# The (Co)homology of the Deligne-Mumford Moduli Spaces
of Marked Rational Curves

Aleksey Zinger

###### Abstract

This informal note collects key results and open problems on the (co)homology
of the Deligne-Mumford moduli spaces of real marked rational curves. The open
problems are both of topological nature, aiming to investigate the
(co)homology of these spaces further, and of algebraic nature, aiming to
describe the structure of the homology of these spaces in operad-like terms.

$\overline{\mathcal{M}}_{\ell}=$ Deligne-Mumford moduli space of (complex)
rational curves with $\ell$ marked points; smooth projective variety of
complex dimension $\ell\\!-\\!3$. An element of this space is the equivalence
class of stable tuples

$\mathcal{C}\equiv\big{(}\Sigma,z_{1},\ldots,z_{\ell}\big{)},$

where $\Sigma$ is a connected, possibly nodal, Riemann surface and
$z_{1},\ldots,z_{\ell}\\!\in\\!\Sigma$ are distinct smooth points.

$\mathbb{R}\overline{\mathcal{M}}_{k,\ell}=$ Deligne-Mumford moduli space of
real rational curves with $k$ real marked points and $\ell$ conjugate pairs of
marked points; compact smooth real analytic variety of dimension
$k\\!+\\!2\ell\\!-\\!3$; orientable if $k\\!=\\!0$ or
$k\\!+\\!2\ell\\!\leq\\!4$. An element of this space is the equivalence class
of stable tuples

$\mathcal{C}\equiv\big{(}\Sigma,\sigma,z_{1},\ldots,z_{k},(z_{1}^{+},z_{1}^{-}),\ldots,(z_{\ell}^{+},z_{\ell}^{-})\\!\big{)},$

where $\Sigma$ is a connected, possibly nodal, Riemann surface, $\sigma$ is an
anti-holomorphic involution on $\Sigma$ and

$z_{1},\ldots,z_{k},z_{1}^{+},z_{1}^{-},\ldots,z_{\ell}^{+},z_{\ell}^{-}\in\Sigma$

are distinct smooth points such that $\sigma(z_{i}^{+})\\!=\\!z_{i}^{-}$ for
$i\\!=\\!1,\ldots,\ell$.

$[\ell]\\!=\\!\\{1,\ldots,\ell\\}$.

##  1 Key Results

Complex case, $\overline{\mathcal{M}}_{\ell}$. This smooth projective variety
contains smooth divisors $D_{J,K}$, with $J\\!\sqcup\\!K\\!=\\![\ell]$ and
$|J|,|K|\\!\geq\\!2$, whose generic element is a two-component curve with the
marked points on the two components indexed by $J$ and $K$; see Figure 1 for
some examples. It is a now classical result of Keel [7] that

  1. ($\mathbb{C}1$)

these divisors generate $H^{*}(\overline{\mathcal{M}}_{\ell};\mathbb{Z})$ as
an algebra,

  2. ($\mathbb{C}2$)

subject only to some obvious relations;

see [1, Theorem 2.1] for a precise statement. The relations arise from

  1. ($\mathbb{C}\ref{Crel_it}a$)

certain divisors $D_{J,K}\\!\subset\\!\overline{\mathcal{M}}_{\ell}$ being
disjoint and

  2. ($\mathbb{C}\ref{Crel_it}b$)

all divisors $D_{J,K}$ in
$\overline{\mathcal{M}}_{4}\\!\approx\\!\mathbb{C}\mathbb{P}^{1}$ being points
and thus equivalent in $H^{2}(\overline{\mathcal{M}}_{4};\mathbb{Z})$, as
indicated in Figure 1.

The above relations in $H^{2}(\overline{\mathcal{M}}_{4};\mathbb{Z})$ pull
back to relations in $H^{2}(\overline{\mathcal{M}}_{\ell};\mathbb{Z})$ with
$\ell\\!\geq\\!4$ via the forgetful morphisms
$\overline{\mathcal{M}}_{\ell}\\!\longrightarrow\\!\overline{\mathcal{M}}_{4}$
dropping all but four of the marked points (every 4-element subset of $[\ell]$
determines such a morphism).

$1$$2$$3$$4$$=$$D_{\\{1,2\\},\\{3,4\\}}$$1$$3$$2$$4$$=$$D_{\\{1,3\\},\\{2,4\\}}$$1$$4$$2$$3$$D_{\\{1,4\\},\\{2,3\\}}$
Figure 1: Relations between the divisors
$D_{J,K}\\!\subset\\!\overline{\mathcal{M}}_{4}$ in
$H^{2}(\overline{\mathcal{M}}_{4};\mathbb{Z})$.

The proof in [7] presents $\overline{\mathcal{M}}_{\ell+1}$ as a sequence of
holomorphic blowups of
$\overline{\mathcal{M}}_{\ell}\\!\times\\!\overline{\mathcal{M}}_{4}$ and
determines $H^{*}(X_{\varrho};\mathbb{Z})$ inductively for every intermediate
blowup $X_{\varrho}$, assuming the claimed result for
$H^{*}(\overline{\mathcal{M}}_{\ell};\mathbb{Z})$. The proof of the main
result of [7] given in [1] still uses this sequence of blowups to obtain
(CDM1), but establishes (CDM2) directly, without determining
$H^{*}(X_{\varrho};\mathbb{Z})$ for the intermediate blowups $X_{\varrho}$.

For any commutative ring $R$ with unity 1, the modules
$H_{*}(\overline{\mathcal{M}}_{\ell};R)$ can be arranged into an operad of
graded $R$-modules

$\mathcal{O}_{\mathbb{C}}(\ell)\equiv\begin{cases}R,&\hbox{if}~{}\ell\\!=\\!1;\\\
H_{*}(\overline{\mathcal{M}}_{\ell+1};R),&\hbox{if}~{}\ell\\!\geq\\!2;\end{cases}$

as follows. For integers $k,\ell\\!\geq\\!2$ and $i\\!\in\\![k]$, let

$\iota_{k\ell
i}\\!:\overline{\mathcal{M}}_{k+1}\\!\times\\!\overline{\mathcal{M}}_{\ell+1}\longrightarrow\overline{\mathcal{M}}_{k+\ell}$

be the immersion identifying the $i$-th marked point of a marked curve
$[\mathcal{C}_{1}]\\!\in\\!\overline{\mathcal{M}}_{k+1}$ with the last marked
point of a marked curve
$[\mathcal{C}_{2}]\\!\in\\!\overline{\mathcal{M}}_{\ell+1}$, keeping the
indices of the first $i\\!-\\!1$ marked points of $[\mathcal{C}_{1}]$ the
same, increasing the indices of the last $k\\!+\\!1\\!-\\!i$ marked points of
$[\mathcal{C}_{1}]$ by $\ell\\!-\\!1$, and increasing the indices of the first
$\ell$ marked points of $[\mathcal{C}_{2}]$ by $i\\!-\\!1$. This induces
homomorphisms

$\iota_{k\ell
i*}\\!:\mathcal{O}_{\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell)\\!\approx\\!H_{*}\big{(}\overline{\mathcal{M}}_{k+1}\\!\times\\!\overline{\mathcal{M}}_{\ell+1};R\big{)}\longrightarrow\mathcal{O}_{\mathbb{C}}(k\\!+\\!\ell\\!-\\!1).$

For $k,\ell\\!\in\\!\mathbb{Z}^{+}$ and $i\\!\in\\![k]$, define

$\displaystyle\iota_{k1i*}\\!:\mathcal{O}_{\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(1)$
$\displaystyle\longrightarrow\mathcal{O}_{\mathbb{C}}(k),$
$\displaystyle\qquad\iota_{k1i*}(\eta,r)$ $\displaystyle=r\eta,$ (1)
$\displaystyle\iota_{1\ell
1*}\\!:\mathcal{O}_{\mathbb{C}}(1)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell)$
$\displaystyle\longrightarrow\mathcal{O}_{\mathbb{C}}(\ell),$
$\displaystyle\qquad\iota_{1\ell 1*}(r,\eta)$ $\displaystyle=r\eta.$

For $k,\ell_{1},\ldots,\ell_{k}\\!\in\\!\mathbb{Z}^{+}$, define

$\displaystyle\circ\\!:\mathcal{O}_{\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell_{1})\\!\times\\!\ldots\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell_{k})\longrightarrow\mathcal{O}_{\mathbb{C}}\big{(}\ell_{1}\\!+\\!\ldots\\!+\\!\ell_{k}\big{)},$
(2)
$\displaystyle\eta\\!\circ\\!(\eta_{1},\ldots,\eta_{k})=\iota_{(1+\ell_{2}+\ldots+\ell_{k})\ell_{1}1*}\big{(}\ldots\iota_{(k-1+\ell_{k})\ell_{k-1}(k-1)*}\big{(}\iota_{k\ell_{k}k*}(\eta,\eta_{k}),\eta_{k-1}\big{)},\ldots,\eta_{1}\big{)}.$

The resulting operad
$(\\!(\mathcal{O}_{\mathbb{C}}(\ell))_{\ell\in\mathbb{Z}^{+}},\circ)$ is the
operad of hypercommutative algebras; see [5].

Real case with real points only, $\mathbb{R}\overline{\mathcal{M}}_{k,0}$. By
the main result of [8],
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Z}_{2})$ is described in
the same way as $H^{2*}(\overline{\mathcal{M}}_{k};\mathbb{Z})$ above with the
complex subvarieties $D_{J,K}\\!\subset\\!\overline{\mathcal{M}}_{k}$ replaced
by the analogous real subvarieties
$\mathbb{R}D_{J,K}\\!\subset\\!\mathbb{R}\overline{\mathcal{M}}_{0,k}$; see
the proof of [3, Theorem 5.6]. Since $\mathbb{R}\overline{\mathcal{M}}_{k,0}$
is not orientable for $k\\!\geq\\!5$, the homology and cohomology of
$\mathbb{R}\overline{\mathcal{M}}_{k,0}$ with coefficients in a ring of
characteristic larger than 2 are quite distinct. For any commutative ring $R$
with unity and distinct $a,b,c,d\\!\in\\![k]$,
$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{k,0};R)$ contains the pullback
$\omega_{abcd}$ of the Poincare dual of a point in
$\mathbb{R}\overline{\mathcal{M}}_{4,0}\\!\approx\\!\mathbb{R}\mathbb{P}^{1}$
by the forgetful morphism
$\mathbb{R}\overline{\mathcal{M}}_{k,0}\\!\longrightarrow\\!\mathbb{R}\overline{\mathcal{M}}_{4,0}$
sending the marked indexed by $a,b,c,d$ to the marked points indexed by
$1,2,3,4$, respectively, and dropping the remaining marked points. This class
is Poincare dual to a linear combination of the real hypersurfaces
$\mathbb{R}D_{J,K}$.

