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\newtheorem{Theorem}[equation]{$BDjM}(B}
\newtheorem{Corollary}[equation]{$B7O(B}
\newtheorem{Lemma}[equation]{$BJdBj(B}
\newtheorem{Proposition}[equation]{$BL?Bj(B}
\newtheorem{Axiom}{$B8xM}(B}
\newtheorem{Fact}[equation]{$B;v<B(B}

\newtheorem{Definition}[equation]{$BDj5A(B}
\newtheorem{Example}[equation]{$BNc(B}
\newtheorem{CounterExample}[equation]{$BH?Nc(B}
\newtheorem{Notation}[equation]{$B5-K!(B}
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\newcommand{\shfO}{{\mathcal O}} % sheaf
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\title{$B6JLL>e$NE@$N(BHilbert$B357?$H(BHeisenberg$BBe?t(B}
\author{$BCfEg(B  $B7<(B\\$B5~ETBg3XM}3X8&5f2J(B}
\begin{document}
\maketitle

$B$3$NO@@b$G$O(B, $BFs$D$N0l8+4XO"$N$J$$BP>]$N4V$K3]$166$rEO$9(B.  $B$R$H$D$O(B,
$B%"%U%!%$%s(BLie$B4D$NI=8=O@$G4pACE*$JLr3d$r2L$?$9L58B<!85(BHeisenberg$BBe?t$G(B
$B$"$j(B, $B$b$&0lJ}$O(B, $BBe?t4v2?$K8=$l$k6JLL$N>e$NE@$N(BHilbert$B357?$G$"$k(B.

$B$9$0$K@bL@$9$k$h$&$K(B, Heisenberg$BBe?t$O%U%)%C%/6u4V$H8F$P$l$kL58BJQ?t$N(B
$BB?9`<04D$KI=8=$r;}$D$,(B, $BB?9`<0$N<!?t(B($BC"$7(B, $B3FJQ?t$K$O0c$&<!?t$E$1$rM?(B
$B$($k(B)$B$K$h$j$3$NI=8=$OD>OBJ,2r(B($B%&%'%$%HJ,2r(B)$B$9$k(B. $B0lJ}(B, Hilbert$B357?$NJ}(B
$B$O(B, $BE@$N?t$K$h$C$FL58B8D$NO"7k@.J,$KJ,2r$5$l$F$$(B
$B$k(B. G\"ottsche\cite{Got}$B$O(B, Hilbert$B357?$N(BBetti$B?t$r7W;;$7$?$,(B,$B7k2L$H$7(B
$B$F$O(B $n$$B8D$NE@$N(BHilbert$B357?$N(BBetti$B?t$O(B, $B%U%)%C%/6u4V$N<!?t(B$n$$B$NItJ,6u(B
$B4V$N<!85$KEy$7$+$C$?(B. $B8@$$49$($l$P(B, $BE@$N?t$rF0$+$9$3$H$K$h$C$F(B
Poincar\'e$BB?9`<0$NJl4X?t$r9M$($k$H(B, Heisenberg$BBe?t$NI=8=$N;XI8$H0lCW$9(B
$B$k$H$$$&$3$H$G$"$k(B. $B$3$N0lCW$*$h$SJ*M}3X<T(BVafa-Witten$B$N(BS-$BAPBP@-$NM=A[(B
\cite{VW}$B$K?(H/$5$l$F(B, $BI.<T$O(BHilbert$B357?$N%[%b%m%8!<72$K(BHeisenberg$BBe?t(B
$B$NI=8=$r4v2?3XE*$K9=@.$9$k$3$H$K$h$C$F(B, G\"ottsche$B$N8x<0$r(B``$BM}2r(B''$B$9$k(B
$B$3$H$K@.8y$7$?(B. (\cite{Na-Hilb}) $B$3$N$H$-Bg@Z$J$3$H$O(B, $BI=8=$O(BHilbert$B35(B
$B7?<+?H$K:n$i$l$k$N$G$O$J$/(B, $B$"$/$^$G$=$N%[%b%m%8!<72$N>e$G$"$k$3$H(B, $B$=(B
$B$7$F$3$l$O$*$=$i$/$h$j=EMW$G$"$m$&$,(B, Hilbert$B357?$NO"7k@.J,$rF1;~$KA4(B
$B$F9M$($k$3$H$G=i$a$F9=@.$5$l$k$3$H$G$"$k(B. $BNc$($P(B, Borel-Weil$BM}O@$N$h$&(B
$B$K6u4V<+?H$K72$,:nMQ$7$FI=8=$,9=@.$5$l$k(B, $B$H$$$C$?9=@.K!$H$O$^$C$?$/<q(B
$B$,0[$J$k$b$N$G$"$k(B.

$B%"%U%!%$%s(BLie$B4D$O(B, $B6&7A>lM}O@$G=EMW$JLr3d$r2L$?$9$N$G(B(cf.\ 
\cite{Tsuchiya})$BDL>o$O<BFs<!85$NBP>]$HM}2r$5$l$F$$$k(B. $B0lJ}(B, Hilbert$B35(B
$B7?$OJ#AGFs<!85(B=$B<B;M<!85$NBP>]$G$"$k(B. $B>e$NI.<T$N9=@.$O(B, Heisenberg$BBe?t(B
$B$N@8@.85$r4v2?3XE*$KDj5A$7$FDj5A4X78<0$r%A%'%C%/$9$k$H$$$&$b$N$@$C$?$N(B
$B$G(B, $B9=@.$r$7$?$"$H$b$J$<Fs<!85$H;M<!85$N4V$K7R$,$j$,$"$k$N$+(B, $B$H$$$&5?(B
$BLd$K$OEz$($k$3$H$,$G$-$J$+$C$?(B. $B$7$+$7(B, $B$=$N8eJ*M}$K$*$$$FBP>N@-$N9M;!(B
$B$,?J$`Cf$G(B, $B$3$l$O89BP>N@-(B($BNc$($P(B, \cite{Vafa}$B$r;2>H(B)$B$NH?1G$G$"$k$HB*(B
$B$($i$l$k$h$&$K$J$C$F$-$?(B. $B89BP>N@-$,$J$<B8:_$9$k$N$+$O$^$@L@$i$+$K$J$C(B
$B$F$*$i$:(B, $B$3$l$rM}2r$9$k$3$H$OMh@$5*$K8~$1$FM}O@J*M}(B, $B?t3XN><T$K2]$;$i(B
$B$l$?Bg$-$J2]Bj$G$"$k$h$&$KI.<T$O;W$C$F$$$k(B.

$B$3$NO@@b$O(B, $B9V5AO?(B\cite{Lecture}$B$rMWLs$7$FF|K\8l$KLu$7$?$b$N$G$"$k(B. $B$3(B
$B$NO@@b$rFI$s$G6=L#$,M/$+$l$?FI<T$O(B, \cite{Lecture}$B$K$bL\$rDL$7$F$$$?$@(B
$B$1$l$P9,$$$G$"$k(B.

\section{$BBe?tE*=`Hw(B: Heisenberg/Clifford$BBe?t$H$=$NI=8=(B}

$B$^$:(BHeisenberg$BBe?t$H$=$NI=8=$r@bL@$9$k(B. $B$^$:I=8=$NJ}$+$i=PH/$7$h$&(B. $B2D(B
$B;;L58B8D$NJQ?t(B $p_1, p_2, \dots$ $B$rMQ0U$7(B, $B$=$NM-M}78?tB?9`<0$NA4BN(B $R
= {\mathbb Q}[p_1,p_2,\dots]$ $B$r9M$($k(B. $B<+A3?t(B $i$ $B$KBP$7$F(B
$i\partial/\partial p_i$ $B$r(B $P[i]$ $B$H$*$-(B, $R$ $B$N85$K(B $p_i$ $B$r3]$1$k:n(B
$BMQAG$r(B $P[-i]$ $B$HDj5A$9$k(B. $B$9$k$H(B, $B<!$N8r494X78<0(B
\begin{equation*}
 \big[\, P[i], P[j]\, \big] = i\delta_{i+j,0} \operatorname{id}_R
\end{equation*}
$B$,@.N)$9$k(B. $B$=$3$G(B, $P[i]$ ($i\neq 0$)$B$H(B $K$ $B$G@8@.$5$l(B, $B4X78<0(B
\begin{equation}\label{eq:defHeis}
 \big[\, P[i], P[j]\, \big] = i\delta_{i+j,0} K
\end{equation}
$B$GDj5A$5$l$k(BLie$B4D$r9M$((B, {\bf ($BL58B<!85$N(B)Heisenberg $BBe?t(B}$B$H8F(B
$B$V(B. $R$$B$O(B, $B$=$NI=8=$K$J$j(B,$BJ*M}$G$O(B{\bf $B%\%=%s$N%U%)%C%/6u4V(B}$B$H8F$P$l(B
$B$k(B. $BDj?tB?9`<0(B $1\in R$ $B$O(B, $B<!$N@-<A$rK~$?$9(B:
\begin{align*}
& P[i] 1 = 0\qquad (i > 0) \\
& R = \operatorname{Span} \{ P[-j_1]\cdots P[-j_k] 1
\mid k\in\mathbb Z_{\ge 0}, j_1,\dots, j_k\in \mathbb Z_{>0} \}
\end{align*}
$B$9$J$o$A(B, $BM-8B<!85(BLie$B4D$NM-8B<!854{LsI=8=$H$NN`;w$G8@$($P(B, $1$ $B$O:G(B
$B9b%&%'%$%H%Y%/%H%k$G$"$k(B. $B$^$?(B, $BJ*M}$G$O(B, $1$ $B$O(B{\bf $B??6u%Y%/%H%k(B},
$P[i]$ ($i < 0$) $B$r(B{\bf $B@8@.1i;;;R(B}, $P[i]$ ($i > 0$) $B$r(B{\bf $B>CLG1i;;(B
$B;R(B}$B$H8F$P$l$k(B. $B??6u$O2?$bL5$$>uBV$rI=$o$7(B, $B@8@.(B/$B>CLG1i;;;R$K$h$C$FN3;R(B
$B$,@8@.(B/$B>CLG$9$k$H$$$&$o$1$G$"$k(B.

$B<!?t:nMQAG(B $D \colon R\to R$ $B$r(B
\begin{equation*}
  D (p_1^{m_1}p_2^{m_2}\cdots) 
    = \left(\sum_i i m_i\right)p_1^{m_1}p_2^{m_2} \cdots
\end{equation*}
$B$GDj5A$9$k(B. ($BBP1~$9$k85$r(BHeisenberg$BBe?t$KIU$12C$($k$3$H$b$"$k(B.) 
$R$ $B$O(B, $D$ $B$K$h$C$F8GM-6u4VJ,2r$5$l$k$,(B, $B8GM-CM(B $n$ $B$N8GM-6u4V$N4p(B
$BDl$H$7$F(B, 
$$
 p_1^{m_1}p_2^{m_2}\cdots \qquad \left( \sum_i im_i = n \right).
$$
$B$r;}$D(B. $B>e$N4pDl$N85$H(B $n$ $B$NJ,3d(B $\lambda =
(1^{m_1}2^{m_2}\cdots)$\footnote{$BJ,3d$NI=$o$7J}$H$7$F$O(B, $B<!$NFs$D$rMQ(B
$B$$$k(B: $\lambda=(1^{m_1}2^{m_2}\cdots)$ $B$H(B 
$\lambda=(\lambda_1,\lambda_2,\dots)$. $BA0<T$G$O(B, $BJ,3d$N@.J,$O(B
$1$ $B$,(B $m_1$ $B8D(B, $2$ $B$,(B $m_2$ $B8D$G$"$k$3$H$rI=$o$7(B, $B8e<T$G$O(B,
$\lambda_1\ge\lambda_2\ge\dots$ $B$G(B, $\lambda_1$, $\lambda_2$,\dots $B$,(B
$B@.J,$G$"$k(B.}
$B$,0lBP0l$KBP1~$9$k$+$i(B, $B8GM-6u4V$N<!85$O(B, $n$$B$NJ,3d?t$KEy$7$/$J$k(B. 
($p_1^{m_1}p_2^{m_2}\cdots$ $B$O(B, $p_\lambda$ $B$H=q$+$l$k(B.)
$B$h$C$F<!85$NJl4X?t(B($BI=8=O@$N8@MU$G$O(B, {\bf $B;XI8(B})$B$O(B,
\begin{equation*}
  \tr_R q^D \defeq \sum_{n\in\mathbb Z} q^n \dim \{v \in R\mid Dv =
  n v\} = \prod_{d=1}^\infty \frac{1}{1-q^d}
\end{equation*}
$B$GM?$($i$l$k$3$H$,J,$+$C$?(B.

