# Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

M Rouleux  Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille,
France<EMAIL_ADDRESS>
###### Abstract

We are interested in decay estimates of the ground state (or the low energy
eigenstates), outside the potential wells, for a semi-classical Magnetic
Schrödinger operator with smooth coefficients $P_{A}(x,hD_{x})=(hD_{x}-\mu
A(x))^{2}+V(x)$ on $L^{2}({\bf R}^{d})$. We shall essentially consider the
case where $\mu$ is large. This kind of estimates, in case of Schrödinger
operator without a magnetic field, have been studied by Agmon [1], also in the
case of a Riemannian manifold $M$. Agmon estimates hold true for any $h$, but
are particularly useful in the limit $h\to 0$ when studying tunneling.

##  1 Agmon estimates and tunneling parameters for the ground states

Given a semi-classical Schrödinger operator
$P_{0}(x,hD_{x})=-h^{2}\Delta+V(x)$ on $L^{2}(M)$, $M$ a smooth manifold, we
consider spectral effects induced by adding a magnetic potential (minimal
coupling). For simplicity we assume here that $P_{0}$ and $P_{A}$ have compact
resolvent.

A lot is known on the mathematical theory of magnetic classical and quantum
Hamiltonians : see e.g. [15], [16] and [8], [9] with references therein. The
link between the classical flow and the spectral asymptotics, e.g. trace
formulas, is examined in [15], [16]. This does not require precise information
on eigenfunctions. Estimates on eigenfunctions and WKB solutions considered in
[8], [9] are in turn of special importance for investigating tunneling between
“potential wells”, related to the decay of eigenfunctions. The fact that
eigenfunctions are real or complex also plays a role. Within our framework (an
application is given in [14]), we can summarize some of the main problems
related to Magnetic Schrödinger operators $P_{A}(x,hD_{x})=(hD_{x}-\mu
A(x))^{2}+V(x)$ as follows.

$\bullet$ Problem No.1: Compare the first (non degenerate) eigenvalue
$\lambda_{0}(h)$ of $P_{0}(x,hD_{x})$ with the first eigenvalue
$\lambda_{A}(h)$ (possibly degenerate) of $P_{A}(x,hD_{x})$.

Kato inequality shows that $\lambda_{0}(h)\leq\lambda_{A}(h)$.

More refined properties rely on the topology of $M$:

If $M$ is an open bounded subset of ${\bf R}^{d}$ with smooth boundary (non
necessarily simply connected), then $\lambda_{0}(h)=\lambda_{A}(h)$ iff
$dA(x)=0$ in $M$ and the cohomology class of $A$ verifies $[A]\in H^{1}(M,2\pi
h{\bf Z})$, see [17], [7]. This reminds of Aharonov-Bohm effect when $M={\bf
R}^{3}\setminus{\bf R}(1,0,0)$ but $[A]\notin H^{1}(M,2\pi h{\bf Z})$, in
which case $P_{A}$ has continuous spectrum.

Another remarkable result (which is not to be used here) gives an upper bound
on $\lambda_{A}(h)$. Namely, if $M$ is a compact manifold without boundary,
then $\lambda_{A}(h)$ is bounded by Mañé constant for the Lagrangian $L_{A}$
associated with $P_{A}$: $\lambda_{A}(h)\leq c(L_{A})$, where

$c(L_{A})=\inf_{\phi\in C^{\infty}(M)}\max_{x\in
M}P_{A}(x,{\partial\phi\over\partial x})$

The minimizer $\phi$ is of Lipschitz class. This is investigated within “weak
KAM” or “weak WKB” theory. We have also the bound $\lambda_{A}(h)\leq\min
c(L_{A}-\omega)$, over $[\omega]\in H^{1}(M,2\pi h{\bf Z})$, expressing in
particular the gauge invariance of the spectrum, see [21].

$\bullet$ Problem No.2: Compare the decay of $u_{A}(h)$ with this of
$u_{0}(h)>0$. Generally, $\lambda_{A}(h)$ is simple, but $u_{A}(h)$ is complex
and may have zeroes (vortices in 2-D, threads in 3-D, …).

When $\lambda_{0}(h)=\lambda_{A}(h)$ as above, then
$u_{A}(x;h)=u_{0}(x;h)e^{i\mu(\int^{x}A(y)\,dy)/h}$, so the magnetic potential
with $dA(x)=0$ induces no additional decay.

Let $M$ be an open subset of ${\bf R}^{d}$ with smooth boundary, and
$U_{E}=\\{x\in M:V(x)\leq E\\}$ be the potential well in $M$ at energy $E$.
First we assume that $U_{E}$ is connected.

We focus to the case where $V(x)\geq E=0$ and $U_{E}=\\{x\in
M:V(x)=E\\}=\\{x_{0}\\}$ is a non degenerate minimum (simplest case). Actually
energy level $E=0$ will be lifted by $\lambda_{0}(h)={\cal O}(h)$.

Decay estimates in some global $L^{2}$ norm with exponential weight
$e^{\Phi_{0}(x)/h}$ for an eigenstate $u_{0}(h)$ of $P_{0}$ outside $U_{E}$
are known as Agmon estimates. They hold in any Sobolev norm and hence also
pointwise.

Let us add the vector potential $A$, with $A(x_{0})=0$, and assume
$\rho_{0}=(x_{0},0)\in T^{*}M$ is a non degenerate elliptic point for
$P_{A}(x,\xi)$, that we call the magnetic well.

In case of $P_{A}$ we know already that Agmon type estimates with the same
weight $e^{\Phi_{0}(x)/h}$ hold for $u_{A}(h)$ [13], [7]. Thus we call them
“basic Agmon estimates”. But generally speaking, magnetic fields are
confining, so we can expect that the eigenfunction $u_{A}(h)$ will decay
faster indeed than $u_{0}(h)$ outside $x_{0}$.

Local WKB constructions of $u_{A}(h)$ near the magnetic well, using quadratic
approximation near the elliptic point, are computed in [20]. When $V,A$ are
analytic, they are of the form

$u_{A}(x;h)=e^{-\Psi_{A}(x)/h}\bigl{(}a(x,h)+{\cal
O}(e^{-{\mathop{\rm\varepsilon}\nolimits}_{A}/h})\bigr{)},\ x\in
B_{A}(x_{0},r_{0})$ (1)

and $u_{A}(h)$ decays generally faster than $u_{0}(h)$ in
$B_{A}(x_{0},r_{0})$. Using almost analytic extensions, we can show that (1)
holds true when $A$ and $V$ are merely $C^{\infty}$, provided we replace
${\cal O}(e^{-{\mathop{\rm\varepsilon}\nolimits}_{A}/h})$ by ${\cal
O}(h^{\infty})$.

However, it is difficult to extend these expansions in the larger domain $M$.
To this end, we introduce here some relative Agmon estimates: namely, at least
for sufficiently large coupling constant $\mu$, $u_{A}(h)$ decays in
$L^{2}(M)$ norm as $e^{-\Phi_{1}(x)/h}$ with $\Phi_{1}>\Phi_{0}$, and
$\Phi_{1}\approx{\rm Re}\Psi_{A}$ in $B_{A}$.

$\bullet$ Problem No.3 (tunneling for the double well $\\{x^{<},x^{>}\\}$).
When $M={\bf R}^{d}$ instead, we assume that $P_{0}$ and $P_{A}$ commute with
some hyperplane symmetry $\sigma(x^{\prime},x_{d})=(x^{\prime},-x_{d})$. Let
$M^{>}$ be a sufficiently large open set containing $x^{>}$, but not $x^{<}\in
M^{<}=\sigma(M^{>})$.

In the presence of magnetic wells, we are interested in an estimate (from
above) of the splitting between the two first eigenvalues
$\lambda^{\pm}_{A}(h)$, which, incidentally, may coincide.

Much is known for the splitting (from above and from below) between
$\lambda_{0}^{+}(h)>\lambda_{0}^{-}(h)$, see [12], [18].

In case $V$ has non-degenerate minima at $x^{>,<}\in\\{\pm x_{d}>0\\}$,
$V(x^{>,<})=E=0$ we can construct (quite accurately from the point of vue of
tunneling) approximate eigenfunctions $u_{0}^{\pm}(h)$ for $P_{0}$ in
$L^{2}({\bf R}^{d})$ from the ground state $u_{0}(h)$ of the one-well problem,
i.e. localized at $x_{0}=0$. Here $u_{0}^{-}(h)$ is even in $x$ and given
roughly by
$u_{0}^{-}(x;h)={1\over\sqrt{2}}\bigl{(}u_{0}(x-x^{>};h)+u_{0}(x-x^{<};h)\bigr{)}$,
while $u_{0}^{+}(h)$ is odd in $x$ and of the form
$u_{0}^{+}(x;h)={1\over\sqrt{2}}\bigl{(}u_{0}(x-x^{>}+;h)-u_{0}(x-x^{<};h)\bigr{)}$
Here we have assumed for simplicity that $u_{0}$ is even in $x$, see [6] for
the correct formula.

Define as a libration $\gamma(h)$ a minimal geodesic for Agmon distance at
energy $\lambda_{0}(h)$, with $\lambda_{0}(h)={\cal O}(h)$, connecting
$U^{>,<}(h)=\\{x\in M^{>,<}:V(x)\leq\lambda_{0}(h)\\}$, see e.g. [3], [2]. An
instanton $\gamma$ is a limiting curve of the family of librations, as $h\to
0$, i.e. as $U^{>,<}(h)$ shrinks to $\\{x^{>,<}\\}$, the well at energy 0.
This is a minimal geodesic for Agmon distance between $x^{<}$ and $x^{>}$ at
energy 0. Instantons form generically a discrete set.

Localizing Agmon estimates near $\gamma$ gives [12], [18]: for all $\delta>0$

$\Delta\lambda_{0}(h)=\lambda^{+}_{0}(h)-\lambda^{-}_{0}(h)={\cal
O}_{\delta}\bigl{(}\exp\bigl{[}-(d_{0}(x^{>},x^{<})-\delta)/h\bigr{]}\bigr{)}$
(2)

when $0<h<h_{\delta}$. Here $d_{0}(x^{>},x^{<})$ is Agmon distance between
$x^{>}$ and $x^{<}$, that satisfies eikonal equality $|\nabla
d_{0}(x^{<},x)|^{2}=|\nabla d_{0}(x,x^{>})|^{2}=V(x)$ (here we use the
stable/unstable manifold theorem). We shall content ourselves with leading
exponential estimates, allowing for arbitrary small loss $\delta$ in the
exponent.

Introduce now the magnetic potential. Again the localization procedure works,
and it suffices to know the ground state $u_{A}(h)$ of the one-well problem.

For very small ($h$ dependent) coupling constant $\mu$, WKB expansions of
$u_{A}(h)$ yield an estimate similar to (2) on the splitting
$\Delta\lambda_{A}(h)$ [13]. For small coupling constant $\mu$ (but
independent of $h$) we need analyticity of $V,A$ to ensure exponential
accuracy on WKB solutions.

Under some suitable hypotheses, we expect that our relative Agmon estimates
approximate $\Delta\lambda_{A}(h)$ in a similar way to (2), and moreover

$\liminf_{h\to
0}\bigl{(}-h\log\Delta\lambda_{A}(h)+h\log\Delta\lambda_{0}(h)\bigr{)}>0$ (3)

This holds actually in 2 regimes: (1) the coupling constant $\mu$ is
sufficiently large, so that the decay of $u_{A}(h)$ is little sensitive to
$V$; (2) $\mu^{2}|A(x)|^{2}-V(x)$ is positive and small enough.

Recent results in the 2-D case of compactly supported and radially symmetric
potential wells, and constant $B$ ([4], [10]), express the rate of exponential
decay in term of the hopping formula. See also [11], and [5] when $V=0$.

##  2 Precise statements and outline of proofs

1) General Agmon identities in the case of a magnetic potential

The scalar and magnetic potentials $V,A$ will be assumed to be $C^{\infty}$.
Let $\Omega\subset{\bf R}^{d}$ be a bounded open set with $C^{2}$ boundary.

We will consider weighted integrals, related with Dirichlet forms, like
$\int_{\Omega}|(hD_{x}-\mu A(x))e^{\Phi(x)/h}u|^{2}\,dx$ or
$\int_{\Omega}|(hD_{x}-\mu A(x)+ihf(x))e^{\Phi(x)/h}u|^{2}\,dx$ for a suitable
real vector field $f$.

When $u$ is a complex $C^{2}$ function on $\Omega$, we define the real vector
field (quantum current with $U(1)$ symmetry)

$J_{u}=-i(\nabla u\cdot\overline{u}-\nabla\overline{u}\cdot u)$

It is not a priori integrable on $\Omega$ when $\Omega$ is not simply
connected but if $u$ is not vanishing, then $J_{u}=|u|^{2}\nabla\arg u^{2}$.
Using Green formula one can show:

Proposition 1: Let $\Phi=\Phi_{R}+i\Phi_{I}$ be a complex Lipschitz function
defined on $\Omega$. For all $u\in C^{2}(\overline{\Omega};{\bf C})$, we have

$\displaystyle\int_{\Omega}|(hD_{x}-\mu A(x))e^{\Phi(x)/h}u|^{2}\,dx+$ (4)
$\displaystyle+\int_{\Omega}e^{2\Phi_{R}(x)/h}\bigl{(}V(x)-|\nabla\Phi_{R}(x)|^{2}-|\nabla\Phi_{I}(x)|^{2}+2\mu\langle\nabla\Phi_{I}(x),A(x)\rangle\bigr{)}|u(x)|^{2}\,dx$
$\displaystyle-h\int_{\Omega}e^{2\Phi_{R}(x)/h}\langle
J_{u}(x),\nabla\Phi_{I}(x)\rangle\,dx=$ $\displaystyle{\rm
Re}\int_{\Omega}e^{2\Phi_{R}(x)/h}P_{A}(x,hD_{x})u(x)\cdot\overline{u}(x)\,dx+h^{2}{\rm
Re}\int_{\partial\Omega}e^{2\Phi_{R}/h}\overline{u}{\partial
u\over\partial\nu}\,d\sigma$

where $\nabla\Phi\in L^{\infty}(\Omega)$ is understood in the usual sense, and
$d\sigma$ is the surface measure on $\partial\Omega$.

