The Quaternions and Bott Periodicity are Quantum Hamiltonian Reductions

The Quaternions and Bott Periodicity
Are Quantum Hamiltonian Reductions\star\star$\star$This paper is a
contribution to the Special Issue “Gone Fishing”. The full collection is
available at http://www.emis.de/journals/SIGMA/gone-fishing2016.html

Theo JOHNSON-FREYD

T. Johnson-Freyd

Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
<EMAIL_ADDRESS>https://perimeterinstitute.ca/personal/tjohnsonfreyd/

Received August 30, 2016, in final form December 09, 2016; Published online
December 11, 2016

We show that the Morita equivalences $\mathrm{Clif\/f}(4)\simeq{\mathbb{H}}$,
$\mathrm{Clif\/f}(7)\simeq\mathrm{Clif\/f}(-1)$, and
$\mathrm{Clif\/f}(8)\simeq{\mathbb{R}}$ arise from quantizing the Hamiltonian
reductions ${\mathbb{R}}^{0|4}//\mathrm{Spin}(3)$,
${\mathbb{R}}^{0|7}//G_{2}$, and ${\mathbb{R}}^{0|8}//\mathrm{Spin}(7)$,
respectively.

Clifford algebras; quaternions; Bott periodicity; Morita equivalence; quantum
Hamiltonian reduction; super symplectic geometry

15A66; 53D20; 16D90; 81Q60

This note provides (super) symplectic origins for the quaternion algebra
$\mathbb{H}$ and for the eight-fold “Bott periodicity” of Clifford algebras
(due originally to Cartan [3]) in terms of quantum Hamiltonian reduction.
Clifford algebras arise in symplectic supergeometry as the Weyl (aka canonical
commutation) algebras of purely-odd symplectic supermanifolds
$\mathbb{R}^{0|n}$. As we explain, Hamiltonian reductions quantize to
bimodules, which are often Morita equivalences. In particular, we will show
that the well-known Morita equivalence $\mathbb{H}\simeq\mathrm{Clif\/f}(4)$
is the quantization of the Hamiltonian reduction
$\mathbb{R}^{0|4}//\mathrm{Spin}(3)$, where $\mathrm{Spin}(3)=\mathrm{SU}(2)$
acts on $\mathbb{R}^{0|4}$ as the underlying real module of the defining
action of $\mathrm{SU}(2)$ on $\mathbb{C}^{2}$, and that the reduction
$\mathbb{R}^{0|8}//\mathrm{Spin}(7)$ coming from the spin representation
quantizes to the “Bott periodicity” Morita equivalence
$\mathrm{Clif\/f}(8)\simeq\mathbb{R}$. We also show that the Morita
equivalence $\mathrm{Clif\/f}(7)\simeq\mathrm{Clif\/f}(-1)$ arises from the
Hamiltonian reduction $\mathbb{R}^{0|7}//G_{2}$, where
$G_{2}\subseteq\mathrm{SO}(7)$ is the exceptional Lie group of automorphisms
of the octonion algebra $\mathbb{O}$.

##  1 Symplectic supermanifolds and Clifford algebras

A superalgebra is a $\mathbb{Z}/2$-graded associative algebra (meaning, in
particular, that the multiplication adds degree modulo $2$); morphisms are
grading-preserving. A supermodule is a $\mathbb{Z}/2$-graded module. If $M$ is
a left $A$-supermodule, the algebra $\operatorname{End}_{A}(M)$ of all
$A$-linear endomorphisms of $M$ is naturally a superalgebra acting on $M$ from
the right (with multiplication $fg=g\circ f$). Two superalgebras $A$ and $B$
are super Morita equivalent if there are $\mathbb{Z}/2$-graded bimodules
${}_{A}M_{B}$ and ${}_{B}N_{A}$ with grading-preserving bimodule isomorphisms
$M\otimes_{B}N\cong A$ and $N\otimes_{A}M\cong B$. We will generally suppress
the word “super”: for example, “module” and “Morita equivalence” will
henceforth always be meant in the super sense.

A superalgebra $A$ is commutative if for homogeneous elements $x$ and $y$
(with degrees $|x|$ and $|y|$), $yx=(-1)^{|x|\cdot|y|}xy$. Note in particular
that for an odd element $x$ in a commutative superalgebra, $x^{2}=-x^{2}$, and
so $x^{2}=0$. By definition, odd $n$-dimensional space $\mathbb{R}^{0|n}$ is
the “spectrum” of the commutative superalgebra
$\mathcal{O}(\mathbb{R}^{0|n})=\mathbb{R}[x_{1},\dots,x_{n}]$ where the
coordinate functions $x_{1},\dots,x_{n}$ are odd, and $\mathbb{R}[\dots]$
denotes the free commutative superalgebra on “$\dots$”. Thus
$\mathcal{O}(\mathbb{R}^{0|n})\cong\bigwedge^{\bullet}\mathbb{R}^{n}$. In
general, a supermanifold is a “space” that looks locally like
$\mathbb{R}^{m|n}=\mathbb{R}^{m}\times\mathbb{R}^{0|n}$. See [5] for details
on superalgebras and supermanifolds.

A symplectic structure on a supermanifold is an even nondegenerate closed de
Rham 2-form. De Rham forms can be defined for commutative superalgebras just
like for commutative algebras, but behave differently in one important way: if
$x$ is an odd coordinate, then $\mathrm{d}x$ is even, and so
$\mathrm{d}x\wedge\mathrm{d}x\neq 0$, and if $x$ and $y$ are both odd, then
$\mathrm{d}x\wedge\mathrm{d}y=\mathrm{d}y\wedge\mathrm{d}x$ with no sign. A
side effect of this is that symplectic structures on odd manifolds behave
somewhat like metrics on even manifolds.

Equip $\mathbb{R}^{0|n}$ with the positive-definite symplectic form
$\omega=\sum_{i}\frac{(\mathrm{d}x_{i})^{2}}{2}$. The corresponding Poisson
structure on $\mathbb{R}^{0|n}$ is given by the Poisson brackets
$\\{x_{i},x_{j}\\}=-2\delta_{ij}$. (The sign depends on an essentially-
arbitrary choice of convention for inverse matrices in superalgebra.) The
symplectic form $\omega$ on $\mathbb{R}^{0|n}$ is translation-invariant and so
admits a canonical quantization to the Weyl algebra
$\mathcal{W}(\mathbb{R}^{0|n})=\mathbb{R}\langle
x_{1},\dots,x_{n}\rangle/([x_{i},x_{j}]=\\{x_{i},x_{j}\\})$, where by
definition in a superalgebra the commutator is defined on homogeneous elements
by $[x,y]=xy-(-1)^{|x|\cdot|y|}yx$. Thus $\mathcal{W}(\mathbb{R}^{0|n})$ is
the Clifford algebra $\mathrm{Clif\/f}(n)=\mathbb{R}\langle
x_{1},\dots,x_{n}\rangle/(x_{i}^{2}=-1,x_{i}x_{j}=-x_{j}x_{i}\text{ for }i\neq
j)$ with its usual $\mathbb{Z}/2$-grading in which all generators $x_{i}$ are
odd. The Weyl algebra of $\mathbb{R}^{0|n}$ equipped with symplectic form
$-\omega$ is $\mathrm{Clif\/f}(-n)=\mathbb{R}\langle
x_{1},\dots,x_{n}\rangle/(x_{i}^{2}=1,x_{i}x_{j}=-x_{j}x_{i})$.

