์์์ฅ๋ก ์์, 2์ฐจ์
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ์ฅ๋ก (ไบๆฌกๅ
N
=
2
{\displaystyle {\mathcal {N}}=2}
่ถ
็ญ่งๅ ด่ซ, ์์ด : two-dimensional
N
=
2
{\displaystyle {\mathcal {N}}=2}
superconformal theory )์ ๋ ๊ฐ์ ์ด๋์นญ ์ ๊ฐ์ง๋ 2์ฐจ์ ๋ฑ๊ฐ ์ฅ๋ก ์ด๋ค. ๋ ์ด๋ก ๋ฐ ๊ฑฐ์ธ ๋์นญ ์์ ์ค์ํ ์ญํ ์ ํ๋ค.
๐ฉ=2 ์ด๋ฑ๊ฐ ๋์ [ ํธ์ง ]
2์ฐจ์
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ๋์์ ์์ฑ์์ ๋ค์๊ณผ ๊ฐ๋ค.[1]
๊ธฐํธ
์ด๋ฆ
๋ฌด๊ฒ
h
{\displaystyle h}
U(1) R๋์นญ ์ ํ
q
{\displaystyle q}
T
(
z
)
{\displaystyle T(z)}
์๋์ง-์ด๋๋ ํ
์
2
0
G
±
(
z
)
{\displaystyle G^{\pm }(z)}
์ด์ ๋ฅ
3/2
ยฑยฝ
J
(
z
)
{\displaystyle J(z)}
R๋์นญ ๋ณด์กด๋ฅ
1
0
c
{\displaystyle c}
์ค์ฌ ์์
0
0
ํต์์ ์ผ๋ก,
c
^
{\displaystyle {\hat {c}}}
๋ฅผ ๋ค์๊ณผ ๊ฐ์ด ์ ์ํ๋ค.
c
^
=
c
/
3
{\displaystyle {\hat {c}}=c/3}
์ด๋ ์นผ๋ผ๋น-์ผ์ฐ ๋ค์์ฒด ์์ ์๊ทธ๋ง ๋ชจํ ์ ๊ฒฝ์ฐ, ์นผ๋ผ๋น-์ผ์ฐ ๋ค์์ฒด์ ๋ณต์์ ์ฐจ์์ ํด๋นํ๋ค.
์ด๋ค์ ์ฐ์ฐ์ ๊ณฑ ์ ๊ฐ ๋ ๋ค์๊ณผ ๊ฐ๋ค.[2] :(2.1) ์ฌ๊ธฐ์
⋯
{\displaystyle \cdots }
๋
z
→
0
{\displaystyle z\to 0}
์์ ๋นํน์ดํญ์ ๋ปํ๋ค.
T
(
z
)
T
(
0
)
=
3
2
c
^
z
−
4
+
2
z
−
2
T
(
0
)
+
z
−
1
∂
T
(
0
)
+
⋯
{\displaystyle T(z)T(0)={\frac {3}{2}}{\hat {c}}z^{-4}+2z^{-2}T(0)+z^{-1}\partial T(0)+\cdots }
T
(
z
)
G
±
(
0
)
=
3
2
z
−
2
G
±
(
0
)
+
z
−
1
∂
G
(
0
)
+
⋯
{\displaystyle T(z)G^{\pm }(0)={\frac {3}{2}}z^{-2}G^{\pm }(0)+z^{-1}\partial G(0)+\cdots }
T
(
z
)
J
(
0
)
=
z
−
2
J
(
0
)
+
z
−
1
∂
J
(
0
)
+
⋯
{\displaystyle T(z)J(0)=z^{-2}J(0)+z^{-1}\partial J(0)+\cdots }
J
(
z
)
J
(
0
)
=
c
^
z
−
2
+
⋯
{\displaystyle J(z)J(0)={\hat {c}}z^{-2}+\cdots }
J
(
z
)
G
±
(
0
)
=
±
z
−
1
G
±
(
0
)
+
⋯
{\displaystyle J(z)G^{\pm }(0)=\pm z^{-1}G^{\pm }(0)+\cdots }
G
±
(
z
)
G
±
(
0
)
=
⋯
{\displaystyle G^{\pm }(z)G^{\pm }(0)=\cdots }
G
±
(
z
)
G
∓
(
0
)
=
c
^
z
−
3
±
z
−
2
J
(
0
)
+
z
−
1
(
T
(
0
)
±
1
2
∂
J
(
0
)
)
+
⋯
{\displaystyle G^{\pm }(z)G^{\mp }(0)={\hat {c}}z^{-3}\pm z^{-2}J(0)+z^{-1}\left(T(0)\pm {\frac {1}{2}}\partial J(0)\right)+\cdots }
๋ชจ๋ ์ ๊ฐ [ ํธ์ง ]
N
=
2
{\displaystyle {\mathcal {N}}=2}
๋์์ ์์ฑ์์ ๋ค์๊ณผ ๊ฐ์ ๋ชจ๋ ์ ๊ฐ๋ฅผ ๊ฐ๋๋ค.[2] :(2.2)
T
(
z
)
=
∑
n
z
n
−
2
L
−
n
{\displaystyle T(z)=\sum _{n}z^{n-2}L_{-n}}
G
±
(
z
)
=
∑
r
∈
Z
±
η
z
r
−
3
/
2
G
−
r
±
{\displaystyle G^{\pm }(z)=\sum _{r\in \mathbb {Z} \pm \eta }z^{r-3/2}G_{-r}^{\pm }}
J
(
z
)
=
∑
n
z
n
−
1
J
−
n
{\displaystyle J(z)=\sum _{n}z^{n-1}J_{-n}}
์ฌ๊ธฐ์ NS ๊ฒฝ๊ณ ์กฐ๊ฑด์ ๊ฒฝ์ฐ
η
=
0
{\displaystyle \eta =0}
์ด๋ฉฐ R ๊ฒฝ๊ณ ์กฐ๊ฑด์ ๊ฒฝ์ฐ
η
=
1
/
2
{\displaystyle \eta =1/2}
์ด๋ค.
์ด๋ค์ ๋ค์๊ณผ ๊ฐ์ ๊ตํ์ ๋ฅผ ๊ฐ๋๋ค.[1] :178, (5.14) [2] :(2.3) [3] :(1.1)
c ๋ ๋ชจ๋ ์์์ ๊ฐํ
[
L
m
,
L
n
]
=
(
m
−
n
)
L
m
+
n
+
1
4
c
^
(
m
3
−
m
)
δ
m
+
n
,
0
{\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {1}{4}}{\hat {c}}(m^{3}-m)\delta _{m+n,0}}
[
L
m
,
J
n
]
=
−
n
J
m
+
n
{\displaystyle [L_{m},J_{n}]=-nJ_{m+n}}
[
J
m
,
J
n
]
=
c
^
m
δ
m
+
n
,
0
{\displaystyle [J_{m},J_{n}]={\hat {c}}m\delta _{m+n,0}}
{
G
r
+
,
G
s
−
}
=
L
r
+
s
+
1
2
(
r
−
s
)
J
r
+
s
+
1
2
c
^
(
r
2
−
1
4
)
δ
r
+
s
,
0
{\displaystyle \{G_{r}^{+},G_{s}^{-}\}=L_{r+s}+{\frac {1}{2}}(r-s)J_{r+s}+{\frac {1}{2}}{\hat {c}}(r^{2}-{1 \over 4})\delta _{r+s,0}}
{
G
r
+
,
G
s
+
}
=
0
=
{
G
r
−
,
G
s
−
}
{\displaystyle \{G_{r}^{+},G_{s}^{+}\}=0=\{G_{r}^{-},G_{s}^{-}\}}
[
L
m
,
G
r
±
]
=
(
m
/
2
−
r
)
G
r
+
m
±
{\displaystyle [L_{m},G_{r}^{\pm }]=(m/2-r)G_{r+m}^{\pm }}
[
J
m
,
G
r
±
]
=
±
G
m
+
r
±
{\displaystyle [J_{m},G_{r}^{\pm }]=\pm G_{m+r}^{\pm }}
๋์ญ์ ๋์ [ ํธ์ง ]
NS ๋์์์,
L
0
{\displaystyle L_{0}}
,
L
±
1
{\displaystyle L_{\pm 1}}
,
J
0
{\displaystyle J_{0}}
,
G
1
/
2
±
{\displaystyle G_{1/2}^{\pm }}
,
G
−
1
/
2
±
{\displaystyle G_{-1/2}^{\pm }}
๋ ๋ถ๋ถ ๋ฆฌ ์ด๋์๋ฅผ ์ด๋ฃฌ๋ค. ์ด๋ ๋์ญ์ ์ผ๋ก ์ ์๋๋ ์ด๋ฑ๊ฐ ๋์นญ
o
s
p
(
2
|
2
)
{\displaystyle {\mathfrak {osp}}(2|2)}
์ด๋ค. ์ด ๋ฆฌ ์ด๋์ ์ ๋ณด์ ์ฑ๋ถ
s
o
(
2
)
⊕
u
s
p
(
2
)
≅
u
(
1
)
⊕
s
o
(
3
)
{\displaystyle {\mathfrak {so}}(2)\oplus {\mathfrak {usp}}(2)\cong {\mathfrak {u}}(1)\oplus {\mathfrak {so}}(3)}
๋ ๊ฐ๊ฐ R๋์นญ ๋ฐ (์ ์น) ๋ฑ๊ฐ ๋์นญ์ ๋์ํ๋ค.