As a particular case $\Phi_{I}=0$, we retrieve the classical formula [7], p.95
which will give the “basic” Agmon estimate :

Corollary 2: Let $\Phi=\Phi_{R}$ be a Lipschitz real function defined on
$\Omega$. For all $u\in C^{2}(\overline{\Omega};{\bf C})$, we have

$\displaystyle\int_{\Omega}$ $\displaystyle|(hD_{x}-\mu
A(x))e^{\Phi(x)/h}u|^{2}\,dx+\int_{\Omega}e^{2\Phi(x)/h}\bigl{(}V(x)-|\nabla\Phi(x)|^{2}\bigr{)}|u(x)|^{2}\,dx$
(5) $\displaystyle={\rm
Re}\int_{\Omega}e^{2\Phi(x)/h}P_{A}(x,hD_{x})u(x)\cdot\overline{u}(x)\,dx+h^{2}{\rm
Re}\int_{\partial\Omega}e^{2\Phi(x)/h}\overline{u}{\partial
u\over\partial\nu}(x)\,d\sigma(x)$

2) Standard Agmon estimate without a magnetic potential: a review

Assume also $A=0$, and consider Dirichlet realization of
$P_{0}(x,hD_{x})=-h^{2}\Delta+V(x)$ in a domain $M\subset{\bf R}^{d}$. For
simplicity we assume $M$ to be bounded. Let $u_{0}(h)$ be a normalized
eigenfunction, associated with the eigenvalue $\lambda_{0}(h)\in]0,Ch]$. We
apply Corollary 2 to $\Omega=M$. Assume here that $V$ has a single potential
well $U_{0}(h)=\\{x\in M:V(x)\leq\lambda_{0}(h)\\}$ at energy
$\lambda_{0}(h)$, let $\Phi_{0}(x)=d_{0}(x,U_{0}(h))$ be Agmon distance of $x$
to $U_{0}(h)$, i.e. satisfy the differential inequality (only when
“$\lambda_{0}(h)=0$” we have an equality)

$|\nabla\Phi_{0}(x)|^{2}\leq\bigl{(}V(x)-\lambda_{0}(h)\bigr{)}_{+}$ (6)

Then Corollary 2 for $A=0$ implies the following standard Agmon estimate.
Namely, there is $a(\delta)>0$, with $a(\delta)\to 0$ as $\delta\to 0$, such
that

$h^{2}\int_{M}|\nabla
e^{\Phi_{0}(x)/h}u_{0}(x;h)|^{2}\,dx+\int_{M}e^{2\Phi_{0}(x)/h}|u_{0}(x;h)|^{2}\,dx\leq
C_{\delta}e^{2a(\delta)/h}$ (7)

for $0<h\leq h_{\delta}$. Using the equation
$P_{0}(x,hD_{x})u_{0}(h)=\lambda_{0}(h)$, this $L^{2}$ estimate carries to any
Sobolev space $H^{s}(M)$, and by Sobolev embedding theorem, also to
$L^{\infty}$ estimates.

For the symmetric double well problem in a “large” bounded domain
$M\subset{\bf R}^{d}$, with a single well $U^{>,<}(h)$ on each side of the
hyperplane $x_{d}=0$. consider instead Dirichlet realization of
$P_{0}(x,hD_{x})=-h^{2}\Delta+V(x)$ , containing sub-domains $M^{>,<}\subset
M$ as above symmetric of each other with respect to $x_{d}=0$,
$U^{>,<}(h)\subset M^{>,<}$, but $U^{>}(h)\cap M^{<}=\emptyset$.

We set
$\widetilde{\Phi}_{0}(x)=\min\bigl{(}d_{0}(x,U^{>}(h)),d_{0}(x,U^{<}(h))\bigr{)}$.

Then (7) holds true for $\widetilde{\Phi}_{0}(x)$ instead of $\Phi_{0}(x)$,
see also [12], [7] when $M={\bf R}^{d}$.

3) Standard Agmon estimates with a magnetic potential

Consider next the Dirichlet realization of $P_{A}(x,hD_{x})=(hD_{x}-\mu
A(x))^{2}+V(x)$ in the domain $M\subset{\bf R}^{d}$, $\lambda_{A}(h)$ its
ground state energy (possibly degenerate), and $u_{A}(h)$ an eigenfunction
associated with $\lambda_{A}(h)$. As before we start with the “magnetic one-
well problem”. Let $\Omega$ be the complement of a (euclidian) ball
$B_{A}\subset M$ (the projection of the “magnetic well”).

We have the “basic” Agmon estimate for $u_{A}(x;h)$, which we express as in
(7). Namely for all $\delta>0$, there is $C_{\delta}>0$ such that, uniformly
for $0<h\leq h_{\delta}$

$\displaystyle\int_{\Omega}|(hD_{x}-\mu
A(x))e^{\Phi_{0}(x)/h}u_{A}(x;h)|^{2}+\int_{\Omega}e^{2\Phi_{0}(x)/h}|u_{A}(x;h)|^{2}\leq$
(8) $\displaystyle
C_{\delta}\sup_{\Omega}e^{2\delta\Phi_{0}/h}\int_{\partial\Omega}e^{2(1-\delta)\Phi_{0}(x)/h}|\overline{u}_{A}{\partial
u_{A}\over\partial\nu}|$

At this point we use the precise local decay estimates (1) of $u_{A}(h)$ to
control the RHS of (8). Recall $\Psi_{A}$ is a (possibly complex) phase with
real part that we shall assume strictly larger than $\Phi_{0}$ on $\partial
B_{A}$, at least for large $\mu$. This assumption is supported by the
following example on $L^{2}({\bf R}^{2})$

$P_{A}(x,hD_{x})=(hD_{x_{1}}+{\mu\over 2}x_{2})^{2}+(hD_{x_{2}}-{\mu\over
2}x_{1})^{2}+x_{1}^{2}+x_{2}^{2}$ (9)

with ground state of the form $u_{A}(x;h)={\rm Const.}\exp[-c(\mu)x^{2}/2h]$,
and ground state energy $\lambda_{A}(h)=c(\mu)h=(1+\mu^{2}/4)^{1/2}h$.
Actually this extends easily to higher dimensions, see [22].

After some manipulation, (8) reduces to an estimate like (7) where the RHS can
be replaced by ${\cal O}(e^{-\mathop{\rm\varepsilon}\nolimits_{1}h})$, for
some $\mathop{\rm\varepsilon}\nolimits_{1}>0$, provided $u_{A}(x;h)$ decreases
indeed faster (as in the latter example) than $e^{-\Phi_{0}(x)/h}$ on
$\partial B_{A}$.

4) R.Lavine and M.O.Carroll’s formula and relative Agmon estimate

Contrary to (5) that suggests to introduce Agmon distance $d_{0}$ as a natural
geometric guideline, (4) encodes no direct geometric information for complex
$\Phi$. This reflects the intrication between the cyclotronic motion and the
magnetic drift at the classical level. We only know [16] that these motions
are relatively well decoupled for large $\mu$. Thus we can expect that in this
regime to factor out the wave function as $u_{A}(h)=u_{0}(h)v(h)$ and find
Agmon type estimates on $v(h)$. Recall the following formula [17]:

Let $A,f$ be real vector fields locally square integrable on ${\bf R}^{d}$,
then for all $\widetilde{u}\in C^{\infty}_{0}({\bf R}^{d};{\bf C})$ we have

$\int_{{\bf R}^{d}}|(hD_{x}-\mu A(x)+ihf)\widetilde{u}|^{2}\,dx=\int_{{\bf
R}^{d}}|(hD_{x}-\mu A(x))\widetilde{u}|^{2}\,dx+h^{2}\int_{{\bf R}^{d}}(\nabla
f+|f|^{2})|\widetilde{u}|^{2}\,dx$ (10)

Let $(\lambda_{0}(h),u_{0}(h))$ be the ground state of the Dirichlet
realization of $P_{0}$ in $M$, then $f=\nabla u_{0}/u_{0}$ is a natural
choice, for

$h^{2}\nabla\cdot{\nabla u_{0}\over u_{0}}=h^{2}{\Delta u_{0}\over
u_{0}}-h^{2}\big{|}{\nabla u_{0}\over
u_{0}}\big{|}^{2}=V-\lambda_{0}(h)-h^{2}\big{|}{\nabla u_{0}\over
u_{0}}\big{|}^{2}$

so that $h^{2}(\nabla f+|f|^{2})|u|^{2}=(V(x)-\lambda_{0}(h))|u|^{2}$.

Let as above $\lambda_{A}(h)$ (we assume to be non degenerate) be the ground
state energy of of $P_{A}(x,hD_{x})$ on $L^{2}(M)$ with Dirichlet boundary
condition, and $u_{A}(h)$ the corresponding normalized eigenfunction, which we
write as $u_{A}(h)=u_{0}(h)v(h)$. Suitably modifying (10) with
$\widetilde{u}=e^{\Phi/h}u_{A}(h)$ to account for boundary terms on
$\Omega\subset M$, we can express $\int_{{\bf R}^{d}}|(hD_{x}-\mu
A(x)+ihf)\widetilde{u}|^{2}\,dx$ in term of a Dirichlet form
$\int_{\Omega}e^{2\Phi_{R}(x)/h}u_{0}^{2}(x;h)|(h\nabla+\nabla\Phi_{R})v(x;h)|^{2}\,dx$
and get :

Proposition 3: Let $\Phi_{R},\Phi_{I}$ be Lipschitz real functions defined on
$\Omega$. If $u_{A}(h)=u_{0}(h)v(h)$, we have

$\displaystyle\int_{\Omega}e^{2\Phi_{R}(x)/h}u_{0}^{2}(x;h)|(h\nabla+\nabla\Phi_{R})v(x;h)|^{2}\,dx+$
(11)
$\displaystyle\int_{\Omega}e^{2\Phi_{R}(x)/h}u_{0}^{2}(x;h)\bigl{(}-|\nabla\Phi_{R}(x)|^{2}+|A(x)|^{2}-\lambda_{A}(h)+\lambda_{0}(h))\bigr{)}|v(x;h)|^{2}\,dx-$
$\displaystyle h\int_{\Omega}e^{2\Phi_{R}(x)/h}u_{0}^{2}(x;h)\langle
J_{v}(x),2\nabla\Phi_{I}(x)-A(x)\rangle\,dx=$ $\displaystyle h^{2}{\rm
Re}\int_{\partial\Omega}e^{2\Phi(x)/h}\overline{u}_{A}{\partial
u_{A}\over\partial\nu}(x)-h^{2}\int_{\partial\Omega}e^{2\Phi_{R}(x)/h}|u_{A}|^{2}\langle
f,\nu\rangle$

To get rid of the extra term on the RHS we try to solve

$\langle J_{v}(x),2\nabla\Phi_{I}(x)-A(x)\rangle=0,\quad x\in M$ (12)

by

$2\langle J_{v}(x),\nabla\Phi_{I}(x)\rangle=\langle J_{v}(x),A(x)\rangle$

This can be done if $J_{v}(x)\neq 0$ is an integrable field in the simply
connected open set $M$. When $u_{A}(h)$ is non vanishing in $M$,
$J_{v}(x)=|u_{A}(x;h)|^{2}\nabla\arg u_{A}^{2}(x;h)$, and $\arg
u_{A}^{2}(x;h)$ is a smooth function. Vanishing of $u_{A}$ maybe an
obstruction for solving (12), in particular when the zeroes $x_{j}$ of
$J_{v}(x)$ are vortices. We can also try to solve (12) approximately along a
magnetic line $\gamma$, namely look for $\Phi_{I}$ such that the 1-form
$2d\Phi_{I}(x)-\mu A(x)$ vanishes on $\gamma$.

Once the contribution of $J_{v}(x)$ in (12) has been removed, we are left with
an expression similar to (5), with $|A(x)|^{2}$ instead of $V(x)$. Provided
$A\neq 0$ outside the well, we solve an eikonal inequality

$|\nabla\Phi_{1}|^{2}\leq\bigl{(}\mu^{2}|A(x)|^{2}-\lambda_{A}(h)+\lambda_{0}(h)\bigr{)}_{+}$
(13)

and put $\Phi_{R}=(1-\delta)\Phi_{1}$. This yields again an Agmon estimate of
the form (8), with $\Phi_{1}$ instead of $\Phi_{0}$. On the other hand, we
know that $u_{0}(x;h)$ is everywhere of the same order of magnitude as
$e^{-\Phi_{0}(x)/h}$. More precisely, if $0\leq V\in C^{\infty}(M)$ and
$u_{0}$ be the normalized ground state of $P_{0}=-h^{2}\Delta+V(x)$ on
$L^{2}(M)$, then for all $\delta>0$, $u_{0}(x;h)\geq
C_{\delta}e^{-(d_{0}(x,U_{h})+\delta)/h}$ locally uniformly on any compact set
$K\subset M\setminus U_{0}(h)$, see e.g. [19]; moreover in a neighborhood of
the quadratic well $x_{0}$, we have $|u_{0}(x;h)|\leq
Ch^{-N}e^{-\Phi_{0}(x)/h}$ (as well as all derivatives), see [7], Proposition
3.3.5.

