์์์ฅ๋ก ์์, 2์ฐจ์
์ด๋ฑ๊ฐ ์ฅ๋ก (ไบๆฌกๅ
่ถ
็ญ่งๅ ด่ซ, ์์ด: two-dimensional
superconformal theory)์ ๋ค ๊ฐ์ ์ด๋์นญ์ ๊ฐ์ง๋ 2์ฐจ์ ๋ฑ๊ฐ ์ฅ๋ก ์ด๋ค.
2์ฐจ์
์ด๋ฑ๊ฐ ๋์์ ์์ฑ์์ ๋ค์๊ณผ ๊ฐ๋ค.
๊ธฐํธ |
์ด๋ฆ |
๋ฌด๊ฒ ![{\displaystyle h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a) |
SU(2) R๋์นญ ํํ |
ํ๋ฅด๋ฏธ์จ ์
|
![{\displaystyle T(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4749f82e035168e816434e3e3e7bf24e1c92e69b) |
์๋์ง-์ด๋๋ ํ
์ |
2 |
1 |
0
|
![{\displaystyle G^{a}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/650b402e1d884633461504b3922cbb0b9338f02e) |
์ด์ ๋ฅ |
3/2 |
![{\displaystyle \mathbf {2} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8db0cc42a494c9891ec4a9c91dc2c88d1fb65f1d) |
+1
|
![{\displaystyle {\bar {G}}^{\bar {a}}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88784901a72ef812c9abf5a515e7980a4bda970c) |
์ด์ ๋ฅ |
3/2 |
![{\displaystyle {\bar {\mathbf {2} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0b44cb6d30106d70ac4bfa0a0ec91b837d6352) |
โ1
|
![{\displaystyle J^{i}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e675c63f798ce7b487ee90eaa5a823474428ebe7) |
R๋์นญ ๋ณด์กด๋ฅ |
1 |
![{\displaystyle \mathbf {3} ={\mathfrak {su}}(2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e38dd965b58eb4f106336a956150a22b5d72307) |
0
|
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40) |
์ค์ฌ ์์ |
0 |
1 |
0
|
์ ํ์์
๋ฅผ ์ ์ธํ ๋ค๋ฅธ ์์ฑ์๋ค์ ๋ชจ๋ ์๋ฅด๋ฏธํธ ์ฅ์ด๋ฉฐ,
์ ์๋ฅด๋ฏธํธ ์๋ฐ์
์ด๋ค.
์ค์ฌ ์์
๋ SU(2) ์ํ ๋ฆฌ ๋์์ ์ค์์ ๊ฐ๋ค. ๋น๋ผ์๋ก ์ค์ฌ ์ ํ
๋ ๋ค์๊ณผ ๊ฐ๋ค.
![{\displaystyle c=6k=0,6,12,18,\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9589bcb91c2ddaa6d526f9cdc88053a0afb89f2)
์ด๋ค์ ์ฐ์ฐ์ ๊ณฑ ์ ๊ฐ๋ ๋ค์๊ณผ ๊ฐ๋ค. ์ฌ๊ธฐ์
๋
์์ ๋นํน์ดํญ์ ๋ํ๋ธ๋ค.
![{\displaystyle T(z)T(0)=3kz^{-4}+2z^{-2}T(0)+z^{-1}\partial T(0)+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/545ab04d044caf83379b667dde429f0a3935aa21)
![{\displaystyle T(z)G(0)={\frac {3}{2}}z^{-2}G(0)+z^{-1}\partial G(0)+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a8093371c41837bc639a74810f59b2097d2c8f)
![{\displaystyle T(z)J(0)=z^{-2}J(0)+z^{-1}\partial J(0)+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/05fff31aed553793421891119c8b03d05f794921)
![{\displaystyle J^{i}(z)J^{j}(0)={\frac {1}{2}}kz^{-2}\delta ^{ij}+iz^{-1}\epsilon ^{ijk}J^{k}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce718b0df95ba1fd2d4ed3a1b6e7336b7f76ea3a)
![{\displaystyle J^{i}(z)G^{a}(0)=-{\frac {1}{2}}\sigma _{ab}^{i}z^{-1}G^{b}(0)+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9598a05e2792ead11b5731fb2a880a3d10d7ad31)
![{\displaystyle G^{a}(z){\bar {G}}^{\bar {b}}(0)=4k\delta ^{ab}z^{-3}-4\sigma _{a{\bar {b}}}^{i}z^{-2}J^{i}(0)+z^{-1}\left(2\delta ^{ab}T(z)-2\sigma _{a{\bar {b}}}^{i}\partial J^{i}(0)\right)+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eeaa82fdf4d79d7ef3d3440c4033a91d6e83a5a)
์ด๋ค์ ๋ค์๊ณผ ๊ฐ์ ๋ชจ๋ ์ ๊ฐ๋ฅผ ๊ฐ๋๋ค.