R ๋์์์,
L
0
{\displaystyle L_{0}}
,
J
0
{\displaystyle J_{0}}
,
G
0
±
{\displaystyle G_{0}^{\pm }}
,
c
{\displaystyle c}
๋ ๋ถ๋ถ ๋ฆฌ ์ด๋์๋ฅผ ์ด๋ฃจ๋ฉฐ, ๋ค์๊ณผ ๊ฐ๋ค.
{
G
0
+
,
G
0
−
}
=
L
0
−
c
/
24
{\displaystyle \{G_{0}^{+},G_{0}^{-}\}=L_{0}-c/24}
[
J
0
,
G
0
±
]
=
±
G
0
±
{\displaystyle [J_{0},G_{0}^{\pm }]=\pm G_{0}^{\pm }}
[
L
0
,
G
0
±
]
=
[
L
0
,
L
0
]
=
[
L
0
,
J
0
]
=
{
G
0
±
,
G
0
±
}
=
[
J
0
,
J
0
]
=
0
{\displaystyle [L_{0},G_{0}^{\pm }]=[L_{0},L_{0}]=[L_{0},J_{0}]=\{G_{0}^{\pm },G_{0}^{\pm }\}=[J_{0},J_{0}]=0}
์ด๋ ์ฝคํฉํธ ๋ฆฌ๋ง ๋ค์์ฒด ์์
Δ
{\displaystyle \Delta }
,
d
{\displaystyle d}
,
d
†
{\displaystyle d^{\dagger }}
๊ฐ ์ด๋ฃจ๋ ๋์์ ๊ฐ๋ค.
๋ฒ ๋ฅด๋ง ๊ฐ๊ตฐ [ ํธ์ง ]
๋น๋ผ์๋ก ๋์ ์ ๊ฒฝ์ฐ์ ๋ง์ฐฌ๊ฐ์ง๋ก,
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ๋์์ ๊ธฐ์ฝ ํํ์ ์ด1์ฐจ์ฅ (่ถ
ไธๆฌกๅ ด, ์์ด : superprimary field ) ์์ ์ฌ๋ค๋ฆฌ ์ฐ์ฐ์ ๋ค์ ์์ฉ์ผ๋ก ๊ตฌ์ฑ๋๋ค. ์ด1์ฐจ์ฅ
|
ϕ
⟩
{\displaystyle |\phi \rangle }
์ ๋ค์ ์กฐ๊ฑด์ ๋ง์กฑ์ํจ๋ค.
L
n
|
ϕ
⟩
=
G
r
|
ϕ
⟩
=
0
∀
n
,
r
>
0
{\displaystyle L_{n}|\phi \rangle =G_{r}|\phi \rangle =0\qquad \forall n,r>0}
์ด๋ฑ๊ฐ ๋์ ๊ธฐ์ฝ ํํ์ ๋๋จธ์ง ์ฅ๋ค์ ๋ค์๊ณผ ๊ฐ์ด ํ์ค์ ์ผ๋ก ๋ํ๋ผ ์ ์๋ค.
L
−
n
1
L
−
n
2
⋯
L
−
n
k
G
−
r
1
+
G
−
r
2
+
⋯
G
−
r
p
+
G
−
s
1
−
G
−
s
2
−
⋯
G
−
s
q
−
|
ϕ
⟩
(
0
<
n
1
≤
n
2
≤
⋯
≤
n
k
,
0
<
r
1
<
r
2
<
⋯
<
r
p
,
0
≤
s
1
<
⋯
<
s
q
{\displaystyle L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}G_{-r_{1}}^{+}G_{-r_{2}}^{+}\cdots G_{-r_{p}}^{+}G_{-s_{1}}^{-}G_{-s_{2}}^{-}\cdots G_{-s_{q}}^{-}|\phi \rangle \qquad (0<n_{1}\leq n_{2}\leq \cdots \leq n_{k},\;0<r_{1}<r_{2}<\cdots <r_{p},\;0\leq s_{1}<\cdots <s_{q}}
์ฌ๊ธฐ์
r
i
<
r
i
+
1
{\displaystyle r_{i}<r_{i+1}}
๋ฐ
s
i
<
s
i
+
1
{\displaystyle s_{i}<s_{i+1}}
์ธ ๊ฒ์
(
G
r
+
)
2
+
=
(
G
s
−
)
2
=
0
{\displaystyle (G_{r}^{+})^{2}+=(G_{s}^{-})^{2}=0}
์ด๊ธฐ ๋๋ฌธ์ด๋ค. ๋ํ,
0
<
r
1
{\displaystyle 0<r_{1}}
์ธ ๊ฒ์ ๋ผ๋ชฝ ๋์ญ ์ด๋ฑ๊ฐ ๋์
(
G
0
+
,
G
0
−
,
L
0
,
J
0
,
c
)
{\displaystyle (G_{0}^{+},G_{0}^{-},L_{0},J_{0},c)}
์ ์์ฉ์ ๋๊ฐํ์์ผ, ์ด1์ฐจ์ฅ
|
h
,
q
⟩
{\displaystyle |h,q\rangle }
์ ๋ํ์ฌ ํญ์ ๋ค์์ด ์ฑ๋ฆฝํ๋๋ก ์ก์ ์ ์๊ธฐ ๋๋ฌธ์ด๋ค.
G
0
+
|
h
,
q
⟩
=
0
{\displaystyle G_{0}^{+}|h,q\rangle =0}
์ ๋ํฐ๋ฆฌ ์ด1์ฐจ์ฅ์ ๋ฒ์ [ ํธ์ง ]
c
=
6
{\displaystyle c=6}
N
=
2
{\displaystyle {\mathcal {N}}=2}
NS ์ด๋ฑ๊ฐ ๋์์ ์ ๋ํฐ๋ฆฌ ์ด1์ฐจ์ฅ๋ค์ ๊ฐ๋ฅํ ๋ฒ์๋ ํ์์ผ๋ก ์น ํด์ง ๋ฒ์ ๋ฐ ๊ตต์ ์ค์ ์ด๋ค. ๊ตต์ ์ค์ ๊ณผ ๊ทธ ์ ์ ์ฐ์ฅ์
g
r
NS
=
0
{\displaystyle g_{r}^{\operatorname {NS} }=0}
์ผ๋ก ์ ์๋๋ฉฐ, ๊ฐ๋ ์ค์ ํฌ๋ฌผ์ ์
f
1
,
2
NS
=
0
{\displaystyle f_{1,2}^{\operatorname {NS} }=0}
์ ์ํ์ฌ ์ ์๋๋ค.
c
=
6
{\displaystyle c=6}
N
=
2
{\displaystyle {\mathcal {N}}=2}
R ์ด๋ฑ๊ฐ ๋์์ ์ ๋ํฐ๋ฆฌ ์ด1์ฐจ์ฅ๋ค์ ๊ฐ๋ฅํ ๋ฒ์๋ ํ์์ผ๋ก ์น ํด์ง ๋ฒ์ ๋ฐ ๊ตต์ ์ค์ ์ด๋ค. ๊ตต์ ์ค์ ๊ณผ ๊ทธ ์ ์ ์ฐ์ฅ์
g
r
R
=
0
{\displaystyle g_{r}^{\operatorname {R} }=0}
์ผ๋ก ์ ์๋๋ฉฐ, ๊ฐ๋ ์ค์ ํฌ๋ฌผ์ ์
f
1
,
2
R
=
0
{\displaystyle f_{1,2}^{\operatorname {R} }=0}
์ ์ํ์ฌ ์ ์๋๋ค.