In turn, this gives again an estimate like (7), so there is a set of two
(independent) Agmon estimates for $u_{A}(h)$. Let
$\Phi_{A}(x)=\max(\Phi_{0}(x),\Phi_{1}(x))$, which amounts to consider the
(Lipschitz) distance $d_{A}(x)=\max(d_{0}(x,U_{0}(h)),d_{1}(x,U_{1}(h))$,
where $U_{0}(h)=\\{x\in M:V(x)-\lambda_{0}(h)\leq 0\\}$, $U_{1}(h)=\\{x\in
M:\mu^{2}|A(x)|^{2}-\lambda_{A}(h)+\lambda_{0}(h)\leq 0\\}$. Thus we proved:

Theorem 4: Let $u_{A}(h)$ be an eigenfunction of $P_{A}$ on $L^{2}(M)$ with
eigenvalue $\lambda_{A}(h)$. Assume we have solved (12). Then

$\int_{\Omega}|\nabla
e^{\Phi_{A}(x)/h}u_{A}(x;h)|^{2}\,dx+\int_{\Omega}e^{2\Phi_{A}(x)/h}|u_{A}(x;h)|^{2}\,dx\leq
C_{\delta}e^{2a(\delta)/h}$ (14)

with $a(\delta)\to 0$ as $\delta\to 0$. Moreover, if the real part of
$\Psi_{A}(x)=\Psi_{A}(x,\mu)$ is increasing with $\mu$ on $\partial
B_{A}(x_{0},r_{0})$ as in Example (9), then the remainder term ${\cal
O}(e^{2a(\delta)/h})$ can be improved to ${\cal
O}(e^{-\mathop{\rm\varepsilon}\nolimits_{2}/h})$ for $\mu$ large enough,
$\mathop{\rm\varepsilon}\nolimits_{2}>0$.

This is consistent with (9) where
$\Psi_{A}=\sqrt{\Phi_{0}^{2}+\Phi_{1}^{2}}\geq\Phi_{A}$. But it is clear that
our method cannot capture the precise decay of (9).

The point is that in general $d_{A}$ is no longer a (degenerate) Riemannian
metric. This situation is met however in the following cases, discarding the
energy shifts $\lambda_{0}(h)$ or $\lambda_{A}(h)-\lambda_{0}(h)$ in the
eikonal inequalities (6) and (13) : (1) $\Phi_{A}=\Phi_{1}$ almost everywhere
(i.e. the magnetic potential is more confining than the scalar potential); (2)
the potentials verify $\mu^{2}|A(x)|^{2}=V(x)$, or $\mu^{2}|A(x)|^{2}-V(x)$ is
sufficiently small. The latter case can be treated by perturbation theory,
since the trajectories for $\xi^{2}-V(x)$ at energy $\lambda_{0}(h)$ and
$\xi^{2}-|A(x)|^{2}$ at energy $\lambda_{A}(h)-\lambda_{0}(h)$ will be
comparable, in particular near the set of minimal geodesics of either system.

##  3 Application to tunneling

We present here the main ideas towards (3). As usual [13] we need first an
assumption on the spectrum (spectral gap). Let $\lambda_{A}^{\pm}(h)$ be the
two first eigenvalues of $P_{A}$ on $L^{2}({\bf R}^{d})$,
$\mu_{A}(h)=\mu_{A}^{>}(h)=\mu_{A}^{<}(h)$ this of the Dirichlet realization
of $P_{A}^{M^{\pm}}$ on the localized domains $M=M^{>,<}$ as above. Assume
$\mu_{A}(h)$ is simple and asymptotically simple, see e.g. [19]. As a rule, we
expect the splitting
$\Delta\lambda_{A}(h)=\lambda_{A}^{+}(h)-\lambda_{A}^{-}(h)$ to be ${\cal
O}(e^{-S_{A}/h})$ for some “Agmon distance” $S_{A}>0$ between the magnetic
wells, to be computed in term of the ground state $u^{>,<}$ of
$P_{A}^{M^{>,<}}$, see [13], Theorem 3.1 and Remark 3.7. The “gap formula” is
thus of the form

$\displaystyle\Delta$
$\displaystyle\lambda_{A}(h)=h^{2}\int_{\Gamma}\bigl{(}\,\overline{u^{>}}(y)\partial_{n}u^{<}(y)-u^{<}(y)\partial_{n}\overline{u^{>}}(y)\bigr{)}\,dS(y)+$
(15) $\displaystyle h\int_{\Gamma}\bigl{(}\,\overline{u^{>}}(y)\langle
A(y),n(y)\rangle u^{<}(y)-\overline{u^{<}}(y)\langle A(y),n(y)\rangle
u^{>}(y)\bigr{)}\,dS(y)$

where $\Gamma\subset\\{x_{d}=0\\}$ is a “geodesic bisector” and $n$ a normal
to $\Gamma$. The eigenvalues $\lambda_{A}^{\pm}(h)$ are exponentially close to
the corresponding $\mu_{A}(h)$, and the functions $u^{>},u^{<}$ are related by
the hyperplane symmetry.

In the case $\Phi_{A}=\Phi_{1}$, we apply Theorem 4 to the functions
$u^{>},u^{<}$, which allows to bound from above the first two traces of
$u^{>},u^{<}$ in $L^{2}(\Gamma)$ with the exponential weight
$e^{-\Phi_{A}/h}$. This is essentially done as in the situation without the
magnetic potential. This gives $\Delta\lambda(h)={\cal O}(e^{-(S_{A}-c)/h})$,
$c$ arbitrary small, where $S_{A}$ is (ordinary) Agmon distance between the
wells $x^{>}$ and $x^{<}$ for the potential $|A(x)|^{2}$.

The case $\mathop{\rm\varepsilon}\nolimits W(x)=\mu^{2}|A(x)|^{2}-V(x)$ small
(where we can set $\mu=1$) is more difficult since we must work directly from
(11). We can replace $\Omega$ by a neighborhood $\Omega_{1}$ of a minimal
geodesic between $x^{>}$ and $x^{<}$ common to $|A(x)|^{2}$ and $V(x)$ when
they are equal, and use perturbation theory in
$\mathop{\rm\varepsilon}\nolimits$. We expect again
$\Delta\lambda_{A}(h)={\cal O}(e^{-(S_{A}-c)/h})$, $c$ arbitrary small, where
$S_{A}$ is, at 0-order in $\mathop{\rm\varepsilon}\nolimits$, Agmon distance
between the wells $x^{>}$ and $x^{<}$ for the potential $|A(x)|^{2}=V(x)$.
Moreover when $W(x)>0$, $\Phi_{R}$ should be strictly larger than $\Phi_{0}$
in $\Omega_{1}$, which implies (3).

Acknowledgements: I thank Bernard Helffer for his useful remarks and
interesting references.

References

## References

  * [1] Agmon S 1982 Lectures on exponential decay of solutions of second order elliptic equation (Princeton: Princeton Univ. Press)
  * [2] Anikin A, Dobrokhotov S and Nosikov 2022 Th. Math. Phys. 213 (1) p.1453-1476
  * [3] Anikin A, Rouleux M 2014 Days of Diffraction 2014 Saint-Petersburg Conference Proc., IEEE, p.17-22
  * [4] Fefferman C, Shapiro J and Weinstein M 2022 SIAM J. Math. Analysis 54 (1), p.1105-1130.
  * [5] Fournais S, Morin L and Raymond N 2023 Purely magnetic tunnelling between radial magnetic wells Preprint arXiv:2308.04315
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  * [9] Helffer B 2009 Introduction to semi-classical methods for the Schrödinger operator with magnetic field Theoretical and applied aspects of some PDEs from geometry or physics, ed A. El Soufi and M. Jazar 17 Seminars and Congresses, Soc. Math. France.
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  * [13] Helffer B, Sjöstrand J 1987 Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (4) p.625-657
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  * [18] Martinez A 1987 J. Math. Pures Appl. 66, p.195-215
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  * [22] Shigekawa I 1987 J. Funct. Analysis 75 p.92-127


# Equivariant $K$-theory of even-dimensional complex quadrics

Bidhan Paul  Department of Mathematics, Indian Institute of Technology,
Madras, Chennai 600036, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to describe the torus equivariant $K$-ring of even-
dimensional complex quadrics by studying the graph equivariant $K$-theory of
their corresponding GKM graphs. This involves providing a presentation for its
graph equivariant $K$\- ring in terms of generators and relations. This
parallels the description of the equivariant cohomology ring of even-
dimensional complex quadrics due to Kuroki.

###### Key words and phrases:

Complex Quadrics, GKM manifold, GKM graph, Equivariant $K$\- theory

###### 1991 Mathematics Subject Classification:

55N15, 14M15, 19L99

##  1\. Introduction

A GKM manifold is defined as an _equivariantly formal_ manifold $M^{2k}$
endowed with an action of a compact torus $T:=(S^{1})^{n}$. This action
satisfies the condition that the set of $0$ and $1$-dimensional orbits have
the structure of a graph. This class of objects was initially introduced by
Goresky, Kottwitz, and MacPherson in [6] as a class of algebraic varieties
(GKM derives from their initials).

Inspired by their work, Guillemin and Zara [9] introduced the concept of an
_abstract GKM graph_ as an abstract graph with edges labeled by vectors in the
dual of Lie algebra of $T$. In particular, an abstract GKM graph is a
combinatorial counterpart of GKM manifolds and helps to study various
topological and geometric properties of GKM manifolds (see e.g. [9], [7], [5],
[11]) using combinatorial properties of GKM graph. On the other hand, abstract
GKM graphs themselves have gathered attention beyond their geometric
motivations (see e.g. [3, 4, 12, 14, 17, 20]). These concepts gave rise to a
rich field in mathematics commonly referred to as _GKM theory_. This theory
has been explored from both geometric and combinatorial perspectives and has
found applications in various areas, such as representation theory (e.g [2]).

In [6], Goresky, Kottwitz, and MacPherson showed that the equivariant
cohomology ring $H_{T}(M)$ of a GKM pair $(M,\leavevmode\nobreak\ T)$ can be
computed by studying the graph equivariant cohomology of the associated GKM
graph $(\Gamma,\alpha)$. Later, Knutson and Rosu proved the much harder part
[18, Theorem A.4, Corollary A.5] that the results remain true for the
equivariant $K$\- theory ring $K_{T}(M)$ as well (see, (2.4)). It is
noteworthy that although Knutson and Rosu primarily focused on equivariant
$K$-theory with complex coefficients in [18], the validity of (2.4) persists
with integer coefficients due to the works of Vezzosi and Vistoli [19,
Corollary 5.10], which is an algebraic analogue of [18, Theorem A.4] (this
fact is also remarked in [18, p.447]).

A complex quadric in dimension $2n$ is defined to be the zero locus of a non-
degenerate quadratic form in $2n+2$ complex variables and can be represented
as

$Q_{2n}:=\Big{\\{}[x_{1}:\cdots:x_{2n+2}]\in\mathbb{CP}^{2n+1}\leavevmode\nobreak\
|\leavevmode\nobreak\ \sum_{j=1}^{n+1}x_{i}x_{2n+3-i}=0\Big{\\}}.$

There exists the natural $T^{n+1}$\- action on $Q_{2n}$

$[x_{1}:\cdots:x_{2n+2}]\cdot(t_{1},\ldots,t_{n+1}):=[x_{1}t_{1}:\cdots:x_{n+1}t_{n+1}:t_{n+1}^{-1}x_{n+2}:\cdots:t_{1}^{-1}x_{2n+2}].$
(1.1)

where $(t_{1},\ldots,t_{n+1})\in T^{n+1}$. Since, $Q_{2n}$ is diffeomorphic
with the Grassmannian $\mathbb{R}_{2,2n}\cong SO(2n+2)/(SO(2n)\times SO(2))$
of oriented 2-planes through the origin in $\mathbb{R}^{2n+2}$, the above
action is equivalent to a maximal torus action $T^{n+1}$ which is induced by
restricting the transitive $SO(2n+2)$\- action. Further, since $SO(2n)\times
SO(2)\subset SO(2n+2)$ is a subgroup of maximal rank (i.e $T^{n+1}$ is also a
maximal torus of $SO(2n+2)$), the set of $0$ and $1$-dimensional orbits of
$T^{n+1}$ action have the structure of a graph (see, [7]). Therefore, the GKM
graph of $Q_{2n}$ with $T^{n+1}$\- action (1.1) can be determined by labeling
the edges with tangential representations. Note that, the torus action in
(1.1) is not effective as it has a non-trivial kernel $\mathbb{Z}_{2}=\\{\pm
1\\}$. Alternatively, one can consider the
$T^{n+1}/\mathbb{Z}_{2}\leavevmode\nobreak\ (\cong T^{n+1})$-action on
$Q_{2n}$, which is effective (see, Section 3).

In [15, 16], H. Lai studied the ordinary cohomology ring (over $\mathbb{Z}$)
and homotopy groups of even-dimensional complex quadrics $Q_{2n}$ explicitly.
Since $H^{odd}(Q_{2n})=0$ by [16, Theorem 3.2], $Q_{2n}$ is an equivariantly
formal GKM manifold (see, [6]). Therefore, the equivariant $K$-theory
$K_{T^{n+1}}(Q_{2n})$ (resp. equivariant cohomology $H_{T^{n+1}}(Q_{2n})$) of
the effective $T^{n+1}$ -action can be determined from the graph equivariant
$K$\- theory (resp. graph equivariant cohomology) of its GKM graph .

Recently, in [13], S. Kuroki studied the GKM graph of $Q_{2n}$ under the
effective $T^{n+1}$\- action and computed the graph equivariant cohomology of
their GKM graphs. In particular, the author has provided a ring structure for
the equivariant cohomology, presenting it in the form of generators and
relations.

Our main aim of this paper is to determine the equivariant $K$\- theory of the
effective $T^{n+1}$\- action on $Q_{2n}$ by explicitly describing its
generators and relations in terms of the GKM graphs. This parallels the
description of the equivariant cohomology ring of even-dimensional complex
quadrics due to Kuroki [13].