![{\displaystyle T(z)=\sum _{n}z^{n-2}L_{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d879a7a443fd94f742f27503e68ed6f25cbe7397)
![{\displaystyle G(z)=\sum _{r\in \mathbb {Z} +\eta }z^{r-3/2}G_{-r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/071fdd023cd4e5f2a8ca779bc1fb40cec0ba273b)
![{\displaystyle J^{i}(z)=\sum _{n}z^{n-1}J_{-n}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df90a6515f4179e7ff39d69fd4bf63831cbe7e4)
์ฌ๊ธฐ์ NS ๊ฒฝ๊ณ ์กฐ๊ฑด์ ๊ฒฝ์ฐ
์ด๋ฉฐ R ๊ฒฝ๊ณ ์กฐ๊ฑด์ ๊ฒฝ์ฐ
์ด๋ค.
๊ทธ๋ ๋ค๋ฉด ๋ชจ๋ ์ ๊ฐ์ ๋ฆฌ ๊ดํธ๋ ๋ค์๊ณผ ๊ฐ๋ค.[1]
![{\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {1}{2}}k(m^{3}-m)\delta _{m+n,0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a886b938d5489e14faef26017591d01e8732b2)
![{\displaystyle [L_{m},G_{r}^{a}]=(m/2-r)G_{m+r}^{a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3178621307c24ab982352f9d7d360e11ffd5e4c0)
![{\displaystyle [L_{m},J_{n}^{i}]=-nJ_{m+n}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e11ef33471ebeab59ea2e1a66c00bd926f6917)
![{\displaystyle [J_{m}^{i},J_{n}^{j}]=i\epsilon ^{ijk}J_{m+n}^{k}+{\frac {1}{2}}mk\delta ^{ij}\delta _{m+n,0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a418cb6dd11a256a6c3ad1219533a473e0fc767)
![{\displaystyle [J_{m}^{i},G_{r}^{a}]=-{\frac {1}{2}}\sigma _{ab}^{i}G_{m+r}^{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1ec6c137ed41078080666272729d6553453d56)
![{\displaystyle \{G_{r}^{a},G_{s}^{b}\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd664ca8d24e9f951d15b11f85c29559d84c82f0)
![{\displaystyle \{G_{r}^{a},{\bar {G}}_{s}^{\bar {b}}\}=2\delta ^{a{\bar {b}}}L_{r+s}-2(r-s)\sigma _{a{\bar {b}}}^{i}J_{r+s}^{i}+{\frac {1}{2}}k(4r^{2}-1)\delta _{r+s,0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d6d035f909494a72ef041c342050a2f551e58d)
๋์ญ์ ๋์[ํธ์ง]
NS ๋์์์,
,
,
,
,
๋ ๋ค์๊ณผ ๊ฐ์ด ๋ถ๋ถ ๋ฆฌ ์ด๋์๋ฅผ ์ด๋ฃฌ๋ค. ์ด๋ ๋์ญ์ ์ผ๋ก ์ ์๋๋ ์ด๋ฑ๊ฐ ๋ณํ๋ค์ ๋ฆฌ ์ด๋์์ด๋ค.