๋ค์๊ณผ ๊ฐ์ ํจ์๋ค์ ์ ์ํ์.
g
r
NS
(
c
,
h
,
q
)
=
2
(
h
−
r
q
)
+
(
c
/
3
−
1
)
(
r
2
−
1
/
4
)
(
r
∈
Z
+
1
/
2
)
{\displaystyle g_{r}^{\text{NS}}(c,h,q)=2(h-rq)+(c/3-1)(r^{2}-1/4)\quad (r\in \mathbb {Z} +1/2)}
g
r
R
(
c
,
h
,
q
)
=
2
(
h
−
r
q
)
+
(
c
/
3
−
1
)
(
r
2
−
1
/
4
)
−
1
/
4
(
r
∈
Z
)
{\displaystyle g_{r}^{\text{R}}(c,h,q)=2(h-rq)+(c/3-1)(r^{2}-1/4)-1/4\quad (r\in \mathbb {Z} )}
f
1
,
2
NS
(
c
,
h
,
q
)
=
2
(
c
/
3
−
1
)
h
−
q
2
+
1
4
(
c
/
3
+
1
)
2
−
1
4
(
c
/
3
−
1
)
2
{\displaystyle f_{1,2}^{\text{NS}}(c,h,q)=2(c/3-1)h-q^{2}+{\frac {1}{4}}(c/3+1)^{2}-{\frac {1}{4}}(c/3-1)^{2}}
f
1
,
2
R
(
c
,
h
,
q
)
=
2
(
c
/
3
−
1
)
(
h
−
c
/
24
)
−
q
2
+
1
4
(
c
/
3
+
1
)
2
{\displaystyle f_{1,2}^{\text{R}}(c,h,q)=2(c/3-1)(h-c/24)-q^{2}+{\frac {1}{4}}(c/3+1)^{2}}
q
k
,
l
,
m
NS
=
−
m
k
+
2
{\displaystyle q_{k,l,m}^{\text{NS}}=-{\frac {m}{k+2}}}
q
k
,
l
,
m
R
=
−
m
k
+
2
+
1
2
{\displaystyle q_{k,l,m}^{\text{R}}=-{\frac {m}{k+2}}+{\frac {1}{2}}}
h
k
,
l
,
m
NS
=
l
(
l
+
2
)
−
m
2
4
(
k
+
2
)
{\displaystyle h_{k,l,m}^{\text{NS}}={\frac {l(l+2)-m^{2}}{4(k+2)}}}
h
k
,
l
,
m
R
=
l
(
l
+
2
)
−
m
2
4
(
k
+
2
)
+
1
8
{\displaystyle h_{k,l,m}^{\text{R}}={\frac {l(l+2)-m^{2}}{4(k+2)}}+{\frac {1}{8}}}
g
r
=
0
{\displaystyle g_{r}=0}
์
(
q
,
h
)
{\displaystyle (q,h)}
๋ฐํ๋ฉด์ ์ง์ ์ ์ ์ํ๋ฉฐ,
f
1
,
2
=
0
{\displaystyle f_{1,2}=0}
์
(
q
,
h
)
{\displaystyle (q,h)}
๋ฐํ๋ฉด์ ํฌ๋ฌผ์ ์ ์ ์ํ๋ค.
๊ทธ๋ ๋ค๋ฉด,
N
=
2
{\displaystyle {\mathcal {N}}=2}
๋์์ ์ ๋ํฐ๋ฆฌ ํํ์ ๋ค์๊ณผ ๊ฐ์ ์ธ ๊ฐ์ง๊ฐ ์๋ค.[4] [5] [6] [7]
c
{\displaystyle c}
๋ฒ์
NS ์กฐ๊ฑด
R ์กฐ๊ฑด
์ ์ง๋ ํํ
c
≥
3
{\displaystyle c\geq 3}
∀
r
∈
Z
+
1
2
:
g
r
NS
>
0
{\displaystyle \forall r\in \mathbb {Z} +{\tfrac {1}{2}}\colon g_{r}^{\text{NS}}>0}
∀
r
∈
Z
:
g
r
R
>
0
{\displaystyle \forall r\in \mathbb {Z} \colon g_{r}^{\text{R}}>0}
๋ฌด์ง๋ ํํ
c
≥
3
{\displaystyle c\geq 3}
f
1
,
2
NS
≥
0
,
∃
r
∈
Z
+
1
2
:
g
r
NS
=
0
,
g
r
+
sgn
r
≤
0
{\displaystyle f_{1,2}^{\operatorname {NS} }\geq 0,\;\exists r\in \mathbb {Z} +{\tfrac {1}{2}}\colon g_{r}^{\text{NS}}=0,\;g_{r+\operatorname {sgn} r}\leq 0}
f
1
,
2
R
≥
0
,
∃
r
∈
Z
:
g
r
R
=
0
,
g
r
+
sgn
r
≤
0
{\displaystyle f_{1,2}^{\operatorname {R} }\geq 0,\;\exists r\in \mathbb {Z} \colon g_{r}^{\text{R}}=0,\;g_{r+\operatorname {sgn} r}\leq 0}
์ต์ ๋ชจํ ์ ํํ
c
=
3
k
/
(
k
+
2
)
<
3
,
(
k
=
1
,
2
,
…
)
{\displaystyle c=3k/(k+2)<3,\quad (k=1,2,\dots )}
∃
l
,
m
:
(
h
,
q
)
=
(
h
k
,
l
,
m
NS
,
q
k
,
l
,
m
NS
)
{\displaystyle \exists l,m\colon (h,q)=(h_{k,l,m}^{\text{NS}},q_{k,l,m}^{\text{NS}})}
,
0
≤
l
≤
k
{\displaystyle 0\leq l\leq k}
,
|
m
|
≤
l
{\displaystyle |m|\leq l}
∃
l
,
m
:
(
h
,
q
)
=
(
h
k
,
l
,
m
R
,
q
k
,
l
,
m
R
)
{\displaystyle \exists l,m\colon (h,q)=(h_{k,l,m}^{\text{R}},q_{k,l,m}^{\text{R}})}
,
0
≤
l
≤
k
{\displaystyle 0\leq l\leq k}
,
m
∈
[
1
−
l
,
l
−
1
]
∪
{
l
+
1
}
{\displaystyle m\in [1-l,l-1]\cup \{l+1\}}
c
≥
3
{\displaystyle c\geq 3}
์ผ ๊ฒฝ์ฐ, ์ ์ง๋ ยท ๋ฌด์ง๋ ์ ๋ํฐ๋ฆฌ ํํ์ด ์กด์ฌํ๋ค. ์ ์ง๋ ํํ์ ๊ฒฝ์ฐ ์นด์ธ ํ๋ ฌ์์ด ์์์ด๋ฉฐ, ๋ฐ๋ผ์ ์ ๋ํฐ๋ฆฌ ํํ์ด๋ค. ๋ฌด์ง๋ ํํ์ ๊ฒฝ์ฐ ์นด์ธ ํ๋ ฌ์์ด 0์ด๋ค.
c
<
3
{\displaystyle c<3}
์ผ ๊ฒฝ์ฐ, ๊ฐ๋ฅํ ์ ๋ํฐ๋ฆฌ ํํ๋ค์
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ต์ ๋ชจํ ์ ๋ฑ์ฅํ๋ ๊ฒ๋ค์ด๋ค.
ํนํ,
G
r
+
{\displaystyle G_{r}^{+}}
์์
r
∈
Z
+
ϵ
{\displaystyle r\in \mathbb {Z} +\epsilon }
์ด๋ผ๊ณ ํ๋ค๋ฉด,
0
≤
⟨
h
,
q
|
G
ϵ
±
G
−
ϵ
∓
|
h
,
q
⟩
+
⟨
h
,
q
|
G
−
ϵ
∓
G
+
ϵ
±
|
h
,
q
⟩
=
⟨
h
,
q
|
{
G
ϵ
±
,
G
−
ϵ
∓
}
|
h
,
q
⟩
=
⟨
h
,
q
|
(
L
0
±
ϵ
J
0
+
c
(
ϵ
2
−
1
/
4
)
/
6
)
|
h
,
q
⟩
=
h
±
ϵ
q
+
c
4
ϵ
2
−
1
24
{\displaystyle 0\leq \langle h,q|G_{\epsilon }^{\pm }G_{-\epsilon }^{\mp }|h,q\rangle +\langle h,q|G_{-\epsilon }^{\mp }G_{+\epsilon }^{\pm }|h,q\rangle =\langle h,q|\{G_{\epsilon }^{\pm },G_{-\epsilon }^{\mp }\}|h,q\rangle =\langle h,q|(L_{0}\pm \epsilon J_{0}+c(\epsilon ^{2}-1/4)/6)|h,q\rangle =h\pm \epsilon q+c{\frac {4\epsilon ^{2}-1}{24}}}
์ด๋ฏ๋ก, ํญ์
h
+
c
4
ϵ
2
−
1
24
≥
|
ϵ
q
|
{\displaystyle h+c{\frac {4\epsilon ^{2}-1}{24}}\geq |\epsilon q|}
๊ฐ ์ฑ๋ฆฝํ๋ค. ์ฆ, NS ๊ฒฝ๊ณ ์กฐ๊ฑด์์๋
h
≥
2
|
q
|
{\displaystyle h\geq 2|q|}
์ด๋ฉฐ (
g
±
1
/
2
NS
≥
0
{\displaystyle g_{\pm 1/2}^{\text{NS}}\geq 0}
), R ๊ฒฝ๊ณ ์กฐ๊ฑด์์๋
h
≥
c
/
24
{\displaystyle h\geq c/24}
์ด๋ค (
g
0
R
≥
0
{\displaystyle g_{0}^{\text{R}}\geq 0}
). NS (๋ฐ)์์ง๊ธฐ์ฅ ๋๋ R ๋ฐ๋ฅ ์ํ๋ ์ ๋ถ๋ฑ์์ ํฌํ์ํจ๋ค.