$B0J>e$N(BHeisenberg$BBe?t$NDj5A$K$O(B, $P[0]$ $B$r4^$a$J$+$C$?$,(B, $BDj5A<0(B
\eqref{eq:defHeis}$B$r8+$l$P(B, $P[0]$ $B$rCf?4$K4^$a$F$*$$$F$bLdBj$O$J$$(B.
$B0J2<(B $P[0]$ $B$r4^$a$k$H$-$O(B, $R$ $B$X$O(B $0$ $B$G:nMQ$9$k$HLsB+$9$k(B.

$B<!$K(B, $BHsB`2=$JBP>NAP0l<!7A<0(B $(\ ,\ )$ $B$r;}$D(B$\mathbb Q$ $B>e$NM-8B<!85%Y(B
$B%/%H%k6u4V(B $V$ $B$KBP$7$F(B, $B$=$l$KIU?o$9$k(BHeisenberg$BBe?t$rDj5A$9$k(B. $W =
\left( V\otimes t{\mathbb Q}[t]\right) \oplus \left( V \otimes
    t^{-1}{\mathbb Q}[t^{-1}]\right)$ $B>e$KODBP>NAP0l<!7A<0$r(B
\begin{equation*}
  (v\otimes t^i, w\otimes t^j) = i\delta_{i+j,0}(v,w)
\end{equation*}
$B$K$h$C$FDj5A$9$k(B. $W$ $B$N%F%s%=%kBe?t(B $T(W)$ $B$r(B $[v, w] - (v,w) 1$
($v,w\in W$) $B$G@8@.$5$l$kN>B&%$%G%"%k(B $I$ $B$G3d$C$?>&Be?t(B $T(W)/I$ $B$r9M(B
$B$((B, {\bf ($V$ $B$KIU?o$7$?(B)Heisenberg$BBe?t(B}$B$H8F$V(B. $V = \mathbb Q$ $B$H$7$?(B
$B$b$N$,>e$G=R$Y$?(BHeisenberg$BBe?t$KB>$J$i$J$$(B. $V$ $B$ND>8r4pDl(B 
$\{ v_1, v_2, \dots, v_p \}$ $B$r<h$l$P(B, $V$ $B$KIU?o$7$?(BHeisenberg$BBe?t$O(B
$B>e$G=R$Y$?(BHeisenberg$BBe?t$N(B$p$$B8D$N%F%s%=%k@Q$KF17?$H$J$k(B.

$B$3$N9=@.$r(B`$B%9!<%Q!<2=(B'$B$9$l$P(B, super-Heisenberg$BBe?t$H$G$b8F$Y$kBP>]$,<!(B
$B$N$h$&$K<+A3$KDj5A$5$l$k(B. $B:G=i$KM?$($i$l$k%G!<%?$O(B, $BD>OBJ,2r$r;}$C$?%Y(B
$B%/%H%k6u4V(B $V = V_{\mathrm{even}}\oplus V_{\mathrm{odd}}$ $B$G$"$j(B,$BHsB`(B
$B2=$JAP0l<!7A<0$O(B$(v, w) = (-1)^{|v||w|}(w,v)$ $B$rK~$?$9$b$N$H$9$k(B. $B$3$3(B
$B$G(B, $v, w$ $B$O(B, $V_{\mathrm{even}}$ $B$+(B$V_{\mathrm{odd}}$ $B$N85$G$"$j(B,
$$
|v| = 
\begin{cases}
1 & \text{if $v\in V_{\mathrm{even}}$} \\
0 & \text{if $v\in V_{\mathrm{odd}}$}
\end{cases}
$$$B$GDj5A$5$l$k(B. $B$3$N>e$G(B$W$$B$H$=$N>e$NFb@Q(B, $B$=$7$F(B $T(W)/I$ $B$O>e$HA4$/(B
$BF1MM$KDj5A$9$l$P$h$$(B. $BC"$7(B, Lie$B3g8L(B $[\ , \ ]$ $B$O%9!<%Q!<(BLie$B3g8L$KCV$-(B
$B49$($k(B.
$B$9$J$o$A(B, $v, w\in V_{\mathrm{odd}}$$B$N$H$-$K$O(B $[v\otimes t^i,
w\otimes t^j] = (v\otimes t^i)(w\otimes t^j) + (w\otimes t^j)(v\otimes
t^i)$ $B$GDj5A$9$k(B. $B$^$?(B, $B>e$NL58BJQ?tB?9`<0$N6u4V$X$NI=8=$N<+A3$J3HD%$K(B
$B$h$j(B, $B@5<!?t$NItJ,(B $V\otimes t{\mathbb Q}[t]$ $B$N%9!<%Q!<$N0UL#$NBP>NBe(B
$B?t(B $R = S^*(V\otimes t{\mathbb Q}[t])$ $B$K(Bsuper-Heisenberg$BBe?t$NI=8=$,(B
$B<+A3$KDj$^$k(B. $B<!?t:nMQAG(B $D$ $B$b>e$HF1MM$KDj$a$k$H(B, $B<!$N;XI88x<0$,F3$+(B
$B$l$k(B:
\begin{equation}
  \tr_R q^D = \prod_{d=1}^\infty \frac{(1 + q^d)^{\dim
  V_{\mathrm{odd}}}}{(1-q^d)^{\dim V_{\mathrm{even}}}}
\label{eq:char}\end{equation}
$BC"$7(B, $B$3$3$G:8JU$ODL>o$N0UL#$N<!85$r9M$($?$,(B, $B4q?t<!$NItJ,$O(B$-$$B$G4*Dj(B
$B$9$k$3$H$K$9$k(B(supertrace$B$H8F$P$l(B, $\operatorname{str}$$B$GI=$o$9(B)$B$H(B,
\begin{equation*}
  \operatorname{str}_R q^D =
\prod_{d=1}^\infty \left(\frac{1}{1-q^d}\right)^{\dim
V_{\mathrm{even}} - \dim V_{\mathrm{odd}}}
\end{equation*}
$B$H$$$&7A$K$J$k(B.

$V$ $B$H$7$F4q?t<!$NItJ,$,(B$1$$B<!85$G(B, $B6v?t<!ItJ,$,$J$$$b$N$r<h$C$?$H$-$K(B
$B$G$-$k(Bsuper-Heisenberg$BBe?t$O(B, {\bf ($BL58B<!85$N(B)Clifford$BBe?t(B}$B$H8F$P$l$k(B.
$B$^$?J*M}$G$O(B, $B$=$N$H$-$N>e$NI=8=6u4V$O(B{\bf $B%U%'%k%_%*%s$N%U%)%C%/6u4V(B}
$B$H8F$P$l$k(B.

$B6v?t<!$NItJ,$NI=8=$O(B, $B2D;;L58B8D$NJQ?t$NB?9`<0$NA4BN$G$"$C$?$,(B, $B4q?t<!(B
$B$NItJ,$O(B, $B2D;;L58B8D$N4pDl$r;}$C$?%Y%/%H%k6u4V$N30@QBe?t(B $R =
\bigwedge^* (\mathbb Q dp_1\oplus \mathbb Q dp_2\oplus\cdots)$ $B$N>e$K(B
$B<B8=$5$l$k(B. $B$9$J$o$A(B, $BAP0l<!7A<0$K4X$9$kItJ,$rL5;k$7$F>]D'E*$K8@$($P(B,
$i > 0$ $B$N$H$-(B, $t^{-i}$ $B$O(B $dp_i$ $B$r30@Q$9$k$3$H(B, $t^i$ $B$O(B $i
\partial/\partial p_i$$B$rFbIt@Q$9$k$3$H$HDj5A$9$l$P$h$$$o$1$G$"$k(B. $B>e$N(B
$B;XI88x<0$G(B, odd$B$JItJ,$N4sM?$,(B $1/(1-q^d)=1+q^d+q^{2d}+\cdots$ $B$G$J$/(B
$1+q^d$ $B$H$J$C$?$N$O(B, $dp_i\wedge dp_i = 0$ $B$+$iMh$k5"7k$G$"$k(B. $BJ*M}$G(B
$B$O(B, $B%\%=%s$OF1$8>l=j$K$$$/$i$G$bF~$l$k$,(B, $B%U%'%k%_%*%s$OFs8D0J>e$OF~$l(B
$B$J$$$H@bL@$5$l$k(B.

Heisenberg$BBe?t$O(B, $BNL;RNO3X$K$*$$$F@5=`6&Lr4X78(B($BC"$7:#$N>l9g$O(B,$BL58B8D$N(B
$B@.J,(B $(p_1,p_2,\dots; q_1,q_2,\dots)$ $B$r;}$D(B)$B$rNL;R2=$7$F8=$l$k$,(B, $B?t(B
$B3X$G$b$$$m$$$m$J6ILL$K$*$$$F<+A3$K8=$l$k(B.
$B$"$H$G(BHilbert$B357?$N%[%b%m%8!<72$K8=$l$k$N$r8+$k$,(B, $B$=$NA0$KE/3XE*$K$h(B
$B$/;w$?>u67(B, $BBP>N72$NI=8=4D$K8=$l$k$3$H$r8+$h$&(B. $S_n$ $B$r(B$n$$B<!BP>N72$H(B
$B$7(B, $R(S_n)$ $B$r$=$NI=8=4D$H$9$k(B. ($B0J2<(B, $BI=8=4D$O(B, $BM-M}?t(B$\mathbb Q$ $B$r(B
$B%F%s%=%k$7$?$b$N$r9M$($k(B.) $B$=$ND>OB(B $\bigoplus_n R(S_n)$$B$K@Q$r<!$K$h$j(B
$BDj5A$9$k(B.
\begin{align*}
 \bullet & \colon R(S_m)\otimes R(S_n) \to R(S_{m+n}) \\
  & \rho\bullet \rho' \defeq 
   \operatorname{Ind}_{S_m\times S_n}^{S_{m+n}}\rho\times\rho'
\end{align*}
$BC"$7(B, $\operatorname{Ind}$ $B$O(B$S_m\times S_n$ $B$+$i(B $S_{m+n}$ $B$X$NM6F3I=(B
$B8=$G$"$k(B. $BF1MM$KM>@Q$,(B
\begin{align*}
 \Delta & \colon R_p \to \bigoplus_{m+n=p} R(S_m)\otimes R_(S_n) \\
 & \Delta\rho \defeq\bigoplus\operatorname{Res}_{S_{m+n}}^{S_m\times S_n}
  (\rho)
\end{align*}
$B$GDj5A$5$l$k(B. $BC"$7(B, $\operatorname{Res}$ $B$O@)8BI=8=$G$"$k(B. $B$^$?(B 
$R(S_n)$ $B>e$KFb@Q$r(B
\begin{equation*}
  (\rho, \rho') = \dim \operatorname{Hom}_{S_n}(\rho, \rho')
\end{equation*}
$B$GDj$a(B, $\bigoplus_n R(S_n)$ $B>e$K$O(B, $R(S_n)$ $BC#$,8_$$$KD>8r$9$k$h$&$K(B
$BDj5A$9$k(B. $\bigoplus_n R(S_n)\otimes \bigoplus_n R(S_n)$ $B$K<+A3$KFb@Q(B
$B$rF~$l$F$*$1$P(B, Frobenius$B$NAj8_N'$K$h$C$F(B
\begin{equation*}
  (\rho\bullet\rho', \rho'') = (\rho\otimes\rho', \Delta\rho'')
\end{equation*}
$B$,@.$jN)$D(B. $B$5$i$K(B, $n$$B$NJ,3d(B $\lambda=(1^{m_1}2^{m_2}\dots)$ $B$r(B $S_n$
$B$N6&LrN`$HF10l;k$9$k$H$-(B, $S_n$ $B$N(B virtual$B$JI=8=$H$7$F(B, $B$=$N;XI8$,(B
$\lambda$ $B$N>e$G(B $z_\lambda = \prod i^{m_i}m_i!$ $B$G$"$k$b$N$r<h$k(B. $B$3(B
$B$N(B $z_\lambda$ $B$O(B, $\lambda$ $B$KB0$9$k85$N?t$,(B $n!/z_\lambda$ $B$K$J$k$h(B
$B$&$KDj$a$i$l$F$$$k(B. $BDj5A$+$i(B $(p_\lambda,p_\mu) = \delta_{\lambda\mu}
z_\lambda$$B$,@.$jN)$D(B. $BFC$K(B, $\lambda=(n)$ $B$KBP1~$9$k85$r(B $p_n$ $B$GI=$o(B
$B$9$3$H$K$9$k$H(B, $\bigoplus_n R(S_n)$ $B$O(B, ${\mathbb Q}[p_1,p_2,\dots]$
$B$HF17?$H$J$k$3$H$,J,$+$k(B. $B$=$3$G(B, $i > 0$$B$KBP$7$F(B, $p_i$ $B$r3]$1$k:nMQ(B
$BAG$r(B $P[-i]$$B$HDj5A$7(B, $B>e$NFb@Q$K4X$9$k?oH<:nMQAG$r(B, $P[i]$ $B$HDj5A$9$k(B.
$B>e$NFb@Q$N@-<A$rMQ$$$F(B $P[i]p_\lambda$ $B$r7W;;$9$k$3$H$K$h$C$F(B, $BF17?(B
$\bigoplus_n R(S_n)\cong{\mathbb Q}[p_1,p_2,\dots]$ $B$rDL$8$F(B, $B$3$l$i$N(B
$B:nMQ$,(BHeisenberg$BBe?t$N4X78<0$rK~$?$9$3$H$,%A%'%C%/$G$-$k(B. $B$^$H$a$k$H(B,
$B<!$N$h$&$K8@$($k(B: $\bigoplus_n R(S_n)$ $B$O%\%=%s$N%U%)%C%/6u4V$HF17?$G(B
$B$"$k(B.