This article is structured as follows : In Section 2, we provide an overview
of fundamental concepts related to GKM manifolds, integral GKM graphs, and
their $K$-ring. Moving on to Section 3, we define even-dimensional complex
quadrics $Q_{2n}$ and present the associated GKM graph $\mathscr{Q}_{2n}$
arising from the effective $T^{n+1}$-action.

In Section 4, we introduce the generators $M_{v}$ and $\Delta_{P}$ of
$K(\mathscr{Q}_{2n})$ and establish four types of relations among them.
Additionally, we explore various properties of $M_{v}$ and $\Delta_{P}$ in
this section. Subsequently, in Section 5, we provide a proof for the main
theorem of this paper (Theorem 5.1).

##  2\. Preliminaries

In this section, we review some basic facts of GKM manifolds (see, [10]),
integral GKM graphs (see, [9]), and the $K$-ring of integral GKM graphs (see,
[8]).

Let $M$ be a compact manifold of real dimension $2k$. Let $T$ be a compact
$n$-dimensional torus of dimension $n$, $\mathfrak{t}$ be its Lie algebra
($\mathfrak{t}^{*}$ be the dual of $\mathfrak{t}$), which acts effectively on
$M$. A pair $(M,T)$ is said to be a _GKM manifold_ if the following conditions
are satisfied :

  * •

the fixed point set $M^{T}$ is finite.

  * •

$M$ possess a $T$\- invariant almost complex structure.

  * •

for each element $p\in M^{T}$, the weights
$\\{\alpha_{i,p}\in\mathfrak{t}^{*}\,:\,1\leq i\leq k\\}$ of the isotropy
representation $T_{p}M$ of $T$ are pairwise linearly independent.

###  2.1. An integral GKM graph and its $K$\- ring

Let $\Gamma=({\mathcal{V}},\leavevmode\nobreak\ {\mathcal{E}})$ be a regular
graph of degree $d$, where ${\mathcal{V}}$ and ${\mathcal{E}}$ are the set of
vertices and the set of oriented edges respectively. For an oriented edge
$e\in\mathcal{E}$, we define $i(e)$ (resp, $t(e)$) as the initial (resp,
terminal) vertex of $e$. Moreover, we let $\overline{e}$ to be the edge with
direction opposite to $e$. Let ${\mathcal{E}}_{p}\subset{\mathcal{E}}$ be the
set of edges whose initial vertex is $p$.

######  Definition 2.1 (Connection).

Let $e=(pq)\in{\mathcal{E}}$ be an oriented edge. A bijection
$\nabla_{e}:{\mathcal{E}}_{p}\to{\mathcal{E}}_{q}$ with
${\nabla_{e}}(e)=\overline{e}$ is said to be a connection along $e$.
Furthermore, we define a _connection_ on graph $\Gamma$ to be a collection
$\nabla:=\\{\nabla_{e}\\}_{e\in{\mathcal{E}}}$ satisfying
$\nabla_{e}^{-1}=\nabla_{\overline{e}}$ for every $e\in{\mathcal{E}}$.

Let $(\mathfrak{t}_{\mathbb{Z}})^{*}$ be the weight lattice of torus $T$.

######  Definition 2.2 (Integral axial function).

Let $\nabla$ be a connection on graph $\Gamma$. A map
$\alpha:{\mathcal{E}}\to(\mathfrak{t}_{\mathbb{Z}})^{*}$ is called a _$\nabla$
-compatible integral axial function_ on $\Gamma$ if it satisfies the following
conditions :

  1. (1)

$\alpha(\overline{e})=-\alpha(e)$ for all $e\in{\mathcal{E}}$.

  2. (2)

the vectors $\\{\alpha(e)\,:\,e\in{\mathcal{E}}_{p}\\}$ are pairwise linearly
independent for all $p\in{\mathcal{V}}$.

  3. (3)

for every edge $e=(pq)$ and every $e^{\prime}\in{\mathcal{E}}_{p}$, there
exists $m(e,e^{\prime})\in\mathbb{Z}$ depending on $e,\,e^{\prime}$ such that

$\alpha(\nabla_{e}(e^{\prime}))-\alpha(e^{\prime})=m(e,e^{\prime})\alpha(e).$

An _integral axial function_ on $\Gamma$ is a $\nabla$-compatible integral
axial function for some connection $\nabla$ on $\Gamma$.

######  Definition 2.3 (Integral GKM graph).

An integral GKM graph is a pair $(\Gamma,\alpha,\nabla)$ or $(\Gamma,\alpha)$
if ($\nabla$ is obviously determined) which consists of a graph
$\Gamma=({\mathcal{V}},{\mathcal{E}})$ and an integral axial function
$\alpha:{\mathcal{E}}\to(\mathfrak{t}_{\mathbb{Z}})^{*}$.

######  Remark 2.4.

For an integral GKM graph $(\Gamma,\alpha,\nabla)$, if every three-tuple of
elements in $\alpha({\mathcal{E}}_{p})$ is linearly independent for all
vertices $p\in{\mathcal{V}}$, then $\nabla$ is uniquely determined (see [17,
9]).

Let $R(T)$ be the representation ring of $T$ which can be identified with the
ring of finite sums

$\sum_{k=1}^{l}m_{k}e^{2\pi\sqrt{-1}\alpha_{k}}$

for $m_{k}\in\mathbb{Z}$ and $\alpha_{k}\in(\mathfrak{t}_{\mathbb{Z}})^{*}$.
Let $Maps({\mathcal{V}},\,R(T))$ be the ring of maps from the set of vertices
to the representation ring of $T$.

######  Definition 2.5 ($K$\- ring of an integral GKM graph).

An element $f\in Maps({\mathcal{V}},\,R(T))$ is said to be a $K$\- class if
for every edge $e=(xy)\in{\mathcal{E}}$,

$f(x)-f(y)=\beta(1-e^{2\pi\sqrt{-1}\alpha(e)})$ (2.2)

for some $\beta\in R(T)$. The collection of all $K$\- classes is called $K$\-
ring of $(\Gamma,\alpha)$ and is denoted by $K_{\alpha}(\Gamma)$ (see, [8] for
further details).

Note that $K_{\alpha}(\Gamma)$ has a $R(T)$ module structure which is induced
by the injective homomorphism $i:R(T)\to K_{\alpha}(\Gamma)$, where the image
of $x\in R(T)$, denoted as $i(x)$, is defined by

$i(x)(v)=x\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all $v\in{\mathcal{V}}$.}$ (2.3)

###  2.2. $K$-ring of GKM graphs induced from GKM manifolds

For a given GKM manifold $(M,T)$, one can define a graph
$\Gamma_{M}:=({\mathcal{V}},{\mathcal{E}})$ with vertices
${\mathcal{V}}=M^{T}$. For $p,\,q\in M^{T}$, there is an oriented edge
$e=(pq)\in{\mathcal{E}}$ if and only if there exists a $T$-invariant embedded
2-sphere $S^{2}_{e}$ containing $p$ and $q$ as north and south poles
respectively. The edges are labeled by a function
$\alpha_{M}:{\mathcal{E}}\to(\mathfrak{t}_{\mathbb{Z}})^{*}$, known as _axial
function_ , which takes an edge $e=(pq)\in{\mathcal{E}}$ to the weight of the
isotropy representation $T_{p}S^{2}_{e}$ of $T$.

Further, since $(M,T)$ is a GKM manifold, a _connection_ $\nabla$ can be
canonically defined (see, Remark 2.4) such that the above
$\alpha_{M}:{\mathcal{E}}\to(\mathfrak{t}_{\mathbb{Z}})^{*}$ is an integral
axial function. In particular, the pair $(\Gamma_{M},\alpha_{M})$ is an
integral GKM graph associated to the GKM manifold $(M,T)$ (see, [9, Section
1.1] for further details).

Kuntson and Rosu [18, Theorem A.4, Colollary A.5] have proven that, for a GKM
graph $(\Gamma_{M},\alpha_{M})$ associated to the GKM pair $(M,T)$, the
following isomorphisim holds

$K_{T}(M)\simeq K_{\alpha_{M}}(\Gamma_{M}),$ (2.4)

where $K_{T}(M)$ is the $T$\- equivariant $K$\- ring of the manifold $M$.

######  Remark 2.6.

Knutson and Rosu have studied equivariant $K$\- theory with complex
coeffecients in [18]. However, (2.4) remains true over $\mathbb{Z}$ as well
due to the works of Vezzosi and Vistoli [19, Corollary 5.10], which is an
algebraic analogue of [18, Theorem A.4] (this fact is also remarked in [18,
p.447]).

##  3\. Even dimensional complex quadrics and its GKM graph

In this section we define even dimensional complex quadrics and briefly recall
their GKM graphs from [13]. We broadly follow the notations and conventions
from that article.

######  Definition 3.1.

An even dimensional complex quadrics is defined to be the following manifold

$Q_{2n}:=\Big{\\{}[x_{1}:\cdots:x_{2n+2}]\in\mathbb{CP}^{2n+1}\leavevmode\nobreak\
|\leavevmode\nobreak\ \sum_{j=1}^{n+1}x_{i}x_{2n+3-i}=0\Big{\\}}.$

There exists a natural action of $T^{n+1}$ on $Q_{2n}$ :

$Q_{2n}\times T^{n+1}\longrightarrow Q_{2n}$
$[x_{1}:\cdots:x_{2n+2}]\cdot(t_{1},\ldots,t_{n+1}):=[x_{1}t_{1}:\cdots:x_{n+1}t_{n+1}:t_{n+1}^{-1}x_{n+2}:\cdots:t_{1}^{-1}x_{2n+2}].$
(3.5)

Note that, $Q_{2n}\cong SO(2n+2)/(SO(2n)\times SO(2))$. Hence the above action
is equivalent to a maximal torus $T^{n+1}$\- action, induced by restricting
the transitive action of $SO(2n+2)$. Additionally, $T^{n+1}$ is also a maximal
torus of $SO(2n)\times SO(2)$. Therefore from [7], the fixed points and one-
dimensional orbits under $T^{n+1}$\- action have the structure of a graph.

Since, the above action (3.5) is not effective due to its non-trivial kernel
$\mathbb{Z}_{2}=\\{\pm 1\\}$, we consider
$T^{n+1}/\mathbb{Z}_{2}\leavevmode\nobreak\ (\simeq T^{n+1})$\- action on
$Q_{2n}$. In particular, we obtain the GKM graph of $Q_{2n}$ (denoted by
$\mathscr{Q}_{2n})$ by labeling edges with tangential representations on
(effective) $T^{n+1}$-fixed points.

######  Remark 3.2.

Throughout the paper, we consider the cohomology ring

$H^{*}(BT^{n+1})\cong\mathbb{Z}[x_{1},\ldots,x_{n+1}].$ (3.6)

The generators $x_{k}$, for $k=1,\ldots,n+1$, has degree 2 and can be regarded
as the $k$-th projection $\rho_{k}:T^{n+1}\to S^{1}$. In particular, we
commonly use the following equivalences :

$H^{2}(BT^{n+1})\cong(\mathfrak{t}^{n+1}_{\mathbb{Z}})^{*}\cong\mbox{Hom}(T^{n+1},S^{1})\cong\mathbb{Z}^{n+1}.$

Further, we consider the representation ring of $T^{n+1}$ as the following
Laurent polynomial ring

$R(T^{n+1})\cong\mathbb{Z}[y_{1}^{\pm 1},\ldots,y_{n+1}^{\pm 1}],$ (3.7)

generated by 1-dimensional representations
$y_{k}=e^{2\pi\sqrt{-1}x_{k}},\leavevmode\nobreak\
y_{k}^{-1}=e^{-2\pi\sqrt{-1}x_{k}}$ for all $k=1,\ldots,n+1$. For further
details, interested readers are referred to [1, Section 4.3].

###  3.1. The GKM graph of the $T^{n+1}$ action on $Q_{2n}$

We first recall the GKM graph of non-effective $T^{n+1}$ action (3.5) on
$Q_{2n}$. One can check that the fixed points of $Q_{2n}$ are

$Q_{2n}^{T}=\\{[e_{i}]\,:\,1\leq i\leq 2n+2\\},$

where $[e_{i}]=[0:\cdots:0:1:0:\cdots:0]\in\mathbb{CP}^{2n+1}$ which has 1 in
the $i$-th place and 0 elsewhere. Furthermore, the invariant two spheres of
$Q_{2n}$ are denoted by the following symbol :

$[w_{i}:w_{j}]:=[0:\cdots:0:w_{i}:0:\cdots:0:w_{j}:0:\cdots:0]$ (3.8)

where $i+j\neq 2n+3$.

