Gauge/gravity duality: an overview Z. Bajnok
Gauge/gravity duality: an overview Z. Bajnok
Gauge/gravity duality: an overview Z. Bajnok
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CFT: Integrablity + Bethe Ansatz<br />
Mixing matrix on the subspace Tr [ Z i1 . . . Z iJ<br />
]<br />
of dim 2 J : Minah<strong>an</strong>-Zarembo 2002<br />
∆ = H 0 + λH 1 + λ 2 H 2 + . . . = J I +<br />
λ<br />
8π 2H XXX + λ 2 H 2 + . . .<br />
H 2 : next-to-nearest neighbour integrable! → use Bethe <strong>an</strong>satz<br />
1. choose a groundstate: Z = Z 1 → Tr [ Z J] = Tr [ZZZZZ . . . ZZZZ] ↔| ↑ . . . ↑〉<br />
2. excitations Z...ZXZ...X with SUSY multiplet X = Z 2 , Z 3 , Ψ α a, Ψa, ˙α D µ<br />
n<br />
{ }} {<br />
3. pl<strong>an</strong>e wave: ∑ n e ipn Tr( Z...Z XZ . . . ZZ)<br />
4. scattering states: ∑ n 1 n 2 a 1 a 2<br />
e ip 1n 1 +ip 2 n 2 Tr( Z...Z } {{ } X a1 Z...Z X a2 Z . . . Z)+S(12) b 1b 2<br />
n 1<br />
n 2<br />
{ }} {<br />
symmetry completely fixes the S-matrix for <strong>an</strong>y λ (satisfies unitarity, crossing, Y<strong>an</strong>g-Baxter)<br />
Bethe <strong>an</strong>satz follows from S-matrix: Shastry’s Hubbard S-matrix<br />
a 1 a 2<br />
∑<br />
12
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