\section{$B4v2?$+$i$N=`Hw(B: Hilbert$B357?$K$D$$$FCN$i$l$F$$$?$3$H(B}\label{sec:Hilb}

$X$ $B$rJ#AG?tBN(B $\C$ $B>e$N=`<M1FE*$JHsFC0[Be?t6JLL$H$7(B, $\HilbX{n}$ $B$r$=(B
$B$N>e$ND9$5$,(B $n$$B$N(B$0$$B<!85ItJ,357?$NA4BN$r%Q%i%a%H%i%$%:$9$k(BHilbert$B357?(B
$B$H$9$k(B. Hilbert$B357?$H$$$&8@MU$rJ9$$$?$3$H$,$J$$FI<T$bB?$$$H;W$&$N$G(B,$B>/(B
$B$7@bL@$7$h$&(B. $X$ $B$N>e$NAj0[$J$k(B $n$ $B8D$NE@(B $x_1,\dots, x_n$ ($B=gHV$N0c(B
$B$$$O9M$($J$$(B)$B$NA4BN$N$J$9G[0U6u4V$r9M$($k(B. $BE@$,$V$D$+$k$H$-$^$G4^$a$F(B
$B2?$i$+$N4v2?3XE*BP>]$r9M$($k$3$H$K$h$C$F(B, $B$3$NG[0L6u4V$r(B`$B40Hw2=(B'$B$9$k$3(B
$B$H$r9M$($h$&(B. $B0lHV0BD>$J40Hw2=$O(B, $BE@$,$V$D$+$C$F$bC1$K=89g$H$7$F$N9=B$(B
$B$@$1$7$+9M$($J$$$3$H$K$7$?$b$N$G(B, $B7k2L$O(B $X$ $B$N(B$n$$B<!$NBP>N@Q(B $S^n X =
X^n/S_n$ $B$H$J$k(B. ($S_n$$B$O(B, $X^n$ $B$K@.J,$NF~$lBX$($H$7$F:nMQ$9$k(B.) $B0lJ}(B,
Hilbert$B357?$O(B, $BC1$K=89g$H9M$($k$N$G$J$/$=$N>e$N4X?t$N6u4V$N9=B$$^$G9M(B
$B$($F40Hw2=$7$?$b$N$G$"$k(B. $B@53N$K8@$($P(B, $B9=B$AX(B$\shfO_X$ $B$N%$%G%"%kAX(B
$\idl\subset\shfO_X$ $B$G$"$C$F(B, $B$=$N(BHilbert$BB?9`<0$,(B, $BDj?t4X?t(B $n$ $B$H$J(B
$B$k$b$N$NA4BN$,(B $\HilbX{n}$ $B$G$"$k(B.

$BNc$($P(B, $n=2$ $B$N$H$-$r9M$($h$&(B.  $x_1, x_2$ $B$,(B $X$ $B>e$NAj0[$J$kE@(B
$B$G$"$k$H$9$k$H(B, $Z = \{x_1, x_2\}$ $B>e$N@5B'4X?t$N$J$9AX$O(B, $X$ $B$N9=B$(B
$BAX(B $\shfO_X$ $B$r(B, $Z$ $B>e$G>C$($k@5B'4X?t(B, $B$9$J$o$A(B $x_1$ $B$H(B $x_2$ $B$G>C(B
$B$($k@5B'4X?t$NAX$G3d$C$?AX$KB>$J$i$J$$(B. $x_2$ $B$,(B $x_1$ $B$K6a$E$$$F$$$C(B
$B$?$H$-(B, $x_1$ $B$H(B $x_2$ $B$G>C$($k@5B'4X?t$NA4BN$O(B, $B$I$&$J$k$@$m$&$+!)(B
$BMF0W$KJ,$+$k$h$&$K(B, $BEz$($O(B, $x_1$ $B$G>C$($F(B, $B$5$i$K(B $x_2$ $B$,(B $x_1$ $B$K6a(B
$B$E$$$?J}8~(B $v\in T_x X$ $B$GHyJ,$7$?$b$N$,>C$($k@5B'4X?t$NA4BN(B
\begin{equation}\label{eq:infnear}
  \{f\in\shfO_X \mid f(x_1) = 0, d f_{x_1}(v) = 0 \}
\end{equation}
$B$K$J$k(B. $B=>$C$F(B, $\HilbX{2}$ $B$K$O(B, $\{x_1, x_2\}$ ($x_1\neq x_2$)
$B$H$$$&7A$N$b$N$H(B, $x\in X$ $B$H@\6u4V(B $T_xX$ $B$N0l<!85ItJ,6u4V(B $L$ $B$NAH$_(B
$B$NFs<oN`$,8=$l$F$/$k(B. $B>e$NNc$G$O(B, $v$ $B$,D%$kItJ,6u4V$,(B $L$ $B$G$"$k(B. $B$3(B
$B$N(B $L$ $B$^$G9M$($?$H$3$m$,(B, $BBP>N@Q$H(BHilbert$B357?$N0c$$$G$"$k(B. 

Hilbert$B357?$+$iBP>N@Q$X(BHilbert-Chow$B<LA|$H8F$P$l$k@5B'<LA|(B$\pi\colon
\HilbX{n}\to S^n X$ $B$,<!$N$h$&$K$7$FDj5A$G$-$k(B. $\idl\in\HilbX{n}$ $B$N(B
$B$H$-(B, $B>&AX(B $\shfO_X/\idl$ $B$r9M$($k$H(B, $B$3$l$O9b!9M-8B8D$NE@$N>e$K$N$_Bf(B
$B$r;}$DAX$G$"$k(B.  $BBf$NE@(B $x$ $B$K$*$$$F(B $\shfO_X/\idl$ $B$ND9$5$r(B
$\operatorname{length}(\shfO_X/\idl)_x$ $B$H$*$/$H(B, Hilbert-Chow$B<LA|$O(B,
\begin{equation*}
  \pi(\idl) = \sum_{x\in\operatorname{Supp}(\shfO_X/\idl)}
  \operatorname{length}(\shfO_X/\idl)_x [x]
\end{equation*}
$B$GM?$($i$l$k(B. $B1&JU$G$O(B, $BBP>N@Q$NE@$rOB$N5-9f(B $\sum m_i [x_i]$ $B$GI=$o$7(B
$B$F$$$k(B. $B$3$N5-K!$O:#8e$b;H$o$l$k(B.

$BNc$($P(B, $B>e$G=R$Y$?(B $n=2$ $B$N$H$-$O(B, $Z = \{ x_1, x_2 \}$ $B$N$H$-$O(B 
$\pi(Z) = [x_1]+[x_2]$ $B$G$"$j(B, $Z$ $B$,(B\eqref{eq:infnear}$B$N$H$-$K$O(B,
$\pi(Z) = 2[x_1]$ $B$G$"$k(B.

$B$A$J$_$K(B, $B$3$3$^$G$N9=@.$O(B $X$ $B$N<!85$,2?<!85$G$"$C$F$b4X78$J$/$G$-$k(B.
$BFC$K(B, $X$ $B$,(BRiemann$BLL(B, $B$9$J$o$A(B $\dim X = 1$ $B$N$H$-$K$O(B $S^n X$ $B$H(B 
$\HilbX{n}$ $B$O(B, Hilbert-Chow$B<LA|$K$h$C$FF17?$G$"$k$3$H$,MF0W$KJ,$+$k(B.

$B<!85$r$b$&0l<!85>e$2$k$H<!$,@.$jN)$D$3$H$,CN$i$l$F$$$k(B.

\begin{Fact}[Fogarty~\cite{Fog}]
$X$ $B$,=`<M1FE*$JHsFC0[Be?t6JLL$N$H$-(B, Hilbert-Chow$B<LA|(B
$\pi\colon \HilbX{n}\to S^n X$ $B$O(B, $BBP>N@Q(B $S^nX$ $B$NFC0[E@2r>C$G$"$k(B.
$BFC$K(B, $\HilbX{n}$ $B$O(B, $B<!85$,(B $2n$ $B$NHsFC0[$JBe?tB?MMBN$G$"$k(B.
\end{Fact}

$B$^$?(B, $X$ $B$N<!85$,(B $3$$B0J>e$N$H$-$O0lHL$K$O(B, Hilbert$B357?$OHsFC0[$H$O8B(B
$B$i$:(B, $B4{Ls$G$9$i$J$$(B. $B$3$N0UL#$G(B, Hilbert$B357?$O<!85$,(B$2$$B$N$H$-$,Hs>o$K(B
$BFC<l$G$"$k(B.

$BBP>N@Q(B $S^n X$ $B$O(B, $BE@$N=EJ#EY$K$h$C$FJ,3d$9$k$3$H$,=PMh$k(B. $B$3$N$H$-(B
$n$ $B$NJ,3d(B$\lambda = (\lambda_1,\lambda_2,\dots)$ $B$KBP$7$F(B,
\begin{equation*}
  S^n_\lambda X = \left.\left\{ \sum_i \lambda_i [x_i] \right|
  \text{$x_i$ $BC#$O(B, $B8_$$$KAj0[$J$k(B $X$ $B$NE@(B} \right\}
\end{equation*}
$B$HDj5A$9$k$H(B, 
\begin{equation*}
  S^n X = \bigcup_{\text{$\lambda$ $B$O(B $n$ $B$NJ,3d(B}} S^n_\lambda X
\end{equation*}
$B$H$J$k(B. $B$3$N$H$-(B, $B<!$N;v<B$bCN$i$l$F$$$k(B.

\begin{Fact}[Brian\c{c}on]\label{Brian}{\rm
{\rm (1)} Hilbert-Chow $B<LA|(B $\pi\colon\HilbX{n}\to S^nX$ $B$O(B, $B>e$N(B
stratification$B$K4X$7$F(B, semi-small $B$G$"$k(B. $B$9$J$o$A(B, $B3F(B stratum
$S^n_\lambda X$ $B>e$G(B $\pi$ $B$O(B, fiber bundle $B$G$"$j(B, $C\in S^n_\lambda
X$ $B$N$H$-(B
\begin{equation*}
  2\dim \pi^{-1}(C) \le \codim S^n_\lambda X
\end{equation*}
$B$,@.$jN)$D(B. $B$h$j6/$/(B, $B>e$N(B $\le$ $B$O(B $=$ $B$,@.$jN)$D(B.