Therefore, we associate the following graph
$\Gamma_{2n}:=(\mathcal{V}_{2n},\leavevmode\nobreak\ \mathcal{E}_{2n})$ with
the pair $(Q_{2n},T^{n+1})$ where the vertices are $Q_{2n}^{T}$ i.e

$\mathcal{V}_{2n}=[2n+2]:=\\{1,2,\ldots,2n+2\\},$

and, $[e_{i}],[e_{j}]\in Q_{2n}^{T}$ are connected by an edge if and only if
there is an $T^{n+1}$-invariant embedded 2-sphere containing $[e_{i}],[e_{j}]$
as its poles. Hence the set of edges of $\Gamma_{2n}$ is given by

$\mathcal{E}_{2n}:=\\{ij\,:\,i,j\in[2n+2]\text{ such that }i\neq j,\,i+j\neq
2n+3\\}.$

543261 Figure 1. The above graph is $\Gamma_{4}$ for $n=2$ induced from the
$T^{3}$ -action.

######  Remark 3.3.

For simplicity, we shall write $j$ as $\overline{i}$ when $i+j=2n+3$. Hence
the set of vertices and edges in $\Gamma_{2n}$ can be re-written as

$\mathcal{V}_{2n}=\\{1,2,\ldots,n+1,\overline{n+1},\overline{n+2},\ldots,\overline{1}\\}.$
(3.9) $\mathcal{E}_{2n}:=\\{ij\,:\,i,j\in\mathcal{V}_{2n}\text{ such that
}j\neq i,\overline{i}\\}.$ (3.10)

In order to define an axial function $\widehat{\alpha}:{\mathcal{E}}_{2n}\to
H^{2}(BT^{n+1})$, it is enough to compute the tangential representation of
each invariant 2-spheres $[w_{i}:w_{j}]\in Q_{2n}$. Note that, an element
$t=(t_{1},\ldots,t_{n+1})\in T^{n+1}$ acts on $[w_{i}:w_{j}]$ by the following
:

$[w_{i}:w_{j}]\mapsto[\rho_{i}(t)w_{i}:\rho_{j}(t)w_{j}],$

where $\rho_{i}:T^{n+1}\to S^{1}$ is the surjective homomorphism defined by

$\rho_{i}(t)=\begin{cases}t_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &\text{if
$i\in\\{1,\ldots,n+1\\}$},\\\ t_{\overline{i}}^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ &\text{if
$i\in\\{n+2,\ldots,2n+2\\}$}.\end{cases}$

Therefore, the tangential representations around the fixed points of
$[w_{i}:w_{j}]$ are given by

$[1:w_{j}]\mapsto[1:\rho_{i}(t)^{-1}\rho_{j}(t)w_{j}],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
[w_{i}:1]\mapsto[\rho_{j}(t)^{-1}\rho_{i}(t)w_{1}:1].$

Consequently, we define the axial function
$\widehat{\alpha}:{\mathcal{E}}_{2n}\to H^{2}(BT^{n+1})$ by the equation

$\widehat{\alpha}(ij)=x_{j}-x_{i},$ (3.11)

where $x_{i}\in H^{2}(BT^{n+1})$ corresponds to the $i$-th coordinate
projection $\rho_{i}:T^{n+1}\to\mathbb{C}\setminus\\{0\\}$ and satisfies the
following :

  * •

$\\{x_{i}:i\in[n+1]\\}$ generates $H^{2}(BT^{n+1})$.

  * •

for $i\in\\{n+2,\ldots,2n+2\\}$, $x_{i}:=-x_{\overline{i}}$.

However, since the $T^{n+1}$ action on $Q_{2n}$ in (3.5) is not effective, it
can be verified that for any vertex $i_{0}\in\mathcal{V}_{2n}$, the collection

$\\{\widehat{\alpha}(i_{0}j)\leavevmode\nobreak\ |\leavevmode\nobreak\
i_{0}+j\not=2n+3\\}$ (3.12)

does not span $H^{2}(BT^{n+1})$. In particular, $\widehat{\alpha}$ is not an
_effective_ axial function (see, [12, Section 2.1]).

For example, if we consider $1\in{\mathcal{V}}_{2n}$, the vectors in (3.12)
i.e

$x_{2}-x_{1},\ldots,x_{n+1}-x_{1},-x_{n+1}-x_{1},\ldots,-x_{2}-x_{1}$ (3.13)

span the lattice $\langle
x_{2}-x_{1},\ldots,x_{n+1}-x_{1},-x_{n+1}-x_{1}\rangle_{\mathbb{Z}}$, which is
a proper subspace of $H^{2}(BT^{n+1})$. In order to avoid this problem, we
will identify $\langle
x_{2}-x_{1},\ldots,x_{n+1}-x_{1},-x_{n+1}-x_{1}\rangle_{\mathbb{Z}}$ as
$H^{2}(BT^{n+1})$. This defines the axial function induced by the effective
$T^{n+1}$\- action on $Q_{2n}$. Further, we replace

  * •

$x_{i}-x_{1}$ by $x_{i-1}$ for all $i=2,\ldots,n+1$.

  * •

$-x_{n+1}-x_{1}$ by $x_{n+1}$.

Therefore, we can rewrite the elements in (3.13) as the following elements
respectively

$x_{1},\ldots,x_{n},x_{n+1},-x_{n-1}+x_{n}+x_{n+1},\ldots,-x_{1}+x_{n}+x_{n+1},$
(3.14)

which spans $H^{2}(BT^{n+1})$. The similar procedure applies to the other
vertices as well.

######  Definition 3.4.

Set $h:{\mathcal{V}}_{2n}\to H^{2}(BT^{n+1})$ as

$h(j)=\begin{cases}x_{j-1}-x_{n+1}\text{ if }j=1,\ldots,n+2,\\\
x_{n}-x_{2n+2-j}\text{ if }j=n+3,\ldots,2n+2,\end{cases}$ (3.15)

where $x_{0}=0$. And, we define the axial function
$\alpha:{\mathcal{E}}_{2n}\to H^{2}(BT^{n+1})$ by

$\alpha(ij)=h(j)-h(i)\text{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ for $j\neq\\{i,\overline{i}\\}$.}$
(3.16)

From the above definition, the following holds,

######  Lemma 3.5.

[13, Lemma 2.6] For every vertex $i\in{\mathcal{V}}_{2n}$, and for any
distinct three vertices
$k_{1},k_{2},k_{3}\in{\mathcal{V}}_{2n}\setminus\\{i,\overline{i}\\}$, the
axial functions $\alpha(ik_{1}),\leavevmode\nobreak\
\alpha(ik_{2}),\leavevmode\nobreak\ \alpha(ik_{3})$ are linearly independent.
In other words, the GKM graph $(\Gamma_{2n},\alpha)$ is three-independent.

######  Remark 3.6.

We also make the following replacements in (3.7)

  * •

$y_{i}y_{1}^{-1}$ by $y_{i-1}$ for all $i=2,\ldots,n+1$.

  * •

$y_{n+1}^{-1}y_{1}^{-1}$ by $y_{n+1}$.

######  Remark 3.7.

Set $f:{\mathcal{V}}_{2n}\to R(T^{n+1})$ by

$f(j)=e^{2\pi\sqrt{-1}h(j)}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for $j\in{\mathcal{V}}_{2n}$.}$ (3.17)

Therefore, for $j=1,\ldots,n+1$,

$f(j+1)f(1)^{-1}=e^{2\pi\sqrt{-1}x_{j}}=y_{j}.$ (3.18)

######  Notation 3.8.

Hereafter, we denote the GKM graph $(\Gamma_{2n},\alpha)$ by
$\mathscr{Q}_{2n}$ and, the $K$\- ring $K_{\alpha}(\Gamma_{2n})$ (see,
Definition 2.5) by $K(\mathscr{Q}_{2n})$.

##  4\. Generators and relations of $K(\mathscr{Q}_{2n})$

In this section, we introduce two types of $K$\- classes of $\mathscr{Q}_{2n}$
and four types of relations among them. In Section 5, we show that the
abovementioned $K$\- classes generate $K(\mathscr{Q}_{2n})$.

###  4.1. Generators of type I

######  Definition 4.1.

For a vertex $v\in\mathcal{V}_{2n}$, we define the function
$M_{v}:\mathcal{V}_{2n}\to R(T^{n+1})$ by

$M_{v}(l)=\begin{cases}1;&\text{ if $l=v$},\\\
e^{2\pi\sqrt{-1}\alpha(\overline{v}k)}\cdot
e^{2\pi\sqrt{-1}\alpha(\overline{v}\,\overline{k})}=y_{n}y_{n+1}^{-1}f(\overline{v})^{-2};&\text{
if $l=\overline{v}$},\\\ e^{2\pi\sqrt{-1}\alpha(lv)}=f(v)f(l)^{-1};&\text{ if
$l\neq v,\overline{v}$}.\end{cases}$

Further, we define $M_{v}^{-1}:\mathcal{V}_{2n}\to R(T^{n+1})$ by

$M_{v}^{-1}(l):=(M_{v}(l))^{-1}.$

######  Remark 4.2.

Note that the above definition does not depend on the choice of
$k\in{\mathcal{V}}_{2n}\setminus\\{v,\overline{v}\\}$ since for any
$k\in{\mathcal{V}}_{2n}\setminus\\{v,\overline{v}\\}$, we have

$\alpha(\overline{v}k)+\alpha(\overline{v}\,\overline{k})=h(k)-h(\overline{v})+h(\overline{k})-h(\overline{v})=h(k)+h(\overline{k})-2h(\overline{v}).$

Further, from Definition 3.4,

$h(k)+h(\overline{k})=x_{n}-x_{n+1}.$ (4.19)

In particular,

$e^{2\pi\sqrt{-1}\alpha(\overline{v}k)}\cdot
e^{2\pi\sqrt{-1}\alpha(\overline{v}\,\overline{k})}=y_{n}y_{n+1}^{-1}\leavevmode\nobreak\
e^{-4\pi\sqrt{-1}h(\overline{v})}=y_{n}y_{n+1}^{-1}f(\overline{v})^{-2}.$
(4.20)

######  Proposition 4.3.

For any $v\in\mathcal{V}_{2n}$, $M_{v}\in K(\mathscr{Q}_{2n})$.

###### Proof.

From Definition 2.5, it is enough to show that for any edge
$(jk)\in\mathcal{E}_{2n}$ (i.e $k\not=j,\overline{j}$), we have

$M_{v}(j)-M_{v}(k)=0\,\,(\text{mod }1-e^{2\pi\sqrt{-1}\alpha(jk)}).$

We verify the above condition on a case-by-case basis :

  1. (1)

If $j=v$ : For any $k\not=v,\overline{v}$, we have

$M_{v}(j)-M_{v}(k)=1-(e^{2\pi\sqrt{-1}\alpha(kj)})\equiv 0\,\,(\text{mod
}1-e^{2\pi\sqrt{-1}\alpha(jk)}),$

since $\alpha(kj)=-\leavevmode\nobreak\ \alpha(jk)$ from (3.16).

  2. (2)

If $j=\overline{v}$ : For $k\not=v,\overline{v}$, we have

$\begin{split}M_{v}(j)-M_{v}(k)&=e^{2\pi\sqrt{-1}\alpha(\overline{v}k)}e^{2\pi\sqrt{-1}\alpha(\overline{v}\,\overline{k})}-e^{2\pi\sqrt{-1}\alpha(kv)}\\\
&=e^{2\pi\sqrt{-1}\alpha(\overline{v}k)}e^{2\pi\sqrt{-1}\alpha(\overline{v}\,\overline{k})}-e^{2\pi\sqrt{-1}\alpha(\overline{v}\overline{k})}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{(see, Remark \ref{alphavbarkbar})}\\\
&=\big{(}e^{2\pi\sqrt{-1}\alpha(\overline{v}k)}-1\big{)}e^{2\pi\sqrt{-1}\alpha(\overline{v}\,\overline{k})}\equiv
0\,\,(\text{mod }1-e^{2\pi\sqrt{-1}\alpha(jk)}).\end{split}$

  3. (3)

If $j\not=v,\,\overline{v}$ : We break this case into two more cases.

    1. (a)

If $k=\overline{v}$ or $k={v}$: We follow the similar arguements as above.

    2. (b)

If $k\not=v,\overline{v}$ : We have

$\begin{split}M_{v}(j)-M_{v}(k)&=e^{2\pi\sqrt{-1}\alpha(jv)}-e^{2\pi\sqrt{-1}\alpha(kv)}\\\
&=e^{2\pi\sqrt{-1}\alpha(kv)}(e^{2\pi\sqrt{-1}\alpha(jk)}-1)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{[as, $\alpha(jv)=\alpha(kv)+\alpha(jk)$]}\\\
&\equiv 0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\text{mod }1-e^{2\pi\sqrt{-1}\alpha(jk)}).\end{split}$

Therefore, $M_{v}\in K(\mathscr{Q}_{2n})$ for any $v\in\mathcal{V}_{2n}$. ∎

######  Remark 4.4.

Note that

$\begin{split}\alpha(\overline{v}\overline{k})&=h(\overline{k})-h(\overline{v})=(x_{n}-x_{n+1}-h(k))-(x_{n}-x_{n+1}-h(v))\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{(see, \eqref{hiibar})}\\\
&=h(v)-h(k)=\alpha(kv).\end{split}$

######  Remark 4.5.

Similarly, one can show $M_{v}^{-1}\in K(\mathscr{Q}_{2n})$, for any
$v\in\mathcal{V}_{2n}$.

######  Proposition 4.6.

The generator $y_{i}\in R(T^{n+1})$ for each $i=1,\ldots,n+1$ satisfies the
following equality

$y_{i}=M_{i+1}M_{1}^{-1}.$

###### Proof.

For $i\in\\{1,\ldots,n+1\\}$, we evaluate
$M_{i+1}M_{1}^{-1}(j)=M_{i+1}(j)(M_{1}(j))^{-1}$ for all
$j\in{\mathcal{V}}_{2n}$.