g
r
=
g
r
+
sgn
r
=
0
{\displaystyle g_{r}=g_{r+\operatorname {sgn} r}=0}
์ด ๋๋ ์ ๋ค์ ๋ค์๊ณผ ๊ฐ์ ํฌ๋ฌผ์ ์์ ์์นํ๋ค.
q
2
2
(
c
/
3
−
1
)
=
{
h
NS
h
−
1
/
8
R
{\displaystyle {\frac {q^{2}}{2(c/3-1)}}={\begin{cases}h&{\text{NS}}\\h-1/8&{\text{R}}\end{cases}}}
์ฆ, ์ ์ง๋ ์ด1์ฐจ์ฅ๋ค์ ์ด ํฌ๋ฌผ์ ์ ์๋ถ
h
≥
q
2
/
(
2
(
c
/
3
)
−
1
)
+
{
0
,
1
/
8
}
{\displaystyle h\geq q^{2}/(2(c/3)-1)+\{0,1/8\}}
์๋ง ์กด์ฌํ ์ ์๋ค.
c =3์ธ ๊ฒฝ์ฐ[ ํธ์ง ]
์ ๋ํฐ๋ฆฌ ํํ์ ๋ถ๋ฅ๋
c
=
3
{\displaystyle c=3}
์ธ ๊ฒฝ์ฐ์ ๋จ์ํด์ง๋ค. ์ด ์ค์ฌ ์ ํ๋ ์์ ๋ณด์ ยท ํ๋ฅด๋ฏธ์จ ์ด๋ก ์ ์ค์ฌ ์ ํ์ ๊ฐ๋ค. ์ด ๊ฒฝ์ฐ,
g
r
NS
(
c
,
h
,
q
)
=
2
(
h
−
r
q
)
(
r
∈
Z
+
1
/
2
)
{\displaystyle g_{r}^{\text{NS}}(c,h,q)=2(h-rq)\quad (r\in \mathbb {Z} +1/2)}
g
r
R
(
c
,
h
,
q
)
=
2
(
h
−
r
q
)
−
1
/
4
(
r
∈
Z
)
{\displaystyle g_{r}^{\text{R}}(c,h,q)=2(h-rq)-1/4\quad (r\in \mathbb {Z} )}
f
1
,
2
NS
(
c
,
h
,
q
)
=
f
1
,
2
R
(
c
,
h
,
q
)
=
1
−
q
2
{\displaystyle f_{1,2}^{\text{NS}}(c,h,q)=f_{1,2}^{\text{R}}(c,h,q)=1-q^{2}}
์ด๋ฏ๋ก, ์ ์ง๋ ํํ์ ์กด์ฌํ์ง ์์ผ๋ฉฐ, ๋ฌด์ง๋ ํํ๋ค์ ๋ค์๊ณผ ๊ฐ๋ค.
|
q
|
≤
1
{\displaystyle |q|\leq 1}
h
=
{
|
q
|
/
2
,
3
|
q
|
/
2
,
5
|
q
|
/
2
,
…
NS
1
/
8
,
|
q
|
+
1
/
8
,
2
|
q
|
+
1
/
8
,
…
R
{\displaystyle h={\begin{cases}|q|/2,3|q|/2,5|q|/2,\dots &{\text{NS}}\\1/8,|q|+1/8,2|q|+1/8,\dots &{\text{R}}\end{cases}}}
NS ์ด1์ฐจ์ฅ ๊ฐ์ด๋ฐ ๋ง์ฝ
q
=
±
2
h
{\displaystyle q=\pm 2h}
์ธ ์ํ
|
h
,
±
2
h
⟩
{\displaystyle |h,\pm 2h\rangle }
๊ฐ ์๋ค๋ฉด,
G
−
1
/
2
±
|
h
,
±
2
h
⟩
=
0
{\displaystyle G_{-1/2}^{\pm }|h,\pm 2h\rangle =0}
์ด ๋๋ค. ์ด๋ BPS ์ํ ์ ์ผ์ข
์ด๋ฉฐ, + ๋ถํธ์ผ ๊ฒฝ์ฐ ์์ง๊ธฐ์ฅ (์์ด : chiral field ), โ ๋ถํธ์ผ ๊ฒฝ์ฐ ๋ฐ์์ง๊ธฐ์ฅ (์์ด : antichiral field )์ด๋ผ๊ณ ํ๋ค. ์ ๋ํฐ๋ฆฌ NS (๋ฐ)์์ง๊ธฐ์ฅ์ ๊ฒฝ์ฐ
f
1
,
2
NS
≥
0
{\displaystyle f_{1,2}^{\text{NS}}\geq 0}
์กฐ๊ฑด์ ์ํ์ฌ
2
h
=
|
q
|
≤
c
/
3
{\displaystyle 2h=|q|\leq c/3}
์ด๋ค.[8] :378
ํน๋ณํ ๊ฒฝ์ฐ๋ก, ์ง๊ณต
h
=
q
=
0
{\displaystyle h=q=0}
์ผ ๊ฒฝ์ฐ์๋
L
−
1
|
0
,
0
⟩
=
G
−
1
/
2
±
|
0
,
0
⟩
=
0
{\displaystyle L_{-1}|0,0\rangle =G_{-1/2}^{\pm }|0,0\rangle =0}
์ด๋ค.
๋ผ๋ชฝ ์ด1์ฐจ์ฅ ๊ฐ์ด๋ฐ, ๋ง์ฝ
h
=
c
/
24
{\displaystyle h=c/24}
์ผ ๊ฒฝ์ฐ,
G
0
±
|
c
/
24
,
q
⟩
=
0
{\displaystyle G_{0}^{\pm }|c/24,q\rangle =0}
์ด ์ฑ๋ฆฝํ๋ค. ์ด๋
g
0
R
=
0
{\displaystyle g_{0}^{\text{R}}=0}
๊ณผ ๊ฐ์ ์กฐ๊ฑด์ด๋ค. ์ด ์ญ์ BPS ์ํ ์ ์ผ์ข
์ด๋ฉฐ, ์ด๋ฌํ ์ํ๋ฅผ ๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ (์์ด : Ramond ground state )๋ผ๊ณ ํ๋ค.
์ ๋ํฐ๋ฆฌ ๋ผ๋ชฝ ์ด1์ฐจ์ฅ์ ๊ฒฝ์ฐ
f
1
,
2
R
≥
0
{\displaystyle f_{1,2}^{\text{R}}\geq 0}
์กฐ๊ฑด์ ์ํ์ฌ
|
q
|
≤
c
/
6
+
1
/
2
{\displaystyle |q|\leq c/6+1/2}
์ด๋ค.
2์ฐจ์
N
=
(
2
,
2
)
{\displaystyle {\mathcal {N}}=(2,2)}
์ด๋์นญ์ ์ด๊ณต๊ฐ
R
2
|
4
{\displaystyle \mathbb {R} ^{2|4}}
๋ฅผ ์ฌ์ฉํ์ฌ ๋ํ๋ผ ์ ์๋ค.[9] :271โ276 ์ด ๊ฒฝ์ฐ, ๋ณด์ ์ขํ
R
2
{\displaystyle \mathbb {R} ^{2}}
๋ ๋ณต์ํ๋ฉด
C
{\displaystyle \mathbb {C} }
๋ก ์ฌ๊ฒจ, ๋ณต์์ ์ขํ
(
z
,
z
¯
)
{\displaystyle (z,{\bar {z}})}
๋ก ์ ์ ์ ์์ผ๋ฉฐ, ํ๋ฅด๋ฏธ์จ ์ขํ๋ ๋ ๊ฐ์ ๋ฐ๊ฐํ ๋ณต์์
(
θ
±
,
θ
¯
∓
)
{\displaystyle (\theta ^{\pm },{\bar {\theta }}^{\mp })}
๋ก ๋ํ๋ผ ์ ์๋ค. ์ด ๋
(
θ
±
)
∗
=
θ
¯
∓
{\displaystyle (\theta ^{\pm })^{*}={\bar {\theta }}^{\mp }}
์ด๋ค. ์ด ๊ฒฝ์ฐ,
θ
±
{\displaystyle \theta ^{\pm }}
๋ฐ
θ
¯
±
{\displaystyle {\bar {\theta }}^{\pm }}
์ ๊ฐ๊ฐ ์คํ
±
1
/
2
{\displaystyle \pm 1/2}
๋ฅผ ๊ฐ๋๋ค. ์ฆ, ์ด๋ ๊ตญ์์ ์ผ๋ก ๋ณต์์ ์ธ๋์
C
ω
(
C
)
⊗
C
⋀
(
C
2
)
{\displaystyle {\mathcal {C}}^{\omega }(\mathbb {C} )\otimes _{\mathbb {C} }\bigwedge (\mathbb {C} ^{2})}
์ ์ธต ์ ๊ฐ๋ ํ ๋ฌ๋ฆฐ ๊ณต๊ฐ ์ด๋ค.