By [3, Theorem 2.9],

  1. ($\mathbb{R}\mathbb{R}1$)

the classes $\omega_{abcd}$ generate
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Q})$ as an algebra,

  2. ($\mathbb{R}\mathbb{R}2$)

subject only to some expected relations.

The relations arise from

  1. ($\mathbb{R}\mathbb{R}\ref{RRrel_it}b)$

the diffeomorphism of $\mathbb{R}\overline{\mathcal{M}}_{4,0}$ induced by the
interchange of two marked points being orientation-reversing and

  2. ($\mathbb{R}\mathbb{R}\ref{RRrel_it}c)$

$\mathbb{R}\overline{\mathcal{M}}_{5,0}$ being diffeomorphic to the blowup of
$\mathbb{R}\mathbb{P}^{1}\\!\times\\!\mathbb{R}\mathbb{P}^{1}$ at three points
of the diagonal.

The resulting relations in
$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{4,0};\mathbb{Q})$,
$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{5,0};\mathbb{Q})$, and
$H^{2}(\mathbb{R}\overline{\mathcal{M}}_{5,0};\mathbb{Q})$ pull back to
relations in $H^{1}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Q})$ and
$H^{2}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Q})$ via the forgetful
morphisms

$\mathbb{R}\overline{\mathcal{M}}_{k,0}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{4,0}\qquad\hbox{and}\qquad\mathbb{R}\overline{\mathcal{M}}_{k,0}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{5,0}$

dropping all but four (resp. five) of the marked points. By [3, Corollary
6.9], certain natural subvarieties of $\mathbb{R}\overline{\mathcal{M}}_{k,0}$
form a linear basis for
$H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Q})$. By [9, Corollary
3.8], $H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Z})$ and
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{k,0};\mathbb{Z})$ have only
2-torsion; it is described in [3, Section 5].

Similarly to the complex case, there are node-identifying immersions

$\iota_{k\ell
i}^{\mathbb{R}}\\!:\mathbb{R}\overline{\mathcal{M}}_{k+1,0}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{\ell+1,0}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{k+\ell,0}.$

They again determine an operad structure on the sequence

$\mathcal{O}_{\mathbb{R}\mathbb{R}}(k)\equiv\begin{cases}\mathbb{Q},&\hbox{if}~{}k\\!=\\!1;\\\
H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k+1,0};\mathbb{Q}),&\hbox{if}~{}k\\!\geq\\!2;\end{cases}$

of graded vector spaces. By [3, Theorem 2.14], this operad is the operad of
unital 2-Gerstenhaber algebras.

The proof of [9, Corollary 3.8] is based on presenting
$\mathbb{R}\overline{\mathcal{M}}_{k,0}$ as a sequence of real blowups of
$(\mathbb{R}\mathbb{P}^{1})^{k-3}$ and shows that
$H_{*}(X_{\varrho};\mathbb{Z})$ has only 2-torsion for every intermediate
blowup $X_{\varrho}$. Theorems 2.9 and 2.12 and Corollary 6.9 in [3] are
proved in parallel, by combining the aforementioned topological results of [7]
and [8] with algebraic considerations. The same results should be obtainable
by presenting $\mathbb{R}\overline{\mathcal{M}}_{k+1,0}$ as a sequence of real
blowups of
$\mathbb{R}\overline{\mathcal{M}}_{k,0}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{4,0}$
as in [2] and relating the (co)homologies of the intermediate blowups as in
[1, Section 3.2].

Real case with conjugate points only,
$\mathbb{R}\overline{\mathcal{M}}_{0,\ell}$. This orientable compact smooth
manifold contains compact real hypersurfaces $\mathbb{R}E_{J,K}$, with
$J\\!\sqcup\\!K\\!=\\![\ell]$, and $\mathbb{R}H_{J,K}$, with
$J\\!\sqcup\\!K\\!=\\![\ell]$ and $J,K\\!\neq\\!\emptyset$, whose generic
elements are curves with two irreducible components that are interchanged by
the involution in the first case and are preserved by it in the second case.
This manifold also contains compact submanifolds $\mathbb{R}D_{I;J,K}$, with
$I\\!\sqcup\\!J\\!\sqcup\\!K\\!=\\![\ell]$, $I\\!\neq\\!\emptyset$, and
$|J|\\!+\\!|K|\\!\geq\\!2$, of (real) codimension 2 whose generic elements are
curves with three components two of which are interchanged by the involution;
see Figure 2 for some examples. The submanifolds $\mathbb{R}E_{J,K}$ and
$\mathbb{R}D_{I;J,K}$ are orientable, while $\mathbb{R}H_{J,K}$ is not if
$k\\!\geq\\!3$.

By [1, Theorem 2.2],

  1. ($\mathbb{R}\mathbb{C}1$)

the submanifolds $\mathbb{R}E_{J,K}$ and $\mathbb{R}D_{I;J,K}$ generate
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Q})$ as an algebra,

  2. ($\mathbb{R}\mathbb{C}2$)

subject only to some expected relations.

The relations arise from

  1. ($\mathbb{R}\mathbb{C}\ref{RCrel_it}a$)

certain submanifolds
$\mathbb{R}E_{J,K},\mathbb{R}D_{I;J,K}\\!\subset\\!\mathbb{R}\overline{\mathcal{M}}_{0,\ell}$
being disjoint,

  2. ($\mathbb{R}\mathbb{C}\ref{RCrel_it}b$)

all real hypersurfaces $\mathbb{R}E_{J,K}$ in
$\mathbb{R}\overline{\mathcal{M}}_{0,2}\\!\approx\\!\mathbb{R}\mathbb{P}^{1}$
being points and thus equivalent
in$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{0,2};\mathbb{Q})$, as indicated by
the left identity in Figure 2, and

  3. ($\mathbb{R}\mathbb{C}\ref{RCrel_it}c$)

the relation in $H_{1}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Q})$, or
equivalently in $H^{2}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Q})$, of
Remark 3.4 in [4], as indicated by the right identity and the bottom diagram
in Figure 2.

The above relations in
$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{0,2};\mathbb{Q})$ and
$H^{2}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Q})$ pull back to
relations in $H^{1}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Q})$ and
$H^{2}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Q})$ via the forgetful
morphisms

$\mathbb{R}\overline{\mathcal{M}}_{0,\ell}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{0,2}\qquad\hbox{and}\qquad\mathbb{R}\overline{\mathcal{M}}_{0,\ell}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{0,3}$

dropping all but two (resp. three) conjugate pairs of marked points.

$1^{-}$$2^{+}$$1^{+}$$2^{-}$$\sigma$$=~{}-$$\mathbb{R}E_{\\{1\\},\\{2\\}}$$1^{-}$$2^{-}$$1^{+}$$2^{+}$$\sigma$$\mathbb{R}E_{\\{1,2\\},\emptyset}$$1^{+}$$2^{+}$$3^{+}$$3^{-}$$1^{-}$$2^{-}$$\sigma$$-$$\mathbb{R}D_{\\{3\\};\\{1,2\\},\emptyset}$$1^{+}$$3^{+}$$2^{+}$$2^{-}$$1^{-}$$3^{-}$$\sigma$$=$$\mathbb{R}D_{\\{2\\};\\{1,3\\},\emptyset}$$2^{+}$$3^{-}$$1^{+}$$1^{-}$$2^{-}$$3^{+}$$\sigma$$\mathbb{R}D_{\\{1\\};\\{2,3\\},\emptyset}$$3^{-}$$3^{+}$$2^{-}$$2^{+}$$1^{+}$$1^{-}$$\emptyset$$3$$2$$23$
Figure 2: The first line represents an equivalence of two points in
$H^{1}(\mathbb{R}\overline{\mathcal{M}}_{0,2};\mathbb{Q})$ and a relation
between three loops in
$H^{2}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Q})$. The bottom diagram
represents the intersection pattern of the six codimension 2 submanifolds
$\mathbb{R}D_{I;J,K}\\!\approx\\!S^{1}$ with the four real hypersurfaces
$\mathbb{R}E_{J^{\prime},K^{\prime}}\\!\approx\\!S^{2}$ in
$\mathbb{R}\overline{\mathcal{M}}_{0,3}$. The former are labeled by the unique
element of $I\\!\subset\\![3]$ and the sign of $(-1)^{|J|}\\!=\\!(-1)^{|K|}$;
the latter are labeled by the subset $J^{\prime},K^{\prime}\\!\subset\\![3]$
not containing 1.

The proof of ($\mathbb{R}\mathbb{C}\ref{RCgen_it}$) in [1] is based on the
presentation of $\mathbb{R}\overline{\mathcal{M}}_{0,\ell+1}$ in [2] as a
sequence of blowups of
$\mathbb{R}\overline{\mathcal{M}}_{0,\ell}\\!\times\\!\overline{\mathcal{M}}_{4}$
of three different types and describes generators for
$H^{*}(X_{\varrho};\mathbb{Q})$ for every intermediate blowup $X_{\varrho}$,
assuming ($\mathbb{R}\mathbb{C}\ref{RCgen_it}$) for
$\mathbb{R}\overline{\mathcal{M}}_{0,\ell}$. This approach can also be used to
describe generators for
$H_{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Z})$ and implies that
this module has only 2-torsion. The proof of
($\mathbb{R}\mathbb{C}\ref{RCrel_it}$) in [1] relates
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell+1};\mathbb{Q})$ to
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell^{\prime}};\mathbb{Q})$ with
$\ell^{\prime}\\!\leq\\!\ell$ and
$H^{*}(\overline{\mathcal{M}}_{\ell^{\prime}};\mathbb{Q})$ with
$\ell^{\prime}\\!\leq\\!\ell\\!+\\!1$ by pairing certain cohomology classes
with the submanifolds $\mathbb{R}E_{J,K}$ and $\mathbb{R}D_{I;J,K}$ to show
the algebraic independence of the former.

