Boundary states, overlaps, nesting and bootstrapping AdS/dCFT
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Springer
Received: July 24, 2020 Accepted: September 1, 2020 Published: October 19, 2020
Tamas Gombor and Zoltan Bajnok Wigner Research Centre for Physics, Konkoly-Thege Mikl´ os u. 29-33, 1121 Budapest, Hungary
E-mail: [email protected], [email protected] Abstract: Integrable boundary states can be built up from pair annihilation amplitudes called K-matrices. These amplitudes are related to mirror reflections and they both satisfy Yang Baxter equations, which can be twisted or untwisted. We relate these two notions to each other and show how they are fixed by the unbroken symmetries, which, together with the full symmetry, must form symmetric pairs. We show that the twisted nature of the K-matrix implies specific selection rules for the overlaps. If the Bethe roots of the same type are paired the overlap is called chiral, otherwise it is achiral and they correspond to untwisted and twisted K-matrices, respectively. We use these findings to develop a nesting procedure for K-matrices, which provides the factorizing overlaps for higher rank algebras automatically. We apply these methods for the calculation of the simplest asymptotic allloop 1-point functions in AdS/dCFT. In doing so we classify the solutions of the YBE for the K-matrices with centrally extended su(2|2)c symmetry and calculate the generic overlaps in terms of Bethe roots and ratio of Gaudin determinants. Keywords: Integrable Field Theories, AdS-CFT Correspondence, Boundary Quantum Field Theory, Bethe Ansatz ArXiv ePrint: 2004.11329
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)123
JHEP10(2020)123
Boundary states, overlaps, nesting and bootstrapping AdS/dCFT
Contents 1
2 Boundary states, K- and boundary Yang-Baxter equations 2.1 Boundary states and KYBE 2.2 Reflection matrices and BYBE
3 3 5
3 Spin chains and asymptotic spectrum 3.1 Spectrum of the su(N ) spin chain 3.2 Spectrum of the so(4) spin chain 3.3 Spectrum of the so(6) spin chain 3.4 Spectrum of the su(2|2)c ⊕ su(2|2)c spin chain
7 8 8 9 9
4 Selection rules for integrable overlaps 4.1 Pair structure in the su(N ) spin chain 4.2 Pair structure in the so(4) spin chain 4.3 Pair structure in the so(6) spin chain 4.4 Pair structure in the su(2|2)c ⊕ su(2|2)c spin chain 4.5 General structures
10 10 11 12 13 13
5 Integrable states from K-matrices and their symmetries 5.1 K-matrices in the so(4) spin chain 5.2 K-matrices in the so(6) spin chain 5.3 General Lie algebras 5.4 K-matrices in the su(2|2)c ⊕ su(2|2)c spin chain 5.4.1 Leading order solution 5.4.2 Leading order symmetry 5.4.3 All loop solutions 5.4.4 All loop symmetry
14 14 15 16 16 17 17 18 19
6 Asymptotic overlaps and nesting for K-matrices 6.1 Overlaps in the XXX spin chain 6.2 Overlaps in su(3) spin chains with so(3) symmetry 6.2.1 Two-site state 6.2.2 Matrix product state with Pauli matrices 6.3 Overlaps in so(6) spin chains with so(3) ⊕ so(3) symmetry 6.3.1 Two-site state 6.3.2 Matrix product state with Pauli matrices 6.4 Overlaps in su(2|2)c spin chains 6.
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