##  2 Quantum Hamiltonian reduction

A moment map for the action of a super Lie group $G$ on a symplectic
supermanifold $M$ is a map $\mu\colon
M\to\mathfrak{g}^{*}=\mathrm{Lie}(G)^{*}$ of Poisson supermanifolds such that
the infinitesimal action of an element $a\in\mathfrak{g}$ is given by the
Hamiltonian vector field for the function $m\mapsto\langle\mu(m),a\rangle$,
where $\langle\,,\,\rangle$ denotes the pairing of the vector space
$\mathfrak{g}$ with its dual; such data is equivalent to a Lie algebra map
$\mu^{*}\colon\mathfrak{g}\to\mathcal{O}(M)$, where the latter is treated as a
super Lie algebra with its Poisson bracket. When certain cohomology groups of
$M$ and $\mathfrak{g}$ vanish, $\mu$ exists and is unique. The Hamiltonian
reduction $M//G$ of this data is the quotient space $\mu^{-1}(0)/G$. This can
be defined in the super case via its algebra of functions
$(\mathcal{O}(M)/\langle\mu^{*}\mathfrak{g}\rangle)^{G}$, where
$\langle\mu^{*}\mathfrak{g}\rangle$ denotes the ideal generated by the image
of $\mu^{*}$. As Marsden and Weinstein explained in the even case [7], when
$0$ is a regular value of $\mu$ and the action of $G$ on $\mu^{-1}(0)$ is free
and proper, the manifold $M//G$ is naturally symplectic. The natural maps
$\mu^{-1}(0)\hookrightarrow M$ and $\mu^{-1}(0)\twoheadrightarrow M//G$ are
together a Lagrangian correspondence between $M$ and $M//G$. Super Hamiltonian
reduction can be cleanly expressed as an example of coisotropic reduction of
super Poisson algebras [4, 8].

Suppose that $G$ acts instead on an associative superalgebra $A$. A comoment
map is a Lie algebra map $\mu^{*}\colon\mathfrak{g}\to A$, where $A$ is
treated as a super Lie algebra with its commutator bracket, such that the
infinitesimal action of $a\in\mathfrak{g}$ is given by the inner derivation
$[\mu^{*}(a),-]$. Corresponding to the zero section $\mu^{-1}(0)$ is the
quotient module $A/\langle\mu^{*}\mathfrak{g}\rangle$, where
$\langle\mu^{*}\mathfrak{g}\rangle$ denotes the left ideal generated by the
image of $\mu^{*}$. Corresponding to the quotient $M//G=\mu^{-1}(0)/G$ is the
quantum Hamiltonian reduction
$A//G=(A/\langle\mu^{*}\mathfrak{g}\rangle)^{G}$. This is naturally an algebra
because it is isomorphic to
$\operatorname{End}_{A}(A/\langle\mu^{*}\mathfrak{g}\rangle)$. When $G$ is
compact, $A//G\cong
A^{G}/\bigl{(}A^{G}\cap\langle\mu^{*}\mathfrak{g}\rangle\bigr{)}$, where
$A^{G}$ denotes the $G$-invariant subalgebra of $A$. From this perspective,
the algebra structure on $A//G$ arises because, although
$\langle\mu^{*}\mathfrak{g}\rangle$ is merely a left ideal in $A$, its
intersection with $A^{G}$ is a two-sided ideal, as $\mu^{*}\mathfrak{g}$ is
central in $A^{G}$. The module $A/\langle\mu^{*}\mathfrak{g}\rangle$ is by
construction a bimodule between $A$ and $A//G$.

######  Example 2.1.

Suppose that $M$ is a linear symplectic supermanifold and $C\subseteq M$ is a
coisotropic submanifold cut out by linear equations $r_{1}=\dots=r_{p+q}=0$,
where $r_{1},\dots,r_{p}$ are even and $r_{p+1},\dots,r_{p+q}$ are odd. The
Hamiltonian flows for $r_{1},\dots,r_{p+q}$ define an action on $M$ of the
abelian Lie supergroup $\mathbb{R}^{p|q}$. Let $C^{\perp}\subseteq C$ denote
the symplectic orthogonal to $C$. The Hamiltonian reduction
$M//\mathbb{R}^{p|q}$ is then canonically linearly symplectomorphic to
$C/C^{\perp}$.

Since $M$ is linear, it admits a canonical quantization to the Weyl algebra
$\mathcal{W}(M){=}\mathcal{T}(M^{*}){/}([a{,}b]\\!$ $=\\{a,b\\},a,b\in
M^{*})$. The quotient $\mathcal{W}(C)=\mathcal{W}(M)/\langle
r_{1},\dots,r_{c}\rangle$ is the canonical quantization of $C$, and
$\mathcal{W}(M)//\mathbb{R}^{p|q}\cong\mathcal{W}(C/C^{\perp})$. In the
purely-odd case, which is the only case of concern in this paper,
$\mathcal{W}(C)$ is a Morita equivalence between $\mathcal{W}(M)$ and
$\mathcal{W}(C/C^{\perp})$: it suffices to consider the case
$M=\mathbb{R}^{0|2}$ with “split” symplectic form
$\frac{(\mathrm{d}x)^{2}}{2}-\frac{(\mathrm{d}y)^{2}}{2}$ and Lagrangian
$C\cong\mathbb{R}^{0|1}$ spanned by the vector $(1,1)$; then
$\mathcal{W}(M)\cong\mathrm{Mat}(1|1)$ is the algebra of $2\times 2$ matrices
in which $\bigl{(}\begin{smallmatrix}1&0\\\ 0&0\end{smallmatrix}\bigr{)}$ and
$\bigl{(}\begin{smallmatrix}0&0\\\ 0&1\end{smallmatrix}\bigr{)}$ are even and
$\bigl{(}\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\bigr{)}$ and
$\bigl{(}\begin{smallmatrix}0&0\\\ 1&0\end{smallmatrix}\bigr{)}$ are odd, and
$\mathcal{W}(C)$ is the defining $(1|1)$-dimensional module.

(When there are even coordinates, $\mathcal{W}(C)$ is not a Morita
equivalence. The Stone–von Neumann theorem can be understood as saying that
for purely even $M$, $\mathcal{W}(C)$ becomes a Morita equivalence after
appropriate functional analytic completions. The mixed case can be handled by
decomposing $M$ and $C$ into even and odd parts.)

In particular, linear Lagrangians provide Morita equivalences
$\mathcal{W}(M)\simeq\mathbb{R}$. This does not explain why
$\mathrm{Clif\/f}(8)=\mathcal{W}(\mathbb{R}^{0|8})\simeq\mathbb{R}$, because
the positive-definiteness of the symplectic form prevents $\mathbb{R}^{0|n}$
from admitting Lagrangian sub-supermanifolds, linear or not.

######  Lemma 2.2.

If the Hamiltonian reduction $\mathrm{Clif\/f}(n)//G$ is not the zero algebra,
then $\mathrm{Clif\/f}(n)/\langle\mu^{*}\mathfrak{g}\rangle$ is a Morita
equivalence between $\mathrm{Clif\/f}(n)$ and $\mathrm{Clif\/f}(n)//G$.