Let $R$ be a commutative ring with unity 1 and

$\mathcal{O}_{\mathbb{R}\mathbb{C}}(k)=H_{*}\big{(}\mathbb{R}\overline{\mathcal{M}}_{0,k+1};R\big{)}\qquad\forall~{}k\\!\in\\!\mathbb{Z}^{+}.$

For integers $k\\!\geq\\!1$, $\ell\\!\geq\\!2$, and $i\\!\in\\![k]$, let

$\iota_{k\ell
i}^{+}\\!:\mathbb{R}\overline{\mathcal{M}}_{0,k+1}\\!\times\\!\overline{\mathcal{M}}_{\ell+1}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{0,k+\ell}$

be the immersion identifying the first marked point in the $i$-th conjugate
pair of a marked curve
$[\mathcal{C}_{1}]\\!\in\\!\mathbb{R}\overline{\mathcal{M}}_{0,k+1}$ with the
last marked point of a marked curve
$[\mathcal{C}_{2}]\\!\in\\!\overline{\mathcal{M}}_{\ell+1}$ and the second
marked point in this pair with the last marked point of the conjugate marked
curve $[\overline{\mathcal{C}_{2}}]$, keeping the indices of the first
$i\\!-\\!1$ conjugate pairs of marked points of $[\mathcal{C}_{1}]$ the same,
increasing the indices of the last $k\\!+\\!1\\!-\\!i$ conjugate pairs of
marked points of $[\mathcal{C}_{1}]$ by $\ell\\!-\\!1$, and turning the first
$\ell$ marked points of $[\mathcal{C}_{2}]$ into the first marked of the
conjugate pairs with the indices increased by $i\\!-\\!1$; see Figure 3 for an
example. This induces homomorphisms

$\iota_{k\ell
i*}^{+}\\!:\mathcal{O}_{\mathbb{R}\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell)\\!\approx\\!H_{*}\big{(}\mathbb{R}\overline{\mathcal{M}}_{0,k+1}\\!\times\\!\overline{\mathcal{M}}_{\ell+1};R\big{)}\longrightarrow\mathcal{O}_{\mathbb{R}\mathbb{C}}(k\\!+\\!\ell\\!-\\!1).$

For $k\\!\in\\!\mathbb{Z}^{+}$ and $i\\!\in\\![k]$, define

$\iota_{k1i*}^{+}\\!:\mathcal{O}_{\mathbb{R}\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(1)\longrightarrow\mathcal{O}_{\mathbb{R}\mathbb{C}}(k),\qquad\iota_{k1i*}^{+}(\eta,r)=r\eta.$

For $k,\ell_{1},\ldots,\ell_{k}\\!\in\\!\mathbb{Z}^{+}$, define

$\displaystyle\circ\\!:\mathcal{O}_{\mathbb{R}\mathbb{C}}(k)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell_{1})\\!\times\\!\ldots\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell_{k})\longrightarrow\mathcal{O}_{\mathbb{R}\mathbb{C}}\big{(}\ell_{1}\\!+\\!\ldots\\!+\\!\ell_{k}\big{)},$
$\displaystyle\eta\\!\circ\\!(\eta_{1},\ldots,\eta_{k})=\iota_{(1+\ell_{2}+\ldots+\ell_{k})\ell_{1}1*}^{+}\big{(}\ldots\iota_{(k-1+\ell_{k})\ell_{k-1}(k-1)*}^{+}\big{(}\iota_{k\ell_{k}k*}^{+}(\eta,\eta_{k}),\eta_{k-1}\big{)},\ldots,\eta_{1}\big{)}.$

This determines a right module structure on
$(\mathcal{O}_{\mathbb{R}\mathbb{C}}(k))_{k\in\mathbb{Z}^{+}}$ over the operad
$(\\!(\mathcal{O}_{\mathbb{C}}(\ell)\\!)_{\ell\in\mathbb{Z}^{+}},\circ)$.

$3^{+}$$2^{+}$$1^{+}$$3^{-}$$2^{-}$$1^{-}$$\times$$4$$3$$2$$1$$5^{+}$$1^{+}$$5^{-}$$1^{-}$$4^{+}$$3^{+}$$2^{+}$$4^{-}$$3^{-}$$2^{-}$$\sigma$$\sigma$
Figure 3: The action of the immersion $\iota_{232}^{+}$ on a typical element
in its domain; each double-headed arrow labeled $\sigma$ indicates the
involution on the corresponding real curve.

##  2 Open Problems

Real case with conjugate points only,
$\mathbb{R}\overline{\mathcal{M}}_{0,\ell}$. The approach of [2, 1] to
($\mathbb{R}\mathbb{C}\ref{RCgen_it}$) on page 1 implies that
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Z})$ is linearly
generated by orientable intersections of the submanifolds $\mathbb{R}E_{J,K}$,
$\mathbb{R}H_{J,K}$, and $\mathbb{R}D_{I;J,K}$ and contains only 2-torsion.
For example, the torsion of
$H_{1}(\mathbb{R}\overline{\mathcal{M}}_{0,3};\mathbb{Z})$ is generated by
$\mathbb{R}H_{\\{1,2\\},\\{3\\}}\\!\cap\\!\mathbb{R}H_{\\{1\\},\\{2,3\\}}$, as
shown in [4, Section 3]. However, it still remains to determine the amount of
2-torsion and the ring structure of
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Z})$ for
$\ell\\!\geq\\!4$. The algebraic generators for the torsion of
$H_{*}(\mathbb{R}\overline{\mathcal{M}}_{0,\ell};\mathbb{Z})$ should
correspond to trees, with the univalent and bivalent vertices labeled by
nonempty subsets of $[\ell]$ forming a partition of $[\ell]$. A more algebraic
question is to describe the structure of the right module structure
$(\mathcal{O}_{\mathbb{R}\mathbb{C}}(k))_{k\in\mathbb{Z}^{+}}$ over the operad
$(\\!(\mathcal{O}_{\mathbb{C}}(\ell)\\!)_{\ell\in\mathbb{Z}^{+}},\circ)$
explicitly, in the spirit of [5] and [3, Theorem 2.14].

The general real case, $\mathbb{R}\overline{\mathcal{M}}_{k,\ell}$ with
$k,\ell\\!\in\\!\mathbb{Z}^{+}$. The approach of [2, 1] to
($\mathbb{R}\mathbb{C}\ref{RCgen_it}$) also implies that
$H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k,\ell};\mathbb{Z})$ with
$k\\!\geq\\!1$ is linearly generated by orientable intersections of the
submanifolds $\mathbb{R}H_{J,K}$ and $\mathbb{R}D_{I;J,K}$ and contains only
2-torsion. The former are now indexed by pairs
$J\\!\equiv\\!(J_{\mathbb{R}},J_{\mathbb{C}})$ and
$K\\!\equiv\\!(K_{\mathbb{R}},K_{\mathbb{C}})$ with

$[k]=J_{\mathbb{R}}\\!\sqcup\\!K_{\mathbb{R}},\qquad[\ell]=J_{\mathbb{C}}\\!\sqcup\\!K_{\mathbb{C}},\qquad\hbox{and}\quad\big{|}J_{\mathbb{R}}\big{|}\\!+\\!2\big{|}J_{\mathbb{C}}\big{|},\big{|}K_{\mathbb{R}}\big{|}\\!+\\!2\big{|}K_{\mathbb{C}}\big{|}\geq
2\,,$

and the latter by the partitions $[\ell]\\!=\\!I\\!\sqcup\\!J\\!\sqcup\\!K$
with $|J|\\!+\\!|K|\\!\geq\\!2$. Similarly,
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{k,\ell};\mathbb{Z})$ is generated by
some linear combinations of these intersections. In fact, the most delicate
type of blowups appearing in [2, 1] no longer enters into consideration.
However, it remains to determine the amount of 2-torsion in
$H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k,\ell};\mathbb{Z})$ and the ring
structure even of
$H^{*}(\mathbb{R}\overline{\mathcal{M}}_{k,\ell};\mathbb{Q})$.

For $k\\!\in\\!\mathbb{Z}^{+}$ and $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$, let

$\mathcal{O}_{\mathbb{R}}(k,\ell)=\begin{cases}\mathbb{Q},&\hbox{if}~{}k\\!+\\!\ell\\!=\\!1;\\\
H_{*}(\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell};\mathbb{Q}),&\hbox{if}~{}k\\!+\\!\ell\\!\geq\\!2;\end{cases}\qquad\mathcal{O}_{\mathbb{R}}(k)=\bigoplus_{\ell=0}^{\infty}\mathcal{O}_{\mathbb{R}}(k,\ell).$

Similarly to the previous cases, there are node-identifying immersions

$\begin{split}\iota_{k\ell
k^{\prime}\ell^{\prime}i}^{\mathbb{R}}\\!:\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{k^{\prime}+1,\ell^{\prime}}&\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{k+k^{\prime},\ell+\ell^{\prime}}\qquad\hbox{and}\\\
\iota_{k\ell\ell^{\prime}i}^{+}\\!:\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell}\\!\times\\!\overline{\mathcal{M}}_{\ell^{\prime}+1}&\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell+\ell^{\prime}-1},\end{split}$

with $k\\!+\\!\ell\\!\geq\\!2$ in both cases, $i\\!\in\\![k\\!+\\!1]$ and
$k^{\prime}\\!+\\!\ell^{\prime}\\!\geq\\!2$ in the first case, and
$i\\!\in\\![\ell]$ and $\ell^{\prime}\\!\geq\\!2$ in the last case. They
induce homomorphisms

$\begin{split}\iota_{k\ell
k^{\prime}\ell^{\prime}i*}^{\mathbb{R}}\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{R}}(k^{\prime},\ell^{\prime})\\!\approx\\!H_{*}\big{(}\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{k^{\prime}+1,\ell^{\prime}};\mathbb{Q}\big{)}&\longrightarrow\mathcal{O}_{\mathbb{R}}\big{(}k\\!+\\!k^{\prime}\\!-\\!1,\ell\\!+\\!\ell^{\prime}\big{)}\quad\hbox{and}\\\
\iota_{k\ell\ell^{\prime}i*}^{+}\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{C}}(\ell^{\prime})\\!\approx\\!H_{*}\big{(}\mathbb{R}\overline{\mathcal{M}}_{k+1,\ell}\\!\times\\!\overline{\mathcal{M}}_{\ell^{\prime}+1};\mathbb{Q}\big{)}&\longrightarrow\mathcal{O}_{\mathbb{R}}\big{(}k,\ell\\!+\\!\ell^{\prime}\\!-\\!1\big{)}.\end{split}$

We define the homomorphisms

$\displaystyle\iota_{k\ell
10i*}^{\mathbb{R}}\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{R}}(1,0)\longrightarrow\mathcal{O}_{\mathbb{C}}(k,\ell),\qquad\iota_{10k^{\prime}\ell^{\prime}1*}^{\mathbb{R}}\\!:\mathcal{O}_{\mathbb{R}}(1,0)\\!\times\\!\mathcal{O}_{\mathbb{R}}(k^{\prime},\ell^{\prime})\longrightarrow\mathcal{O}_{\mathbb{C}}(k^{\prime},\ell^{\prime}),$
$\displaystyle\hbox{and}\qquad\iota_{k\ell
1i*}^{+}\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{C}}(1)\longrightarrow\mathcal{O}_{\mathbb{C}}(k,\ell)$

similarly to (1). These bilinear maps induce multilinear maps

$\begin{split}\circ\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{R}}(m_{1},\ell_{1}^{\prime})\\!\times\\!\ldots\\!\times\\!\mathcal{O}_{\mathbb{R}}(m_{k},\ell_{k}^{\prime})&\longrightarrow\mathcal{O}_{\mathbb{R}}\big{(}m_{1}\\!+\\!\ldots\\!+\\!m_{k},\ell\\!+\\!\ell_{1}^{\prime}\\!+\\!\ldots\\!+\\!\ell_{k}^{\prime}\big{)}\qquad\hbox{and}\\\
\circ\\!:\mathcal{O}_{\mathbb{R}}(k,\ell)\\!\times\\!\mathcal{O}_{\mathbb{C}}(m_{1})\\!\times\\!\ldots\\!\times\\!\mathcal{O}_{\mathbb{C}}(m_{\ell})&\longrightarrow\mathcal{O}_{\mathbb{R}}\big{(}k,m_{1}\\!+\\!\ldots\\!+\\!m_{\ell}\big{)}\end{split}$

similarly to (2). The last two maps determine an operad
$(\mathcal{O}_{\mathbb{R}}(k))_{k\in\mathbb{Z}^{+}}$ of graded vector spaces
and a right module structure on
$(\mathcal{O}_{\mathbb{R}}(k,\ell)\\!)_{\ell\in\mathbb{Z}^{+}}$ over the
operad
$(\\!(\mathcal{O}_{\mathbb{C}}(\ell)\\!)_{\ell\in\mathbb{Z}^{+}},\circ)$ for
each $k\\!\in\\!\mathbb{Z}^{+}$.