If $j\in{\mathcal{V}}_{2n}\setminus\\{\overline{1},\overline{i+1}\\}$,

$\begin{split}M_{i+1}M_{1}^{-1}(j)&=f(i+1)f(j)^{-1}(f(1)f(j)^{-1})^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (\text{by
Definition \ref{defmv}})\\\ &=f(i+1)f(1)^{-1}=y_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (\text{see,
\eqref{yjwithfj}})\end{split}$

If $j=\overline{1}=2n+2$,

$\begin{split}M_{i+1}M_{1}^{-1}(2n+2)&=f(i+1)f(2n+2)^{-1}(y_{n}y_{n+1}^{-1}f(2n+2)^{-2})^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (\text{by
Definition \ref{defmv}})\\\
&=f(i+1)f(2n+2)\big{(}y_{n}y_{n+1}^{-1}\big{)}^{-1}\\\
&=f(i+1)f(2n+2)\big{(}f(i+1)f(\overline{i+1})\big{)}^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (\text{see
\eqref{deffj}, \eqref{hiibar}})\\\
&=f(2n+2)f(\overline{i+1})^{-1}=y_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\text{see \eqref{deffj},
\eqref{defhj}})\\\ \end{split},$

If $j=\overline{i+1}=2n+2-i$,

$\begin{split}M_{i+1}M_{1}^{-1}(2n+2-i)&=y_{n}y_{n+1}^{-1}f(2n+2-i)^{-2}(f(1)f(2n+2-i)^{-1})^{-1}\\\
&=y_{n}f(2n+2-i)^{-1}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (as,
$f(1)=y_{n+1}^{-1}$)}\\\ &=y_{i}\end{split},$

Note that $f(2n+2-i)=\begin{cases}y_{n}\cdot y_{i}^{-1};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ &i=1,\ldots,n-1,\\\ y_{2n+1-i}\cdot
y_{n+1}^{-1}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
&i=n,n+1.\end{cases}$

Hence, the proposition follows. ∎

######  Example 4.7.

For $n=2$, we consider $\mathscr{Q}_{4}$ (see, Figure 1) and
$1\in{\mathcal{V}}_{4}$. Then $M_{1}\in K(\mathscr{Q}_{4})$ is given by

$\begin{split}M_{1}(1)=1,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &M_{1}(2)=f(1)f(2)^{-1}=y_{1}^{-1}\\\
M_{1}(3)=f(1)f(3)^{-1}=y_{2}^{-1},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
&M_{1}(4)=f(1)f(4)^{-1}=y_{3}^{-1}\\\
M_{1}(5)=f(1)f(5)^{-1}=y_{1}y_{2}^{-1}y_{3}^{-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
&M_{1}(6)=M_{1}(\overline{1})=y_{2}y_{3}^{-1}f(\overline{1})^{-2}=y_{2}^{-1}y_{3}^{-1}.\end{split}$

###  4.2. Generators of type II

######  Definition 4.8.

Let $P$ be any non-empty subset of $\mathcal{V}_{2n}=[2n+2]$ such that the
full subgraph $\Gamma_{P}$ is a complete subgraph of $\Gamma_{2n}$
or1,1111Notice that these are equivalent due to the definition of
$\Gamma_{2n}$ $\\{i,\overline{i}\\}\not\subset P$ for all
$i\in\mathcal{V}_{2n}$.

Then we define the map $\Delta_{P}:{\mathcal{V}}_{2n}\to R(T^{n+1})$ by

$\Delta_{P}(l)=\begin{cases}\displaystyle\prod_{k\notin
P\cup\\{\overline{l}\\}}\big{(}1-e^{2\pi\sqrt{-1}\alpha(lk)}\big{)}=\prod_{k\notin
P\cup\\{\overline{l}\\}}(1-f(k)f(l)^{-1}),&\text{if $l\in P$,}\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 0&\text{if $l\notin P$.}\end{cases}$
(4.21)

######  Remark 4.9.

Note that $\Delta_{P}$ in (4.21) is basically the _Thom class_ (equivariant
$K$\- theoretic) of the GKM subgraph $\Gamma_{P}$. Geometrically, the Thom
class (equivariant $K$\- theoretic) of a codimension $2m$ GKM submanifold $N$
of a GKM manifold $Y$, for effective $T$-action, with an almost complex
structure can be defined as follows :

Consider the normal bundle $\nu$ of the submanifold $N$, and define its _Thom
space_ as $Th(\nu):=Y/D(\nu)^{c}$, where $D(\nu)^{c}$ denotes the complement
of the unit disc bundle embedded in $Y$. Since $\nu$ inherits its orientation
from the almost complex structure (in fact, this becomes a complex
$m$-dimensional vector bundle), we have the following isomorphism known as the
_Thom isomorphism_ :

$K_{T}(N)\to K_{T}(Th(\nu))$

On the other hand, we have the homomorphism $K_{T}(Th(\nu))\to K_{T}(Y)$,
induced from the collapsing map $Y\to Th(\nu)$. Therefore, we obtain the
following homomorphism by composing the two aforementioned homomorphisms :

$\psi_{N}:K_{T}(N)\to K_{T}(Y)$

Then, we can define the Thom class of the codimension $2m$ GKM submanifold $N$
by

$\psi_{N}({1})=\tau_{N}\in K_{T}(Y).$

Because of the definition of a GKM graph, the GKM subgraph
$\Gamma_{P}\subset\mathscr{Q}_{2n}$ is induced from some GKM submanifold $N$
of $Q_{2n}$; therefore, the Thom class $\Delta_{P}$ of $\Gamma_{P}$ is the
combinatorial analogue of the Thom class $\tau_{N}$ of $N$.

######  Lemma 4.10.

Let $P\subset\mathcal{V}_{2n}$ such that $\\{i,\overline{i}\\}\not\subset P$
for all $i\in\mathcal{V}_{2n}$. Then

$\Delta_{P}\in K(\mathscr{Q}_{2n}).$ (4.22)

###### Proof.

Let $e=(ij)\in\mathcal{E}_{2n}$ be an edge in $\Gamma_{2n}$ (i.e
$j\neq\overline{i},\leavevmode\nobreak\ i\neq\overline{j}$ ) for
$i,j\in\mathcal{V}_{2n}$. We verify the condition (2.2) on a case-by-case
basis :

  1. (i)

If $i,j\notin K$, then $\Delta_{P}(i)=\Delta_{P}(j)=0$. Hence

$\Delta_{P}(i)-\Delta_{P}(j)=0\,\,\,(mod\,\,1-e^{2\pi\sqrt{-1}\alpha(ij)}).$

  2. (ii)

If $i\in K$, $j\notin K$, then from (4.21), we have

$\displaystyle\Delta_{P}(i)-\Delta_{P}(j)=\prod_{k\notin
P\cup\\{\overline{i}\\}}\big{(}1-e^{2\pi\sqrt{-1}\alpha(ik)}\big{)}-0=0\,\,\,(mod\,\,1-e^{2\pi\sqrt{-1}\alpha(ij)}).$

The last equality holds as $j\notin P\cup\\{\overline{i}\\}$. This implies
$(1-e^{2\pi\sqrt{-1}\alpha(ij)})$ is a factor of $\Delta_{P}(i)$.

  3. (iii)

Finally, we assume $i,j\in P$. Let $e^{\prime}=(ik)$ be an edge with initial
point $i$ such that $k\notin P$. Therefore from (4.21), we deduce that
$\big{(}1-e^{2\pi\sqrt{-1}\alpha(ik)}\big{)}$ divides $\Delta_{P}(i)$.

Since $P$ is invariant under the connection $\nabla_{e}$ at $e=(ij)$, we have
$\nabla_{e}(ik)$ is not an edge in $\Gamma_{P}$ i.e the terminal point of
$\nabla_{e}(ik)$ is not in $P$. Therefore,
$\big{(}1-e^{2\pi\sqrt{-1}\alpha(\nabla_{e}(ik))}\big{)}$ is one of the
factors in $\Delta_{P}(j)$.

Furthermore, from the third property of Definition 2.2, we have the following

$1-e^{2\pi\sqrt{-1}\alpha(\nabla_{e}(e^{\prime}))}\equiv
1-e^{2\pi\sqrt{-1}\alpha(e^{\prime})}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\text{mod}\leavevmode\nobreak\
1-e^{2\pi\sqrt{-1}\alpha(e)}).$ (4.23)

Since, one can have the similar relations for every other factors in
$\Delta_{P}(i)$, we obtain

$\Delta_{P}(i)-\Delta_{P}(j)\equiv
0\,\,\,(mod\,\,1-e^{2\pi\sqrt{-1}\alpha(ij)}).$

Hence, the lemma follows. ∎

######  Example 4.11.

For the GKM graph $\mathscr{Q}_{4}$ and $P=\\{2,4,6\\}$ such that $\Gamma_{P}$
is a complete subgraph (see, Figure 1), $\Delta_{P}\in K(\mathscr{Q}_{4})$ is
given by

$\begin{split}\Delta_{P}(2)=(1-y_{1}^{-1})(1-y_{2}y_{1}^{-1}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ &\Delta_{P}(4)=(1-y_{3}^{-1})(1-y_{2}y_{1}^{-1}),\\\
\Delta_{P}(6)=(1-y_{3}^{-1})(1-y_{1}^{-1}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
&\Delta_{P}(1)=\Delta_{P}(3)=\Delta_{P}(5)=0.\end{split}$

###  4.3. Relations between $M_{v}$ and $\Delta_{P}$’s

Within this section, we present four types of relations between $M_{v}$ and
$\Delta_{P}$’s.

In the results below, we frequently use $1\in R(T^{n+1})$ as
$1:{\mathcal{V}}_{2n}\to R(T^{n+1})$ (see (2.3)) which takes any vertices
$v\in{\mathcal{V}}_{2n}$ to $1\in R(T^{n+1})$.

####  4.3.1. Relation 1

Let $J$ be a subset of $\mathcal{V}_{2n}$. We consider

$F_{J}:=\begin{cases}&1-M_{v}\,\text{ if $J=\mathcal{V}_{2n}\setminus\\{v\\}$
for a vertex $v\in\mathcal{V}_{2n}$},\\\ &\Delta_{J}\,\text{ if
$\\{i,\overline{i}\\}\not\subset J$ for every
$i\in\mathcal{V}_{2n}$}.\end{cases}$ (4.24)

######  Lemma 4.12.

The following relation holds :

$\displaystyle\prod_{\cap J=\emptyset}F_{j}=0.$ (4.25)

###### Proof.

The lemma follows from Definition 4.1 and Definition 4.8. ∎

####  4.3.2. Relation 2

Let $I\subset{\mathcal{V}}_{2n}=[2n+2]$ be a subset with $|I|=n$ such that the
full subgraph $\Gamma_{I}$ is a complete subgraph of $\Gamma_{2n}$. Hence,
there exists an unique pair $\\{b,\overline{b}\\}\in
I^{c}={\mathcal{V}}_{2n}\setminus I$ such that

$\Delta_{P},\,\Delta_{L}\in K(\mathscr{Q}_{2n}),$

where $P=(I\cup\\{b\\})^{c}=I^{c}\setminus\\{b\\}$ and
$L=(I\cup\\{\overline{b}\\})^{c}=I^{c}\setminus\\{\overline{b}\\}$.

######  Lemma 4.13.

Let $I\subset{\mathcal{V}}_{2n}=[2n+2]$ be a subset which satisfies the above
conditions. Then the following holds :

$\prod_{i\in
I}(1-M_{i})=\Delta_{(I\cup\\{b\\})^{c}}+M_{b}\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}.$
(4.26)

###### Proof.

We verify the claim for every vertices in ${\mathcal{V}}_{2n}$.

Let $v\in I$. Hence,
$v\notin(I\cup\\{b\\})^{c},(I\cup\\{\overline{b}\\})^{c}$. Therefore from
(4.21),

$\Delta_{(I\cup\\{b\\})^{c}}(v)=0=\Delta_{(I\cup\\{\overline{b}\\})^{c}}(v).$

Also, $\displaystyle\prod_{i\in I}(1-M_{i})(v)=0$ as $(1-M_{v})$ is a factor
of LHS in (4.26).

For the vertex $b\leavevmode\nobreak\ (\notin I)$, we have

$\begin{split}\Big{(}\Delta_{(I\cup\\{b\\})^{c}}+M_{b}\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}\Big{)}(b)&=0+M_{b}(b)\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}(b)\\\
&=0+1\cdot\prod_{i\in
I}\big{(}1-e^{2\pi\sqrt{-1}\alpha(bi)}\big{)}=\prod_{i\in
I}(1-M_{i})(b).\end{split}$

For the vertex $\overline{b}\leavevmode\nobreak\ (\notin I)$, we have

$\begin{split}\Big{(}\Delta_{(I\cup\\{b\\})^{c}}+M_{b}\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}\Big{)}(\overline{b})&=\Delta_{(I\cup\\{b\\})^{c}}(\overline{b})+M_{b}(\overline{b})\cdot
0\\\ &=\prod_{i\in
I}\big{(}1-e^{2\pi\sqrt{-1}\alpha(\overline{b}i)}\big{)}+0=\prod_{i\in
I}(1-M_{i})(\overline{b}).\end{split}$

If $v\notin I\cup\\{b,\overline{b}\\}$, then $\overline{v}\in I$. Thus we have

$\begin{split}\prod_{i\in
I}(1-M_{i})(v)&=(1-M_{\overline{v}})(v)\cdot\prod_{i\in
I\setminus\\{\overline{v}\\}}(1-e^{2\pi\sqrt{-1}\alpha(vi)})\\\
&=(1-e^{2\pi\sqrt{-1}\alpha(vb)}e^{2\pi\sqrt{-1}\alpha(v\,\overline{b})})\cdot\prod_{i\in
I\setminus\\{\overline{v}\\}}(1-e^{2\pi\sqrt{-1}\alpha(vi)})\\\
&=\big{(}(1-e^{2\pi\sqrt{-1}\alpha(vb)})+e^{2\pi\sqrt{-1}\alpha(vb)}(1-e^{2\pi\sqrt{-1}\alpha(v\,\overline{b})})\big{)}\cdot\prod_{i\in
I\setminus\\{\overline{v}\\}}(1-e^{2\pi\sqrt{-1}\alpha(vi)})\\\
&=\prod_{i\in(I\setminus\\{\overline{v}\\})\cup\\{b\\}}(1-e^{2\pi\sqrt{-1}\alpha(vi)})+e^{2\pi\sqrt{-1}\alpha(vb)}\cdot\prod_{i\in(I\setminus\\{\overline{v}\\})\cup\\{\overline{b}\\}}(1-e^{2\pi\sqrt{-1}\alpha(vi)})\\\
&=\big{(}\Delta_{(I\cup\\{b\\})^{c}}+M_{b}\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}\big{)}(v)\end{split}$