์ผ๋ฐ์ ์ด์ฅ ์
Φ
(
z
,
z
¯
,
θ
±
,
θ
¯
∓
)
=
ϕ
(
z
,
z
¯
)
+
θ
+
f
+
(
z
,
z
¯
)
+
θ
−
f
−
(
z
,
z
¯
)
+
θ
¯
+
g
+
(
z
,
z
¯
)
+
θ
¯
−
g
−
(
z
,
z
¯
)
+
θ
+
θ
−
h
(
z
,
z
¯
)
{\displaystyle \Phi (z,{\bar {z}},\theta ^{\pm },{\bar {\theta }}^{\mp })=\phi (z,{\bar {z}})+\theta ^{+}f_{+}(z,{\bar {z}})+\theta ^{-}f_{-}(z,{\bar {z}})+{\bar {\theta }}^{+}g_{+}(z,{\bar {z}})+{\bar {\theta }}^{-}g_{-}(z,{\bar {z}})+\theta ^{+}\theta ^{-}h(z,{\bar {z}})}
์ ๊ฐ์ด, ์ด 6๊ฐ์ ์ฅ์ผ๋ก ๊ตฌ์ฑ๋๋ค. ์ฌ๊ธฐ์ ๊ณต๋ณ ์ด๋ฏธ๋ถ
D
±
{\displaystyle D^{\pm }}
,
D
¯
±
{\displaystyle {\bar {D}}^{\pm }}
์ ์ ์ํ๋ฉด ์์ง๊ธฐ ์ด์ฅ (์์ด : chiral superfield )
D
¯
±
Φ
=
0
{\displaystyle {\bar {D}}_{\pm }\Phi =0}
๋ฐ ๋ฐ์์ง๊ธฐ ์ด์ฅ (์์ด : antichiral superfield
D
¯
±
Φ
=
0
{\displaystyle {\bar {D}}_{\pm }\Phi =0}
๋ฐ ๋คํ๋ฆฐ ์์ง๊ธฐ ์ด์ฅ (์์ด : twisted chiral superfield )
D
¯
+
Φ
=
D
−
Φ
=
0
{\displaystyle {\bar {D}}_{+}\Phi =D_{-}\Phi =0}
๋ฐ ๋คํ๋ฆฐ ๋ฐ์์ง๊ธฐ ์ด์ฅ (์์ด : twisted antichiral superfield )
D
+
Φ
=
D
¯
−
Φ
=
0
{\displaystyle D_{+}\Phi ={\bar {D}}_{-}\Phi =0}
์ด ์กด์ฌํ๋ค. ์ด๋ค์
N
=
(
2
,
2
)
{\displaystyle {\mathcal {N}}=(2,2)}
์ด๋ฑ๊ฐ ๋์นญ์์, ์ ์น ์ด๋ฑ๊ฐ ๋์์ ๋ํ์ฌ (๋ฐ)์์ง๊ธฐ์ฅ์ด์ ๋ฐ์ ์น ์ด๋ฑ๊ฐ ๋์์ ๋ํ์ฌ ์์ง๊ธฐ์ฅ์ธ ์ด๋ฑ๊ฐ ๋์ ๊ธฐ์ฝ ํํ์ ๋ํ๋ธ๋ค.
์์ ๋คํ๋ฆผ [ ํธ์ง ]
์ ์น
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ด ์ฃผ์ด์ก์ ๋, ๊ทธ ํ๋ฒ ๋ฅดํธ ๊ณต๊ฐ
H
{\displaystyle {\mathcal {H}}}
์์
Q
2
=
(
G
−
1
/
2
±
)
2
=
0
{\displaystyle Q^{2}=(G_{-1/2}^{\pm })^{2}=0}
์ด๋ฏ๋ก, ์ด๋ฅผ ์ฌ์ฉํ์ฌ ์ฝํธ๋ชฐ๋ก์ง ๋ฅผ ์ ์ํ ์ ์๋ค. ์ฆ,
G
−
1
/
2
±
{\displaystyle G_{-1/2}^{\pm }}
๋ฅผ BRST ์ฐ์ฐ์ ๋ก ๊ฐ์ฃผํ ์ ์๋ค. ์ด๋ ๊ฒ ํ๋ฉด ์์ ์์์ฅ๋ก ์ ์ป๊ฒ ๋๊ณ , ์ด ๊ฒฝ์ฐ ์ด์๋จ๋ ์ํ๋ค์
h
=
±
q
/
2
{\displaystyle h=\pm q/2}
์ธ ๊ฒ๋ค, ์ฆ (
Q
=
G
−
1
/
2
+
{\displaystyle Q=G_{-1/2}^{+}}
์ธ ๊ฒฝ์ฐ) ์์ง๊ธฐ์ฅ ๋๋ (
Q
=
G
−
1
/
2
−
{\displaystyle Q=G_{-1/2}^{-}}
์ธ ๊ฒฝ์ฐ) ๋ฐ์์ง๊ธฐ์ฅ์ด๋ค. ์ด๋ค์ BPS ์ํ ์ด๋ฉฐ, ๊ฐ๊ฐ ์์ง๊ธฐํ(์์ด : chiral ring ) ๋ฐ ๋ฐ์์ง๊ธฐํ(์์ด : antichiral ring )์ด๋ผ๋ ๋ฑ๊ธ ๊ฐํํ ์ ์ด๋ฃฌ๋ค.
๋น์ ์น
N
=
(
2
,
2
)
{\displaystyle {\mathcal {N}}=(2,2)}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ด ์ฃผ์ด์ก์ ๋์๋ ์๋ก ๋์น์ด์ง ์๋ ๋ ๊ฐ์ ์์ ๋คํ๋ฆผ์ด ์กด์ฌํ๋ฉฐ, ๋ค์๊ณผ ๊ฐ๋ค.[9] :403
Q
A
=
G
¯
−
1
/
2
+
+
G
−
1
/
2
−
{\displaystyle Q_{A}={\bar {G}}_{-1/2}^{+}+G_{-1/2}^{-}}
Q
B
=
G
¯
−
1
/
2
+
+
G
−
1
/
2
+
{\displaystyle Q_{B}={\bar {G}}_{-1/2}^{+}+G_{-1/2}^{+}}
์ด ๊ฒฝ์ฐ, A-๋คํ๋ฆผ์
(
h
,
h
¯
)
=
(
−
q
/
2
,
q
¯
/
2
)
{\displaystyle (h,{\bar {h}})=(-q/2,{\bar {q}}/2)}
์ธ ๊ฒ๋ค์ ๋จ๊ธฐ๊ณ (acํ ์์ด : ac ring ), B-๋คํ๋ฆผ์
(
h
,
h
¯
)
=
(
q
/
2
,
q
¯
/
2
)
{\displaystyle (h,{\bar {h}})=(q/2,{\bar {q}}/2)}
์ธ ๊ฒ๋ค์ ๋จ๊ธด๋ค (ccํ ์์ด : cc ring ). ์ฌ๊ธฐ์ โaโ์ โcโ๋ (๋ฐ)์์ง๊ธฐ(์์ด : (anti)chiral )์ ์์ด ๋จธ๋ฆฟ๊ธ์์ด๋ค. aaํ๊ณผ ccํ์ด ์๋ก ๋ํ์ด๋ฉฐ, acํ๊ณผ caํ์ด ์๋ก ๋ํ์ด๋ค.
์คํํธ๋ผ ํ๋ฆ [ ํธ์ง ]
N
=
2
{\displaystyle {\mathcal {N}}=2}
NS ๋์ ๋ฐ R ๋์๋ ์ฌ์ค ์๋ก ๋ํ ์ด๋ฉฐ, ์ด ๋ํ์ ์คํํธ๋ผ ํ๋ฆ (์์ด : spectral flow )์ด๋ผ๊ณ ํ๋ค.