######  Proof 2.3.

For any superalgebra $A$, an $A$-module $X$ is a Morita equivalence between
$A$ and $\operatorname{End}_{A}(X)$ if and only if $X$ is a finitely-generated
projective generator of the supercategory of $A$-modules. The holomorphic
symplectic supermanifold $\mathbb{C}^{0|n}=\mathbb{R}^{0|n}\otimes\mathbb{C}$
admits a linear Lagrangian $L$ if $n$ is even and an $(n+1)/2$-dimensional
linear coisotropic $C$ if $n$ is odd. Via Example 2.1, these linear
coisotropics provide Morita equivalences
$\mathrm{Clif\/f}(n)\otimes\mathbb{C}\simeq\mathbb{C}=\mathcal{W}(L/L^{\perp})$
or $\mathrm{Clif\/f}(1)\otimes\mathbb{C}=\mathcal{W}(C/C^{\perp})$. For
$\mathbb{C}$ and $\mathrm{Clif\/f}(1)\otimes\mathbb{C}$, any non-zero
finitely-generated module is a projective generator. But “non-zero”,
“finitely-generated”, and “projective” are Morita-invariant notions, so these
properties hold also for $\mathrm{Clif\/f}(n)\otimes\mathbb{C}$ and hence for
$\mathrm{Clif\/f}(n)$.

##  3 $\boldsymbol{\mathrm{Clif\/f}(4)}$ and $\boldsymbol{\mathbb{H}}$

Corresponding to the exceptional isomorphism
$\mathrm{SO}(4)\cong\mathrm{Spin}(3)\times_{\mathbb{Z}/2}\mathrm{Spin}(3)$ are
two commuting actions of $\mathrm{Spin}(3)$ on $\mathbb{R}^{0|4}$ by linear
symplectic automorphisms. (Odd symplectic groups are even orthogonal groups;
metaplectic groups correspond to spin groups.) Denote the coordinates on
$\mathbb{R}^{0|4}$ by $\\{w,x,y,z\\}$ and the bases for two copies of
$\mathfrak{so}(3)$ by $\\{a_{+},b_{+},c_{+}\\}$ and $\\{a_{-},b_{-},c_{-}\\}$,
normalized so that their brackets are $[a_{\pm},b_{\pm}]=\pm 2c_{\pm}$,
$[b_{\pm},c_{\pm}]=\pm 2a_{\pm}$, $[c_{\pm},a_{\pm}]=\pm 2b_{\pm}$. The
comoment maps for the actions are:

$\displaystyle a_{\pm}\mapsto\tfrac{1}{2}(wx\pm yz),\qquad
b_{\pm}\mapsto\tfrac{1}{2}(wy\pm zx),\qquad c_{\pm}\mapsto\tfrac{1}{2}(wz\pm
xy).$

Together these six elements are a basis for the space of homogeneous-quadratic
functions on $\mathbb{R}^{0|4}$. The quadratic Casimir for both
$\mathfrak{so}(3)$s is $\theta=\pm 2a_{\pm}^{2}=\pm 2b_{\pm}^{2}=\pm
2c_{\pm}^{2}=wxyz$. Completing the basis for $\mathcal{O}(\mathbb{R}^{0|4})$
are the unit $1$ and $xyz$, $wyz$, $wzx$, and $wxy$. Basis vectors $1$,
$a_{\pm}$, $b_{\pm}$, $c_{\pm}$, and $\theta$ are even, and $x$, $y$, $z$,
$w$, $xyz$, $wyz$, $wzx$, and $wxy$ are odd.

We now consider the quantization
$\mathrm{Clif\/f}(4)=\mathcal{W}(\mathbb{R}^{0|4})$, for which we can use the
same basis
$\\{1,w,x,y,z,a_{+},b_{+},c_{+},a_{-},b_{-},c_{-},xyz,wyz,wzx,wxy,\theta\\}$
(with the same grading). Note that, whereas in $\mathcal{O}(\mathbb{R}^{0|4})$
we had $a^{2}_{\pm}=b^{2}_{\pm}=c^{2}_{\pm}=\pm\theta/2$, in
$\mathrm{Clif\/f}(4)$ we have
$a^{2}_{\pm}=b^{2}_{\pm}=c^{2}_{\pm}=\frac{1}{2}(\pm\theta-1)$. Since the
actions of $\mathrm{Spin}(3)$ on $\mathbb{R}^{0|4}$ are linear, they lift to
$\mathrm{Clif\/f}(4)$, and the same assignments $a_{+},\dots,c_{-}$ provide
the quantum comoment maps.

######  Theorem 3.1.

The quantum Hamiltonian reduction of either of the $\mathfrak{so}(3)$-actions
on $\mathbb{R}^{0|4}$ produces a Morita equivalence
$\mathrm{Clif\/f}(4)\simeq\mathbb{H}=\mathbb{R}\langle
i,j,k\rangle/(i^{2}=j^{2}=k^{2}=ijk=-1)$ $($where the quaternion algebra
$\mathbb{H}$ is purely even$)$.

######  Proof 3.2.

There is a manifest symmetry interchanging the two $\mathfrak{so}(3)$-actions;
we will work with the action of the $a_{-}$, $b_{-}$, and $c_{-}$. It is not
hard to see that $[a_{-},-]$, $[b_{-},-]$, and $[c_{-},-]$ preserve polynomial
degree. We have observed already that the quadratic elements $a_{+}$, $b_{+}$,
$c_{+}$ commute with the generators $a_{-}$, $b_{-}$, $c_{-}$ of the action,
as well as with $\theta=wxyz$. The subspace of $\mathrm{Clif\/f}(4)$ spanned
by $\\{a_{-},b_{-},c_{-}\\}$ is a submodule for the action of
$\\{a_{-},b_{-},c_{-}\\}$ isomorphic to the adjoint action. The subspaces
spanned by $\\{w,x,y,z\\}$ and $\\{xyz,wyz,wzx,wxy\\}$ are each isomorphic to
the underlying real module of the defining module of $\mathfrak{su}(2)$. Thus
a basis for the $\mathfrak{so}(3)$-fixed subalgebra
$\mathrm{Clif\/f}(4)^{\mathrm{Spin}(3)}$ is given by the five even elements
$\\{1,a_{+},b_{+},c_{+},\theta\\}$.

The left ideal $\langle\mu^{*}\mathfrak{so}(3)\rangle$ in
$\mathrm{Clif\/f}(4)$ generated by $\\{a_{-},b_{-},c_{-}\\}$ is eight-
dimensional with basis
$\\{a_{-},b_{-},c_{-},w-xyz,x+wyz,y+wzx,z+wxy,\theta+1\\}$. This ideal
intersects $\mathrm{Clif\/f}(4)^{\mathrm{Spin}(3)}$ only in the one-
dimensional space spanned by $\theta+1$. It follows that in the quotient
$\mathrm{Clif\/f}(4)^{\mathrm{Spin}(3)}/$
$\bigl{(}\mathrm{Clif\/f}(4)^{\mathrm{Spin}(3)}\cap\langle\mu^{*}\mathfrak{so}(3)\rangle\bigr{)}$
we have $a_{+}^{2}=\frac{1}{2}(\theta-1)\equiv\frac{1}{2}(-2)=-1$, and so the
map $(a_{+},b_{+},c_{+})\mapsto(i,j,k)$ identifies the quantum Hamiltonian
reduction $\mathrm{Clif\/f}(4)//\mathrm{Spin}(3)$ with the quaternion algebra
$\mathbb{H}$. Lemma 2.2 completes the proof.