{\rm (2)} $B3F(B $C\in S^n_\lambda X$ $B$KBP$7$F(B, $\pi^{-1}(C)$ $B$O4{Ls$G$"$k(B.
}\end{Fact}

$BFC$K(B, $BAj0[$J$k(B $n$ $BE@C#$N$J$9(B stratum $S^n_{(1^n)} X$ $B$O(B open stratum
$B$G$"$j(B, $B$=$N>e$G$O(B $\pi$ $B$OF17?$G$"$k(B. $S^n_{(1^n)} X$ $B$O(B $S^n X$ $B$NHs(B
$BFC0[E@$N=89g$G$"$k$+$i(B,$B$3$N<gD%$O(B $\pi$ $B$,FC0[E@2r>C$G$"$k$3$H$K4^$^$l(B
$B$F$$$k(B. $B5U$K0lHV>.$5$$(Bstratum$B$O(B, $n[x]\in S^n_{(n)} X$ $B$H0lE@$K=8Cf$7(B
$B$F$$$k$b$N$G$"$j(B, $B$=$l$O(B $X$ $B<+?H$KF17?$GJ#AG(B$2$$B<!85$G$"$k(B. $BM><!85$O(B
$2(n-1)$ $B$G$"$k$+$i(B, $B>e$N<gD%$K$h$j(B, $B$=$N(Bfiber $\pi^{-1}(C)$ $B$O(B $n-1$
$B<!85$H$$$&$3$H$K$J$k(B. $B$=$7$F>e$N(B(2)$B$h$j4{Ls$G$"$k(B. $B$h$j0lHL$N(Bstratum$B$N(B
$B>e$N(Bfiber$B$O(B, $B$3$N(B $S^n_{(n)} X$ $B$N>e$N(Bfiber$B$N@Q(B($BC"$7(B, $n$ $B$O0[$J$k(B)$B$N(B
$B7A$K=q$1$k$N$G(B, $S^n_{(n)} X$ $B$N>l9g$K>ZL@$7$F$*$1$P0lHL$N>l9g$b>e$N(B
$B;v<B$,@.$jN)$D$3$H$,=>$&(B.

Weyl$B72$N(BSpringer$BI=8=$K>\$7$$J}$O(B, Springer$BFC0[E@2r>C(B
\begin{equation*}
  P\colon \text{$B4zB?MMBN$NM>@\B+(B} \to \text{$B6RNmB?MMBN(B}
\end{equation*}
$B$,(Bsemi-small$B$G$"$j(B, $B$=$N@-<A$rMQ$$$F(BBorho-MacPherson\cite{BM}$B$,(B $P$ $B$N(B
fiber $B$N%[%b%m%8!<72$H6RNm50F;$N8r:5%[%b%m%8!<72$N4V$N4X78$r=q$$$?$3$H(B
$B$O$h$/$4B8$8$N$3$H$@$H;W$&(B. Borho-MacPherson$B$N5DO@$rMQ$$$k$3$H$K$h$C$F(B
G\"ottsche-Soergel\cite{GS}$B$O(B, Hilbert-Chow$B<LA|$,(Bsemi-small$B$G$"$k$3$H(B
$B$H(B, fiber $B$,4{Ls$G$"$k$3$H(B, $B$=$7$F(B $S^n_\lambda X$ $B$NJDJq$N8r:5%[%b%m(B
$B%8!<72$,DL>o$N%[%b%m%8!<72$HF17?$G$"$k$3$H(B, $B$=$7$FBP>N@Q$N%[%b%m%8!<72(B
$B$N(BPoincar\'e$BB?9`<0$N(BMacdonald$B$K$h$k8x<0$rAH$_9g$o$;$k$3$H$K$h$C$F(B,$B<!$N(B
Poincar\'e$BB?9`<0$K4X$9$k8x<0$rF3$$$?(B. ($B:G=i$K(BG\"ottsche\cite{Got}$B$,(B
Weil$BM=A[$rMQ$$$F>ZL@$7$?(B.)

\begin{Fact}[\cite{Got,GS}]\label{Poincare}{\rm
$\HilbX{n}$ $B$N(BPoincar\'e$BB?9`<0(B $P_t(\HilbX{n})$ $B$NJl4X?t$O<!$N<0$GM?$((B
$B$i$l$k(B.
\begin{equation*}
    \sum_{n=0}^\infty q^n P_t(\Hilbn{X}) = \prod_{m=1}^\infty
  \frac{(1 + t^{2m-1}q^m)^{b_1(X)}(1 + t^{2m+1}q^m)^{b_3(X)}}
       {(1 - t^{2m-2}q^m)^{b_0(X)}(1 - t^{2m}q^m)^{b_2(X)}
                 (1 - t^{2m+2}q^m)^{b_4(X)}}\, .
\end{equation*}
$BC"$7(B, $b_i(X)$ $B$O(B $X$ $B$NBh(B$i$$B<!(BBetti$B?t$G$"$k(B.
}\end{Fact}

$B$3$NDjM}$N(B`$B=iEyE*(B'$B$J>ZL@$OCN$i$l$F$$$J$$(B. `$B=iEyE*(B'$B$H$$$&0UL#$O(B, $B@5I8?t(B
$B$N4v2?$d:XF#@9I'$N(BHodge$B2C72$NM}O@$rMQ$$$J$$$H$$$&0UL#$K9M$($F$b$$$$$,(B,
$B$3$3$G$O(B, $BBe?tE*$H$O8B$i$J$$0lHL$NJ#AG6JLL(B $X$ $B$KBP$7$F$bE,MQ$G$-$k>Z(B
$BL@(B, $B$H8@$C$F$b$$$$$@$m$&(B. $B$A$J$_$K(B, \secref{sec:HilbHeis}$B$N5DO@$rMQ$$(B
$B$l$P(B, $B:8JU$,1&JU$h$j$bBg$-$$$+Ey$7$$$3$H$O8e<T$N0UL#$G(B`$B=iEyE*(B'$B$K>ZL@$G(B
$B$-$k(B.

Springer$BFC0[E@2r>C$H(BHilbert-Chow$B<LA|$NN`;w$K$D$$$F=R$Y$?$,(B, $B4zB?MMBN$N(B
$BM>@\B+$O(B, $BM>@\B+$H$7$F%7%s%W%l%/%F%#%C%/7A<0(B\footnote{$B$3$3$G8@$&%7%s%W(B
$B%l%/%F%#%C%/7A<0$O(B, $B<BB?MMBN>e$N$=$l$G$O$J$/(B, $BJ#AGB?MMBN>e$N$b$N$G$"$k(B.
$B$9$J$o$A(B, $B@5B'(B$2$$B7A<0$GJD$G$"$j(B, $BHsB`2=$7$F$$$k$b$N$N$3$H$G$"$k(B. $B6hJL(B
$B$9$k$H$-$K$O(B, $B@5B'%7%s%W%l%/%F%#%C%/7A<0$H8F$P$l$k$3$H$b$"$k$,(B, $B$3$3(B
$B$G$OC1$K%7%s%W%l%/%F%#%C%/7A<0$H8F$V$3$H$K$9$k(B.}$B$r;}$D(B. $B$=$NN`;w$H$7$F(B
Hilbert$B357?>e$K%7%s%W%l%/%F%#%C%/7A<0$,B8:_$9$k$3$H$,CN$i$l$F$$$k(B.

\begin{Fact}[Fujiki~\cite{Fuj} {\rm ($n=2$)}, Beauville~\cite{Bea}
{\rm ($n$ $B0lHL(B)}]{\rm
$X$ $B$,%7%s%W%l%/%F%#%C%/7A<0$r;}$D$H$-(B, $\HilbX{n}$ $B$b%7%s%W%l%/%F%#%C(B
$B%/7A<0$r;}$D(B.
}\end{Fact}

$B>ZL@$O(B, $B<!$N$h$&$K$9$l$P$h$$(B: $BBP>N@Q(B $S^nX$ $B$NHsFC0[$JItJ,(B
$S^n_{(1^n)}X$ $B$O(B, $X$ $B$N%7%s%W%l%/%F%#%C%/7A<0$+$iM6F3$5$l$k<+A3$J%7(B
$B%s%W%l%/%F%#%C%/7A<0$r;}$D$,(B, $B$3$l$r(BHilbert-Chow$B<LA|(B $\pi$ $B$G0z$-La$7(B
$B$?$b$N$r9M$($k(B. $B$b$7$b(B, $B$3$l$,(B$\pi^{-1}(S^n_{(1^{n-2}2)})$ $B$rB-$7$?=8(B
$B9g$^$G?-$S$k$3$H$,>ZL@$G$-$l$P(B, $B$=$NJd=89g$NM><!85$,(B$2$$B0J>e$J$N$G(B,
$\HilbX{n}$ $BA4BN$K?-$S$k$3$H$,J,$+$k(B. $B$5$i$K(B,
$\pi^{-1}(S^n_{(1^{n-2}2)})$ $B$NMM;R$O(B, $B:G=i$K(B $\HilbX{2}$ $B$r@bL@$7$?$3(B
$B$H$+$iA[A|$,$D$/$h$&$K(B, $B6qBNE*$KJ,$+$k(B. $B=>$C$F(B, $B$3$3$^$G?-$S$F$$$k$3$H(B
$B$r6qBNE*$K%A%'%C%/$9$l$P$h$$(B.

$B$3$N;v<B$O$"$H$GI,MW$K$J$k$H$$$&$b$N$G$O$J$$$N$@$,(B, $\HilbX{n}$ $B$OBg@Z(B
$B$J6u4V$G$"$k$H$$$&?4$N;Y$($rM?$($k(B. $B$^$?(B, $B%7%s%W%l%/%F%#%C%/B?MMBN$NCf(B
$B$N%i%0%i%s%8%"%sItJ,B?MMBN$OBg@Z$G$"$k$,(B, \secref{sec:cot}$B$K$*$$$F%i%0(B
$B%i%s%8%"%sItJ,B?MMBN$NNc$,M?$($i$l$k(B. $B$^$?(BRiemann$B4v2?$N4QE@$+$i$O(B, $B%[(B
$B%m%N%_!<72$NJ,N`$K8=$l$k(Bhyper-K\"ahler$BB?MMBN$NNc$r<!$N$h$&$KM?$($k$H$$(B
$B$&0UL#$GBg@Z$G$"$k(B: $X$ $B$O<M1FE*$G(B($B$h$j0lHL$K%3%s%Q%/%H(BK\"ahler$B$G==J,(B
$B$G$"$k(B)$B%7%s%W%l%/%F%#%/7A<0$r;}$D$H$9$k$H(B($BJ#AG6JLL$NJ,N`O@$K$h$j(B$X$ $B$O(B
$BJ#AG%H!<%i%9$+(B$K3$$B6JLL$G$"$k(B), $\HilbX{n}$ $B$O(Bhyper-K\"ahler$B9=B$$r;}$D(B.
$B$3$3$G(B, $BB?MMBN(B $M$ $B>e$N(B{\bf hyper-K\"ahler$B9=B$(B}$B$H$O(B, Riemann$B7WNL(B $g$
$B$H;0$D$N35J#AG9=B$(B$I$, $J$, $K$ $B$NAH$_$G<!$rK~$?$9$b$N$N$3$H$G$"$k(B.
\begin{eqnarray*}
& g(Iv,Iw) = g(Jv, Jw) = g(Kv, Kw) = g(v,w) \\
& I^2 = J^2 = K^2 = IJK = -\operatorname{Id} \\
& \nabla I = \nabla J = \nabla K = 0
\end{eqnarray*}
$BC"$7(B, $\nabla$ $B$O(B, $g$ $B$N(BLevi-Civita$B@\B3$G$"$k(B. $B??$sCf$N<0$O(B, $M$ $B$N@\(B
$B6u4V$,;M85?tBN>e$N%Y%/%H%k6u4V$N9=B$$r;}$D$3$H$r<gD%$7$F$$$k$3$H$KCm0U(B
$B$9$k$H(B, $M$$B>e$K(Bhyper-K\"ahler$B9=B$$rM?$($k$3$H$H(B, $M$ $B$N(Bframe bundle$B$H(B
Levi-Civita$B@\B3$NAH$_$r(B$\Sp(n/4)$-bundle$B$H$=$N>e$N@\B3$X$N(Breduction$B$r(B
$BM?$($k$3$H$OF1CM$G$"$k(B. ($B$3$3$G(B, $n$ $B$O(B $M$ $B$N<!85$G$"$k(B.) $B$3$N$h$&$K(B
$BDj5A$OMF0W$G$"$k$,(B, $BHs<+L@$JNc(B, $BFC$K%3%s%Q%/%H$JNc$r:n$k$3$H$OD9$i$/L$(B
$B2r7h$G$"$C$?(B. $B:G=i$NNc$O(BK3$B6JLL$N>e$N(BRicci-flat K\"ahler$B7WNL$G$"$j(B, Yau$B$K(B
$B$h$k(BCalabi$BM=A[$N2r7h(B\cite{Yau}$B$K$h$jB8:_$,J]>Z$5$l$?(B. Hilbert$B357?>e$K(B
Calabi$BM=A[$N2r7h$K$h$C$FB8:_$9$k(BRicci-flat K\"ahler$B7WNL$rM?$($k$H(B, $B<!(B
$B$N$h$&$K9b<!85$NHs<+L@$J(Bhyper-K\"ahler$BB?MMBN$NNc$rM?$($k(B.