์คํํธ๋ผ ํ๋ฆ์ ์ด๋ค ์ฐ์ ๋งค๊ฐ๋ณ์
η
∈
R
{\displaystyle \eta \in \mathbb {R} }
์ ๋ํ ์ ๋ํฐ๋ฆฌ ๋ณํ์ผ๋ก ๊ตฌํ๋๋ฉฐ,
η
=
1
/
2
{\displaystyle \eta =1/2}
๋ผ๋ฉด ์ด๋ NS ๊ฒฝ๊ณ ์กฐ๊ฑด์์ R ๊ฒฝ๊ณ ์กฐ๊ฑด์ผ๋ก ๊ฐ๋ ๋ณํ์ด๋ฉฐ,
η
=
1
{\displaystyle \eta =1}
์ด๋ผ๋ฉด ์ด๋ NS ๋๋ R ๊ฒฝ๊ณ ์กฐ๊ฑด์์์ ์๊ธฐ ๋ํ์ ์ด๋ฃฌ๋ค. ์ด์ ๋ฐ๋ผ, ๋ฑ๊ฐ ๋ฌด๊ฒ์ R๋์นญ ์ ํ๋ ๋ค์๊ณผ ๊ฐ์ด ๋ณํํ๋ค.[1] :186, (5.35)
h
↦
h
−
η
q
+
c
η
2
/
6
{\displaystyle h\mapsto h-\eta q+c\eta ^{2}/6}
q
↦
q
−
c
η
/
3
{\displaystyle q\mapsto q-c\eta /3}
์คํํธ๋ผ ํ๋ฆ ์๋
h
−
3
2
c
q
2
{\displaystyle h-{\frac {3}{2c}}q^{2}}
๋ ๋ถ๋ณ๋์ด๋ค. ์ฆ, ๋ค์๊ณผ ๊ฐ์ด ๋ก๋ฐ์ธ ๋ณํ ๊ณผ ์ ์ฌํ๊ฒ ์์ฉํ๋ค.
(
2
c
h
/
3
q
)
↦
(
cosh
θ
sinh
θ
sinh
θ
cosh
θ
)
(
2
c
h
/
3
q
)
{\displaystyle {\binom {\sqrt {2ch/3}}{q}}\mapsto {\begin{pmatrix}\cosh \theta &\sinh \theta \\\sinh \theta &\cosh \theta \end{pmatrix}}{\binom {\sqrt {2ch/3}}{q}}}
์ฌ๊ธฐ์
2
c
h
/
3
d
θ
=
c
3
d
η
{\displaystyle {\sqrt {2ch/3}}\,d\theta ={\frac {c}{3}}\,d\eta }
์ด๋ฏ๋ก,
η
=
6
h
/
c
(
sinh
θ
)
+
3
q
c
(
cosh
θ
−
1
)
{\displaystyle \eta ={\sqrt {6h/c}}(\sinh \theta )+{\frac {3q}{c}}(\cosh \theta -1)}
์ด๋ค.
์ด์ ๋ฐ๋ผ,
η
=
1
/
2
{\displaystyle \eta =1/2}
์ผ ๋ ์์ง๊ธฐ์ฅ
h
=
q
/
2
{\displaystyle h=q/2}
์
(
h
′
,
q
′
)
=
(
c
/
24
,
q
−
c
/
6
)
{\displaystyle (h',q')=(c/24,q-c/6)}
์ธ R ๋ฐ๋ฅ ์ํ๋ก ๋์๋๋ค. ๋ง์ฐฌ๊ฐ์ง๋ก,
η
=
1
{\displaystyle \eta =1}
์ธ ๊ฒฝ์ฐ, ์์ง๊ธฐ์ฅ
h
=
q
/
2
{\displaystyle h=q/2}
๋ ๋ฐ์์ง๊ธฐ์ฅ
(
h
′
,
q
′
)
=
(
c
/
6
−
q
/
2
,
q
−
c
/
3
)
{\displaystyle (h',q')=(c/6-q/2,q-c/3)}
์ผ๋ก ๋์๋๋ค.
η
=
−
1
{\displaystyle \eta =-1}
η
=
−
1
/
2
{\displaystyle \eta =-1/2}
η
=
0
{\displaystyle \eta =0}
η
=
1
/
2
{\displaystyle \eta =1/2}
η
=
1
{\displaystyle \eta =1}
์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
h
,
2
h
)
{\displaystyle (h,q)=(h,2h)}
๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ
(
h
,
q
)
=
(
c
/
24
,
2
h
−
c
/
6
)
{\displaystyle (h,q)=(c/24,2h-c/6)}
๋ฐ์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
c
/
6
−
h
,
2
h
−
c
/
3
)
{\displaystyle (h,q)=(c/6-h,2h-c/3)}
์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
c
/
12
+
q
/
2
,
c
/
6
+
q
)
{\displaystyle (h,q)=(c/12+q/2,c/6+q)}
๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ
(
h
,
q
)
=
(
c
/
24
,
q
)
{\displaystyle (h,q)=(c/24,q)}
๋ฐ์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
c
/
12
−
q
/
2
,
q
−
c
/
6
)
{\displaystyle (h,q)=(c/12-q/2,q-c/6)}
์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
c
/
6
−
h
,
c
/
3
−
2
h
)
{\displaystyle (h,q)=(c/6-h,c/3-2h)}
๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ
(
h
,
q
)
=
(
c
/
24
,
−
2
h
+
c
/
6
)
{\displaystyle (h,q)=(c/24,-2h+c/6)}
๋ฐ์์ง๊ธฐ์ฅ
(
h
,
q
)
=
(
h
,
−
2
h
)
{\displaystyle (h,q)=(h,-2h)}
๋ชจ๋๋ฌ ๋ถ๋ณ์ฑ [ ํธ์ง ]
N
=
(
2
,
2
)
{\displaystyle {\mathcal {N}}=(2,2)}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ด ์ฃผ์ด์ก์ ๋, R๋์นญ์ ๋ํ ํจ๊ฐ์ํฐ
y
=
exp
(
2
π
i
z
)
{\displaystyle y=\exp(2\pi iz)}
y
¯
=
exp
(
−
2
π
i
z
¯
)
{\displaystyle {\bar {y}}=\exp(-2\pi i{\bar {z}})}
q
=
exp
(
2
π
i
τ
)
{\displaystyle q=\exp(2\pi i\tau )}
q
¯
=
exp
(
−
2
π
i
τ
¯
)
{\displaystyle {\bar {q}}=\exp(-2\pi i{\bar {\tau }})}
๋ฅผ ์ฝ์
ํ์ฌ, ๋ค์๊ณผ ๊ฐ์ ๋ถ๋ฐฐ ํจ์๋ฅผ ์ ์ํ ์ ์๋ค.
Z
(
q
,
q
¯
,
y
,
y
¯
)
=
tr
NS
(
q
L
0
−
c
/
24
q
¯
L
¯
0
−
c
/
24
y
J
0
y
¯
J
¯
0
)
=
∑
(
h
,
h
¯
,
q
,
q
¯
)
q
h
−
c
/
24
q
¯
h
¯
−
c
/
24
y
j
y
¯
j
¯
{\displaystyle Z(q,{\bar {q}},y,{\bar {y}})=\operatorname {tr} _{\operatorname {NS} }\left(q^{L_{0}-c/24}{\bar {q}}^{{\bar {L}}_{0}-c/24}y^{J_{0}}{\bar {y}}^{{\bar {J}}_{0}}\right)=\sum _{(h,{\bar {h}},q,{\bar {q}})}q^{h-c/24}{\bar {q}}^{{\bar {h}}-c/24}y^{j}{\bar {y}}^{\bar {j}}}
์ด๋ ๋ชจ๋๋ฌ ๊ตฐ ์ S๋ณํ
S
:
(
τ
,
z
)
↦
(
τ
′
,
z
′
)
=
(
−
1
/
τ
,
z
/
τ
)
{\displaystyle S\colon (\tau ,z)\mapsto (\tau ',z')=(-1/\tau ,z/\tau )}
์๋ ๋ค์๊ณผ ๊ฐ์ด ๋ณํํ๋ค.[10]
Z
(
τ
,
z
)
=
exp
(
−
π
i
c
z
2
/
3
τ
)
exp
(
π
i
c
z
¯
2
/
3
τ
¯
)
Z
(
τ
′
,
z
′
)
{\displaystyle Z(\tau ,z)=\exp(-\pi icz^{2}/3\tau )\exp(\pi ic{\bar {z}}^{2}/3{\bar {\tau }})Z(\tau ',z')}
์ฆ, S๋ณํ์ ๋ํ์ฌ ๋ณต์์ ์์์ ์ป๋๋ค.