As suggested by A. Voronov, the above structures can be combined into an
object resembling a bicolored operad; see [6, Section 1.1]. For
$n\\!\in\\!\mathbb{Z}^{+}$, let

$\mathcal{O}_{\mathbb{R}}^{\prime}(n)=\mathcal{O}_{\mathbb{C}}(n)\sqcup\bigsqcup_{\begin{subarray}{c}k+\ell=n\\\
k\geq 0,\ell\geq
1\end{subarray}}\\!\\!\\!\\!\\!\mathcal{O}_{\mathbb{R}}(k\\!+\\!1,\ell\\!-\\!1)\,.$

For each fixed $n\\!\in\\!\mathbb{Z}^{+}$, auxiliary grading is provided by
$k\\!\in\\!\mathbb{Z}^{\geq 0}$ above, with the elements of
$\mathcal{O}_{\mathbb{C}}(n)$ being of the auxiliary degree 0. For
$x\\!\in\\!\mathcal{O}_{\mathbb{R}}^{\prime}(n)$, let $|x|\\!=\\!n$ and

$|x|_{+}=\begin{cases}n,&\hbox{if}~{}x\\!\in\\!\mathcal{O}_{\mathbb{C}}(n);\\\
\ell\\!-\\!1,&\hbox{if}~{}x\\!\in\\!\mathcal{O}_{\mathbb{R}}(k\\!+\\!1,\ell\\!-\\!1).\end{cases}$

We take the set of colors to be $\mathcal{C}\\!\equiv\\!\\{+,\mathbb{R}\\}$
and define

$\textnormal{out}\\!:\mathcal{O}_{\mathbb{R}}^{\prime}(n)\longrightarrow\mathcal{C},\qquad\textnormal{out}(x)=\begin{cases}+,&\hbox{if}~{}x\\!\in\\!\mathcal{O}_{\mathbb{C}}(n);\\\
\mathbb{R},&\hbox{if}~{}x\\!\in\\!\mathcal{O}_{\mathbb{R}}(k\\!+\\!1,\ell\\!-\\!1).\end{cases}$

For $i\\!\in\\![n]$, define

$\textnormal{in}_{i}\\!:\mathcal{O}_{\mathbb{R}}^{\prime}(n)\longrightarrow\mathcal{C},\qquad\textnormal{in}_{i}(x)=\begin{cases}+,&\hbox{if}~{}i\\!\leq\\!|x|_{+};\\\
\mathbb{R},&\hbox{if}~{}i\\!>\\!|x|_{+}.\end{cases}$

Let $1_{+}\\!=\\!1\\!\in\\!\mathcal{O}_{\mathbb{C}}(1)$ and
$1_{\mathbb{R}}\\!=\\!1\\!\in\\!\mathcal{O}_{\mathbb{R}}(1,0)$. The above
auxiliary grading is preserved by the maps

$\circ_{i}\\!:\big{\\{}(x,y)\\!\in\\!\mathcal{O}_{\mathbb{R}}^{\prime}(m)\\!\times\\!\mathcal{O}_{\mathbb{R}}^{\prime}(n)\\!:\textnormal{in}_{i}(x)\\!=\\!\textnormal{out}(y)\\!\big{\\}}\longrightarrow\mathcal{O}_{\mathbb{R}}^{\prime}(m\\!+\\!n\\!-\\!i),\quad
i\\!\in\\![m],$

induced by the homomorphisms $\iota_{k\ell
k^{\prime}\ell^{\prime}i*}^{\mathbb{R}}$ and
$\iota_{k\ell\ell^{\prime}i*}^{+}$ defined in the previous paragraph. These
maps satisfy a modification of [6, (1.1.4a)],

$x\\!\circ_{i}\\!(y\\!\circ_{j}\\!z)=\begin{cases}(x\\!\circ_{i}\\!y)\\!\circ_{j+i-1}\\!z,&\hbox{if}~{}\textnormal{out}(y)\\!=\\!\textnormal{out}(z);\\\
(x\\!\circ_{i}\\!y)\\!\circ_{j+|x|_{+}}\\!z,&\hbox{if}~{}\textnormal{out}(y)\\!\neq\\!\textnormal{out}(z);\end{cases}\quad\hbox{if}~{}1\leq
i\leq|x|,\,1\leq j\leq|y|.$ (3)

These maps also satisfy a modification of [6, (1.1.4b)],

$(x\\!\circ_{i}\\!y)\\!\circ_{j+|y|-1}\\!z=\begin{cases}(x\\!\circ_{j}\\!z)\\!\circ_{i}\\!y,&\hbox{if}~{}\textnormal{out}(y)\\!=\\!+;\\\
(x\\!\circ_{j}\\!z)\\!\circ_{i+|z|_{+}}\\!y,&\hbox{if}~{}\textnormal{out}(y)\\!=\\!\mathbb{R};\end{cases}\quad\hbox{if}~{}1\leq
i<j\leq|x|.$ (4)

The identities (3) and (4) hold whenever both sides are defined. They agree
with [6, (1.1.4a)] and [6, (1.1.4b)] whenever
$\textnormal{out}(y)\\!=\\!\textnormal{out}(x)$ and
$\textnormal{out}(y)\\!=\\!\mathbb{R}$, respectively. The identity [6,
(1.1.4c)] holds without any modifications.

A natural question is to describe the above algebraic structures explicitly
and perhaps to find a more succinct way of presenting them.

Acknowledgment. The author would like to thank A. Voronov for discussions on
operads.

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794
<EMAIL_ADDRESS>

## References

  * [1] X. Chen, P. Georgieva, and A. Zinger, The cohomology ring of the Deligne-Mumford space of real rational curves with conjugate marked points, math/2305.08798
  * [2] X. Chen and A. Zinger, Blowdowns of the Deligne-Mumford spaces of real rational curves, math/2305.08811
  * [3] P. Etingof, A. Henriques, J. Kamnitzer, and E. Rains, The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points, Ann. of Math. 171 (2010), no. 2, 731-–777
  * [4] P. Georgieva and A. Zinger, Enumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a WDVV relation for real Gromov-Witten invariants, Duke Math. J. 166 (2017), no. 17, 3291-–3347
  * [5] E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, in The Moduli Space of Curves, Progr. Math. 129 (1995), Birkhäuser, 199–-230
  * [6] S. Giraudo, Colored operads, series on colored operads, and combinatorial generating systems, Discrete Math. 342 (2019), no. 6, 1624–-1657
  * [7] S. Keel, Intersection theory of moduli spaces of stable $n$-pointed curves of genus zero, Trans. AMS 330 (1992), no. 2, 545–574
  * [8] V. Krasnov, Real algebraic maximal varieties, Mat. Zametki 73 (2003), 853–-860
  * [9] E. Rains, The homology of real subspace arrangements, J. Topol. 3 (2010), no. 4, 786-–818


# Disaster Tweets Classification using BERT-Based Language Model

Anh Duc Le  International School
Vietnam National University, Hanoi
Building C, HACINCO Student Village, 79 Nguy Nhu Kon Tum, Thanh Xuan, Hanoi,
Vietnam
<EMAIL_ADDRESS>
###### Abstract

Social networking services have became an important communication channel in
time of emergency. The aim of this study is to create a machine learning
language model that is able to investigate if a person or area was in danger
or not. The ubiquitousness of smartphones enables people to announce an
emergency they’re observing in real-time. Because of this, more agencies are
interested in programatically monitoring Twitter (i.e. disaster relief
organizations and news agencies). Design a language model that is able to
understand and acknowledge when a disaster is happening based on the social
network posts will become more and more necessary over time.

###### Index Terms:

Applied NLP, Text Classification, Disaster Detection and Fine-tuning

##  I Introduction

###  I-A Overview

The exponential growth of the Internet, social media, and online platforms has
enabled people all over the world to communicate instantly. Every second, on
average, around 6,000 tweets are tweeted on Twitter, which corresponds to over
350,000 tweets sent per minute, 500 million tweets per day and around 200
billion tweets per year [1]. This communication channel has been a concern for
agencies since new information is constantly being updated. By combining the
rapid update pace from social network and the commodities development of
Machine Learning technologies, specifically in Natural Language Processing
(NLP), determining when an individual is in danger using posts from Twitter is
a binary classification task. Therefore, building a machine learning model to
solve this problem by using input text, and provide the result in binary form
(Disaster or Not Disaster) automatically can be the effective method to
acknowledge and take action to the society.

###  I-B Related works

The primary objective of this research is to identify the people who in need
of assistance based on their social network posts. Recent developments in the
field of NLP have taken effect for this specific type of problem with the
followings common approaches:

####  I-B1 Traditional approach

There have been several works proposed for text classification on this labeled
dataset with the well-established Logistic Regression (LR) [2], Support Vector
Machine (SVM) [3], Naive Bayes (NB) [4], Gradient Boosting (XGB) [5] and other
fundamental machine learning models. However, these methods require manual
feature engineering process to encode text sequences in vector form, therefore
can be fed into classifiers, which might suffer from the problem of typing
error and generates noisy words to the corpus.

####  I-B2 Deep learning approach

Figure 1: The Transformer - model architecture.

For this approach, researchers have repeatedly shown the value of deep
learning for NLP over time. There have been recent developments in the field
of NLP, Machine Translation and most of the State Of The Art (SOTA) results
have been achieved using Attention models. Topmost models on the GLUE
Benchmark Leader board use self-attention models Transformers.

The Transformer neural network architecture proposed by Vaswani et al. [6]
marked one of the major breakthroughs of the decade in the NLP field. The
multi-head self-attention layer in Transformer aligns words in a sequence with
other words in the sequence, thereby calculating a representation of the
sequence. It is not only more effective in representation, but also more
computationally efficient compared to convolution and recursive operations.