$B$^$:(BBochoner-Weitzenb\"ock$B8x<0$K$h$j%7%s%W%l%/%F%#%C%/7A<0(B $\omega$ $B$,(B
Ricci-flat K\"ahler$B7WNL(B $g$ $B$N(BLevi-Civita $B@\B3$K4X$7$FJ?9T$G$"$k$3$H$r(B
$B8@$&(B. $B$=$3$G%7%s%W%l%/%F%#%C%/7A<0$r(B
\begin{equation*}
  \omega(v,w) = g(Jv, w) + i g(Kv, w)
\end{equation*}
$B$HJ,2r$9$k$3$H$K$h$C$F(B, $J$, $K$ $B$rDj$a$k$H(B, $B$3$l$,;M85?t$N4X78<0$rK~(B
$B$?$9$3$H$,MF0W$K%A%'%C%/$G$-$k(B. $B$5$i$K(B, $B%7%s%W%l%F%#%C%/7A<0$,J?9T$G$"(B
$B$k$3$H$+$i(B, $J$, $K$ $B$,J?9T$G$"$k$3$H$,=>$&(B. $B@h$K=R$Y$?$h$&$K(B, $B%7%s%W(B
$B%l%/%F%#%C%/7A<0$O(B($B>/$J$/$H$bM><!85(B$2$$B$NItJ,B?MMBN$N30$G$O(B)$B6qBNE*$K9=(B
$B@.$5$l$k$N$G$"$k$,(B, Yau$B$N(BRicci-flat K\"ahler$B7WNL$O(B, $BITF0E@DjM}$GB8:_$,(B
$BJ]>Z$5$l$k$b$N$G(B, $B6qBNE*$K=q$-2<$5$l$k$b$N$G$O$J$$(B. $BFC$K(B, $X$ $B$N85!9$N(B
Ricci-flat K\"ahler$B7WNL$rMQ$$$F=q$-I=$9$h$&$J$3$H$OL$2r7hLdBj$G$"$k(B.

$B$3$N@a$N:G8e$K(B, $X = \C^2$ $B>e$NE@$N(BHilbert$B357?$N9TNs$K$h$k5-=R$r@b(B
$BL@$7$h$&(B. $B>e$K=R$Y$?@-<A$NB?$/$O(B, $X = \C^2$ $B$N>l9g$O(B, $B$3$N5-=R$rMQ$$(B
$B$k$3$H$K$h$C$F6qBNE*$K%A%'%C%/$G$-$k$N$G(B, $B6=L#$"$kFI<T$X$N1i=,LdBj$H$7(B
$B$F$*$3$&(B. $B$^$?@h$r5^$0FI<T$O(B, $B$3$3$rHt$P$7$F$bLdBj$O$J$$(B.

$\C^2$ $B$O(B, $B%"%U%!%$%sB?MMBN$J$N$GItJ,357?$O(B, $B:BI84D$9$J$o$AFsJQ?tB?9`(B
$B<04D(B $\C[z_1,z_2]$ $B$N%$%G%"%k$HF10l;k$5$l$k(B. $BFC$K(B, $\Hilbn{(\C^2)}$ $B$N(B
$BE@$O(B, $B%$%G%"%k(B $I\subset\C[z_1, z_2]$ $B$G(B $\C[z_1, z_2]/I$ $B$N(B $\C$ $B>e$N(B
$B%Y%/%H%k6u4V$H$7$F$N<!85$,(B $n$ $B$G$"$k$b$N$NA4BN$K0lCW$9$k(B. $B$=$3$G(B, $n$
$B<!85%Y%/%H%k6u4V(B$\C[z_1,z_2]/I$ $B$r(B $V$ $B$HCV$-(B, $z_1$, $z_2$ $B$r3]$1$k$3(B
$B$H$,(B $V$ $B$KM6F3$9$k@~7A<LA|$r(B $B_1$, $B_2$ $B$HDj5A$9$k(B. $B$5$i$K(B
$i\colon\C\to V$ $B$r(B $i(1) = 1\bmod{I}$ $B$K$h$C$FDj5A$9$k(B.  $B$3$N$H$-<!$,(B
$B@.$jN)$D(B.
\begin{Proposition}{\rm
$\Hilbn{(\C^2)}$ $B$H(B, $B<!$N@-<A$rK~$?$9(B $(B_1, B_2, i)$ $B$N;0$DAH$_$NA4BN(B
$B$r(B $\GL(V)$ $B$N<+A3$J:nMQ$G3d$C$?>&6u4V$OF17?$G$"$k(B.
\begin{enumerate}
\item $[B_1, B_2] = 0$
\item $B_1$, $B_2$ $B$GITJQ$J(B $V$ $B$NItJ,6u4V(B $S$ $B$G(B $i(1)$ $B$r4^$`$h$&$J(B
$B$b$N$O(B, $V$ $B<+?H$@$1$G$"$k(B. $B$9$J$o$A(B, $i(1)$ $B$O(B cyclic vector $B$G$"$k(B.
\end{enumerate}
}\end{Proposition}

$B>e$N$h$&$K$7$FDj5A$5$l$?(B $(B_1, B_2, i)$ $B$,L?Bj$N@-<A$rK~$?$9$3$H$O(B, 
$BL@$i$+$G$"$k(B. $B5U$KL?Bj$N@-<A$rK~$?$9(B $(B_1, B_2, i)$ $B$,M?$($i$l$?$H(B
$B$-(B,
\begin{equation*}
  I = \{ f(z_1, z_2)\in\C[z_1, z_2]\mid f(B_1, B_2)i(1) = 0 \}
\end{equation*}
$B$K$h$C$F%$%G%"%k(B $I$ $B$rDj$a$l$P(B, $\Hilbn{(\C^2)}$ $B$NE@$,Dj$^$j(B, $B$3$l$G(B
$B0lBP0lBP1~$,Dj5A$5$l$k(B. $B$5$i$K>e$N>&6u4V$,3j$i$+$G$"$k$3$H$r%A%'%C%/$7(B, 
$B>e$NBP1~$NHyJ,$rD4$Y$k$3$H$K$h$C$F(B, $B$3$l$,F17?$G$"$k$3$H$,J,$+$k(B.

$B$3$N5-=R$rMQ$$$F(BHilbert$B357?$N@-<A$r>/$7D4$Y$F8+$h$&(B. $BAj0[$J$k(B $n$ $B8D$NE@(B
\begin{equation*}
  \{ (z_1^1, z_2^1), \dots, (z_1^n, z_2^n) \}
\end{equation*}
$B$KBP1~$9$k$N$O(B, $B_1$, $B_2$ $B$,F1;~BP3Q2=2DG=$J$b$N$G(B
\begin{equation*}
  B_1 = 
  \begin{pmatrix}
  z_1^1 & \hdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \hdots & z_1^n
  \end{pmatrix}, \qquad
  B_2 = 
  \begin{pmatrix}
  z_2^1 & \hdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \hdots & z_2^n
  \end{pmatrix}, \qquad
  i = 
  \begin{pmatrix}
    1 \\ \vdots \\ 1
  \end{pmatrix}
\end{equation*}
$B$G$"$k(B. $B0lHL$K(B, $B_1$, $B_2$ $B$,8_$$$K8r492DG=$G$"$k$H$9$k$H(B, $B$b$7$=$l(B
$B$i$,BP3Q2=2DG=$G$"$l$P(B, $BF1;~$KBP3Q2=$G$-$k(B. $B$3$l$,(B, $B>e$N>u67$KBP1~$7$F(B
$B$$$k(B. $B0lHL$K$O(B, $BF1;~$K(BJordan$BI8=`7?$KD>$9$3$H$O=PMh$J$$$,(B, $n=2$ $B$N(B
$B>l9g$K$O$I$s$J$b$N$,=P$F$/$k$+$OMF0W$KJ,$+$C$F(B, $B>e$N%?%$%W$N$b$N0J30$O(B,
\begin{equation*}
  B_1 = 
  \begin{pmatrix}
   z_1 & \alpha \\ 0 & z_1
  \end{pmatrix}, \qquad
  B_2 =
  \begin{pmatrix}
   z_2 & \beta \\ 0 & z_2
  \end{pmatrix}, \qquad
  i = 
  \begin{pmatrix}
   0 \\ 1
  \end{pmatrix}
\end{equation*}
$B$H=q$1$k(B. $B$3$N$H$-(B, $[\alpha:\beta]$ $B$,(B, $(z_1, z_2)$ $B$K$*$1$k@\6u4V(B
$B$N0l<!85ItJ,6u4V$rDj$a$F$$$k$N$G(B, $B:G=i$K@bL@$7$?(B $\Hilb{(\C^2)}{2}$ $B$N(B
$B5-=R$,:F3NG'$5$l$k(B. $B$^$?(B, $BF1;~;03Q2=$OI,$:=PMh$F(B, $B$=$N$H$-$N8GM-CM$rJB(B
$B$Y$k$3$H$G(B, Hilbert-Chow$B<LA|$,M?$($i$l$k(B.

$B$3$N$h$&$K(BHilbert$B357?$NJ#;($5$O(B, $B8r492DG=$J9TNs$NJ#;($5$HF1$8$b$N$G(B, 
$B$"$^$j62$,$i$J$/$F$bBg>fIW$@(B, $B$H8@$($k(B.

$B$^$?(B, $B>e$N$h$&$J9TNs$K$h$kI=<($O(B, $Bd9B?MMBN(B\cite{Na-Duke}$B$HHs>o$K;w$F$$(B
$B$k(B. $B<B:](B, $B>e$N5-=R$O(B, $B0l$D$ND:E@$H(B, $BD:E@$H$=$l<+?H$r7k$VJU$+$i$J$k?^(B
\ref{fig:quiver}$B$KBP1~$9$kd9B?MMBN$H8@$&$3$H$,$G$-$k(B. $B$b$H$b$H$Nd9B?MM(B
$BBN$NDj5A$G$O(B, Kac-Moody Lie$B4D$KBP1~$9$k$b$N$@$1$r9M$($k$?$a$K(B, $B$3$N$h(B
$B$&$J?^<0$O=|30$5$l$F$$$?$N$@$,(B, $B4v2?3XE*$J@-<A$O(B, $BDL>o$Nd9B?MMBN$HF1MM(B
$B$KD4$Y$k$3$H$,$G$-$k$N$G$"$j(B, $BI.<T$,(BHilbert$B357?$K6=L#$r;}$C$?M}M3$N0l(B
$B$D$O$3$3$K$"$k$N$G$"$k(B.

\begin{figure}[htbp]
\begin{center}
\leavevmode
\setlength{\unitlength}{0.006in}
\begin{picture}(133,145)(0,-10)
\path(8.318,54.086)(6.000,62.000)(4.321,53.926)
\put(68.500,64.500){\arc{125.100}{3.1816}{9.3848}}
\put(6,67){\blacken\ellipse{10}{10}}
\put(6,67){\ellipse{10}{10}}
\end{picture}
\caption{Hilbert$B357?$KBP1~$9$k?^<0(B}
\label{fig:quiver}
\end{center}
\end{figure}

$B6qBNE*$J5-=R$,LrN)$D$3$H$NNc$H$7$F(B, $\Hilbn{(\C^2)}$ $B$N(BPoincar\'e$BB?9`(B
$B<0$r7W;;$7$F$_$h$&(B. $B$3$N>l9g$O(B, G\"ottsche$B$h$j0JA0$K(BEllingsrud-Str\o
mme~\cite{ES}$B$,7W;;$7$?(B.  $B%]%$%s%H$O(B, $BFs<!85%H!<%i%9(B $T^2$ $B$N(B
$\Hilbn{(\C^2)}$$B$X$N:nMQ$r9M$($k$3$H$G$"$k(B. $B$3$N:nMQ$O(B, $T^2$ $B$N(B$\C^2$
$B$X$N:nMQ$,M6F3$9$k$b$N$G$"$k(B. $B$=$N8GDjE@$O(B, $\C[z_1,z_2]$ $B$N%$%G%"%k$H(B
$B$7$FC19`<0$G@8@.$5$l$k$b$N$G$"$k(B.  $B$3$l$r(BYoung$B?^<0$r;H$C$FI=$o$9(B. ($B?^(B
\ref{fig:Young}) $B$h$C$F8GDjE@$O(B $n$ $B$NJ,3d$G%Q%i%a%H%i%$%:$5$l$k(B. $B$5$i(B
$B$K(B, Young$B?^<0(B $D$ $B$KBP1~$9$k8GDjE@$K$*$1$k(B $\Hilbn{(\C^2)}$ $B$N@\6u4V$N(B
$B%H!<%i%92C72$H$7$F$N;XI8$O(B
\begin{equation*}
  \sum_{s\in D} (T_1^{l(s)+1}T_2^{-a(s)} + T_1^{-l(s)}T_2^{a(s)+1})
\end{equation*}
$B$GM?$($i$l$k(B. $B$3$3$G(B, $T_i$ ($i = 1,2$) $B$O(B, $(t_1,t_2) \mapsto t_i$ $B$G(B
$BM?$($i$l$k;XI8$G$"$j(B, $l(s)$, $a(s)$ $B$O(Bleg-length, arm-length$B$H8F$P$l(B
$B$F$$$k(BYoung$B?^<0>e$N4X?t$G(B, $BH"(B $s$ $B$KBP$7$F(B, $B?^$N(B$\spadesuit$$B$H(B
$\heartsuit$$B$NH"$N8D?t$G$"$k(B.