![{\displaystyle [L_{1},L_{-1}]=2L_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/533852b850adeb1767d17f2179651a36fd84e2c5)
![{\displaystyle [L_{\pm 1},L_{0}]=\pm L_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9a22c0f29592ec8e05cb78bfb500523bf72a09)
![{\displaystyle [G_{\pm 12}^{a},L_{0}]=\pm {\frac {1}{2}}G_{\pm 12}^{a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6496dbeb09ba962f8af6e06bb96a821096ab790d)
![{\displaystyle [L_{1},G_{1/2}^{a}]=[L_{-1},G_{-1/2}^{a}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1f3801b1c8f3835f9564e093e57332bac6ffe2)
![{\displaystyle [L_{\pm 1},G_{\mp 1/2}^{a}]=\pm G_{\pm 1/2}^{a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d6af3328cd6ca24da3e8145123496c2b051ebc)
![{\displaystyle [J_{0}^{i},L_{0}]=[J_{0}^{i},L_{\pm 1}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/795db98d0705807fb912c957d9b3119a73baff1f)
![{\displaystyle [J_{0}^{i},J_{0}^{j}]=i\epsilon ^{ijk}J_{0}^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a4f4ca01475e98d982b393c6b74610ac0ac256)
![{\displaystyle \{G_{\pm 1/2}^{a},G_{\pm 1/2}^{b}\}=\{G_{\pm 1/2}^{a},G_{\mp 1/2}^{b}\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe213f483fbcb56a599c5568dff38d60f76a67c)
![{\displaystyle \{G_{\pm 1/2}^{a},{\bar {G}}_{\pm 1/2}^{b}\}=2\delta ^{a{\bar {b}}}L_{\pm 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeb061bec0d9e53ceef31003313f06c7718f81)
![{\displaystyle \{G_{\pm 1/2}^{a},{\bar {G}}_{\mp 1/2}^{b}\}=2\delta ^{a{\bar {b}}}L_{0}\mp \sigma _{a{\bar {b}}}^{i}J_{0}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d5a382cd1118331dd39984ed4d0846e80ee609)
![{\displaystyle [J_{0}^{i},G_{\pm 1/2}^{a}]=-{\frac {1}{2}}\sigma _{ab}^{i}G_{\pm 1/2}^{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3455bf802d7f657f8bc2de4730cfaead8b159b)
๋ง์ฐฌ๊ฐ์ง๋ก, R ๋์์์,
,
,
,
,
๋ ๋ค์๊ณผ ๊ฐ์ด ๋ถ๋ถ ๋ฆฌ ์ด๋์๋ฅผ ์ด๋ฃฌ๋ค.
![{\displaystyle [L_{0},L_{0}]=[G_{0}^{a},L_{0}]=[{\bar {G}}_{0}^{\bar {a}},L_{0}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e75a9e40d2d1e3297011bc2ee9a2ecd8d4a64ac)
![{\displaystyle [J_{0}^{i},L_{0}]=[J_{0}^{i},L_{\pm 1}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/795db98d0705807fb912c957d9b3119a73baff1f)
![{\displaystyle [J_{0}^{i},J_{0}^{j}]=i\epsilon ^{ijk}J_{0}^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a4f4ca01475e98d982b393c6b74610ac0ac256)
![{\displaystyle \{G_{0}^{a},G_{0}^{b}\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1fe115fe914290416ac7333eb1b6798362719c)
![{\displaystyle \{G_{0}^{a},{\bar {G}}_{0}^{b}\}=2\delta ^{a{\bar {b}}}L_{0}-{\frac {1}{2}}k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/818ef497107fed95c8787207eb3bdf6c570a412e)
![{\displaystyle [J_{0}^{i},G_{0}^{a}]=-{\frac {1}{2}}\sigma _{ab}^{i}G_{0}^{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a037af5e5371ff9fba29cc017eb32b7cb869f3f0)
2์ฐจ์
์ด๋ฑ๊ฐ ๋์์ ์ ๋ํฐ๋ฆฌ ํํ์ ์ด1์ฐจ์ฅ์ ๋ฑ๊ฐ ๋ฌด๊ฒ
๋ฐ SU(2) ์์ด์์คํ
์ ๋ฐ๋ผ ๋ถ๋ฅ๋๋ค. ์ด๋ ์ ์ง๋ ํํ(์์ด: massive representation)๊ณผ ๋ฌด์ง๋ ํํ(์์ด: massless representation)์ผ๋ก ๋๋๋ค.[2]
|
NS ๊ฒฝ๊ณ ์กฐ๊ฑด |
R ๊ฒฝ๊ณ ์กฐ๊ฑด
|
๋ฌด์ง๋ ํํ
|
, ![{\displaystyle 0\leq l\leq k/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3a98b721f57af120d719429003bd4025f8eec8) |
,
|
์ ์ง๋ ํํ
|
, ![{\displaystyle 0\leq l\leq k/2-1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4a7dee72dd3cc3626d090d2508ce04f674fd39) |
,
|
์ ๋ํฐ๋ฆฌ ์ด๋ก ์ ๊ฒฝ์ฐ, ํ๋ฒ ๋ฅดํธ ๊ณต๊ฐ์ ๋ชจ๋ ์ํ๋ ๋ค์ BPS ๋ถ๋ฑ์์ ๋ง์กฑ์ํจ๋ค.