##  II Solution methods

###  II-A Data Preprocessing Methods

There are some mechanism we will manually attempt to our experiment:

####  II-A1 Count Vector

is a Frequency-based Embedding, each input document will be one-hot encoded to
an $N-dimension$ vector containing count for each word’s occurrence, where $N$
is the size of vocabulary.

####  II-A2 Term Frequency – Inverse Document Frequency (TF-IDF)

is a Frequency-based Embedding, it uses weighting factor to reflect how
important a word is to a document in a collection or corpus.

####  II-A3 Continuous Bag-of-Word (CBoW)

is a Prediction-based Embedding, it tends to predict the probability of a word
given a context. A context may be a single word or a group of words.

####  II-A4 Skip-gram Vector

is a Prediction-based Embedding, it follows the same topology as of CBOW. It
just flips CBOW’s architecture on its head. The aim of skip-gram is to predict
the context given a word.

Recent studies are also paying great attention to deep learning approaches. It
not only boosts performance considerably, but also requires less effort on
feature engineering. The input of a deep learning model can simply be an one-
hot encoding of text sequences, then meaningful features are learned by
Convolutional Neural Networks (CNN), Long Short-Term Memory (LSTM), or even a
combination of CNN and LSTM. However, one-hot representation suffers from
issues of high dimensionality as its length equals to the vocabulary size.
Therefore, a more convenient way is to embed the input into a low-dimensional
space. This can be character embeddings, comment embeddings or text embeddings
from a two-phase deep learning model.

###  II-B BERT-based Network for Classification

BERT (Bidirectional Encoder Representations from Transformers) [7] is a recent
document published by Google AI Language researchers. It aroused interest in
the Machine Learning community by providing state-of-the-art findings in a
wide range of NLP practices, including Question Answering (SQuAD v1.1),
Natural Language Inference (MNLI), and others. It utilizes Transformer, an
attention mechanism that learns the contextual relationship between words (or
sub-words) in a text. In its original type, Transformer supports the following
mechanisms — the encoder that reads the text input and the decoder that
outputs a mission predictor. Since the aim of BERT is to generate a language
model, only the encoder mechanism is required. We’ll be using $BERT_{LARGE}$
uncased architecture to implement our customized classification task with
$L=24$ hidden layers (i.e., Transformer blocks), a hidden size of $H=1024$,
$A=16$ attention heads and a total of 340M parameters.

###  II-C Proposed Fine-Tuning Strategies

After constructing the network shown in Fig. 2, we will be optimizing some of
the hyper-parameters to minimize the loss function and propose the best result
possible using stochastic gradient descent algorithms with momentum: Root Mean
Square Propagation (RMSProp) [9] and Adaptive Moment Estimation (Adam) [10].

Some of the hyper-parameters in our model used for tuning procedure:

  * •

Random state split

  * •

Dropout rate

  * •

Learning rate

  * •

Validation size

  * •

Epochs

  * •

Batch size

  * •

Optimizer

Figure 2: Our customized network for BERT

##  III Experiments

###  III-A Dataset

To perform experiments on disaster detection task, we use the a dataset given
by Kaggle from the Natural Language Processing with Disaster Tweets
competition [8]. The dataset contains 7,613 raw samples on the training set
and 3,263 raw samples on the test set; all were crawled from posts on Twitter,
and annotated to one of two classes: Not Disaster (0) or Disaster (1) by many
annotators. The training data consists of 7,613 items with label distribution
is showed in Table I.

TABLE I: Training data distribution | NOT DISASTER | DISASTER
---|---|---
Number of sample | 4,305 | 3,198

In general, the porpotion of data for each target is equally distributed and
the dataset contains minority of duplicated records which gives less action
needed for data augmentation. However, it contains a lot of noisy data so some
preprocessing techniques for this problem has also been concerned.

Because the dataset was crawled directly from users’ posts on social networks,
it has some notable properties. This has led to a significant challenge
because BERT or some other pre-trained language model are often trained on
normal clean data such as Wikipedia data or Book corpus.

###  III-B Data Cleansing

As previously stated, the collection of posts was taken from social network.
Hence, the quality of each sentences might be noisy for machine learning model
to acquire. There were some of the efforts has to be made to create a
immaculate form of text beforehand. In this case, we’ll be working on common
substances such as abbreviations, emojis, special characters, foreign
language, teen code, hashtags and urls, typing errors, etc …

The general cleansing steps for each Tweets is constructed as below:

  * •

Case normalization

  * •

Remove emails

  * •

Remove URLs

  * •

Remove HTML tags

  * •

Remove emojis

  * •

Replace abbreviations

  * •

Remove stopwords

  * •

Remove special characters, non-text characters

  * •

Remove repeated punctuations / words

After cleansing, we took each samples and perform word representation to
transfer the unstructured alphabetical data into numerical representation for
words (i.e. High-dimension matrix or vector) in vector space so that if the
word vectors are close to one another means that those words are highly
correlated to one other.

###  III-C Experimental Settings

We use the binary cross entropy loss function for training process, which is
calculated as follows:

$\displaystyle\mathcal{L}(\mathbf{x}^{\prime},\mathbf{x})$
$\displaystyle=-\sum_{i=1}^{C^{\prime}=2}x^{\prime}_{i}\log(x_{i})$
$\displaystyle=-x^{\prime}_{1}\log(x_{1})-(1-x^{\prime}_{1})\log(1-x_{1})$

where ${C^{\prime}=2}$ is the number of classes for binary classification in
the experiment, $\mathbf{x}^{\prime}$ is the true distribution of any
particular data point (one-hot encoded, possibly with label smoothing
applied), and $\mathbf{x}$ is the model’s predicted distribution. The loss for
the whole dataset (or batch) is the sum (or mean) of the losses of individual
data points.

##  IV Experimental Results

###  IV-A Evaluation metrics

F1-score is a common evaluation metrics for classification tasks and is
defined as the harmonic mean of $Precision$ and $Recall$.

F1 score: performance measure for classification

$F_{1}=\frac{2}{Recall^{-1}+Precision^{-1}}$ (1)

###  IV-B Final results

In order to get the best result by traditional machine learning approach, we
have been fitting the data with 4 different vectorizers each, here are top 6
models that perform best:

Model | Vectorizer | F1-Score
---|---|---
MultinomialNB | TF-IDF | 0.80465
LogisticRegression | TF-IDF | 0.80177
MultinomialNB | Count Vector | 0.79969
LogisticRegression | Count Vector | 0.79746
RandomForest | Count Vector | 0.79418
SVMClassifier | TF-IDF | 0.79288

Several text encoder has been implemented and tested with various traditional
machine learning models mentioned above for our experiment. We will use the
Mean F1-Score to evaluate performance for each encoding methods applied to
this problem with the same training parameters respectively.

Vectorizer | Mean F1-Score
---|---
TF-IDF | 0.71902
Count Vector | 0.70751
Skip-gram Vector | 0.62456
CBoW | 0.612

Both of TF-IDF and Count Vector for word representation have outperformed
other techniques in this particular task. For explicit comparision, the TF-IDF
model contains information on the more important words and the less important
ones as well, while CBoW just creates a set of vectors containing the count of
word occurrences in the document. This proves Frequency-based Embedding is
taking its advantage over Prediction-based Embedding for this type of
classification problem.

Figure 3: Confusion matrix for our fine-tuned model

For deep learning approach using BERT - a State-of-the-art Language Model for
NLP, the classification result is significantly improved. We would also want
to propose our tuned model using BERT, some hyper-parameters giving the best
result to our experiment in Fig. 3 with overall score $F_{1}=0.8867$:

  * •

Random state split $=2$

  * •

Dropout rate $=0$

  * •

Learning rate $=6\times 10^{-6}$

  * •

Validation size $=0.15$

  * •

Epochs $=3$

  * •

Batch size $=16$

  * •

Optimizer $=Adam$

##  V Conclusion

In this paper, we have explored and proposed our pipeline to solve the Natural
Language Processing task by using traditional machine learning models as well
as pre-trained universal language model. By intensively conducting experiments
using BERT, we have demonstrated that BERT and our fine-tuning strategy is
highly effective in text classification tasks. We have also made comparision
about the effectiveness of some text encoders. For future work, we would like
to explore different classification head architectures. Instead of only using
BERT, we will experiment with other improved version of it. For instance,
Enhanced Language Representation with Informative Entities (ERNIE) [11] or
Robustly Optimized BERT Pretraining Approach (RoBERTa) [12].

## Acknowledgment

This work was supported by Office of Research & Partnership Development from
International School - Vietnam National University for The $13^{th}$ Student
Research Conference, VNU-IS. Special thanks to PhD. Tran Duc Quynh for giving
advices and guidance for this research.

## References

  * [1] ”Twitter Usage Statistics”. Accessed on: Apr. 1, 2021. [Online]. Available: https://www.internetlivestats.com/twitter-statistics/
  * [2] Bewick, Viv & Cheek, Liz & Ball, Jonathan. (2005). Statistics review 14: Logistic regression. Critical care (London, England). 9. 112-8. 10.1186/cc3045.
  * [3] ”1.4. Support Vector Machines — scikit-learn 0.20.2 documentation”. Archived from the original on Nov. 8, 2017.
  * [4] Rish, Irina. (2001). An Empirical Study of the Naïve Bayes Classifier. IJCAI 2001 Work Empir Methods Artif Intell. 3.
  * [5] Boehmke, Brad & Greenwell, Brandon. (2019). Gradient Boosting. 10.1201/9780367816377-12.
  * [6] Vaswani, Ashish & Shazeer, Noam & Parmar, Niki & Uszkoreit, Jakob & Jones, Llion & Gomez, Aidan & Kaiser, Lukasz & Polosukhin, Illia. (2017). Attention Is All You Need.
  * [7] Devlin, Jacob & Chang, Ming-Wei & Lee, Kenton & Toutanova, Kristina. (2018). BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding.
  * [8] ”Natural Language Processing with Disaster Tweets ”, Kaggle. Accessed on: Jan. 11, 2021. [Online]. Available: https://www.kaggle.com/c/nlp-getting-started
  * [9] G. Hinton, ”Overview of mini-batch gradient descent - Neural Networks for Machine Learning”, University of Toronto Computer Science. Accessed on: Jan. 17, 2021. [Online]. Available: https://www.kaggle.com/c/nlp-getting-started
  * [10] Kingma, Diederik & Ba, Jimmy. (2014). Adam: A Method for Stochastic Optimization. International Conference on Learning Representations.
  * [11] Zhang, Zhengyan & Han, Xu & Liu, Zhiyuan & Jiang, Xin & Sun, Maosong & Liu, Qun. (2019). ERNIE: Enhanced Language Representation with Informative Entities. 1441-1451. 10.18653/v1/P19-1139.
  * [12] Liu, Yinhan & Ott, Myle & Goyal, Naman & Du, Jingfei & Joshi, Mandar & Chen, Danqi & Levy, Omer & Lewis, Mike & Zettlemoyer, Luke & Stoyanov, Veselin. (2019). RoBERTa: A Robustly Optimized BERT Pretraining Approach.