\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{0.001in}
%
{\renewcommand{\dashlinestretch}{30}
\begin{picture}(1224,1539)(0,-10)
\path(312,912)(312,1212)(12,1212)
        (12,912)(312,912)
\path(312,612)(312,912)(12,912)
        (12,612)(312,612)
\path(612,612)(612,912)(312,912)
        (312,612)(612,612)
\path(312,312)(312,612)(12,612)
        (12,312)(312,312)
\path(612,312)(612,612)(312,612)
        (312,312)(612,312)
\path(612,12)(612,312)(312,312)
        (312,12)(612,12)
\path(912,12)(912,312)(612,312)
        (612,12)(912,12)
\path(312,12)(312,312)(12,312)
        (12,12)(312,12)
\path(1212,12)(1212,312)(912,312)
        (912,12)(1212,12)
\path(1212,312)(1212,612)(912,612)
        (912,312)(1212,312)
\path(912,312)(912,612)(612,612)
        (612,312)(912,312)
\path(912,612)(912,912)(612,912)
        (612,612)(912,612)
\path(612,912)(612,1212)(312,1212)
        (312,912)(612,912)
\path(312,1212)(312,1512)(12,1512)
        (12,1212)(312,1212)
\put(162,162){\makebox(0,0)[cc]{$1$}}
\put(462,162){\makebox(0,0)[cc]{$z_1$}}
\put(162,462){\makebox(0,0)[cc]{$z_2$}}
\put(162,1062){\makebox(0,0)[cc]{$z_2^3$}}
\put(162,762){\makebox(0,0)[cc]{$z_2^2$}}
\put(762,162){\makebox(0,0)[cc]{$z_1^2$}}
\put(462,462){\makebox(0,0)[cc]{$z_1z_2$}}
\end{picture}}
\qquad\qquad
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\caption{Young$B?^<0(B}
\label{fig:Young}
\end{center}
\end{figure}

$B0lHL$K%3%s%Q%/%H$J(B K\"ahler$BB?MMBN(B $M$ $B$K%H!<%i%9(B $T$ $B$,EyD9$+$D@5B'E*(B
$B$K:nMQ$7$F$$$k$H$-(B, $BBP1~$9$k1?F0NL<LA|(B $\mu\colon M
\to\operatorname{Lie}(T)^*$ $B$r9M$((B, generic $B$J(BLie$B4D$N85(B
$\zeta\in\operatorname{Lie}(T)$ $B$H(Bcoupling$B$7$F(B, $f = \langle \mu,
\zeta\rangle$ $B$O(B, Bott$B$N0UL#$G$N(BMorse$B4X?t$K$J$j(B, $B40A4$G$"$k$3$H$,CN$i(B
$B$l$F$$$k(B. $B$=$N$H$-(BMorse$B;X?t$O(B, $BNW3&E@$K$*$1$k@\6u4V$X$N%H!<%i%9$N:nMQ(B
$B$N;XI8$+$i7hDj$5$l$k(B. ($BNc$($P(B, \cite{Kirwan}$B$r8+$h(B.) $B:#$N>l9g$K$3$N%"(B
$B%$%G%"$rMQ$$$F(BMorse$B4X?t$r9M$($k$H(B, $\Hilbn{(\C^2)}$ $B$O%3%s%Q%/%H$G$O$J(B
$B$$$N$@$,8GM-$G$"$k$3$H$,J,$+$k$N$G(B, $B%3%s%Q%/%H$N$H$-$HF1$85DO@$rE,MQ=P(B
$BMh$k(B. $B$h$j$$$$$3$H$K(B, $B$3$N>l9g$ONW3&E@$OM-8B8D$NE@$G$"$k$+$i>u67$O$h$j(B
$B4JC1$K$J$k(B. $B7k6I(B, $B8x<0(B(\ref{Poincare})$B$N(B $X = \C^2$ $B$N>l9g$N>ZL@$,M?$((B
$B$i$l$k(B.

\section{Hilbert$B357?$N%[%b%m%8!<72$H(BHeisenberg$BBe?t(B}\label{sec:HilbHeis}

G\"ottsche$B$N(BPoincar\'e$BB?9`<0$NJl4X?t$N8x<0(B(\ref{Poincare})$B$r8+D>$7$F(B
$t = 1$$B$H$*$$$F$_$k$H(B, $B:G=i$K=R$Y$?(BHeisenberg$BBe?t$NI=8=$N;XI88x<0(B
\eqref{eq:char} $B$r(B$V = H_{\mathrm{even}}(X)\oplus H_{\mathrm{odd}}(X)$
$B$KE,MQ$7$?$b$N$K0lCW$7$F$$$k$3$H$K5$IU$/(B. ($B0J2<$G$O(B, ($B%3(B)$B%[%b%m%8!<72(B
$B$O(B, $BM-M}?t78?t$HLsB+$9$k(B.) $B$3$l$O(B, $B%"%U%!%$%s!&(BLie$B4D$NI=8=O@$r@lLg$H$7(B
$B$F$$$k?M$J$i$P$9$0$K5$IU$$$?$H;W$&$N$G$"$k$,(B, $B;d$OJ*M}3X<T(BVafa-Witten
$B$NO@J8(B\cite{VW}$B$G=i$a$F$3$l$rCN$C$?(B. $B$I$3$GCN$k$+$O$H$b$+$/$3$N;v<B$K(B
$B5$IU$1$P(B, $B<+A3$K!V(BHeiseberg$BBe?t$r4v2?3XE*$K9=@.$9$k$3$H$,$G$-$k$+!W$H(B
$B$$$&LdBj$K9T$-Ce$/(B. $B;d$O(B, $B$3$l$rCN$kD>A0$Kd9B?MMBN$N%[%b%m%8!<72$K(B
Kac-Moody Lie$B4D$NI=8=$r9=@.$7$F$$$?$N$G(B, $BF1$8$3$H$r$d$C$F$_$l$P(BHilbert
$B357?$N%[%b%m%8!<72$K(B, Heisenberg$BBe?t$NI=8=$,:n$l$k$O$:$@$H(B, $B$9$0$KL\@1(B
$B$,N)$C$?(B. $B$3$l$r>ZL@$7$?$N$,(B\cite{Na-Hilb}$B$G$"$k$,(B, $B0J2<$G$O(B,$BF1;~4|$K(B
$BFHN)$KF1MM$N7k2L$rF@$?(BGrojnowski~\cite{Gr}$B$K$h$kDj<02=$K=>$C$F@bL@$9$k(B.

G\"ottsche$B$N8x<0$HBP>N72$NI=8=4D$H%\%=%s$N%U%)%C%/6u4V$NBP1~(B, $B$=$7$F;X(B
$BI88x<0Ey$r8+Hf$Y$F$_$k$H(B, $H_*(\HilbX{n})$ $B$H(B $R(S_n)$ $B$,BP(B
$B1~$7$F$$$k$3$H$,J,$+$k(B.\footnote{Hilbert$B357?$G$J$/(B, $BBP>N72$N(B$X^n$$B$X$N(B
$B:nMQ$K4X$9$kF1JQ(B$K$$B72(B $K_{S_n}(X^n)$ $B$r;H$&$HN`;w$,$h$j$O$C$-$j$9$k(B.
(\cite{Gr}$B$N(Bp.283$B$N5SCm;2>H(B)}
$B=>$C$F(B, $\bigoplus_n R(S_n)$ $BDj5A$7$?@Q$NN`;w(B
$B$r(BHilbert$B357?$K9=@.$9$l$P8@$$Lu$G$"$k(B. $B$3$l$O(B, $B<!$N$h$&$K$9$l$P$h$$(B.
$B0J2<(B, $B4JC1$N$?$a(B $X$ $B$O%3%s%Q%/%H$H$9$k(B.
$\HilbX{m}\times\HilbX{n}$ $B$NCf$N3+ItJ,=89g(B
\begin{equation*}
  \left(\HilbX{m}\times\HilbX{n}\right)^{\circ}
\defeq \{ (Z_1, Z_2)\in \HilbX{m}\times\HilbX{n} \mid
\operatorname{Supp}(Z_1)\cap\operatorname{Supp}(Z_2)=\emptyset \}
\end{equation*}
$B$r9M$($k(B. $B$=$NCf$NE@(B $(Z_1, Z_2)$ $B$KBP$7$F(B $Z_1\cup Z_2$ ($BBP1~$9$k%$%G(B
$B%"%k$G$O(B $\idl_{Z_1}\cap\idl_{Z_2}$) $B$O(B, $\HilbX{m+n}$ $B$NE@$rDj$a$k(B.
$B$=$3$G(B, 
\begin{equation*}
\left\{ (Z_1,Z_2,Z_1\cup Z_2)\left| (Z_1,Z_2)\in
\left(\HilbX{m}\times\HilbX{n}\right)^{\circ} \right\}\right.
\end{equation*}
$B$N(B $\HilbX{m}\times\HilbX{n}\times\HilbX{m+n}$ $BFb$G$NJDJq$r9M$((B, $B$=$l(B
$B$r(B $\Lambda_{m,n}$ $B$GI=$o$=$&(B. $B$3$N$H$-(B, $BI=8=4D$N@Q$NN`;w$H$7$F@Q$r(B
\begin{align*}
 \bullet & \colon H_*(\HilbX{m})\otimes 
H_*(\HilbX{n}) \to H_*(\HilbX{m+n}) \\
  & c\bullet c' \defeq 
   p_{3*}\left(p_1^*c\cup p_2^*c'\cap[\Lambda_{m,n}]\right)
\end{align*}
$B$K$h$jDj5A$9$k(B. $BC"$7(B, $p_i$ $B$O(B,
$\HilbX{m}\times\HilbX{n}\times\HilbX{m+n}$ $B$NBh(B$i$$B@.J,$X$N<M1F$G$"$k(B.
$B$^$?M>@Q(B $\Delta$ $B$N(B $\HilbX{m}\times\HilbX{n}$$B@.J,(B $\Delta_{m,n}$ $B$b(B
\begin{align*}
 \Delta & \colon H_*(\HilbX{m+n}) \to 
H_*(\HilbX{m})\otimes H_*(\HilbX{n}) \\
 & \Delta c \defeq (p_1\times p_2)_*(p_3^* c\cap[\Lambda_{m,n}])
\end{align*}
$B$K$h$C$FDj5A$G$-$k(B.