์ด์ ์ ์ฌํ๊ฒ, R ๊ฒฝ๊ณ ์กฐ๊ฑด์์ ํ๋ฅด๋ฏธ์จ ์
(
−
1
)
F
=
exp
(
i
π
(
J
0
−
J
¯
0
)
)
{\displaystyle (-1)^{F}=\exp(i\pi (J_{0}-{\bar {J}}_{0}))}
๋ฅผ ์ฝ์
ํ์ฌ, ๋ค์๊ณผ ๊ฐ์ ํ์ ์ข
์ (ๆฅๅ็จฎๆธ, ์์ด : elliptic genus )๋ฅผ ์ ์ํ ์ ์๋ค.[11] [12] :(2.1)
Z
(
τ
,
z
)
=
tr
R
(
(
−
1
)
F
q
L
0
−
c
/
24
q
¯
L
¯
0
−
c
/
24
y
J
0
)
{\displaystyle Z(\tau ,z)=\operatorname {tr} _{\operatorname {R} }\left((-1)^{F}q^{L_{0}-c/24}{\bar {q}}^{{\bar {L}}_{0}-c/24}y^{J_{0}}\right)}
์ด๋ ์ํผ ์งํ ์ ์ผ๋ฐํ์ด๋ฉฐ,
τ
{\displaystyle \tau }
๋ฐ
z
{\displaystyle z}
์ ๋ํ์ฌ ์ ์น ํจ์ ์ด๋ค. ์ด๋ ๋ค์๊ณผ ๊ฐ์ ์ฑ์ง๋ค์ ๋ง์กฑ์ํจ๋ค.[12] :(2.4), (2.5), (2.6)
Z
(
τ
,
z
)
=
Z
(
τ
,
−
z
)
=
Z
(
τ
+
1
,
z
)
{\displaystyle Z(\tau ,z)=Z(\tau ,-z)=Z(\tau +1,z)}
Z
(
τ
,
z
)
=
exp
(
−
π
i
c
z
2
/
3
τ
)
Z
(
−
1
/
τ
,
z
/
τ
)
{\displaystyle Z(\tau ,z)=\exp(-\pi icz^{2}/3\tau )Z(-1/\tau ,z/\tau )}
๋ฐ๋ผ์, ํ์ ์ข
์๋ ๋ฌด๊ฒ 0, ์งํ
c
^
/
2
{\displaystyle {\hat {c}}/2}
์ ์ฝํ ์ผ์ฝ๋น ํ์ ์ ์ด๋ฃฌ๋ค.
์์ ์ด๋ก [ ํธ์ง ]
2์ฐจ์
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ ๊ฐ์ฅ ๊ฐ๋จํ ์๋ ์์ ์
์๋ก ๊ตฌ์ฑ๋ ์ด๋ก ์ด๋ค. ์ด ์ด๋ก ์
c
=
3
{\displaystyle c=3}
์ ๊ฐ์ง๋ฉฐ, ์ ์น ์ด๋ก ์ ๊ฒฝ์ฐ, ์กด์ฌํ๋ ๋น๋ผ์๋ก 1์ฐจ์ฅ์ ๋ค์๊ณผ ๊ฐ๋ค.
๊ฒฝ๊ณ ์กฐ๊ฑด
์ฅ
๋ฌด๊ฒ
h
{\displaystyle h}
R๋์นญ ์ ํ
q
{\displaystyle q}
์ค๋ช
1
0
1
์ง๊ณต
NS
Q
+
=
ψ
+
∂
ϕ
∗
{\displaystyle Q^{+}=\psi ^{+}\partial \phi ^{*}}
3/2
+1
์ด์ ๋ฅ
NS
Q
−
=
ψ
−
∂
ϕ
{\displaystyle Q^{-}=\psi ^{-}\partial \phi }
3/2
โ1
๋ฐ์ด์ ๋ฅ
NS
J
=
ψ
+
ψ
−
{\displaystyle J=\psi ^{+}\psi ^{-}}
1
0
R๋์นญ ๋ณด์กด๋ฅ
NS
ψ
+
{\displaystyle \psi ^{+}}
ยฝ
+1
ํ๋ฅด๋ฏธ์จ
NS
∂
ϕ
=
G
−
1
/
2
−
∂
ψ
+
{\displaystyle \partial \phi =G_{-1/2}^{-}\partial \psi ^{+}}
1
0
๋ณด์
NS
ψ
−
{\displaystyle \psi ^{-}}
ยฝ
โ1
๋ฐํ๋ฅด๋ฏธ์จ
NS
∂
ϕ
∗
=
G
−
1
/
2
+
∂
ψ
−
{\displaystyle \partial \phi ^{*}=G_{-1/2}^{+}\partial \psi ^{-}}
1
0
๋ฐ๋ณด์
R
U
1
/
2
1
=
U
−
1
/
2
ψ
−
{\displaystyle U_{1/2}1=U_{-1/2}\psi ^{-}}
โ
โยฝ
๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ
R
U
1
/
2
ψ
+
=
U
−
1
/
2
1
{\displaystyle U_{1/2}\psi ^{+}=U_{-1/2}1}
โ
ยฝ
๋ผ๋ชฝ ๋ฐ๋ฅ ์ํ
์ด ์ด๋ก ์ ์ง๊ณต ๋ฐ์, ํ๋์ ์์ง๊ธฐ ์ด1์ฐจ์ฅ
ψ
+
{\displaystyle \psi ^{+}}
๊ณผ ํ๋์ ๋ฐ์์ง๊ธฐ ์ด1์ฐจ์ฅ
ψ
−
{\displaystyle \psi ^{-}}
๋ฅผ ๊ฐ๋๋ค. ์ฆ, ์์ง๊ธฐํ์
Z
/
(
2
)
{\displaystyle \mathbb {Z} /(2)}
์ ๋ํ ์ด๋ค.
์ด๋ค๋ก๋ถํฐ ์ ์๋๋
N
=
2
{\displaystyle {\mathcal {N}}=2}
๋์๋ ๊ตฌ์ฒด์ ์ผ๋ก ๋ค์๊ณผ ๊ฐ๋ค.[13] :240 ๋ณต์์ ๋ณด์
ϕ
{\displaystyle \phi }
์ ์ค์ ยท ํ์ ์ฑ๋ถ์ ๋ชจ๋ ์ ๊ฐ๋ฅผ ๊ฐ๊ฐ
a
n
{\displaystyle a_{n}}
,
b
n
{\displaystyle b_{n}}
์ด๋ผ๊ณ ํ๊ณ , ๋ณต์์ ํ๋ฅด๋ฏธ์จ
ψ
{\displaystyle \psi }
์ ๋ชจ๋ ์ ๊ฐ๋ฅผ
e
r
{\displaystyle e_{r}}
๋ผ๊ณ ํ์. ์ด๋ค์ ๋ค์๊ณผ ๊ฐ์ ๊ตํ ๊ด๊ณ๋ฅผ ๋ง์กฑ์ํจ๋ค.
[
a
m
,
a
n
]
=
m
2
δ
m
+
n
,
0
,
[
b
m
,
b
n
]
=
m
2
δ
m
+
n
,
0
,
a
n
∗
=
a
−
n
,
b
n
∗
=
b
−
n
{\displaystyle \displaystyle {[a_{m},a_{n}]={m \over 2}\delta _{m+n,0},\,\,\,\,[b_{m},b_{n}]={m \over 2}\delta _{m+n,0}},\,\,\,\,a_{n}^{*}=a_{-n},\,\,\,\,b_{n}^{*}=b_{-n}}
{
e
r
,
e
s
∗
}
=
δ
r
,
s
,
{
e
r
,
e
s
}
=
0.
{\displaystyle \displaystyle {\{e_{r},e_{s}^{*}\}=\delta _{r,s},\,\,\,\,\{e_{r},e_{s}\}=0.}}
๊ทธ๋ ๋ค๋ฉด ๋ค์๊ณผ ๊ฐ์ด
c
^
=
1
{\displaystyle {\hat {c}}=1}
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ๋์๋ฅผ ๊ตฌ์ฑํ ์ ์๋ค.
L
n
=
∑
m
:
a
−
m
+
n
a
m
:
+
∑
m
:
b
−
m
+
n
b
m
:
+
∑
r
(
r
+
n
2
)
:
e
r
∗
e
n
+
r
:
{\displaystyle L_{n}=\sum _{m}:a_{-m+n}a_{m}:+\sum _{m}:b_{-m+n}b_{m}:+\sum _{r}(r+{n \over 2}):e_{r}^{*}e_{n+r}:}
J
n
=
∑
r
:
e
r
∗
e
n
+
r
:
{\displaystyle J_{n}=\sum _{r}:e_{r}^{*}e_{n+r}:}
G
r
+
=
∑
(
a
−
m
+
i
b
−
m
)
⋅
e
r
+
m
{\displaystyle G_{r}^{+}=\sum (a_{-m}+ib_{-m})\cdot e_{r+m}}
G
r
−
=
∑
(
a
r
+
m
−
i
b
r
+
m
)
⋅
e
m
∗
{\displaystyle G_{r}^{-}=\sum (a_{r+m}-ib_{r+m})\cdot e_{m}^{*}}
์ต์ ๋ชจํ [ ํธ์ง ]
c
^
<
1
{\displaystyle {\hat {c}}<1}
์ธ ์ผ๋ จ์
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ต์ ๋ชจํ๋ค์ด ์กด์ฌํ๋ค. ์ด๋ค์ ์ ์น ์ด๋ก ๋ค์ด๋ฉฐ, ๋น์ ์น
N
=
2
{\displaystyle {\mathcal {N}}=2}
๋ชจ๋๋ฌ ๋ถ๋ณ ์ต์ ๋ชจํ๋ค์ ์ผ์ข
์ ADE ๋ถ๋ฅ์ ๋ฐ๋ผ ๋ถ๋ฅ๋๋ค.[14]
์์ฌ๋ฅ ๋ชจํ [ ํธ์ง ]
๋จ์ ๋ฆฌ ๊ตฐ
G
{\displaystyle G}
๋ฐ ๋ซํ ๋ถ๋ถ๊ตฐ
H
≤
G
{\displaystyle H\leq G}
๊ฐ ์ฃผ์ด์ก์ผ๋ฉฐ,
rank
G
=
rank
H
{\displaystyle \operatorname {rank} G=\operatorname {rank} H}
์ด๋ฉฐ,
H
{\displaystyle H}
์ ์ค์ฌ ์ ์ฐจ์์ด ์์์ผ ๋, ์์ฌ๋ฅ ๊ณต๊ฐ
G
/
H
{\displaystyle G/H}
๋ ์ฝคํฉํธ ์ผ๋ฌ ๋ค์์ฒด ์ด๋ฉฐ, ๊ทธ ์์ ๊ฐ์๋ง-์ค์ฆํค ๋ชจํ (์์ด : KazamaโSuzuki model )์ด๋ผ๋
N
=
2
{\displaystyle {\mathcal {N}}=2}
๋ฑ๊ฐ ์ฅ๋ก ์ ์ ์ํ ์ ์๋ค.[15] ์ด๋ ๊ฐ์๋ง ์์ด์น(์ผ๋ณธ์ด : ้ขจ้ ๆดไธ )์ ์ค์ฆํค ํ์ฌ์ค(์ผ๋ณธ์ด : ้ดๆจ ไน
็ท )๊ฐ ๋์
ํ์๋ค.