# Sensing directional forces in levitated optomechanics

J.M.H. Gosling  Department of Physics and Astronomy, University College
London, Gower Street, WC1E 6BT London, United Kingdom.  A. Pontin
<EMAIL_ADDRESS>Department of Physics and Astronomy, University College
London, Gower Street, WC1E 6BT London, United Kingdom.  J. H. Iacoponi
Department of Physics and Astronomy, University College London, Gower Street,
WC1E 6BT London, United Kingdom.  P.F. Barker  Department of Physics and
Astronomy, University College London, Gower Street, WC1E 6BT London, United
Kingdom.  T.S. Monteiro<EMAIL_ADDRESS>Department of Physics and
Astronomy, University College London, Gower Street, WC1E 6BT London, United
Kingdom.

###### Abstract

Optomechanical devices are being harnessed as sensors of ultraweak forces for
applications ranging from inertial sensing to the search for the elusive dark
matter; For the latter, the focus is on detection of either higher energy
single recoils or ultralight, narrowband sources; a directional signal is
expected. However, the possibility of searching for a directional broadband
signal need not be excluded; with this and other applications in mind, we
apply a stochastic signal with a well defined direction, $\Psi$, to a trapped
and cooled levitated nanosphere. We find that cross-correlation power spectra
offer a calibration-free distinctive signature of the presence of a
directional force, and its orientation quadrant, unlike normal power spectral
densities (PSDs). With calibration we are able to accurately measure the angle
$\Psi$, akin to a force compass in a plane.

Experiments on levitated nanoparticles, cooled and controlled in cavities or
by active feedback [1], are mainly aimed at two important goals. The first is
the experimental realisation of quantum coherence in systems of mesoscopic or
macroscopic size. The second is the ultra-sensitive detection of forces and
accelerations, with applications ranging from fundamental physics, like
detection of dark matter, to real world applications.

The two goals are by no means exclusive. Indeed, combining them for quantum
force sensing is itself an overarching aim. Recent progress has been
remarkable: cooling to quantum (or near) ground state of a single mode of the
motion has been achieved, either by quantum control [2, 3] or via the optical
mode of a cavity [4] using the coherent-scattering (CS) set-up, where a
nanoparticle is held by a tweezer beam along the $z$ axis and cooled by a
cavity axial mode in the $x-y$ plane. Sensing of forces at the atto or
zeptoNewton scale has already been investigated experimentally [5, 6, 7, 8, 9,
10] with levitated nanoparticles.

Here we investigate directional force sensing. For instance, if we wish to
measure the orientation of a vector force ${\bf f}$, with unknown components
$f_{x},f_{y}$ we might seek to measure $f_{x}$ and $f_{y}$ independently.
Optomechanical force-displacement sensitivity is well understood in 1D. The
question of what microscopic force one might investigate arises. Most studies
focus on resonant or static forces. Yet stochastic forces are ubiquitous.
Nanoparticles experience thermal forces ${\bf f}^{th}(t)$, such as collisions
with surrounding gas molecules undergoing Brownian motion. These represent
stochastic noise baths and are Markovian, so $\langle
f^{th}_{x,y}(t)f^{th}_{x,y}(t^{\prime})\rangle\propto\delta(t-t^{\prime})$.
Studies generally assume $\langle
f^{th}_{x}(t)f^{th}_{y}(t^{\prime})\rangle=0$, i.e. the impulses along $x,y$
(and $z$) are uncorrelated.

Figure 1:  (a) A levitated nanoparticle is exposed to stochastic forces ${\bf
f}^{th}$ such as Brownian thermal motion, uncorrelated in any plane e.g.
$\langle f^{th}_{x}(t)f^{th}_{y}(t^{\prime})\rangle=0$ (upper image). Consider
detection of a stream of stochastic impulses, too weak to be individually
detected but from fixed orientation $\Psi$. (b) By applying a directed
stochastic force, we show that with normal PSDs (solid lines) one must first
subtract the thermal component; and they are insensitive to quadrant:
$\Psi=\pm 45^{\circ}$ are indistinguishable. (Hatched areas show the PSDs
without directed noise; solid lines show experimental data with directed
noise; dashed lines use Eq.4 (for $\beta\simeq\sqrt{2}$, see text). (c) Cross-
correlated spectra, $S_{xy}(\omega)$, in contrast, offer a calibration-free
signature of a directed force and calibration-free detection of orientation
quadrant. Figure 2: (a) Simplified layout of the experimental setup (upper)
and schematic view of the trapped nanoparticle with the geometrical
configuration of the 4 electrodes exploited for injecting the directional
force (lower panel). (b) Experimental data showing the cross-correlation for
six different orientations (right panel) and a 2D map of the cross-spectral
density $S_{xy}(\omega)$ obtained from a detailed sweep of $\Psi$ (left
panel). Even without any calibration, the traces have the ‘smoking gun’
characteristic shape indicating the presence of a directed force; and the flip
in sign that discriminates between orientation quadrants. (c) Shows
theoretical simulation, with the analytical expression Eq.6, in excellent
qualitative and quantitative agreement with experiment, showing that we may
also measure arbitrary values of $\Psi$ to within under a degree. We take
$\beta\simeq\sqrt{2}$.

We propose to detect directionality via the cross-correlation spectrum
$S_{xy}(\omega)=\frac{1}{2}\left(\langle[\hat{x}]^{\dagger}\hat{y}\rangle+\langle[\hat{y}]^{\dagger}\hat{x}\rangle\right)$.
For example, we consider an incident beam of microscopic particles from a
particular direction $\Psi$ in the $x-y$ plane: even if the stochastic
impulses are individually undetectable, integrated over a macroscopic
timescale, the cumulative effect may generate a measurable correlation
spectrum $S_{xy}$. Searches for dark matter largely focus on detectable single
impacts, or weak but narrowband sources [11, 12]. But efforts have not yet
borne fruit and the possibility of detecting correlations from a weak, but
broad-spectrum source by exploiting its directionality has not been excluded.
The cross-correlations may also be useful for detection of weak gas currents
or temperature anisotropies [13].

We demonstrate directional force detection by applying an electrically
generated stochastic force at orientation $\Psi$ to a charged nanosphere, 3D
cooled by feedback control. We show the $S_{xy}$ signal provides an
unambiguous and calibration-free signature not only of the presence of the
directed force, but also of its orientation quadrant. With calibration, we
find accurate values of $\Psi$ are measurable. This is illustrated in Fig. 1.

Theoretical model: We consider a spherical nanoparticle trapped in an optical
tweezer. Its centre of mass motion corresponds to a set of harmonic
oscillators of frequencies $\omega_{u}$ with $u\equiv x,y,z$, under-damped by
collisions with background gas that also drive the particle motion with a
stochastic force $f_{\text{th}}$. The nanoparticle is driven by a feedback
force to pre-cool and stabilize its motion. In addition, we introduce a
directional force $f_{\text{dir}}$. The equations of motion can then be
written as

$\ddot{u}+\gamma_{u}\dot{u}+\omega_{u}^{2}u=(f_{\text{th},u}(t)+f_{\text{dir},u}(t)+f_{\text{fb},u}(t))/m$
(1)

where $u\equiv x,y,z$ and $\gamma$ is the gas damping [14, 15] while $m$ is
the particle mass. For a spherical particle, normally gas damping rates are
isotropic $\gamma_{u}\equiv\gamma$. The feedback force $f_{\text{fb,u}}(t)$ ,
which for the cold-damping scheme corresponds to a force proportional to the
measured velocity $v(t)$. The equations can readily be solved in the frequency
domain. In particular, the feedback force is then approximated
$f_{\text{fb},u}(\omega)/m={\text{H}}_{u}(\omega)v(\omega)\approx-i\omega\Gamma^{(u)}_{eff}x(\omega)$
hence:

$u(\omega)=\chi_{u}(\omega)(f_{\text{th},u}(\omega)+f_{\text{dir},u}(\omega)).$
(2)

Hence the transfer function FT$[{\text{H}}(t)]={\text{H}}(\omega)$ was
replaced by an effective constant damping allowing a simpler analysis, but
noting that the full stochastic numerics can simulate the experimental
transfer function. In this case we obtain the effective susceptibility
$\chi_{u}=[m(\omega_{u}^{2}-\omega^{2}+i\omega\Gamma_{u})]^{-1}$ where we take
$\Gamma_{u}=\gamma+\Gamma^{(u)}_{eff}$ . For simplicity we also neglected the
effect of the imprecision noise on the feedback scheme since it is not
relevant at the pressure the experiment is carried out at [16].

The power spectral densities (PSDs) may be obtained once the properties of the
stochastic force noises are defined. For thermal noise we have zero mean and
correlation $\langle
f_{\text{th},u}(t)f_{\text{th},u^{\prime}}(t^{\prime})\rangle=2k_{B}Tm\gamma\,\delta(t-t^{\prime})\delta_{u,u^{\prime}}$
where $u,u^{\prime}\equiv x,y,z$, with an isotropic power spectrum
$S_{\text{th},u}\equiv S_{\text{th}}$.

We restrict our analysis to the nanoparticle motion in the 2D tweezer
polarization plane identified by the $x$ and $y$ axes in Fig. 2; the $z$
motion remains largely decoupled due to the significant difference in the trap
frequencies. However all calculations and numerics are fully 3D.

Directional forces: For the directional force noises, we assume zero mean and
correlation functions $\langle
f_{\text{dir},x}(t)f_{\text{dir},x}(t^{\prime})\rangle=S_{\text{ff}}^{dir}\cos^{2}\Psi\delta(t-t^{\prime})$,
$\langle
f_{\text{dir},y}(t)f_{\text{dir},y}(t^{\prime})\rangle=S_{\text{ff}}^{dir}\sin^{2}\Psi\delta(t-t^{\prime})$,
and finally a cross correlation $\langle
f_{\text{dir},x}(t)f_{\text{dir},y}(t^{\prime})\rangle=S_{\text{ff}}^{dir}\cos\Psi\sin\Psi\delta(t-t^{\prime})$.
Here, the angle $\Psi$ represents the direction of the stochastic force. It is
convenient to quantify the relative magnitude of the directed and background
thermal forces, especially for the case where both have white noise spectra.
Thus we introduce:

$\beta^{2}=S_{\text{ff}}^{dir}/S_{\text{th}}$ (3)

The experimental PSDs are then given by:

$\displaystyle S_{xx}(\omega)$ $\displaystyle=$
$\displaystyle|\chi_{x}(\omega)|^{2}S_{\text{th}}(1+\beta^{2}\cos^{2}\Psi)$
$\displaystyle S_{yy}(\omega)$ $\displaystyle=$
$\displaystyle|\chi_{y}(\omega)|^{2}S_{\text{th}}(1+\beta^{2}\sin^{2}\Psi)$
$\displaystyle S_{xy}(\omega)$ $\displaystyle=$
$\displaystyle\text{Re}[\chi_{x}^{*}(\omega)\chi_{y}(\omega)]S_{\text{th}}\beta^{2}\cos\Psi\sin\Psi.$
(4)

One readily sees that the cross correlation can isolate the effect of the
directional force from the thermal noise since Eq. 4 does not depend on
$S_{\text{th}}$. Thus $\beta\ll 1$ corresponds to the regime where the
directional force is very weak compared with the thermal component; while for
$\beta\gg 1$, the directed force is dominant. For $\beta=\sqrt{2}$ and
$\Psi=\pi/4$ (which gives the maximal cross-correlation),
$\beta^{2}\cos^{2}\Psi=\beta^{2}\sin^{2}\Psi=1$ thus the directed contribution
to the PSDs is equal to the thermal contribution; the experiments presented in
Fig.1 and Fig.2 correspond to $\beta\simeq 1.4\sim\sqrt{2}$.