Theorem 3.1 suggests that the “classical limit” of $\mathbb{H}$ is the
Hamiltonian reduction $\mathbb{R}^{0|4}//\mathrm{Spin}(3)$. Since $0$ is not a
regular value of the classical moment map,
$\mathbb{R}^{0|4}//\mathrm{Spin}(3)$ is not a supermanifold. It does make
sense as an affine super scheme: its algebra of functions is the purely even
Poisson algebra $\mathbb{R}[a,b,c]/(a^{2}=b^{2}=c^{2}=ab=bc=ca=0)$ with
Poisson brackets $\\{a,b\\}=2c$, $\\{b,c\\}=2a$, and $\\{c,a\\}=2b$. Thus the
“classical limit” of $\mathbb{H}$ is the first-order neighborhood of $0$ in
$\mathfrak{so}(3)^{*}$.

##  4 $\boldsymbol{\mathrm{Clif\/f}(7)}$ and $\boldsymbol{G_{2}}$

The 14-dimensional exceptional Lie group $G_{2}$ is the subgroup of
$\mathrm{SO}(7)$ preserving the alternating 3-form $\epsilon$ on
$\mathbb{R}^{7}$ defined by identifying $\mathbb{R}^{7}$ with the pure-
imaginary octonions and setting $\epsilon(a,b,c)=(ab)c-a(bc)\in\mathbb{R}$
[2]. Since $\mathrm{SO}(7)$ acts by linear symplectic automorphisms of
$\mathbb{R}^{0|7}$, we get an induced symplectic action of $G_{2}$.

######  Theorem 4.1.

The quantum Hamiltonian reduction $\mathrm{Clif\/f}(7)//G_{2}$ provides the
Morita equivalence $\mathrm{Clif\/f}(7)\simeq\mathrm{Clif\/f}(-1)$.

######  Proof 4.2.

By Lemma 2.2, it suffices to compute $\mathrm{Clif\/f}(7)//G_{2}$. As in
Theorem 3.1, the Poincaré–Birkoff–Witt isomorphism
$\mathcal{O}(\mathbb{R}^{0|7})=\bigwedge^{\bullet}(\mathbb{R}^{7})\cong\mathrm{Clif\/f}(7)$
is $\mathrm{SO}(7)$-equivariant by the functoriality of the Weyl algebra
construction. The $G_{2}$-fixed algebra has as its basis a set of the form
$\\{1,\epsilon,\bar{\epsilon},\theta\\}$, where $\epsilon$ is the cubic 3-form
defining $G_{2}$, $\bar{\epsilon}$ is its dual quartic, and $\theta$ is the
generator of $\bigwedge^{7}\mathbb{R}^{7}$. In $\mathrm{Clif\/f}(7)$,
$\bar{\epsilon}=\theta\epsilon$ and $\theta^{2}=1$.

An explicit presentation of the action of
$\mathrm{Lie}(G_{2})=\mathfrak{g}_{2}$ is given in [1] as follows. Denote the
coordinates on $\mathbb{R}^{0|7}$ by $x_{1},x_{2},\dots,x_{7}$. The cubic
function $\epsilon$ is

$\displaystyle\epsilon=x_{1}x_{2}x_{3}+x_{1}x_{4}x_{5}+x_{1}x_{6}x_{7}+x_{2}x_{4}x_{6}+x_{2}x_{7}x_{5}+x_{3}x_{7}x_{4}+x_{3}x_{6}x_{5}.$

Consider the quadratic functions $e_{1},\dots,e_{7}$ defined by
$e_{i}=\frac{\partial}{\partial x_{i}}\epsilon$. For example,
$e_{1}=x_{2}x_{3}+x_{4}x_{5}+x_{6}x_{7}$. Orthogonal to each $e_{i}$ is a two-
dimensional vector space of commuting quadratics given by the differences of
the monomials in $e_{i}$. For example, orthogonal to $e_{1}$ are
$x_{2}x_{3}-x_{4}x_{5}$, $x_{4}x_{5}-x_{6}x_{7}$, and their sum
$x_{2}x_{3}-x_{6}x_{7}$. A basis for the image of $\mathfrak{g}_{2}$ under
$\mu^{*}$ is given by choosing for each $i=1,\dots,7$ two quadratics
orthogonal to $e_{i}$. For example:

$\displaystyle x_{2}x_{3}-x_{4}x_{5},\qquad x_{4}x_{5}-x_{6}x_{7},\qquad
x_{3}x_{1}-x_{4}x_{6},\qquad x_{4}x_{6}-x_{7}x_{5},\qquad
x_{1}x_{2}-x_{7}x_{4},$ $\displaystyle x_{7}x_{4}-x_{6}x_{5},\qquad
x_{5}x_{1}-x_{6}x_{2},\qquad x_{6}x_{2}-x_{3}x_{7},\qquad
x_{1}x_{4}-x_{2}x_{7},\qquad x_{2}x_{7}-x_{3}x_{6},$ $\displaystyle
x_{7}x_{1}-x_{2}x_{4},\qquad x_{2}x_{4}-x_{5}x_{3},\qquad
x_{1}x_{6}-x_{5}x_{2},\qquad x_{5}x_{2}-x_{4}x_{3}.$

We wish to compute
$\operatorname{End}_{\mathrm{Clif\/f}(7)}(\mathrm{Clif\/f}(7)/\langle\mu^{*}\mathfrak{g}_{2}\rangle)=\mathrm{Clif\/f}(7)^{G_{2}}/(\langle\mu^{*}\mathfrak{g}_{2}\rangle\cap\mathrm{Clif\/f}(7)^{G_{2}})$,
where $\langle\mu^{*}\mathfrak{g}_{2}\rangle$ is the left ideal generated by
these $14$ elements.

Note that
$x_{1}x_{4}x_{5}x_{6}x_{7}(x_{2}x_{3}-x_{4}x_{5})=\theta+x_{1}x_{6}x_{7}$,
where $\theta=x_{1}x_{2}\cdots x_{7}\in\mathrm{Clif\/f}(7)^{G_{2}}$. The
numerics of the second summand are: $x_{2}x_{3}-x_{4}x_{5}$ was orthogonal to
$e_{1}$; $x_{6}x_{7}$ is the unused monomial in $e_{1}$. Similarly, for each
monomial $\mu$ in the cubic $\epsilon$ one can find
$\theta+\mu\in\langle\mu^{*}\mathfrak{g}_{2}\rangle$, and summing shows that
$7\theta+\epsilon\in\langle\mu^{*}\mathfrak{g}_{2}\rangle\cap\mathrm{Clif\/f}(7)^{G_{2}}$;
hence also
$7+\bar{\epsilon}\in\langle\mu^{*}\mathfrak{g}_{2}\rangle\cap\mathrm{Clif\/f}(7)^{G_{2}}$.
It follows that
$\mathrm{Clif\/f}(7)^{G_{2}}/(\langle\mu^{*}\mathfrak{g}_{2}\rangle\cap\mathrm{Clif\/f}(7)^{G_{2}})$
is a quotient of the copy of $\mathrm{Clif\/f}(-1)$ spanned by the classes of
$1$ and $\theta$.