$R(S_i)$ $B$K$*$1$k(B $p_i$ $B$NN`;w$H$7$F<!$N(B$\HilbX{i}$$B$NItJ,B?MMBN(B($B0lHL$K(B
$B$OFC0[E@$r;}$D(B)$B$r9M$($k(B:
\begin{equation*}
\frak P_i = \bigcup_{x\in X} \pi^{-1}(i[x])
     = \pi^{-1}(S^i_{(i)}X)
\end{equation*}
$BC"$7(B, $\pi$ $B$O(B, Hilbert-Chow$B<LA|$G$"$k(B. $B$9$J$o$A(B, \secref{sec:Hilb}$B$G(B
$B=P$F$-$?Bf$,0lE@$K=8Cf$7$?%$%G%"%k$NA4BN$N$J$9ItJ,B?MMBN$G$"$k(B. $BDj5A$+(B
$B$i(B, $B<M1F(B $\Pi\colon \frak P_i\to X$ $B$,$"$k(B. $X$ $B$N%[%b%m%8!<72$r(B
$H_*(X)$ $B$H$7(B,
\begin{equation*}
\frak P_i(\alpha) = \Pi^*(\operatorname{PD}\alpha)\cap [\frak P_i]
\end{equation*}
$B$H$*$/(B. $B$3$3$G(B$\operatorname{PD}$ $B$O(B Poincar\'e$BAPBP<LA|(B
$H_*(X)\to H^{4-*}(X)$ $B$G$"$k(B. $\alpha$ $B$N6v4q$K=>$C$F(B $\frak
P_i(\alpha)$ $B$K$b<!?t$rF3F~$7$F$*$/(B.

$B$=$3$G(B, $i > 0$ $B$N$H$-(B, $\frak P_{i}(\alpha)$ $B$r3]$1$k<LA|$r(B
$P_\alpha[-i]$ $B$GI=$o$7(B, $BFb@Q$K4X$9$k(B(super$B$N0UL#$N(B)$B?oH<:nMQAG$r(B
$P_\alpha[i]$ $B$GI=$o$9$3$H$K$9$k(B. $B$3$N$H$-(B, $\frak P_i$ $B$N<!85$N8x<0(B
(\ref{Brian})$B$r;H$C$F%[%b%m%8!<$N<!?t$N:9$rD4$Y$F$_$k$H(B,
\begin{equation*}
 P_{\alpha}[i]\colon H_{2n+k}(\HilbX{n}) \to
 H_{2(n-i)+k+\deg\alpha-2}(\HilbX{n+i})
\end{equation*}
$B$H$J$k$3$H$,J,$+$k(B. $B$D$^$jCf4V<!85$+$i$N$:$l(B
$k$ $B$,(B $k + \deg\alpha-2$ $B$HBe$o$k(B. $BFC$K(B $\deg\alpha = 2$ $B$J$i$PCf(B
$B4V<!85$+$i$N$:$l$OJ]$?$l$k(B. $B$^$?(B, $n$ $B$K4X$7$FD>OB$r<h$j(B,
\begin{equation*}
   \bigoplus_n H_*(\HilbX{n})
\end{equation*}
$B$N@~7AJQ49$,Dj5A$5$l$k$,(B, $B$3$l$b5-9f(B $P_\alpha[i]$ $B$GI=$o$7$F$7$^$&(B
$B$3$H$K$9$k(B. $B$3$N$H$-(B, $B2f!9$N<g7k2L$O<!$N4X78<0$,@.$jN)$D$3$H$G$"$k(B.

\begin{Theorem}[Nakajima~\cite{Na-Hilb}, Grojnowski~\cite{Gr}]
\begin{equation*}
  P_\alpha[i] P_\beta[j] -
  (-1)^{\deg\alpha\deg\beta}P_\beta[j]P_\alpha[i] 
  = (-1)^{i-1}i\delta_{i+j,0}\langle\alpha,\beta\rangle\operatorname{id}
\end{equation*}
\end{Theorem}

$B$3$3$G(B, $\langle\alpha,\beta\rangle$ $B$O(B $\alpha$ $B$H(B $\beta$ $B$N8r:5?t$G(B
$B$"$k(B.

$B:G=i$K$3$NDjM}$N>ZL@$rM?$($?$H$-$K$O(B, $(-1)^{i-1}i$ $B$NItJ,$O(B, $i$$B$@$1(B
($BFC$K(B $X$ $B$d(B $\alpha$, $\beta$ $B$K$OL54X78(B)$B$G7h$^$kDj?t$G$"$k$3$H$@$1$7(B
$B$+J,$+$i$J$+$C$?$,(B, $B8e$K(BEllingsrud-Str\o mme \cite{ES2} $B$,5a$a$?(B.

$B$3$NDjM}$N>ZL@$O(B, $B$=$NItJ,$r=|$1$P=iEyE*$GC1$K8r:5$r7W;;$9$l$P=P$k(B.
$(-1)^{i-1}i$ $B$r7h$a$k$H$3$m$K$D$$$F$O(B, $B<!$N>O$GJL>ZL@$rM?$($k(B.

$B>e$N7k2L$K$h$j(B, $(-1)^{i-1}$ $B$rL5;k$9$l$P(B, $\bigoplus_n
H_*(\HilbX{n})$ $B$,(B $V = H_*(X)$ $B$K(BHeisenberg$BBe?t$NI=8=6u4V$G$"$k$3$H$,(B
$BJ,$+$C$?(B. G\"ottsche$B$N8x<0(B(\ref{Poincare})$B$H9g$o$;$"$l$P(B, $B$=$l$,%U%)%C%/(B
$B6u4V$G$"$k$3$H$bJ,$+$C$?(B.\footnote{$B:#$N=j(B, G\"ottsche$B$N8x<0$rMQ$$$J$$(B
$B$G(B, $BI=8=$N4{Ls@-$r>ZL@$G$-$F$$$J$$(B.}

\section{$BBP>NB?9`<0$H(BHilbert$B357?(B}\label{sec:cot}

$B$3$N@a$G$O(B, $BBP>N72$NI=8=4D(B $R(S_n)$ $B$H(B Hilbert$B357?$N%[%b%m%8!<(B 
$H_*(\HilbX{n})$ $B$H$NN`;w$r$5$i$K?J$a$k(B.

$B$^$:(B, $BBP>N<0$NM}O@$r4JC1$KI|=,$9$k(B. ($B$b$C$H>\$7$$$3$H$,CN$j$?$$FI<T$O(B, 
\cite{Macdonald}$B$r;2>H$9$k$H$h$$(B.)
$N$$B8D$NJQ?t(B $x_1,\dots, x_N$ $B$NM-M}78?tBP>NB?9`<0$NA4BN$r(B $\Lambda_N$
$B$H$7(B, $B<!?t$K$h$C$F(B $\Lambda_N = \bigoplus_n \Lambda_N^n$ $B$HJ,2r$9$k(B.
$M \le N$ $B$N$H$-(B, $B:G8e$N(B $M-N$$B8D$NJQ?t(B $x_{M+1}, \dots, x_N$ $B$r(B $0$ $B$H(B
$B$*$/$3$H$K$h$C$F(B, $B<M1F(B $\Lambda_N^n \to \Lambda_M^n$ $B$,Dj5A$5$l$k$,(B,
$B5U6K8B(B $\projlim_{N\to\infty} \Lambda_N^n$ $B$r(B $\Lambda^n$ $B$H$*$-(B, $B$=$N(B
$BD>OB(B $\Lambda = \bigoplus_n \Lambda^n$ $B$r9M$($k(B. $B0J2<$G$O(B, $\Lambda$
$B$N85$rC1$KBP>NB?9`<0$H8F$V$3$H$K$9$k(B.

$x_1^n+x_2^n+\cdots$ $B$,Dj$a$k(B $\Lambda^n$ $B$N85$r(B $p_n$ $B$H=q$/$3$H$K$9(B
$B$k(B. $B$h$/CN$i$l$F$$$k$h$&$K(B, $\Lambda$ $B$O(B, $p_n$ $BC#$NB?9`<0$N4D(B
${\mathbb Q}[p_1,p_2,\dots]$ $B$HF17?$K$J$k(B. $BJ,3d(B $\lambda =
(1^{m_1}2^{m_2}\cdots)$ $B$KBP$7$F(B, $p_\lambda =
p_1^{m_1}p_2^{m_2}\cdots$ $B$H$*$/$H(B, $p_\lambda$ $B$O(B $\Lambda$ $B$N4pDl$H(B
$B$J$k(B. $B0lJ}$G(B, $n$$B$NJ,3d(B $\lambda$ $B$rC1D4HsA}2C$N<+A3?tNs(B$(\lambda_1\ge
\lambda_2\ge\dots)$ $B$GI=$o$7(B, $BC19`<0(B$x^{\alpha_1}x_2^{\alpha_2}\dots$
$B$G$=$N78?t(B $\alpha_i$ $B$rBg$-$$=g$KJB$SJQ$($?$b$N$,(B, $\lambda_i$ $B$G$"$k(B
$B$h$&$J$b$N$NOB$,Dj$a$k(B $\Lambda^n$$B$N85$r(B $m_\lambda$ $B$GI=$o$9$3$H$K$9(B
$B$k(B. ($B@53N$K$O(B, $\Lambda_N^n$ $B$N85$r9M$((B, $B$=$l$,(B $\Lambda^n$ $B$N85$rDj5A(B
$B$7$F$$$k$3$H$r%A%'%C%/$9$k(B.) $BJ,3d$rA4$FF0$+$9$H(B, $m_\lambda$ $B$b$d$O$j(B 
$\Lambda$ $B$N4pDl$H$J$k(B.

$B$5$F(B, $BBP1~$9$k(BHilbert$B357?$N4v2?$r@bL@$7$h$&(B. $B6JLL(B $X$ $B$NCf$K6J@~(B $C$$B$,(B
$BF~$C$F$$$k$H$7(B, $C$ $B$NDj$a$k%[%b%m%8!<N`(B $[C]$ $B$KBP1~$9$k:nMQAG(B
$P_{[C]}[i]$ $B$r9M$($k(B. $BA0$N@a$N7k2L$h$j(B, $P_{[C]}[i]$ ($i < 0$) $BC#$r??(B
$B6u(B $1\in H_0(\HilbX{0})$ $B$K2?2s$b:nMQ$5$;$F$G$-$k%Y%/%H%kC#$GD%$i$l$k(B
$BItJ,6u4V(B
\begin{equation*}
  \operatorname{Span}\{ P_{[C]}[-j_1]\cdots P_{[C]}[-j_k] 1
\mid k\in\mathbb Z_{\ge 0}, j_1,\dots, j_k\in \mathbb Z_{>0} \}
\end{equation*}
$B$O(B, $BBP1~(B
\begin{equation*}
P_{[C]}[-j_1]\cdots P_{[C]}[-j_k] 1 \longleftrightarrow
p_{j_1}\cdots p_{j_k}
\end{equation*}
$B$K$h$C$F(B, $B2D;;L58B8D$NJQ?t$NB?9`<04D(B ${\mathbb Q}[p_1,p_2,\dots]$ $B=>$C(B
$B$FBP>NB?9`<0$N4D(B $\Lambda$ $B$HF17?$G$"$k(B. $B$3$N$H$-(B, $m_\lambda$ $B$rI=$o(B
$B$9$h$&$J%5%$%/%k$r0J2<$N$h$&$KDj5A$9$k$3$H$,$G$-$k(B.
\begin{equation*}
  L^{\lambda}C \defeq
 \overline{\pi^{-1}(S^n_\lambda C)}
\end{equation*}
$BC"$7(B, $S^n_\lambda C$ $B$O(B, $S^nC \subset S^nX$ $B$rDL$8$F(B, $S^n X$ $B$N6I=j(B
$BJD=89g$H9M$($F$*$j(B, $\pi$ $B$O(B Hilbert-Chow$B<LA|$G$"$k(B.

\begin{Theorem}[cf. Grojnowski~\cite{Gr}]
(1) $L^\lambda C$ $B$O(B, $\HilbX{n}$ $BFb$N(B $n$ $B<!85$N4{Ls$JItJ,B?MMBN$G$"$k(B.
($B0lHL$K$OFC0[E@$r;}$D(B.) $B$=$N4pK\N`(B $[L^\lambda C]\in
H_{2n}(\HilbX{n})$ $B$O(B, $B>e$NBP>NB?9`<0$H$NF17?$rDL$8$F(B, $m_\lambda$ $B$K(B
$BBP1~$9$k(B.