![{\displaystyle h\geq k/4\qquad ({\text{NS}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79b6292244a53398ef85023668c7cb2fb59ec4d6)
![{\displaystyle h\geq |l|\qquad ({\text{R}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c261dbdb88c276d504e61aa41e897d1ab62f82e)
๋ฌด์ง๋ ํํ์ ์ด BPS ๋ถ๋ฑ์์ ํฌํ์ํจ๋ค. ์ ์ง๋ ํํ์ ์ํผ ์งํ๊ฐ 0์ด์ง๋ง, ๋ฌด์ง๋ ํํ์ ์ํผ ์งํ๊ฐ 0์ด ์๋๋ค. ์ ์ง๋ ํํ์์ BPS ๋ถ๋ฑ์์ ํฌํ์ํค๋ ๊ทนํ์ ์ทจํ๋ฉด ์ด๋ ๋ฌด์ง๋ ํํ์ผ๋ก ๋ถํด๋๋ค.
๋ถ๋ฐฐ ํจ์[ํธ์ง]
์ด๋ฑ๊ฐ ์ฅ๋ก ์์๋
์ธ ๊ฒฝ์ฐ์ ๋ฌ๋ฆฌ
์ด๋ฉฐ, ๋ฐ๋ผ์ ๋ถ๋ฐฐ ํจ์์๋ ์์ด์์คํ (SU(2) R๋์นญ์ ์นด๋ฅดํ ๋ถ๋ถ๊ตฐ U(1)์ ๋ํ ์ ํ)
์ ๋ํ ํจ๊ฐ์ํฐ
์ ํ๋ฅด๋ฏธ์จ ์
์ ๋ํ ํจ๊ฐ์ํฐ
๋ฅผ ๋ค์๊ณผ ๊ฐ์ด ๋
๋ฆฝ์ ์ผ๋ก ์ฝ์
ํ ์ ์๋ค.[2]
![{\displaystyle Z(q,z,y)=\sum _{(h,q,F)}q^{h-k/4}z^{q}y^{F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2b45e1dc6a7063fd475435eac9d957826296d2)
์ด ํฉ์ NS ๊ฒฝ๊ณ ์กฐ๊ฑด ๋๋ R ๊ฒฝ๊ณ ์กฐ๊ฑด์์ ์ทจํ ์ ์๋ค.
์ด์ผ๋ฌ ๋ค์์ฒด ์์ 2์ฐจ์ ์๊ทธ๋ง ๋ชจํ์
์ด๋ฑ๊ฐ ์ฅ๋ก ์ ์ด๋ฃฌ๋ค. ์ด ๊ฒฝ์ฐ,
์ค์ ์ฐจ์์ ์ด์ผ๋ฌ ๋ค์์ฒด๋ SU(2) ์ํ ๋ฆฌ ๋์ ์ค์๊ฐ
์ธ ์ด๋ฑ๊ฐ ์ฅ๋ก ์ ์ด๋ฃฌ๋ค. K3 ๊ณก๋ฉด ์์ ์๊ทธ๋ง ๋ชจํ์ด ๋ํ์ ์ธ ์์ด๋ฉฐ, ์ด ๊ฒฝ์ฐ
์ด๋ค.
์ด๋ฑ๊ฐ ์ฅ๋ก ์ ์์ ๋คํ์ ๊ฐํ์ฌ
์์ ๋ ์ด๋ก ์ ์ ์ํ ์ ์๋ค.[3]
์ฐธ๊ณ ์๋ฃ[ํธ์ง]
- Sevrin, A.; Troost, W.; Van Proeyen, A. (1988๋
7์ 21์ผ). “Superconformal algebras in two dimensions with
”. 《Physics Letters B》 (์์ด) 208 (3โ4): 447โ450. Bibcode:1988PhLB..208..447S. doi:10.1016/0370-2693(88)90645-4.
- Schwimmer, Adam; Seiberg, Nathan (1987๋
1์ 29์ผ). “Comments on the
superconformal algebras in two dimensions”. 《Physics Letters B》 (์์ด) 184 (2โ3): 191โ196. Bibcode:1987PhLB..184..191S. doi:10.1016/0370-2693(87)90566-1.
- Ooguri, Hirosi (1989๋
10์ 20์ผ). “Superconformal symmetry and geometry of Ricci-flat Kรคhler manifolds” (PDF). 《International Journal of Modern Physics A》 (์์ด) 4 (17): 4303. Bibcode:1989IJMPA...4.4303O. doi:10.1142/S0217751X89001801. [๊นจ์ง ๋งํฌ(๊ณผ๊ฑฐ ๋ด์ฉ ์ฐพ๊ธฐ)]
๊ฐ์ด ๋ณด๊ธฐ[ํธ์ง]