Finally, for any basis element $\alpha\in\mu^{*}\mathfrak{g}_{2}$, we have
$\alpha(\theta-\epsilon)=0$, from which it follows that
$1\not\in\langle\mu^{*}\mathfrak{g}_{2}\rangle$. The ideal cannot mix even and
odd terms without setting both to $0$, and so we find
$\mathrm{Clif\/f}(7)^{G_{2}}/(\langle\mu^{*}\mathfrak{g}_{2}\rangle\cap\mathrm{Clif\/f}(7)^{G_{2}})\cong\mathrm{Clif\/f}(-1)$.

##  5 $\boldsymbol{\mathrm{Spin}(7)}$ and Bott periodicity

We conclude by providing a Hamiltonian reduction whose quantization is the
famous “Bott periodicity” equivalence $\mathrm{Clif\/f}(8)\simeq\mathbb{R}$.
The irreducible real spin representations of all four groups
$\mathrm{Spin}(5)$, $\mathrm{Spin}(6)$, $\mathrm{Spin}(7)$, and
$\mathrm{Spin}(8)$ are eight-real-dimensional. The reduction
$\mathrm{Clif\/f}(8)//\mathrm{Spin}(8)$ vanishes since the image of the
comoment map consists of all quadratic elements of $\mathrm{Clif\/f}(8)$,
including some which are invertible, and so the $\mathrm{Spin}(8)$-action does
not induce a Morita equivalence. The reader is invited to compute
$\mathrm{Clif\/f}(8)//\mathrm{Spin}(5)$ and
$\mathrm{Clif\/f}(8)//\mathrm{Spin}(6)$. We will show:

######  Theorem 5.1.

$\mathrm{Clif\/f}(8)//\mathrm{Spin}(7)\cong\mathbb{R}$.

By Lemma 2.2, Theorem 5.1 establishes that the cyclic module
$\mathrm{Clif\/f}(8)/\langle\mu^{*}\mathfrak{so}(7)\rangle$ is a Morita
equivalence between $\mathrm{Clif\/f}(8)$ and $\mathbb{R}$.

######  Proof 5.2.

The following construction of $\mathrm{Spin}(7)$, and its eight-dimensional
spin representation, are developed in [6]. Consider the octonion algebra
$\mathbb{O}$ and the 4-form $\phi\in\bigwedge^{4}\mathbb{O}^{*}$ defined by
$\phi(a,b,c,d)=\langle a,b\times c\times d\rangle$, where the triple cross
product is by definition $b\times c\times
d=\frac{1}{2}\bigl{(}b(\bar{c}d)-d(\bar{c}b)\bigr{)}$ and $\bar{c}$ is the
octonionic conjugate of $c$. Then $\mathrm{Spin}(7)$ is precisely the subgroup
of $\mathrm{SO}(8)$ fixing $\phi$. In terms of coordinates
$x_{1},x_{2},\dots,x_{8}$ on $\mathbb{R}^{0|8}$, $\phi$ corresponds to the
function

$\displaystyle\phi=x_{1234}+x_{1256}+x_{1278}+x_{1357}-x_{1368}-x_{1458}-x_{1467}$
$\displaystyle\hphantom{\phi=}{}+x_{5678}+x_{3478}+x_{3456}+x_{2468}-x_{2457}-x_{2367}-x_{2358},$

where we have abbreviated $x_{ij\dots k}{=}x_{i}x_{j}\cdots x_{k}$. Let
$\theta{=}x_{12345678}$. The fixed algebra
$\mathrm{Clif\/f}(8)^{\mathrm{Spin}(7)}$ has basis $\\{1,\phi,\theta\\}$ and
multiplication $\theta^{2}=1$, $\theta\phi=\phi\theta=\phi$, and
$\phi^{2}=14\theta+14-12\phi$.

The image
$\mu^{*}\mathfrak{so}(7)\subseteq\bigwedge^{2}\mathbb{R}^{8}\subseteq\mathrm{Clif\/f}(8)$
of the comoment map is spanned by quadratic elements of the form
$\alpha=x_{ij}\pm x_{kl}$ such that $-\frac{1}{2}\alpha^{2}-1=\mp x_{ijkl}$ is
(with the given sign) a monomial in $\phi$. (There are $3\cdot 14$ such
$\alpha$s; given $\\{i,j\\}$, there are three $\alpha$s that include the
monomial $x_{ij}$ and three that are disjoint from $\\{i,j\\}$, and these six
span a three-dimensional space; this counts correctly the $21$-dimensional
space $\mathfrak{so}(7)$.) It follows that $\phi+14$, being a sum of $14$
terms of the form $-\frac{1}{2}\alpha^{2}$, is in the ideal
$\langle\mu^{*}\mathfrak{so}(7)\rangle$, and so
$\mathrm{Clif\/f}(8)//\mathrm{Spin}(7)$ is a quotient of
$\mathbb{R}\\{1,\phi,\theta\\}/\langle\phi+14\rangle\cong\mathbb{R}$.

Each $\alpha=x_{ij}\pm x_{kl}\in\mu^{*}\mathfrak{so}(7)$ determines a
splitting $\phi=x_{ijkl}(1+\theta)+\kappa+\lambda$ where $\kappa$ is a sum of
four quartic monomials each of which has indices containing either $\\{i,j\\}$
or $\\{k,l\\}$ but not both, and $\lambda$ is a sum of eight quartic monomials
each of which has indices intersecting the sets $\\{i,j\\}$ and $\\{k,l\\}$ at
one element each. For example, when $\alpha=x_{13}-x_{57}$, we have

$\displaystyle\phi=\underbrace{x_{1357}+x_{2468}}_{x_{ijkl}(1+\theta)}+\underbrace{x_{1234}-x_{1368}+x_{5678}-x_{2457}}_{\kappa}$
$\displaystyle\hphantom{\phi=}{}+\underbrace{x_{1256}+x_{1278}-x_{1458}-x_{1467}+x_{3478}+x_{3456}-x_{2367}-x_{2358}}_{\lambda}.$

We see that $[\alpha,x_{ijkl}(1+\theta)]=0$ and $[\alpha,\kappa]=0$. Since we
know that $[\alpha,\phi]=0$, we find $[\alpha,\lambda]=0$ as well. Suppose
$\beta$ is a quadratic monomial and $\nu$ a quartic monomial such that the
indices in $\beta$ and $\nu$ overlap at one element. Then
$\beta\nu=\frac{1}{2}[\beta,\nu]$, from which it follows that
$\alpha\lambda=\frac{1}{2}[\alpha,\lambda]=0$. It’s also clear that
$\alpha\kappa=0$, since $\kappa$ factors as $(x_{ij}\mp x_{kl})(\dots)$ where
$\alpha=x_{ij}\pm x_{kl}$ and $(\dots)$ is a sum of two quadratic monomials
that have no overlap with $\alpha$. Finally, $\alpha
x_{ijkl}(1+\theta)=\alpha(-1-\theta)$. All together, we find:

$\displaystyle\alpha(\phi+1+\theta)=\alpha\bigl{(}x_{ijkl}(1+\theta)+\kappa+\lambda+(1+\theta)\bigr{)}=\alpha(-1-\theta)+0+0+\alpha(1+\theta)=0.$

It follows that $1\not\in\langle\mu^{*}\mathfrak{so}(7)\rangle$, and so
$\mathrm{Clif\/f}(8)//\mathrm{Spin}(7)\not\cong 0$, completing the proof.