Hence, the lemma follows. ∎

####  4.3.3. Relation 3

######  Lemma 4.14.

Fix $i\in{\mathcal{V}}_{2n}$. Let $P\subset{\mathcal{V}}_{2n}$ be a subset
containing $i$ such that $|P|>1$ and $\Gamma_{P}$ is a complete subgraph of
$\Gamma_{2n}$. Then we have the following :

$\Delta_{P}\cdot(1-M_{i})=\Delta_{P\setminus\\{i\\}}.$ (4.27)

###### Proof.

Note that, $\Delta_{P}\cdot(1-M_{i})$ is not zero only on
$P\cap({\mathcal{V}}_{2n}\setminus\\{i\\})=P\setminus\\{i\\}$. In particular,
we have

$\Delta_{P}\cdot(1-M_{i})(v)=\begin{cases}\prod_{j\notin(P\setminus\\{i\\})\cup\\{\overline{v}\\}}(1-e^{2\pi\sqrt{-1}\alpha(vj)}&\text{
if $v\in P\setminus\\{i\\}$}\\\ 0&\text{ if $v\notin
P\setminus\\{i\\}$}\end{cases}$

Therefore, the lemma follows from (4.21). ∎

####  4.3.4. Relation 4

We define $X:\mathcal{V}_{2n}\to R(T^{n+1})$ by (see, (3.17))

$X(k)=y_{n}y_{n+1}^{-1}f(k)^{-2},\text{\,\, for all $k\in\mathcal{V}_{2n}$}.$

######  Lemma 4.15.

For every $v\in\mathcal{V}_{2n}$, we have the following equality :

$M_{v}\cdot M_{\overline{v}}=X.$

###### Proof.

The lemma follows from Definition 4.1 and the fact
$f(v)f(\overline{v})=y_{n}y_{n+1}^{-1}$ (see, (4.19)). ∎

######  Remark 4.16.

Note that $X\in K(\mathscr{Q}_{2n})$ since $M_{v},M_{\overline{v}}\in
K(\mathscr{Q}_{2n})$.

##  5\. Proof of the main theorem

In this section, we present the equivariant $K$\- ring of even dimensional
complex quadrics in terms of generators and relations. To convey our main
theorem (see, Theorem 5.1) precisely, we prepare some notations as follows :

Let $\mathbf{M}$ denote the set of $K$-classes

$\\{M_{v},\leavevmode\nobreak\ M_{v}^{-1}\leavevmode\nobreak\
:\leavevmode\nobreak\ v\in{\mathcal{V}}_{2n}\\},$

and, $\mathbf{D}$ denote the set of $K$-classes

$\\{\Delta_{P}\leavevmode\nobreak\ :\leavevmode\nobreak\
\text{$P\subset{\mathcal{V}}_{2n},\leavevmode\nobreak\ \Gamma_{P}$ is a
complete subgraph of $\Gamma_{2n}$\\}}.$

Let $\mathbb{Z}[\mathbf{M},\mathbf{D}]$ be the polynomial ring generated by
all elements in $\mathbf{M},\mathbf{D}$. Let
$\mathfrak{I}\lhd\mathbb{Z}[\mathbf{M},\mathbf{D}]$ be an ideal generated by
the following types of elements :

  1. (i)

$\prod_{\cap J=\emptyset}F_{j}$ for $F_{J}$ as in (4.24).

  2. (ii)

$\prod_{i\in
I}(1-M_{i})=\Delta_{(I\cup\\{b\\})^{c}}+M_{b}\cdot\Delta_{(I\cup\\{\overline{b}\\})^{c}}$
for $I$ and $b$ as in Lemma 4.13.

  3. (iii)

$\Delta_{P}\cdot(1-M_{i})=\Delta_{P\setminus\\{i\\}}$ for $\\{i\\}\subsetneq
P$.

  4. (iv)

$M_{v}M_{\overline{v}}-M_{w}M_{\overline{w}}$ for every distinct
$v,w\in{\mathcal{V}}_{2n}$.

We define
$\mathbb{Z}[\mathscr{Q}_{2n}]:=\mathbb{Z}[\mathbf{M},\mathbf{D}]/\mathfrak{I}$.
Let $\widetilde{\varphi}:\mathbb{Z}[\mathbf{M},\mathbf{D}]\to
K(\mathscr{Q}_{2n})$ be a ring homomorphism, and let

${\varphi}:\mathbb{Z}[\mathscr{Q}_{2n}]\to K(\mathscr{Q}_{2n})$ (5.28)

be the homomorphism induced from $\widetilde{\varphi}$.

In other words, the following diagram commutes.

${\mathbb{Z}[\mathbf{M},\mathbf{D}]}$${\mathbb{Z}[\mathscr{Q}_{2n}]}$${K(\mathscr{Q}_{2n})}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\widetilde{\varphi}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varphi}$
(5.29)

Now we state the main theorem of this paper, whose proof relies on Lemma 5.2
and Lemma 5.6.

######  Theorem 5.1.

The homomorphism $\varphi$ is an isomorphism of rings, i.e

$\varphi:\mathbb{Z}[\mathscr{Q}_{2n}]\xrightarrow{\cong}K(\mathscr{Q}_{2n}).$
(5.30)

In particular, for the effective $T^{n+1}$\- action on $Q_{2n}$,

$K_{T^{n+1}}(Q_{2n})\cong
K(\mathscr{Q}_{2n})\cong\mathbb{Z}[\mathscr{Q}_{2n}].$ (5.31)

In the proofs below, we frequently use $M_{i}$ and $\Delta_{P}$ instead of
$\widetilde{\varphi}(M_{i})$ and $\widetilde{\varphi}(\Delta_{P})$,
respectively.

######  Lemma 5.2.

The homomorphism $\varphi:\mathbb{Z}[\mathscr{Q}_{2n}]\to K(\mathscr{Q}_{2n})$
is surjective.

###### Proof.

This is enough to prove that
$\widetilde{\varphi}:\mathbb{Z}[\mathbf{M},\mathbf{D}]\to K(\mathscr{Q}_{2n})$
is surjective. Consider an element $f\in K(\mathscr{Q}_{2n})$. For the vertex
$1\in{\mathcal{V}}_{2n}$, one can write $f(1)\in R(T^{n+1})$ (see, Definition
2.5) as

$f(1)=\sum_{\mathbf{j}}c_{\mathbf{j}}y_{1}^{j_{1}}\cdots
y_{n+1}^{j_{n+1}}=h_{1}$

where $c_{\mathbf{j}}\in\mathbb{Z}$ and
$\mathbf{j}=(j_{1},\ldots,j_{n+1})\in\mathbb{Z}^{n+1}$.

Further, from Proposition 4.6, it follows that $M_{i+1}(1)=y_{i}$ for all
$1\leq i\leq n+1$. Hence

$f(1)=\sum_{\mathbf{j}}c_{\mathbf{j}}M_{2}^{j_{1}}\cdots
M_{n+2}^{j_{n+1}}(1)=h_{1}.$

This means that, there exists an element in
$\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\ |\leavevmode\nobreak\ 2\leq
i\leq n+2]\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$ whose image under
$\widetilde{\varphi}$ coincides with $f(1)$ on the vertex
$1\in{\mathcal{V}}_{2n}$.

We next put $f_{2}=f-h_{1}$ and hence $f_{2}(1)=0$. By using the congruence
relation (2.2) on the edge $(21)\in{\mathcal{E}}_{2n}$, we have

$f_{2}(2)-f_{2}(1)\equiv 0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\
1-e^{2\pi\sqrt{-1}\alpha(21)}=(1-M_{1})(2).$

Therefore, $f_{2}(2)=h_{2}(1-M_{1})(2)$ for some $h_{2}\in R(T^{n+1})$. By
Proposition 4.6, it follows that $M_{i+1}M_{1}^{-1}=y_{i}$ for all $1\leq
i\leq n+1$. Hence,

$h_{2}\in\mathbb{Z}[M_{i+1}M_{1}^{-1},M_{i+1}^{-1}M_{1}\leavevmode\nobreak\
|1\leq i\leq n+1]\subset\mathbb{Z}[\mathbf{M},\mathbf{D}].$

This shows that $h_{2}(1-M_{1})$ is in the image of $\widetilde{\varphi}$. Put

$f_{3}=f_{2}-h_{2}(1-M_{1})\leavevmode\nobreak\ (=f-h_{1}-h_{2}(1-M_{1})).$

Notice that

$f_{3}(1)=0=f_{3}(2),$

where the first equality follows due to the fact $f_{2}(1)=0$, $M_{1}(1)=1$,
and the second equality follows by $f_{2}(2)=h_{2}(1-M_{1})(2)$.

By using the congruence relations on the edge $(31)$ and
$(32)\in{\mathcal{E}}_{2n}$, we may write

$f_{3}(3)=h_{3}(1-M_{1})(1-M_{2})(3)$

for some $h_{3}\in R(T^{n+1})\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$. We put

$f_{4}:=f_{3}-h_{3}(1-M_{1})(1-M_{2})$

which satisfies $f_{4}(1)=f_{4}(2)=f_{4}(3)=0$. By iterating the similar
procedure $n+2$ times, we get an element

$f_{n+2}:=f_{n+1}-h_{n+1}(1-M_{1})\cdots(1-M_{n})$

for some $h_{n+1}\in R(T^{n+1})\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$ and
$f_{n+1}\in K(\mathscr{Q}_{2n})$ satisfying $f_{n+1}(i)=0$ for $1\leq i\leq n$
and $f_{n+1}(n+1)=h_{n+1}(1-M_{1})\cdots(1-M_{n})(n+1)$.

Furthermore, we have

$\begin{split}f_{n+2}&=f_{n+1}-h_{n+1}(1-M_{1})\cdots(1-M_{n})\\\
&=f_{n}-h_{n}(1-M_{1})\cdots(1-M_{n-1})-h_{n+1}(1-M_{1})\cdots(1-M_{n})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \vdots\\\
&=f-h_{1}-h_{2}(1-M_{1})-\cdots-
h_{n}(1-M_{1})\cdots(1-M_{n-1})-h_{n+1}(1-M_{1})\cdots(1-M_{n}).\end{split}$
(5.32)

which statisfies $f_{n+2}(i)=0$ for all $1\leq i\leq n+1$.

Therefore by using (2.2) for the edges $(n+2\leavevmode\nobreak\
i)\in{\mathcal{E}}_{2n}$ for all $1\leq i\leq n$, one can observe

$f_{n+2}(n+2)\equiv 0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{mod}\leavevmode\nobreak\
1-e^{2\pi\sqrt{-1}\alpha(n+2\leavevmode\nobreak\ i)}$

for all $1\leq i\leq n$. In particular, from the definition of
$\Delta_{\\{n+2,\ldots,2n+2\\}}$ (see, (4.21)), one can write

$f_{n+2}(n+2)=h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}(n+2).$

for some $h_{n+2}\in R(T^{n+1})\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$. We
put

$f_{n+3}=f_{n+2}-h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}$

which satisfies $f_{n+3}(1)=\cdots=f_{n+3}(n+2)=0$ since
$\Delta_{\\{n+2,\ldots,2n+2\\}}(i)=0$ for all $1\leq i\leq n+1$. Similarly,
for $2\leq j\leq n+2$, there exists $h_{n+j}\in
R(T^{n+1})\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$ such that

$f_{n+j+1}:=f_{n+j}-h_{n+j}\Delta_{\\{n+j,\ldots,2n+2\\}}$ (5.33)

satisfying $f_{n+j+1}(1)=\cdots=f_{n+j+1}(n+j)=0$.

Notice that, when $j=n+2$,

$f_{2n+3}:=f_{2n+2}-h_{2n+2}\Delta_{\\{2n+2\\}}$

satisfying $f_{2n+3}(1)=\cdots=f_{n+j+1}(2n+2)=0$ i.e $f_{2n+3}\equiv 0$.

Therefore, $f_{2n+2}=h_{2n+2}\Delta_{\\{2n+2\\}}$, and we have

$\begin{split}f_{2n+1}&=h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\\\
f_{2n}&=h_{2n}\Delta_{\\{2n,2n+1,2n+2\\}}+h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \vdots\\\
f_{n+2}&=h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}+\cdots+h_{2n}\Delta_{\\{2n,2n+1,2n+2\\}}+h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\end{split}$
(5.34)

Therefore, we have the following from (5.32) and (5.34),

$\begin{split}f=h_{1}+h_{2}(1-M_{1})&+\cdots+h_{n+1}(1-M_{1})\cdots(1-M_{n})\\\
&+h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}+\cdots+h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\end{split}$
(5.35)

where $h_{i}\in R(T^{n+1})\subset\mathbb{Z}[\mathbf{M},\mathbf{D}]$ for each
$1\leq i\leq 2n+2$.

Hence, the lemma follows. ∎

######  Remark 5.3.

As both ${\bf M}$ and ${\bf D}$ consist of $K$-classes, any
$f\in\mathbb{Z}[{\bf M},{\bf D}]$ is also a $K$-class. Consequently, $f$ can
be expressed in the format of (5.35), by similar arguements as in Lemma 5.2.
Further, for any $g\in\mathbb{Z}[\mathscr{Q}_{2n}]=\mathbb{Z}[{\bf
M,\leavevmode\nobreak\ D}]/\mathfrak{I}$, one can choose $f\in\mathbb{Z}[{\bf
M},{\bf D}]$ of the form (5.35) such that $g=f+\mathfrak{I}$.

