  _et al._ [2018] R. Acciarri _et al._ (MicroBooNE), Eur. Phys. J. C 78, 82 (2018).
  * Chen and Guestrin [2016] T. Chen and C. Guestrin (Association for Computing Machinery, New York, NY, USA, 2016) pp. 785–794.
  * Ester  _et al._ [1996] M. Ester _et al._ , KDD’96: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining , 226 (1996).
  * Feldman and Cousins [1998] G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998).
  * Ji  _et al._ [2020] X. Ji _et al._ , Nucl. Instr. and Meth. A 961, 163677 (2020).
  * Abratenko  _et al._ [2019] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 123, 131801 (2019), arXiv:1905.09694 [hep-ex] .
  * Andreopoulos  _et al._ [2010b] C. Andreopoulos _et al._ , Nucl. Instr. and Meth. A 614, 87 (2010b).
  * Agostinelli  _et al._ [2003] S. Agostinelli _et al._ , Nucl. Instr. and Meth. A 506, 250 (2003).
  * Abratenko  _et al._ [2021b] P. Abratenko _et al._ (MicroBooNE), (2021b), arXiv:2111.03556 [hep-ex] .
  * Abratenko  _et al._ [2020a] P. Abratenko _et al._ (MicroBooNE), JINST 15, P07010 (2020a).
  * Abratenko  _et al._ [2020b] P. Abratenko _et al._ (MicroBooNE), JINST 15, P12037 (2020b).
  * Eaton [1983] M. L. Eaton,  _Multivariate statistics : a vector space approach_ , Wiley series in probability and mathematical statistics. Probability and mathematical statistics. (Wiley, New York, 1983 - 1983).
  * Aguilar-Arevalo  _et al._ [2009c] A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 102, 101802 (2009c).
  * Bertuzzo  _et al._ [2018] E. Bertuzzo _et al._ , Phys. Rev. Lett. 121, 241801 (2018).
  * Ballett  _et al._ [2019] P. Ballett _et al._ , Phys. Rev. D 99, 071701 (2019).
  * Abdallah  _et al._ [2020] W. Abdallah, R. Gandhi, and S. Roy, Preprint (2020), arXiv:2010.06159 [hep-ph] .