### Acknowledgements

I would like to thank the referees for their comments and improvements to this
paper. This work was completed during the “Gone Fishing 2016” conference at
University of Colorado, Boulder, which was supported by the NSF grant
DMS-1543812. This research was also supported by the NSF grant DMS-1304054.
The Perimeter Institute for Theoretical Physics is supported by the Government
of Canada through the Department of Innovation, Science and Economic
Development Canada and by the Province of Ontario through the Ministry of
Research, Innovation and Science.

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#  $T=1$ Pairing Along the $N=Z$ Line

B. Alex Brown  The Facility for Rare Isotope Beams and Department of Physics
and Astronomy, Michigan state University, East Lansing, Michigan 48824-1321,
USA  Michio Honma  Center for Mathematical Sciences, University of Aizu,
Tsuruga, Ikki-machi, Aizu-Wakamatsu, Fukushima 965-8580, Japan  Ragnar
Stroberg  Department of Physics and Astronomy, University of Notre Dame, Notre
Dame, IN, 46556, USA

###### Abstract

Pairing energies for the addition of two neutrons on even-even nuclei with
$N=Z$ are studied. The $Z$ dependence is attributed to the number and type of
orbitals that are occupied in the valence shell-model space. Properties in the
region from $Z=60-100$ depend on the location of the $0g_{9/2}$ orbital.

Nuclear masses in the region of $A=70$ were measured and recently reported by
Wang et al. wang . These results were used to extract the quantity $\delta
V^{oo}_{pn}$ for odd-odd nuclei defined in terms of the nuclear binding
energies $B$ by the double-difference equation related to the arrows (a) and
(b) in Fig. 1:

$\delta V^{oo}_{pn}=\Delta B_{a}-\Delta B_{b},$ $None$

where

$\Delta B_{a}=B(Z,N)-B(Z,N-1)$ $None$

and

$\Delta B_{b}=B(Z-1,N)-B(Z-1,N-1).$ $None$

The cases of interest involve an even-even core nucleus, $A_{c}$, with $T=0$
and $N_{c}=Z_{c}$, and the addition of a proton and neutron coupled to $T=1$.
It was shown that the experimental values for this quantity were much higher
than all of the theoretical calculations for 66Ga, 70Br and 74Rb used for Fig.
2 of wang . The reason for this difference was investigated using valence-
space in-medium similarity renormalization group (VS-IMSRG) method in the $fp$
$(0f_{7/2},0f_{5/2},1p_{3/2},1p_{1/2})$ model space. The purpose of this paper
it to put these results in the context of other data and calculations along
the $N=Z$ line of nuclei.

Figure 1: Segment of the nuclear chart near $N=Z$ showing the binding-energy
differences discussed in the text.

The series of odd-odd nuclei 62Ga, 66As, 70Br and 74Rb used for Fig. 2 of wang
all have $J^{\pi}$=0+, $T=1$ ground states. For these cases, the ground states
of the nuclei connected by arrow (c) in Fig. 1 are isobaric analogues of the
ground states connected by arrow (a). Thus, results for $T=1$ pairing can also
be obtained using the binding-energy differences for the nuclei shown by the
arrows (c) and (b) in Fig. 1:

$D_{n}(Z-1,N)=\Delta B_{c}-\Delta B_{b},$ $None$

where

$\Delta B_{c}=B(Z-1,N+1)-B(Z-1,N).$ $None$

This is the pairing-gap equation defined in low . For example, for 74Rb, we
have from Eq. (1), $\delta V^{oo}_{pn}$ = 13.976(40) - 10.682(11) = 3.294(42)
MeV, and from Eq. (4), $D_{n}$ = 13.851(7) - 10.682(11) = 3.169(13) MeV. The
difference between these two results is due to the small charge-symmetry
breaking interaction in the $T=1$ triplet, as well as the small $N$ dependence
in the binding-energy differences between isobaric analogue states.

In Fig. 2a we show $D_{n}$ and $\delta V_{pn}(T=1)$ for the even-even core
nuclei with $N_{c}=Z_{c}$ from $Z_{c}=4$ up to $Z_{c}=50$. The binding
energies are from the 2020 mass table mass  together with the new data from
wang . For the odd-odd nuclei involved in $\delta V_{pn}(T=1)$ we use the
binding energy associated with the 0+, $T=1$ states which in some cases are
excited states. Binding energy for nuclei near $N=Z$ above about $A=76$ are
not measured, but based on mass-value extrapolations with relatively large
uncertainties. The region of data considered in wang  is $A_{c}=56-72$ in Fig.
2. The largest difference between $D_{n}$ and $\delta V_{pn}(T=1)$ is for
$A_{c}=68$. The masses involved in the results for $A_{c}=68$ should be
experimentally confirmed.

Figure 2:  $D_{n}$ for the even-even core-nuclei with $N_{c}=Z_{c}$from
$Z_{c}=2$ up to $Z_{c}=50$. The experimental data are shown by the black
cicles with error bars (same for all panels). In panel (a) the data for
$D_{n}$ are compared to the data for $\delta V_{pn}(T=1)$ as obtained from Eq.
(1). In panel (b) the data for $D_{n}$ are compared to those obtained with
empirical shell-model Hamiltonians in various model space. The red crosses are
the results of calculations in the $ds$ model space with the USDC Hamiltonian
usdc . The blue crosses are the results of calculations in the $fp$ model
space with the GPFX1A Hamiltonian gx1a . The green circles are the results of
calculations in the $jj44$ model space with the JUN45 Hamiltonian jun45 . In
panel (c) the data for $D_{n}$ are compared to the results of VS-IMSRG
calculations shown in red.

In Fig. 2b the experimental data for $D_{n}$ are compared with the results of
configuration-interaction calculations. The results for $A_{c}=16-36$ were
obtained in the $ds$ $(0d_{5/2},0d_{3/2},1s_{1/2})$ model space with the USDC
Hamiltonians usdc  are shown by the red crosses in Fig. 2b. The agreement is
excellent. The same level of agreement would be obtained with the earlier
versions of the ”universal” $ds$ Hamiltonians USD usd , USDA usda  and USDB
usda . This agreement it not surprising since these are empirical Hamiltonians
whose two-body matrix elements (TBME) are obtained by singular-valued
decomposition fits to binding energies and excitation energy data for nuclei
in the region of $Z=8-20$ and $N=8-20$. All energy data in the $ds$ model
space can be described by a unified set of TBME within an rms uncertainty of
about 150 keV, except for those in the region around 32Mg that lie within the
island-or-inversion ioi , where one or more neutrons are in the $fp$ shell in
the ground states.

The results for $A_{c}=40-78$ were obtained in the $fp$ model space with the
GPFX1A Hamiltonians gx1a  are shown by the blue crosses in Fig. 2b. Binding
energies and excitation energies for nuclei in the region $A\geq 47$ and
$Z\leq 32$ were used to determined the universal effective TBME for GPFX1A. It
is not suprising that the calculated $D_{n}$ values are in relatively good
agreement with experiment for this region. Data for $A=40-46$ was not included
since the energies are influenced by mixing with low-lying intruder states
coming from particle-hole states across $Z=20$ and $N=20$. The calculated
pairing for $A=40$ and $A=44$ are significantly smaller than experiment. At
$A=60$ and above theory and experiment start to diverge.