(2) $X$ $B$,%7%s%W%l%/%F%#%C%/7A<0$r;}$D$H$-(B, $L^\lambda C$ $B$O(B,
$\Hilbn{X}$ $B$N%i%0%i%s%8%"%sItJ,B?MMBN$G$"$k(B.
\end{Theorem}

$BNc$($P(B, $BJ,3d(B $\lambda$ $B$,(B, $(n)$ $B$N$H$-(B, $L^{(n)} C$ $B$O(B, $BItJ,%9%-!<%`(B
$Z$ $B$G(B, $B$=$NBf$,(B $C$ $B$N0lE@$G$"$k$b$N$NA4BN$G$"$k(B. $B$^$?(B, $\lambda$ $B$,(B
$(1^n)$ $B$N$H$-$O(B, $C$ $B$NBP>N@Q(B $S^n C$ $B$r(B $C$ $B$N(BHilbert$B357?(B 
$\Hilbn{C}$ $B$H;W$C$F(B $\Hilbn{X}$ $B$NCf$KF~$l$?$b$N$KB>$J$i$J$$(B.

$m_\lambda$ $B$O(B, $p_i$ $BC#$NB?9`<0$G=q$-I=$9$3$H$,$G$-$k$,(B, $BFC$K(B,
$m_{(1^n)}$ $B$O$-$l$$$JI=<(<0$r;}$C$F$$$k(B. $B$3$l$r(B, $B>e$NDjM}$rMQ$$$F(B,
Hilbert$B357?$N%5%$%/%k$N8@MU$KD>$9$H(B, $B<!$rF@$k(B.

\begin{Corollary}
$C$ $B$NBP>N@Q(B $S^nC$ $B$NI=$o$9%[%b%m%8!<N`(B $[S^n C]$ $B$NJl4X?t$O(B,
$P_{[C]}[i]$ $B$rMQ$$$F<!$N<0$GM?$($i$l$k(B.
\begin{equation*}
\sum_{n=0}^\infty z^n [S^n C]
    = \exp\left(\sum_{i=1}^\infty
    \frac{z^i P_{[C]}[-i]}{(-1)^{i-1}i}\right)\cdot 1
\end{equation*}
$BC"$7(B, $1$ $B$O??6u%Y%/%H%k$G$"$k(B.
\end{Corollary}

$B$3$N5DO@$N1~MQ$H$7$F(B, $BA0$N>O$G=R$Y$?(B $(-1)^{i-1}i$ $B$,<!$N$h$&$K7hDj$5(B
$B$l$k(B. $X = \mathbb{C}\text{P}^2$ $B$H$7(B, $C = \text{line}$ $B$H$7$F(B,
$P_{[C]}[i][S^nC]$ $B$r9M$($k(B. $C$ $B$r>/$7F0$+$7$F$b$&0l$DJL$N(Bline $C'$
$B$r<h$C$F$_$k$H$9$0J,$+$k$h$&$K(B, $B$3$l$O(B $[S^{n-i}C]$ $B$GM?$($i$l$k(B. $B$3$N(B
$B<0$H>e$NJl4X?t$r8+Hf$Y$F(B, $B7W;;$9$l$PDj?t$,(B $(-1)^{i-1}i$ $B$G$"$k$3$H$,(B
$BMF0W$KJ,$+$k(B.

$B$5$F(B, $B>e$N<0$N1&JU$O89M}O@(B/$B6&7A>lM}O@$G(B{\bf $BD:E@:nMQAG(B}$B$H8F$P$l$k$b$N(B
$B$N0lHV86;OE*$J7A$G$"$k(B. $BD:E@:nMQAG$O(B, $B$3$l$i$NM}O@$G=EMW$JLr3d$r2L$?$9(B
$B$3$H$,CN$i$l$F$$$k(B. $BNc$($P(B, Heisenberg$BBe?t$N%U%)%C%/6u4V$+$i%"%U%!%$%s(B
Lie$B4D$NI=8=(B($B%Y!<%7%C%/I=8=$H8F$P$l$kI=8=(B)$B$r9=@.$G$-$k$,(B(Frenkel-Kac$B9=(B
$B@.K!(B), $B$=$N$H$-%"%U%!%$%s(BLie$B4D$N@8@.85$,D:E@:nMQAG$GDj5A$5$l$k(B. $B$3$N$h(B
$B$&$JOCBj$NH/E8$H$7$F(B, $BBe?tE*$JJ}LL$G$OD:E@Be?t$d(B, $B0lHL2=$5$l$?(B
Kac-Moody$BBe?t$J$I(B, $BLLGr$$OC$,$"$k$N$@$,(B, $B$3$l$i$r(BHilbert$B357?$N4v2?$H$7(B
$B$FM}2r$9$k$3$H$O:#8e$N2]Bj$H8@$($h$&(B. $BFq$7$$$H$3$m$O(B, $B>e$N1&JU$N$h$&$J(B
$B<0$K4v2?3XE*$J0UL#$r$D$1$k$3$H$K$"$k(B. $B:#$N$H$3$m(B, $B8r:5$r7W;;$7$F$_$k0J(B
$B30$K<j$@$F$,$J$$$N$@$,(B, $B>e$N:8JU$N$h$&$K0UL#$,$"$k$b$N$K$I$&$7$?$i$J$k(B
$B$N$+$rH=Dj$9$k$$$$>r7o$r8+$D$1$k$3$H$,$G$-$F$$$J$$(B. $BD:E@Be?t$d8e$G=P$F(B
$B$/$k(BVirosoro$BBe?t$K$D$$$F$O(B,$B:G6a=PHG$5$l$?(B Kac $B$N652J=q(B\cite{Kac}$B$K$^$H(B
$B$a$i$l$F$$$k(B. $B$3$l$i$r2#L\$G8+$J$,$iBP1~$9$k4v2?3XE*BP>]$rC5$9$3$H$O=E(B
$BMW$JLdBj$G$"$k(B.

$B$^$?(B, $B6&7A>lM}O@$G$O(B, $B>e$GF3F~$5$l$?JQ?t(B $z$ $B$O<M1FD>@~$NE@$N:BI8$H8+(B
$B$J$5$l$k(B. $B$7$+$7(B, Hilbert$B357?$NN)>l$+$i$O(B, $B<M1FD>@~$NLr3d$OL@$i$+$G$J(B
$B$$(B. $B$3$l$b>e$HF1MM(B, $B:#8e$N2]Bj$G$"$k(B. $B$3$N$h$&$J$3$H$r9M$($F$$$k$H(B, $B?t(B
$BNs$+$iJl4X?t$r:n$C$?$h$&$K(B, $B6u4V$NNs$rA4It4]$4$H$&$^$/<h$j07$&5;=Q$r2f!9(B
$B$,$"$^$j;}$C$F$$$J$$$3$H$,:$Fq$N0lHV$N860x$G$"$k$h$&$K;W$($F$/$k(B. $B$=$3(B
$B$G(B, $B$^$:?F$7$_$r;}$H$&$H8@$&$3$H$G(B, $B$3$N$h$&$J9M$(J}$r(B{\bf $BJl6u4V(B}$B$HL>(B
$BA0$rIU$1$F$_$?(B. $B:#$N>u67$@$1$G$J$/MM!9$J>l=j$GJl6u4V$N9M$(J}$NI,MW@-$,(B
$B8=$l$F$$$k$h$&$J5$$,$7$F$J$i$J$$(B. (cf.~\cite{suusemi})

$B$^$?(B, $B%"%U%!%$%s(BLie$B4D$NI=8=O@$d6&7A>lM}O@$G=EMW$JLr3d$r$9$k$b$N$H$7$F(B,
Virasoro$BBe?t$,$"$k(B. $B$3$l$O(B, $L[i]$ ($i\in\mathbb Z$), $K$ $B$G@8@.$5$l(B,
$B4X78<0(B
\begin{equation*}
  \big[ L[i], L[j] \big] = (i-j)L[i+j] + \frac{i^3-i}{12}\delta_{i+j,0}K
\end{equation*}
$B$GDj5A$5$l$k(BLie$B4D$G$"$k(B. $S^1$$B>e$N%Y%/%H%k>l$N$J$9(BLie$B4D$NJ#AG2=$NCf?4(B
$B3HBg$HB*$($k$N$,DL>o$G$"$k(B. ($L[i]$ $B$O(B, $B%Y%/%H%k>l(B $z^{i+1}d/dz$ $B$KBP1~(B
$B$9$k(B.) Heisenberg$BBe?t$+$i<!$N$h$&$K$7$F(BVirasoro$BBe?t$,9=@.$5$l$k$3$H$O(B, 
$B$h$/CN$i$l$F$$$k(B:
\begin{equation*}
  L[i] = \frac12 \sum_{j\in\mathbb Z} \normord P[j] P[i-j] \normord
\end{equation*}
$BC"$7(B, $B$3$3$G(B$\normord\ \normord$ $B$O(B normal ordering $B$G(B,
\begin{equation*}
 \normord P[j] P[i] \normord =
 \begin{cases}
   P[i] P[j] & \text{$i\le j$ $B$N$H$-(B} \\
   P[j] P[i] & \text{$i > j$ $B$N$H$-(B}
 \end{cases}
\end{equation*}
$B$HDj5A$5$l$k(B. $B>e$NDj5A$O$_$+$1>e$OL58BOB$G$"$k$,(B, $B%U%)%C%/6u4V$N85$K:n(B
$BMQ$9$k$H$-$K$O(B, $BM-8B8D$r=|$$$F(B$0$$B$K$J$k(B.

Virasoro$BBe?t$N(BHilbert$B357?$N4v2?$G$N<B8=$O(B, Lehn\cite{Lehn}$B$K$h$C$FM?$((B
$B$i$l$?(B. $B0J2<(B, $B$3$l$r@bL@$9$k(B. $B$^$:>e$N9=@.$O0l8D$N(BHeisenberg$BBe?t$KBP$7(B
$B$F$G$"$C$?$,(B, Hilbert$B357?$N>l9g$O3F%[%b%m%8!<N`(B $\alpha$ $B$KBP$7$F<!$N(B
$B$h$&$K$7$F(B $L_\alpha[i]$ $B$rDj5A$9$k(B. $BBP3Q@~Kd$a9~$_$K$h$k(Bpush-forward
$\delta\colon H_*(X)\to H_*(X\times X)\cong H_*(X)\otimes H_*(X)$ $B$K$h(B
$B$j(B $\delta\alpha = \sum_a \alpha'_a\otimes \alpha''_a$ $B$H$J$C$F$$$k$H(B
$B$7(B,
\begin{equation*}
  L_\alpha[i] = \frac 12 \sum_a \sum_{j\in\mathbb Z}
 \normord P_{\alpha'_a}[j] P_{\alpha''_a}[i-j] \normord
\end{equation*}
$B$H$*$/(B. $B$3$l$r4v2?3XE*$K<B8=$9$k$3$H$,L\I8$H$J$k(B.

$\HilbX{n}$ $B$NItJ,B?MMBN(B
\begin{equation*}
  \partial\HilbX{n} \defeq 
  \HilbX{n}\setminus\pi^{-1}(S^n_{(1^n)}X) =
  \overline{\pi^{-1}(S^n_{(1^{n-1}2)}X)}
\end{equation*}
$B$r9M$($k(B. $BMF0W$K%A%'%C%/$G$-$k$h$&$K(B, $B$3$l$O(B $\HilbX{n}$ $B$N4{Ls$J0x;R(B
$B$G$"$k(B. $B$=$3$G(B, $B$=$N4pK\N`$N(BPoincar\'e$BAPBP(B $[\partial\HilbX{n}]\in
H^2(\HilbX{n},\mathbb Q)$ $B$r9M$((B, $H_*(\HilbX{n})$ $B>e$N:nMQAG(B
\begin{equation*}
  {\frak d}\colon H_*(\HilbX{n})\to H_*(\HilbX{n}); \qquad
  {\frak d}(x) \defeq -\frac12[\partial\HilbX{n}]\cup x
\end{equation*}
$B$r9M$($k(B. $B$3$N$H$-<!$,@.N)$9$k(B.
\begin{Theorem}[Lehn~\cite{Lehn}]
\begin{equation*}
  \big[\, \frak d, P_\alpha[i]\,\big] = 
  i L_\alpha[i] + \frac{i(|i| -1)}{2} P_{K\alpha}[i]
\end{equation*}
$B$3$3$G(B, $K$ $B$O(B $X$ $B$NI8=`B+(B $\in H^2(X,\mathbb Q)$ $B$G$"$k(B.
\end{Theorem}

$B$9$J$o$A(B, Virasoro$BBe?t$N@8@.85$O(B, $BI8=`B+$N>pJs$r4^$`%(%i!<9`$r=|$1$P(B,
$\frak d$ $B$H(B, Heisenberg$BBe?t$N@8@.85$N8r49;R$K$h$C$FM?$($i$l$k(B. 

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\end{document}

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