Experimental set-up: An overview of the experiment is shown in Fig. 2a). A
diode pumped solid state laser of wavelength $\lambda=1064$ nm illuminates a
single aspheric lens of nominal numerical aperture NA$=0.77$ to obtain an
optical tweezer. At its focus, we trap single silica nanoparticles with
typical radii in the range $R=60-90$ nm. A condenser aspheric lens, NA=$0.55$,
re-collimates the beam after the tweezer focus. Subsequently, the beam is
attenuated and divided in two paths by a non-polarizing beam splitter. On each
path, a split detection scheme allows one to measure the particle motion in
the tweezer polarization plane $x$-$y$ along orthogonal directions. The
particle motion is cooled in all three center of mass (CoM) directions by a
cold damping scheme [17, 3, 2, 18] exploiting the Coulomb force. Thus, the
trapped particles are charged with a number of elementary charges typically in
the range $1-50$ \- and this represents an essential requirement for the
feedback scheme.

Hence, the most effective approach is to introduce a directional stochastic
Coulomb force, ${\bf f}_{\text{elec}}$. In order for ${\bf
f}_{\text{elec}}\equiv{\bf f}_{\text{dir}}$ there are specific requirements
for the force components in the $x,y$ directions: (i) vanishing mean values
$\langle f_{\text{elec},x}\rangle=\langle f_{\text{elec},y}\rangle=0$, (ii)
correlation $\langle
f_{\text{elec},x}(t)f_{\text{elec},x}(t^{\prime})\rangle=S_{\text{ff}}^{elec}\cos^{2}\Psi\delta(t-t^{\prime})$
and $\langle
f_{\text{elec},y}(t)f_{\text{elec},y}(t^{\prime})\rangle=S_{\text{ff}}^{elec}\sin^{2}\Psi\delta(t-t^{\prime})$,
and finally a cross correlation $\langle
f_{\text{elec},x}(t)f_{\text{elec},y}(t^{\prime})\rangle=S_{\text{ff}}^{elec}\cos\Psi\sin\Psi\delta(t-t^{\prime})$
(iii) while $S_{\text{ff}}^{elec}=S_{th}\beta^{2}\equiv
2k_{B}Tm\gamma\beta^{2}$.

To implement this experimentally we use four rod electrodes laying in a plane
parallel to the tweezer polarization plane. This is shown in Fig. 2a. The
plane of the electrodes is slightly shifted from the origin which represents
the particle mean position. The symmetry of this geometrical configuration
allows one to have full control of the electric field components in the
$x$-$y$ plane by applying the same voltage signal to 3 electrodes each with
the right scaling factor while keeping the fourth grounded. This can be
intuitively understood by considering that a voltage $V_{s}$ on electrode-1,
with all other grounded, will give an electric field
$\textbf{E}_{1}=V_{s}\,(-T_{x},-T_{y},-T_{z})^{T}$ where $T_{i}$ represents
the inverse of an effective distance. For symmetry reasons, if the same
voltage is applied on electrode-2 the electric field would be
$\textbf{E}_{2}=V_{s}\,(-T_{x},T_{y},-T_{z})^{T}$. Thus, the superposition of
the two electric fields will have a vanishing component on the $y$ direction.
The same approach can be used with electrodes-1 and 4 to obtain an electric
field with vanishing $x$ component.

The last step to obtain an arbitrary electric field direction in the $x$-$y$
plane is to control the relative amplitude of the voltage signal delivered to
electrodes 1-2 and 1-4. We do this with a Wheatstone bridge which given an
input $V_{s}$ provides two outputs $V_{s1}$, $V_{s2}$ whose ratio fully
depends on a variable resistor. By tuning the bridge one can approximate
$V_{s1}\simeq V_{s}\cos\Psi$ and $V_{s2}\simeq V_{s}\sin\Psi$.

In the experiment, $V_{s}$ is a white noise signal, provided by a standard
signal generator, with a spectral density of $S_{VV}=1.4\times 10^{-8}$ V2/Hz
on a bandwidth of $BW=2.5$ MHz, after amplification with gain $G=80$, $V_{s1}$
and $V_{s2}$ drive electrode-2 and 4 respectively while electrode-1 is driven
with their sum.

Results and discussion: Sensitivity analysis for the detection of
optomechanical forces is well studied [1]. We can readily compare the relative
sensitivity of the cross-correlation spectra by considering the ratio (for
$\omega\simeq\omega_{x}$):

$\frac{S_{xy}(\omega)}{S_{xx}(\omega)}=\frac{\text{Re}[\chi_{x}(\omega)\chi_{y}^{*}(\omega)]}{|\chi_{x}(\omega)|^{2}}\frac{\beta^{2}\sin\Psi\cos\Psi}{1+\beta^{2}\cos^{2}\Psi}$
(5)

We approximate
$\frac{\text{Re}[\chi_{x}(\omega)\chi_{y}^{*}(\omega)]}{|\chi_{x}(\omega)|^{2}}\approx\frac{\Gamma_{x}^{2}}{4(\omega_{x}-\omega_{y})^{2}+\Gamma_{x}^{2}}$
near $\omega\sim\omega_{x}$. Hence, we readily see that optimal visibility of
the cross correlation spectrum, relative to the ordinary PSD, requires
$\Gamma_{x}\sim|\omega_{x}-\omega_{y}|$. As the gas damping is very weak, this
means that cooling (such as by active feedback) is essential or else the cross
correlations are strongly suppressed. A similar result is obtained at
$\omega\simeq\omega_{y}$, by comparing $S_{yy}$ and $S_{xy}$. In addition, one
may readily see that maximal values for the cross-correlation are obtained for
$\tan\Psi\simeq 1$.

Experimentally, we infer $\Gamma_{eff}^{(x)}\approx 2.1$ kHz,
$\Gamma_{eff}^{(y)}\approx 2.5$ kHz. To simulate the data in Figs.1 and 2 we
take pressure $P=4\times 10^{-3}$ mbar.

In Fig.2 we show that there is excellent agreement, for all angles $\Psi$,
between the measured cross-correlations and the simple expression,

$S_{xy}(\omega)\simeq\text{Re}[\chi_{x}(\omega)\chi_{y}^{*}(\omega)]2k_{B}Tm\gamma\beta^{2}\sin\Psi\cos\Psi$
(6)

by taking $\beta=1.4$. It is also possible to analyse the imaginary component
of the experimental correlations
$\text{Imag}(\langle[\hat{x}]^{\dagger}\hat{y}\rangle)\propto\text{Re}[\chi_{x}(\omega)\chi_{y}^{*}(\omega)]$.
However this does not have the distinctive signature, and is more easily
confused with misalignment errors.

The accuracy of the value of $\beta=1.4$ for modelling the applied electric
stochastic noise in Fig.2, may be further tested by considering the normal
PSDs. The directed force can affect the usual PSDs quantitatively. If the
directed force provides heating, the final temperature is increased by the
factor $(1+\beta^{2}\cos^{2}\Psi)$ for the $x$ mode, and
$(1+\beta^{2}\sin^{2}\Psi)$ for $y$. In the absence of the directed force, we
measured $T_{y}=215.7\pm 0.6$ mK and $T_{x}=297\pm 2$ mK. In the presence of
the directed noise, for the same pressure, the temperatures were enhanced by a
factor of $\approx 3\approx 1+\beta^{2}$ for $\Psi=\pm\pi/2$ and were measured
to be $T_{y}=641\pm 0.2$ mK and $T_{x}=936\pm 3$ mK respectively, as expected
from theory.

Figure 3:  (a) For 2D quantum cooled motion in the $x-y$ plane, using
cavities, heterodyne detection gives the most sensitivity for displacement
sensing. Using parameters similar to 2D $x-y$ quantum cooling experiments [19]
we show that provided $\beta\gg 1$ the $S_{xy}$ contribution is exposed and
isolated for $\Psi\sim 0^{o}$ and $\Psi\sim 90^{o}$. In the left panel it is
shown that the sharp asymmetry in the $y$ mode ‘flips’ at $\Psi=0$ for the
case of $\beta=10$. The $y$ mode is not heated and remains at quantum phonon
occupancy $n_{y}\simeq 1$. The change in shape yields a characteristic
signature of the presence and orientation of the directional force, even in a
quantum regime. For general orientations, the $S_{xy}$ contribution still
results in a $\Psi$-dependent heterodyne PSD, but the difference between
$\Psi=\pi/4$ and $3\pi/4$ is now only quantitative, rather than a qualitative
change in shape.

Quantum directed force and heterodyne detection: while future technical
advances may allow 2D quantum cooling in the $x-y$ plane without a cavity and
direct measurement of $S_{xy}(\omega)$, to date only cavity setups have
achieved quantum regimes in the $x-y$ plane, with the coherent scattering set-
up [4] that allows very strong light-matter coupling rates $g_{x},g_{y}$.
Unfortunately, these lead also to hybridisation of the $x-y$ modes, [20] and
the formation of dark modes [21, 22] for $g_{x}\sim
g_{y}\gtrsim|\omega_{x}-\omega_{y}|$ that suppress 2D $x-y$ cooling. Analysis
showed that 2D quantum cooling occurs in a ‘Goldilocks’ zone of coupling that
is not large enough to form dark modes, but remains large enough for effective
cooling [21]. Quantum cooling of both $x-y$ modes, in this Goldilocks zone was
shown in [19].