The irregular patterns found in the experimental data are related to the types
of valence orbitals involved in the configurations. Just below $jj$ magic
numbers $A_{c}$=12, 28 and 56, the pairing is dominated by high $j$ orbitals
($0p_{3/2}$, $0d_{5/2}$ and $0f_{7/2}$, respectively) giving a relatively
large pairing energy. Just after these magic number a lower $j$ orbital starts
to be filled ($0p_{1/2}$, $1s_{1/2}$ and $1p_{3/2}$, respectively) and the
pairing energy drops.

For $A_{c}\geq 64$, the $0g_{9/2}$ orbital becomes important. This can be seen
in the spectra of nuclei with $A_{c}+1$. Energies of the lowest 9/2+ states
are 2.40 MeV (61Zn), 1.22 MeV (65Ge), 0.57 MeV (69Se) and 0.43 MeV (73Kr).
Starting in 77Sr the nuclei are more deformed and the ground state has 5/2+.
Calculations for pairing in this region must explicitly take into account the
growing importance of the $0g_{9/2}$ orbital. The $fp$ model space used for
the interpretation of the data in wang  is insufficient.

In a similar way, pairing with the $0f_{7/2}$ orbital in the upper part of the
$ds$ model space must influence the $D_{n}$ values. 7/2- states appear at 3.62
MeV (29Si), 2.93 MeV (33S) and 1.61 MeV (37Ar). In contrast to $0g_{9/2}$ in
the $A=70$ region mixing with $0f_{7/2}$ in the upper $ds$ shell appears to be
small enough to be contained implicitly as a perturbative contribution to the
$ds$ effective TBME. In contrast, in the $A=70$ region, the $0g_{9/2}$ must be
treated explicitly in the pairing. Then one expects $D_{n}$ to become larger
than those obtained in the $fp$ model space due to mixing with this high-$j$
orbital. An interesting feature for $A_{c}>60$ in Fig. 2 is the sharp drop at
$A_{c}=76$. Perhaps this is due to a sudden shape change.

In order to explain the data, the calculations presented in wang  used the
valence-space in-medium similarity renormalization group (VS-IMSRG) method
with and without the 3N force from the chiral EFT interaction in the $fp$
model space. To put this comparison in a broader context, we show in Fig. 2c
results based on binding energies obtained with VS-IMSRG calculations as given
in the supplemental material obtained of vs , for the $0p$ ($A_{c}$=8,12),
$ds$ ($A_{c}$=16-36) and $fp$ ($A_{c}$=40-52) model spaces. The error band
takes into account the estimated 0.8 MeV error in the one-neutron separation
energies found in vs . The error band also includes an estimated systematic
downward shift of 0.2 to 0.4 MeV that takes into account that the interaction
gives rms radii that are systematically too small. For $A_{c}$=56-76 we show
the results of VS-IMSRG calculations using the same method as vs . The error
band for $A_{c}\geq 56$ has not been evaluated, but it should be similar those
shown for $A_{c}\leq 52$. The overall size of the VS-IMSRG results for $D_{n}$
are qualitatively consistent with experiment within the error band, but they
are generally larger than experiment. In order to draw conclusions about the
deficiencies of the VS-IMSRG results for $A_{c}\geq 64$, one must first
understand and improve the results for $A_{c}\leq 64$.

The minimal model space appropriate for the region above $A_{c}=60$ must
involve the orbitals $(0f_{5/2},1p_{3/2},1p_{1/2},0g_{9/2})$ (the $jj44$ model
space) jun45 , jj44 . Calculations within the $jj44$ model space must be able
to reproduce the low-lying energy of the $0g_{9/2}$ orbital and the shape
change around 80Zr. The full deformation of 80Zr also requires the addition of
the $1d_{5/2}$ orbital d5 .

For $A_{c}$=56, 60, 64, and 66 we show the results obtained with the JUN45
hamiltonian in the $jj44$ model space. For $A_{c}$=68 the $jj44$ $D_{n}$ value
is about one MeV larger than the $fp$ model space result. This is correlated
with a change in the occupation of the $0g_{9/2}$ from 0.48 in 66Ge to 1.48 in
70Se. The basis dimensions in the $jj44$ model space for $A_{c}\geq 70$ are
very large, and the consistent set of calculations required to reproduce the
spectroscopy in this region is beyond the scope of this short paper.

In summary, we have used experimental data on binding energies to deduce the
$T=1$ pairing energy related to the addition of two neutrons in nuclei with
$N=Z$ from 8Be to 100Sn. We observe trends that are correlated with the change
of the orbitals that are occupied along this set of nuclei. In the regions of
nuclei described by the $ds$ and $fp$ model spaces, the ”universal” effective
Hamiltonians describe the data, as expected, since they are obtained from fits
to energy data in these model spaces. The results obtained from the VS-IMSRG
method in these same model spaces are not good enough to draw conclusions
about the orbital contributions to pairing. Based on the spectra of odd-even
nuclei for $A_{c}\geq 64$, it is important to use a model space that includes
the $0g_{9/2}$ orbital to obtain the pairing and deformed properties. The
computationaly demanding configuration-mixing calculations for $A_{c}\geq 72$
that include the ”dip” at $A_{c}=76$ remain to be carried out.

Acknowledgements: BAB acknowledges support from NSF grant PHY-2110365. The
IMSRG calculations were performed with the codes imsrg$++$ imsrg  and kshell
kshell . The code imsrg$++$ utilizes the Armadillo C$++$ library armadillo1 ,
armadillo2 .

## References

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# The gravitational redshift of solar-type stars from Gaia DR3 wide binaries

Kareem El-Badry  Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Harvard Society of Fellows, 78 Mount Auburn
Street, Cambridge, MA 02138  Max-Planck Institute for Astronomy, Königstuhl
17, D-69117 Heidelberg, Germany

###### Abstract

Light escaping from a gravitational potential suffers a redshift with
magnitude proportional to the depth of the potential. This “gravitational
redshift” is easily measurable in dense stars such as white dwarfs, but is
much weaker and has evaded unambiguous detection in main-sequence stars. I
show that the effect is directly measurable in the Gaia DR3 radial velocities
(RVs) of the components of wide binary stars. In a sample of $\sim$500 wide
binaries containing a solar-type main-sequence star and a red giant or red
clump companion, the apparent RV of the giant is on average $0.49\pm
0.02\,\,\rm km\,s^{-1}$ lower than that of the main-sequence star. This owes
primarily to the giants’ weaker gravitational fields and is in reasonably good
agreement with the value expected from general relativity.

##  1 Introduction

General relativity predicts that light emitted from the surface of a massive
object undergoes a redshift, mimicking the Doppler shift when light is emitted
from a receding object. For a star of mass $M$ and radius $R$, the amplitude
of the induced apparent RV in the weak-field limit is

$v_{\rm GR}=\frac{GM}{Rc}=0.64\,{\rm
km\,s^{-1}}\left(\frac{M}{M_{\odot}}\right)\left(\frac{R}{R_{\odot}}\right)^{-1}.$
(1)

This velocity is large and easily detectable in white dwarfs, where values of
about $30\,\rm km\,s^{-1}$ are typical (e.g. Falcon et al., 2010). In neutron
stars, it is enormous (of order $0.3c$; e.g. Cottam et al., 2002). In main-
sequence stars, it is smaller, but still comfortably within the capabilities
of modern spectrographs.