Let $v\in{\mathcal{V}}_{2n}=[2n+2]$, we define
$I_{v}\subset[n+2]\subset{\mathcal{V}}_{2n}$ by

$I_{v}=\begin{cases}[n+2]\setminus\\{v\\}&\text{ if }1\leq v\leq n+1\\\
[n+2]\setminus\\{\overline{v}\\}&\text{ if }n+2\leq v\leq 2n+2\\\ \end{cases}$

######  Lemma 5.4.

Let $v\in{\mathcal{V}}_{2n}$ and $\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ v\notin J\rangle$ be an ideal in
$\mathbb{Z}[\mathscr{Q}_{2n}]$ which is generated by $F_{J}$ (see, (4.24)) for
all $v\notin J$. Then we have the following isomorphism :

$\mathbb{Z}[\mathscr{Q}_{2n}]/\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ v\notin
J\rangle\simeq\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}]\simeq R(T^{n+1}).$ (5.36)

###### Proof.

This is enough to prove the lemma for the vertex $v=1\in{\mathcal{V}}_{2n}$.
The proof for other vertices will follow similarly.

When $v=1\in{\mathcal{V}}_{2n}$, (5.36) reduces to the following

$\mathbb{Z}[\mathscr{Q}_{2n}]/\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin
J\rangle\simeq\mathbb{Z}[M_{2},M_{2}^{-1},\ldots,M_{n+2},M_{n+2}^{-1}]\simeq
R(T^{n+1}).$

Claim : The following canonical morphism (see, (5.29)) is surjective :

$p:\mathbb{Z}[M_{2},M_{2}^{-1},\ldots,M_{n+2},M_{n+2}^{-1}]\to\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle.$

Note that the following elements generate
$\mathbb{Z}[\mathscr{Q}_{2n}]/\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin J\rangle$ :

$\\{\overline{M_{v}},\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{M_{v}^{-1}}:v\in{\mathcal{V}}_{2n}\\}\cup\\{\overline{\Delta_{P}}:P\subset{\mathcal{V}}_{2n},\leavevmode\nobreak\
\\{i,\overline{i}\\}\not\subset P\text{ for all $1\leq i\leq 2n+2$}\\},$
(5.37)

where

$\begin{split}\overline{M_{v}}:=M_{v}+(\mathfrak{I}+\langle
F_{J}\leavevmode\nobreak\ &|\leavevmode\nobreak\ 1\notin
J\rangle),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{M_{v}^{-1}}:=M_{v}^{-1}+(\mathfrak{I}+\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle),\\\
\overline{\Delta_{P}}&:=\Delta_{P}+(\mathfrak{I}+\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle).\end{split}$

Let $L\subset{\mathcal{V}}_{2n}$ such that $\\{i,\overline{i}\\}\not\subset L$
for every $1\leq i\leq n+1$. If $1\in L$ and $|L|<n+1$, then from Lemma 4.14,
we have

$\Delta_{L}=\Delta_{L\cup\\{i\\}}\cdot(1-M_{i})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for }i,\overline{i}\notin L$

If $1\notin L$, then $\Delta_{L}=F_{L}\in\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin J\rangle$. Therefore, $\overline{\Delta_{L}}=0$
in $\mathbb{Z}[\mathscr{Q}_{2n}]/\langle F_{J}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\notin J\rangle$.

The above implies that every generators
$\overline{\Delta_{P}}\in\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$ can be
written by $\overline{\Delta_{L}}$’s with $1\in L$ and $|L|=n+1$.

We next assume that $I\subset{\mathcal{V}}_{2n}$ with $1\notin I$ and $|I|=n$.
Further, we assume that there is the unique pair
$\\{b,\overline{b}\\}\not\subset I$. Let $I=\\{i_{1},\ldots,i_{n}\\}$ and
${\mathcal{V}}_{2n}\setminus I=\\{1,b,\overline{b},j_{1},\ldots,j_{n-1}\\}$.
Then by Lemma 4.13, we have

$\Delta_{\\{1,\overline{b},j_{1},\ldots,j_{n-1}\\}}=(1-M_{i_{1}})\cdots(1-M_{i_{n}})-M_{b}\cdot\Delta_{\\{1,b,j_{1},\ldots,j_{n-1}\\}}$

This shows that from the generators $\\{\overline{\Delta_{L}}\\}$ with $1\in
L$ and $|L|=n+1$, if there is a vertex $\overline{b}\in L$ for some
$b=2,\ldots,n+1$, we may replace $\\{\overline{\Delta_{L}}\\}$ by
$\overline{\Delta_{(L\setminus\\{\overline{b}\\})\cup\\{b\\}}}$ by using
$\overline{M_{b}}$.

Therefore, all other generators of type $\overline{\Delta_{P}}$, where
$P\subset{\mathcal{V}}_{2n}$ and $\\{i,\overline{i}\\}\not\subset P$ for all
$1\leq i\leq 2n+2$, can be replaced by
$\overline{\Delta_{\\{1,\ldots,n+1\\}}}$. Moreover, by Lemma 4.13 for
$I=\\{n+2,\ldots,2n+1\\}\subset[2n+2]$, we have

$(1-M_{n+2})\cdots(1-M_{2n+1})=\Delta_{\\{1,2,\ldots,n+1\\}}+M_{1}\cdot\Delta_{\\{2,\ldots,n+1,2n+2\\}}$

Consequently, since $\Delta_{\\{2,\ldots,n+1,2n+2\\}}\in\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$, we have the
following

$(\overline{1-M_{n+2}})\cdots(\overline{1-M_{2n+1}})=\overline{\Delta_{\\{1,2,\ldots,n+1\\}}}\in\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$

This concludes that every element in $\overline{{\bf D}}$ can be written in
terms of the elements in $\overline{{\bf M}}$.

Next, from (4.24), we have $1-M_{1}=F_{{\mathcal{V}}_{2n}\setminus\\{1\\}}$
i.e $\overline{M_{1}}=\overline{1}\in\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$. Therefore,
together with Lemma 4.15, we have

$\overline{1}\cdot\overline{M_{\overline{1}}}=\overline{M_{2}}\cdot\overline{M_{\overline{1}}}=\cdots=\overline{M_{n+1}}\cdot\overline{M_{\overline{n+1}}}=\overline{M_{n+1}}\cdot\overline{M_{n+2}}\in\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle.$

Consequently,

$\overline{M_{\overline{1}}}=\overline{M_{n+1}}\cdot\overline{M_{n+2}},\text{
and
}\overline{M_{\overline{k}}}=\overline{M_{k}^{-1}}\cdot\overline{M_{n+1}}\cdot\overline{M_{n+2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for }k=2,\ldots,n$

This implies that all other generators in (5.37) can be reduced to the
following

$\\{\overline{M_{i}},\leavevmode\nobreak\
\overline{M_{i}^{-1}}\leavevmode\nobreak\ |\leavevmode\nobreak\ 2\leq i\leq
n+2\\}$

Hence,
$p:\mathbb{Z}[M_{2},M_{2}^{-1},\ldots,M_{n+2},M_{n+2}^{-1}]\to\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin J\rangle$ is
surjective.

Finally, we consider the homomorphisms

$\mathbb{Z}[M_{2},M_{2}^{-1},\ldots,M_{n+2},M_{n+2}^{-1}]\xrightarrow{p}\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ 1\notin
J\rangle\xrightarrow{\psi}R(T^{n+1})$

where $\psi$ is induced from the following composition
$\mathbb{Z}[\mathscr{Q}_{2n}]\xrightarrow{\varphi}K(\mathscr{Q}_{2n})\to
R(T^{n+1})$, and is defined by $f\mapsto\varphi(f)(1)$ for
$f\in\mathbb{Z}[\mathscr{Q}_{2n}]$. Notice that

$\psi\circ p(M_{i})=y_{i-1},\leavevmode\nobreak\ \text{ and,
}\leavevmode\nobreak\ \psi\circ p(M_{i}^{-1})=y_{i-1}^{-1}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for }i=2,\ldots,n+2.$

Therefore, $\psi\circ p$ is an isomorphism, hence $p$ is injective.
Consequently, $p$ is an isomorphism.

Hence, the lemma follows. ∎

######  Corollary 5.5.

We have the following injective homomorphism :

$\displaystyle
K(\mathscr{Q}_{2n})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{2n}}R(T^{n+1})\simeq\bigoplus_{v\in{\mathcal{V}}_{2n}}\mathbb{Z}[\mathscr{Q}_{2n}]/\langle
F_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ v\notin
J\rangle\simeq\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}].$

###### Proof.

The first inclusion directly follows from the definition of equivariant
$K$-theory of GKM graphs (see, [18]), while the later part follows from Lemma
5.4. ∎

######  Lemma 5.6.

The homomorphism $\varphi:\mathbb{Z}[\mathscr{Q}_{2n}]\to K(\mathscr{Q}_{2n})$
is injective.

###### Proof.

This is enough to show that the following map $\Phi$ is injective :

$\Phi:\displaystyle\mathbb{Z}[\mathscr{Q}_{2n}]\xrightarrow{\varphi}K(\mathscr{Q}_{2n})\hookrightarrow\bigoplus_{v\in{\mathcal{V}}_{2n}}R(T^{n+1})\simeq\bigoplus_{v\in{\mathcal{V}}_{2n}}\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}].$

Let
$r_{u}:\displaystyle\bigoplus_{v\in{\mathcal{V}}_{2n}}\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in
I_{v}]\to\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\ |\leavevmode\nobreak\
i\in I_{u}]$ be the restriction map at $u\in{\mathcal{V}}_{2n}$. For any
$f\in\mathbb{Z}[\mathscr{Q}_{2n}]$, let $f(u)$ denote the image of $f$ by
$r_{u}\circ\Phi$.

Assume that $\Phi(f)=0$ for an element $f\in\mathbb{Z}[\mathscr{Q}_{2n}]$.
Therefore,

$r_{v}\circ\Phi(f)=f(v)=0\in\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in I_{v}]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all }v\in{\mathcal{V}}_{2n}.$

Note that $f\in\mathbb{Z}[\mathscr{Q}_{2n}]$ can be written as follows (see,
Remark 5.3)

$\begin{split}f=h_{1}+h_{2}(1-M_{1})&+\cdots+h_{n+1}(1-M_{1})\cdots(1-M_{n})\\\
&+h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}+\cdots+h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\\\
\end{split}$ (5.38)

where $h_{i}\in R(T^{n+1})\subset\mathbb{Z}[{\bf M},\leavevmode\nobreak\ {\bf
D}]$ for all $i=1,\ldots,2n+2$.

Note that, for all $i=1,\ldots,2n+2$,
$\varphi(h_{i})\in\mathbb{Z}[y_{1},y_{1}^{-1},\ldots,y_{n+1},y_{n+1}^{-1}]$
(see, (2.3)). This implies that if for some vertex $u\in{\mathcal{V}}_{2n}$,
we have $h_{i}(u)=0$, then $h_{i}\equiv 0$.

Claim 1: $h_{i}=0$ for all $1\leq i\leq n+1$.
Note that

$\Delta_{\\{n+2,\ldots,2n+2\\}}(i)=\cdots=\Delta_{\\{2n+2\\}}(i)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
i=1,\ldots,n+1,$ (5.39)

and $({1}-M_{1})(1)=0$. Therefore, from (5.38), we have

$0=f(1)=h_{1}(1)+0+\cdots+0.$

Hence, $h_{1}=0$. Further, by using $h_{1}=0$ in (5.38) and, for
$({1}-M_{2})(2)=0$, we have

$0=f(2)=h_{2}(\mathfrak{1}-M_{1})(2)+0+\cdots+0.$

Notice that $(\mathfrak{1}-M_{1})(2)\not=0$, hence $h_{2}(2)=0$ as
$\mathbb{Z}[M_{i},M_{i}^{-1}\leavevmode\nobreak\ |\leavevmode\nobreak\ i\in
I_{2}]$ is an integral domain. Therefore $h_{2}=0$. By following the similar
arguements, one can have

$h_{3}=\cdots=h_{n+1}=0.$

Therefore,

$\begin{split}f=h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}+\cdots+h_{2n+1}\Delta_{\\{2n+1,2n+2\\}}+h_{2n+2}\Delta_{\\{2n+2\\}}\end{split}$
(5.40)

Claim 2: $h_{i}=0$ for all $n+2\leq i\leq 2n+2$. From (5.40), we have the
following for the vertex $n+2\in{\mathcal{V}}_{2n}$.

$0=f(n+2)=h_{n+2}\Delta_{\\{n+2,\ldots,2n+2\\}}(n+2)+h_{n+3}\Delta_{\\{n+3,\ldots,2n+2\\}}(n+2)+\cdots+h_{2n+2}\Delta_{\\{2n+2\\}}(n+2).$

Since

$\begin{split}\Delta_{\\{n+3,\ldots,2n+2\\}}&(n+2)=\cdots=\Delta_{\\{2n+2\\}}(n+2)=0,\\\
&\Delta_{\\{n+2,\ldots,2n+2\\}}(n+2)\not=0,\end{split}$

by the similar reason as above we have $h_{n+2}=0$. Iterrating the similar
arguements as above for other vertices $n+3,\ldots,2n+2\in{\mathcal{V}}_{2n}$,
we have $h_{i}=0$ for all $n+3\leq i\leq 2n+2$.

Therefore $f=0$, which shows the injectivity of $\Phi$.

Hence, the lemma follows. ∎

Proof of Theorem 5.1.  The result (5.30) follows from Lemma 5.2 and Lemma 5.6.
Further, since $Q_{2n+1}$ is an equivariantly formal GKM manifold under the
effective $T^{n+1}$-action, we have (5.31) from (2.4). ∎

Acknowledgement : The author is very grateful to Prof. Shintarô Kuroki and
Prof. Vikraman Uma for several valuable discussions and for their
encouragement during the preparation of the manuscript. The author thanks
Prof. Allen Knutson for helpful email exchanges.

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