Heterodyne (or homodyne) detection of the output cavity light is widely used
to detect displacements in quantum regimes and may be written, in 2D:

$S_{het}(\omega)\approx|\chi_{c}(\omega)|^{2}[g_{y}^{2}S_{yy}+g_{x}^{2}S_{xx}+g_{x}g_{y}S_{xy}]+S_{imp}$
(7)

where $S_{imp}$ is the imprecision noise (ideally optical quantum shot noise)
and $\chi_{c}(\omega)$ is the optical susceptibility. It may be seen that the
heterodyne spectrum does not isolate $S_{xy}$ so its contribution is generally
masked by the stronger $S_{xx},S_{yy}$ contributions. In addition, there are
additional artificial correlations induced by back-action induced rotations of
the modes, that for strong coupling yield dark modes [21, 23]. These rotations
can be suppressed by trapping away from the cavity node [23]. To retain the
advantages of low photon heating at the node while allowing 2D cooling implies
$g_{x}\sim g_{y}\ll|\omega_{x}-\omega_{y}|$. However this suppresses cross
correlations as these require cooling widths
$\Gamma_{cool}\sim|\omega_{x}-\omega_{y}|$.

However, we may still detect the cross-correlation spectrum in certain cases.
For $g_{x}\sim g_{y}\equiv g$, and $\beta^{2}\gg 1$, thus in the case of
ultrahigh vacuum where the directed noise is dominant, then $S_{het}\propto
g^{2}(|\chi_{x}(\omega)|^{2}\cos^{2}\Psi+|\chi_{y}(\omega)|^{2}\sin^{2}\Psi+\text{Re}[\chi_{x}^{*}(\omega)\chi_{y}(\omega)]\cos\Psi\sin\Psi$.
Using parameters similar to the quantum cooling experiments in [19], we find
that the $S_{xy}$ spectrum is exposed for $\Psi\sim 0$ for
$\omega\sim\omega_{y}$ and for $\Psi\sim\pi/2$ for $\omega\sim\omega_{x}$. The
reason for this is that e.g., for $\omega\sim\omega_{y}$,
$\frac{S_{xy}(\omega\sim\omega_{y})}{S_{xx}(\omega\sim\omega_{y})}=\frac{\Gamma_{cool}}{|\omega_{x}-\omega_{y}|}\cot\Psi$.
In Fig.3, we show that even for
$\frac{\Gamma_{cool}}{|\omega_{x}-\omega_{y}|}\ll 1$, as $\cot\Psi\to\infty$
near 0, the cross-correlations are exposed.

Conclusions: the search for a suitable dark matter candidate has not yet been
successful and remains one of the major challenges of physics at present.
Searches for dark matter within optomechanics focus on detection of higher
energy single recoils [11], or ultralight, weak but narrowband sources [12].
However, weak but broadband directional sources might also be considered [24]
within optomechanics-based searches. This possibility might encompass an
ultralight source with a very short coherence time; or a source spanning a
band of multiple frequencies. Directionality is expected, due to the Earth’s
motion relative to the galaxy. This directionality significantly increases
background rejection, even for a handful of events [25].

Here we proposed and demonstrated experimentally that the steady state cross-
correlation PSD provides a distinctive and unambiguous signature of the
presence of a directed force and also of its orientation quadrant. The
experiment exploited a singular advantage of levitated nanoparticles wherein
its centre of mass motion modes have a clear geometric interpretation (motion
along $x,y,z$ axes [20, 21]), unlike, say, two breathing modes of a membrane.
The method generalises to other planes, provided the condition
$\Gamma_{i,j}\sim|\omega_{i}-\omega_{j}|$ (where $i,j\equiv x,y,z$) can be
satisfied.

In principle, by measuring relative changes in $S_{xx}$ and $S_{yy}$, one
might infer information on directionality. However as the normal PSDs cannot
discriminate between orientation quadrants (thus between $\Psi=\pm\pi/4$),
some information is lost. However, as measuring $S_{xy}$ implies measurement
of $S_{xx},S_{yy}$, no choice is required: the cross-correlation complements
current methods at no extra cost. There is also the added advantage that it is
not necessary to subtract the thermal non-directional component, as $S_{xy}$
can isolate the directional component, thus avoiding another requirement for
calibration.

Sensitivity may be enhanced by improving the cooling to frequency ratio which
is not optimal as $\Gamma_{x,y}/|\omega_{x}-\omega_{y}|\sim 1/7$; and
operating at lower pressures of $10^{-6}-10^{-8}$ mbar to achieve quantum
limited regimes. The method is also relevant to narrowband directed forces,
provided $\Gamma_{cool}\sim|\omega_{x}-\omega_{y}|$. The study opens the way
to detection of different classes of forces of microscopic origin such as tiny
gas currents, small anisotropies in the background Brownian motion, possibly
due to incoming streams of gas or temperature variations [13].

Acknowledgements: The authors would like to acknowledge helpful discussions
with Markus Rademacher, Hayden Fu, Fiona Alder and Marko Toroš. The authors
acknowledge funding from the Engineering and Physical Sciences Research
Council (EPSRC) Grant No. EP/N031105/1.

## References

  * [1] James Millen, Tania S Monteiro, Robert Pettit, and A Nick Vamivakas.  Optomechanics with levitated particles.  Reports on Progress in Physics, 83(2):026401, 2020.
  * [2] Lorenzo Magrini, Philipp Rosenzweig, Constanze Bach, Andreas Deutschmann-Olek, Sebastian G. Hofer, Sungkun Hong, Nikolai Kiesel, Andreas Kugi, and Markus Aspelmeyer.  Real-time optimal quantum control of mechanical motion at room temperature.  Nature, 595(7867):373–377, July 2021.
  * [3] Felix Tebbenjohanns, Luisa Mattana, Massimiliano Rossi, Martin Frimmer, and Lukas Novotny.  Quantum control of a nanoparticle optically levitated in cryogenic free space.  Nature, 595:378–382, 2021.
  * [4] U Delić, M Reisenbauer, K Dare, D Grass, V Vuletić, N Kiesel, and M Aspelmeyer.  Cooling of a levitated nanoparticle to the motional quantum ground state.  Science, 367(6480):892–895, 2020.
  * [5] Gambhir Ranjit, Mark Cunningham, Kirsten Casey, and Andrew A. Geraci.  Zeptonewton force sensing with nanospheres in an optical lattice.  Phys. Rev. A., 93:053801, 2016.
  * [6] Erik Hebestreit, Martin Frimmer, René Reimann, and Lukas Novotny.  Sensing static forces with free-falling nanoparticles.  Physical review letters, 121(6):063602, 2018.
  * [7] Fernando Monteiro, Gadi Afek, Daniel Carney, Gordan Krnjaic, Jiaxiang Wang, and David C. Moore.  Search for composite dark matter with optically levitated sensors.  Phys. Rev. Lett., 125:181102, Oct 2020.
  * [8] Tao Liang, Shaochong Zhu, Peitong He, Zhiming Chen, Yingying Wang, Cuihong Li, Zhenhai Fu, Xiaowen Gao, Xinfan Chen, Nan Li, Qi Zhu, and Huizhu Hu.  Yoctonewton force detection based on optically levitated oscillator.  Fundamental Research, 3(1):57–62, 2023.
  * [9] David Hempston, Jamie Vovrosh, Marko Toros, George Winstone, Muddassar Rashid, and Hendrik Ulbricht.  Force sensing with an optically levitated charged nanoparticle.  Applied Physics Letters, 111(13):133111, 09 2017.
  * [10] Chris Timberlake, Marko Toroš, David Hempston, George Winstone, Muddassar Rashid, and Hendrik Ulbricht.  Static force characterization with fano anti-resonance in levitated optomechanics.  Applied Physics Letters, 114(2):023104, 2019.
  * [11] Daniel Carney, Gordann Krnjaic, David Moore, and et. al.  Mechanical quantum sensing in the search for dark matter.  Quantum Sci. Technol., 6:024002, 2021.
  * [12] Daniel Carney, Anson Hook, Zhen Liu, J.M. Taylor, and Yue Zhao.  Ultralight dark matter detection with mechanical quantum sensors.  New J.Phys, 23:023041, 2021.
  * [13] J Millen, T. Deesuan, J. Anders, and PF Barker.  Nanoscale temperature measurements using non-equilibrium brownian dynamics of a levitated nanosphere.  Nature Nanotechnology, 9:425, 2014.
  * [14] Paul S. Epstein.  On the resistance experienced by spheres in their motion through gases.  Phys. Rev., 23:710–733, Jun 1924.
  * [15] A. Cavalleri, G. Ciani, R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P.J. Wass, and W.J. Weber.  Gas damping force noise on a macroscopic test body in an infinite gas reservoir.  Physics Letters A, 374(34):3365–3369, July 2010.
  * [16] T. W. Penny, A. Pontin, and P. F. Barker.  Performance and limits of feedback cooling methods for levitated oscillators: A direct comparison.  Phys. Rev. A, 104:023502, Aug 2021.
  * [17] M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar.  Feedback cooling of a cantilever’s fundamental mode below 5 mk.  Phys. Rev. Lett., 99:017201, Jul 2007.
  * [18] M. Kamba, H. Kiuchi, T. Yotsuya, and K. Aikawa.  Recoil-limited feedback cooling of single nanoparticles near the ground state in an optical lattice.  Phys. Rev. A, 103:L051701, May 2021.
  * [19] J. Piotrowski, D. Windey, J. Vijayan, C Gonzalez-Ballestero, A de los RÃos Sommer, N Meyer, R Quidant, O Romero-Isart, R Reimann, and Lukas Novotny.  Simultaneous ground-state cooling of two mechanical modes of a levitated nanoparticle.  Nat. Phys., 2023.
  * [20] M Toroš and TS Monteiro.  Quantum sensing and cooling in three-dimensional levitated cavity optomechanics.  Physical Review Research, 2(2):023228, 2020.
  * [21] M Toroš, U Delic, F Hales, and TS Monteiro.  Coherent-scattering two-dimensional cooling in levitated cavity optomechanics.  Physical Review Research, 3:023071, 2021.
  * [22] A. Ranfagni, P. Vezio, M. Calamai, A. Chowdhury, F. Marino, and F. Marin.  Vectorial polaritons in the quantum motion of a levitated nanosphere.  Nature Physics, 17:1120–1124, 2021.
  * [23] A Pontin, H. Fu, J.H. Iacoponi, P.F. Barker, and T S Monteiro.  Controlling mode orientations and frequencies in levitated cavity optomechanics.  Phys. Rev. Research, 5:013013, 2023.
  * [24] Gary P. Centers, John W. Blanchard, Jan Conrad, Nataniel L. Figueroa, Antoine Garcon, Alexander V. Gramolin, Derek F. Jackson Kimball, Matthew Lawson, Bart Pelssers, Joseph A. Smiga, Alexander O. Sushkov, Arne Wickenbrock, Dmitry Budker, and Andrei Derevianko.  Stochastic fluctuations of bosonic dark matter.  Nature Communications, 12(1):7321, Dec 2021.
  * [25] S Ahlen, N Afshordi, James BR Battat, J Billard, N Bozorgnia, S Burgos, T Caldwell, JM Carmona, S Cebrian, P Colas, et al.  The case for a directional dark matter detector and the status of current experimental efforts.  International Journal of Modern Physics A, 25(01):1–51, 2010.