An obvious challenge in measuring gravitational redshifts is that stars are
not stationary, but have quasi-random velocities along our line of sight, with
typical amplitudes of tens of $\rm km\,s^{-1}$. This stymies detection of weak
gravitational redshifts unless the star’s true radial velocity can be
determined through external information. Wide binary stars offer precisely the
required external information: their orbital velocities are small, such that
to first approximation, the two stars in a wide binary have the same RV, and
any difference between their observed RVs can be attributed to gravitational
redshift. This allowed Greenstein & Trimble (1967) and many subsequent works
to measure gravitational redshifts and masses of white dwarfs in wide binaries
with main-sequence companions.

The situation is more complicated for binaries containing two main-sequence
stars, because (a) the gravitational redshifts of both components are expected
to have similar magnitude, and (b) the orbital velocity and surface convective
shift both have comparable magnitude to the expected gravitational redshift.
This can complicate attempts to precisely measure genuine RV differences in
wide binaries (e.g. Loeb, 2022), but it also suggests that the effects of
gravitational redshift should be detectable in a large sample of wide
binaries. This has recently become possible thanks to the Gaia mission, which
both enables selection of pure wide binaries samples using astrometry, and
measures RVs for the component stars.

The most straightforward approach is to compare the RVs of giants and dwarfs,
as the giants are expected to have much smaller (usually negligible)
redshifts. Pasquini et al. (2011) compared the mean RVs of a large sample of
dwarfs and giants in a star cluster, where the peculiar velocities should
average out. However, they were unable to detect the gravitational redshift
statistically, attributing the non-detection to the confounding effects of
surface convection. More recently, Gutiérrez & Ramos-Chernenko (2022) did
detect a gravitational redshift signal in the RVs of star cluster members, but
only when combining data from many different clusters and fitting a global
model. Similarly, Moschella et al. (2022) recently found tentative evidence of
gravitational redshift in main-sequence wide binaries selected from Gaia DR2
data, but the effect was not obvious because Gaia DR2 contained RVs for both
components of only a few giant/dwarf binaries.

Gaia DR3 (Gaia Collaboration et al., 2022) includes precise RVs to stars
$\approx$ 2 magnitudes fainter than DR2, leading to a much larger usable
sample of giant/dwarf binaries. Here, I leverage these new RVs to measure a
visually obvious signature of gravitational redshift in the velocity
difference of wide binaries’ components.

##  2 Methods

From the wide binary catalog constructed by El-Badry et al. (2021), I selected
binaries satisfying the following cuts:

  * •

The position of the primary (the brighter component) in the color-magnitude
diagram suggests it is a giant or red clump star, satisfying
$M_{G}<2\times(G_{\rm BP}-G_{\rm RP})-1$. I corrected for extinction using the
3D dust map of Green et al. (2019) in the north and that of Lallement et al.
(2019) in the south.

  * •

The position of the secondary suggest it is a dwarf, with
$M_{G}>2\times(G_{\rm BP}-G_{\rm RP})+0.5$.

  * •

The pair is likely to be gravitationally bound, with a chance alignment
probability below 10% (R_chance_align $<$ 0.1). In practice, almost all the
selected binaries have R_chance_align $<$ 0.01.

  * •

Both stars have a RV measurement reported in Gaia DR3, with
radial_velocity_error $<2\,\rm km\,s^{-1}$.

  * •

The pair is separated by at least 5 arcsec, to minimize the effects of
blending on the RVs (Katz et al., 2022). The median projected physical
separation of the sample is 6,000 AU, corresponding to a typical orbital RV
difference of order $0.5\,\rm km\,s^{-1}$ (with no preferred sign).

I next “corrected” the Gaia RVs using the empirical magnitude-dependent RV
zeropoint derived by Katz et al. (2022, their equation 5). The magnitude of
this correction is $<0.1\,\rm km\,s^{-1}$ for a majority of the sample.

These cuts yielded 538 binaries, which are shown on the color-magnitude
diagram in the left panel of Figure 1. The median apparent magnitude of the
giants is $G\approx 8.3$, and that of the dwarfs is $G\approx 12.0$. The
dwarfs are mostly solar-type stars with masses $0.5\lesssim
M/M_{\odot}\lesssim 1.5$. A majority are near the main sequence, but a few
subgiants are also included in the sample. A majority of the “giants” are not
strictly on the giant branch but are core helium burning red clump stars, with
$R\approx 10\,R_{\odot}$ and (most likely) $M\approx 1\,R_{\odot}$. The median
distance to the sample is 350 pc.

Figure 1: Left: color-magnitude diagram of the cleaned sample of 538
giant/dwarf wide binaries. Most of the dwarfs are near the main-sequence with
mass $0.5\lesssim M/M_{\odot}\lesssim 1.5$; most of the “giants” are core
helium burning red clump stars. Center: radial velocity difference between the
giant and dwarf for this full sample. Right: radial velocity difference for
the subset in which both components have radial velocity uncertianties below
$1\,\rm km\,s^{-1}$. The fact that the distributions are not centered on 0
owes to the larger gravitational redshift in the dwarfs.

The main result is shown the center and right panels of Figure 1: there is a
not-very-subtle difference in the mean RV of dwarfs and giants. The inverse-
variance weighted mean RV difference is $\approx 0.49\,\rm km\,s^{-1}$, both
for the full sample (center column) and for the subsample in which both
components have radial_velocity_error $<1\,\rm km\,s^{-1}$ (right column). The
observed RV difference distribution is narrower for latter subsample because
the RV differences for individual binaries are still dominated by measurement
uncertainties.

The expected RV difference due to gravitational redshift is
${\rm\Delta}v_{{\rm GR}}=\frac{G}{c}\left(M_{{\rm giant}}/R_{{\rm
giant}}-M_{{\rm dwarf}}/R_{{\rm dwarf}}\right)$. Assuming $R_{{\rm
dwarf}}\approx R_{\odot}\left(M_{{\rm dwarf}}/M_{\odot}\right)$, $M_{\rm
giant}\approx 1\,M_{\odot}$, and $R_{\rm giant}\approx 10\,R_{\odot}$, this
translates to an expected typical ${\rm\Delta}v_{{\rm GR}}\approx 0.58\,\rm
km\,s^{-1}$, pleasingly close to the observed value.

##  3 Conclusion

I have shown that the different gravitational redshifts of main-sequence and
giant stars leads to an unambiguous shift in their apparent relative RVs in
wide binaries observed by Gaia. I have not attempted to account for star-to-
star variation in $M/R$ within the dwarf and giant samples, or for the effects
of convective shifts that can lead the mean RV of the photosphere to differ
from that of a star’s center of mass. The fact that the observed mean RV
offset is close to the value predicted due to gravitational redshift suggests
that these simplifications are tolerable at the $\lesssim 20\%$ level, at
least in a population-averaged sense, but a more detailed investigation is
certainly warranted.

Given the uncertainties in RV calibration and the still poorly understood
effects of convective shifts, this approach seems unlikely to yield precise
tests of GR in the near future. It is, however, a nice demonstration of the
gravitational redshift phenomenon and of the precision and stability of the
Gaia DR3 RV data.

## Acknowledgements

I thank George Seabroke and David W. Hogg for discussions that prompted this
note. This work was done at the Gaia Hike meeting at the University of British
Columbia in June 2022. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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