์๊ทธ๋ง ๋ชจํ [ ํธ์ง ]
๋ณต์์
d
{\displaystyle d}
์ฐจ์ ์นผ๋ผ๋น-์ผ์ฐ ๋ค์์ฒด ์์ ์ ์๋ 2์ฐจ์ ์๊ทธ๋ง ๋ชจํ ์
c
^
=
c
/
3
=
d
{\displaystyle {\hat {c}}=c/3=d}
์ธ
N
=
(
2
,
2
)
{\displaystyle {\mathcal {N}}=(2,2)}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ ์ด๋ฃฌ๋ค.
์์์ ์ด๋ค์คํญ์ ์ผ์ชฝ ์์ง๊ธฐ์ธ์ง ์ฌ๋ถ์ ์ค๋ฅธ์ชฝ ์์ง๊ธฐ์ธ์ง์ ์ฌ๋ถ์ ๋ฐ๋ผ์, ๋ค์๊ณผ ๊ฐ์ด ์ธ ๊ฐ์ง๋ก ๋ถ๋ฅ๋๋ค.[16] [17]
0-BPS ์ํ. ์ฆ, ์ผ์ชฝ ์์ง๊ธฐ๋, ์ค๋ฅธ์ชฝ ์์ง๊ธฐ๋ ์๋๋ค. ์ด๋ ์นผ๋ผ๋น-์ผ์ฐ ๋ค์์ฒด์ ๊ธฐํํ์ (๋น(้)์์์ํ์ ) ์ฑ์ง๋ก๋ถํฐ ๊ฒฐ์ ๋๋ค.
ยผ-BPS ์ํ. ์ผ์ชฝ ์์ง๊ธฐ์ด์ง๋ง ์ค๋ฅธ์ชฝ ์์ง๊ธฐ๊ฐ ์๋๊ฑฐ๋, ๊ทธ ๋ฐ๋์ด๋ค. ์ด๋ ํ์ ์ข
์๋ก๋ถํฐ ๊ฒฐ์ ๋๋ค.
ยฝ-BPS ์ํ. ์ผ์ชฝ ยท ์ค๋ฅธ์ชฝ ์์ง๊ธฐ์ด๋ค. ์ด๋ ์์ ๋คํ์ผ๋ก ์ป์ด์ง๋ (c,c) ๋๋ (a,c) ํ์ ์ํ๋ฉฐ, ์ด๋ฌํ ์ํ์ ์๋ ์นผ๋ผ๋น-์ผ์ฐ ๋ค์์ฒด์ ํธ์ง ์ ์ ์ํ์ฌ ๊ฒฐ์ ๋๋ค.
ํ์ ์ข
์๋ ์ผ์ฝ๋น ํ์ ์ ์ด๋ฃจ๋๋ฐ, ๋ฎ์ ์ฐจ์ (
d
≤
3
{\displaystyle d\leq 3}
)์ ๊ฒฝ์ฐ ์ด๋ฌํ ์ผ์ฝ๋น ํ์์ ๋ฒกํฐ ๊ณต๊ฐ์ 1์ฐจ์์ด๋ฉฐ, ๋ฐ๋ผ์ ํธ์ง ์๋ก๋ถํฐ ์์ ํ ๊ฒฐ์ก๋๋ค.[17]
N
=
2
{\displaystyle {\mathcal {N}}=2}
์ด๋ฑ๊ฐ ์ฅ๋ก ์ ์ด๋ ์ด๋ก ์์ 4์ฐจ์
N
=
1
{\displaystyle {\mathcal {N}}=1}
์ง๊ณตํด๋ฅผ ์ป๊ธฐ ์ํ์ฌ ์ฌ์ฉ๋๋ค.[1] :191โ192
1976๋
์ ์ด๋ ์ด๋ก ์์ ์ต์ด๋ก ๋ฐ๊ฒฌ๋์๊ณ ,[18] 1977๋
์ ๋น
ํ ๋ฅด ์นด์ธ ๊ฐ ๊ฐ์ ๋์๋ฅผ ๋
์์ ์ผ๋ก ์ฌ๋ฐ๊ฒฌํ์๋ค.[3] [19]
๊ทธ ์ ๋ํฐ๋ฆฌ ํํ๋ค์ 1986๋
์ ๋ถ๋ฅ๋์๋ค.[4] ์ดํ 1988๋
์ NS ๋์์ R ๋์๊ฐ ์คํํธ๋ผ ํ๋ฆ์ผ๋ก ์๋ก ๋ํ์์ด ๋ฐํ์ก๋ค.[20]
๊ฐ์ด ๋ณด๊ธฐ [ ํธ์ง ]
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โ ๊ฐ ๋ Gato-Rivera, Beatriz (2002). 〈Recent results on
N
=
2
{\displaystyle N=2}
superconformal algebras〉. Engin Arฤฑk. 《Proceedings of the Sixth International Wigner Symposium, 16-20 Aug 1999, Istanbul, Turkey》 (์์ด). ์ด์คํ๋ถ : Boฤaziรงi รniversitesi Yayฤฑnevi. arXiv :hep-th/0002081 . Bibcode :2000hep.th....2081G . ISBN 978-975518188-2 . 2017๋
10์ 8์ผ์ ์๋ณธ ๋ฌธ์ ์์ ๋ณด์กด๋ ๋ฌธ์. 2015๋
8์ 31์ผ์ ํ์ธํจ .
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5์ 22์ผ). “Determinant formulae and unitarity for the
N
=
2
{\displaystyle N=2}
superconformal algebras in two dimensions or exact results on string compactification” (PDF) . 《Physics Letters B》 (์์ด) 172 : 316โ322. Bibcode :1986PhLB..172..316B . doi :10.1016/0370-2693(86)90260-1 . ISSN 0370-2693 . Zbl 1174.81319 .
โ Eguchi, Tohru; Taormina, Anne (1988๋
8์ 18์ผ). “On the unitary representations of
N
=
2
{\displaystyle N=2}
and
N
=
4
{\displaystyle N=4}
superconformal algebras” (PDF) . 《Physics Letters B》 (์์ด) 210 (1โ2): 125โ132. Bibcode :1988PhLB..210..125E . doi :10.1016/0370-2693(88)90360-7 . ISSN 0370-2693 .
โ Iohara, Kenji (2010๋
3์). “Unitarizable highest weight modules of the
N
=
2
{\displaystyle N=2}
super Virasoro algebras: untwisted sectors”. 《Letters in Mathematical Physics》 (์์ด) 91 (3): 289โ305. doi :10.1007/s11005-010-0375-7 . Zbl 1228.17022 .
โ Iohara, Kenji (2008). “Modules de plus haut poids unitarisables sur la super-algรจbre de Virasoro
N
=
2
{\displaystyle N=2}
tordue”. 《Annales de lโInstitute Fourier》 (ํ๋์ค์ด) 58 (3): 733โ754. doi :10.5802/aif.2367 . MR 2427508 . Zbl 1174.17021 .
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10์ 30์ผ). “On the LandauโGinzburg description of
N
=
2
{\displaystyle N=2}
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N
=
2
{\displaystyle N=2}
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=
2
{\displaystyle N=2}
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1์ 29์ผ). “Comments on the
N
=
2
,
3
,
4
{\displaystyle N=2,3,4}
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์ธ๋ถ ๋งํฌ [ ํธ์ง ]