Quantum Algebras and Integrable Boundaries in AdS/CFT VIDAS REGELSKIS Dissertation submitted for the Degree of Doctor of Philosophy University of York Department of Mathematics September 2012 i Abstract This thesis studies quantum integrable structures such as Yangians and quantum affine algebras that arise in and are inspired by the AdS/CFT duality, with a primary emphasis on the exploration of integrable boundaries deeply hidden in the duality. The main goal of this thesis is to find novel algebraic structures and methods that could lead to new horizons in the theory of quantum groups and in the exploration of boundary effects in the gauge/gravity dualities. The main thrust of this work is the exploration of the AdS/CFT worlsheet scatter- ing theory and of integrable boundaries that manifest themselves as Dp-branes (p + 1- dimensional Dirichlet submanifolds) which are a necessary part of the superstring the- ory. The presence of these objects breaks some of the underlying symmetries and leads to boundary scattering theory governed by coideal subalgebras of the bulk symmetry. Here the boundary scattering theory for D3-, D5- and D7-branes is considered in detail, and the underlying boundary Yangian symmetries are revealed. The AdS/CFT worldsheet scattering theory is shown to be closely related to that of the deformed Hubbard chain. This similarity allows us to apply the quantum deformed approach to the boundary scattering theory. Such treatment of the system leads to quan- tum affine symmetries that manifest themselves in a very elegant and compact form. In such a way the symmetries of distinct boundaries that previously seemed to be unrelated to each other emerge in a uniform and coherent form. The quantum deformed approach also helps us to better understand the phenomena of the so-called secret symmetry. It is called secret due to its peculiar feature of appear- ing as a level-one generator of the Yangian of the system. However it does not have a Lie algebra (level-zero) analogue. This symmetry is shown to have origins in the quan- tum deformed model, where it manifest itselfs as two, level-one and level-minus-one, generators of the corresponding quantum affine algebra. ii Contents Abstract i Preface vi Acknowledgements vii Declaration viii 1 Introduction 1 1.1 Integrability and the algebraic Bethe ansatz for spin chains . .2 1.2 Hopf algebras and the YBE . .8 1.3 Reflection algebras and the BYBE . 10 1.4 Integrability in AdS/CFT . 14 1.5 Outline . 19 2 Reflection algebras 21 2.1 Preliminaries . 23 2.1.1 Coideal subalgebras for quantum deformed algebras . 24 2.1.2 Coideal subalgebras for Yangian algebras . 32 2.2 Reflection algebras for Uq(slb(2)) ......................... 36 2.2.1 Singlet boundary . 36 2.2.2 Vector boundary . 38 2.3 Reflection algebras for Y(sl(2)) ......................... 41 2.3.1 Singlet boundary . 42 2.3.2 Vector boundary . 43 2.3.3 Yangian limit . 45 2.4 Reflection algebras for Uq(glb(1j1)) ........................ 47 2.4.1 Singlet boundary . 48 2.4.2 Vector boundary . 49 2.5 Reflection algebras for Y(gl(1j1)) ........................ 51 2.5.1 Singlet boundary . 52 2.5.2 Vector boundary . 53 2.5.3 Yangian limit . 55 A The Yangian limit of Uq(slb(2)) .......................... 57 iii iv 3 Integrable boundaries in AdS/CFT 59 3.1 Worldsheet scattering . 61 3.1.1 Underlying symmetries . 61 3.1.1.1 Lie algebra . 61 3.1.1.2 Yangian . 65 3.1.2 S-matrix . 67 3.2 Boundary scattering . 72 3.2.1 The spectrum of boundaries . 72 3.2.2 Reflected algebra . 73 3.3 Giant gravitons . 75 3.3.1 Y=0 giant graviton . 77 3.3.1.1 Boundary symmetries . 77 3.3.1.2 Boundary scattering . 78 3.3.1.3 Mirror model . 83 3.3.2 Z=0 giant graviton . 85 3.3.2.1 Boundary symmetries . 85 3.3.2.2 Boundary scattering . 88 3.4 D3-D7 brane system . 91 3.4.1 Z=0 D7-brane . 91 3.4.1.1 Boundary symmetries . 92 3.4.1.2 Boundary scattering . 93 3.4.2 Y=0 D7-brane . 95 3.5 D3-D5 brane system . 95 3.5.1 Achiral boundary conditions in the bosonic Principal Chiral Model 95 3.5.2 The D5-brane: general considerations . 99 3.5.3 The horizontal D5-brane . 104 3.5.4 The vertical D5-brane . 108 B Reflection matrices . 115 4 q-deformed scattering in AdS/CFT 119 4.1 Quantum affine algebra Qb ............................ 122 4.1.1 Quantum deformation of psu(2j2)C ................... 122 4.1.2 Affine Extension . 123 4.1.3 Quantum oscillators and representations . 125 4.1.3.1 q-Oscillators . 125 4.1.3.2 Affine extension . 128 4.1.3.3 Summary . 130 4.2 q-deformed S-matrix . 132 4.2.1 Scattering in subspace I . 134 4.2.2 Scattering in subspace II . 141 4.2.3 Scattering in subspace III . 143 4.2.4 Special cases of the S-matrix . 145 4.2.4.1 Fundamental S-matrix . 145 4.2.4.2 The S-matrix SQ1 ....................... 147 v 4.3 q-deformed boundary scattering . 148 4.3.1 Quantum affine coideal subalgebras . 148 4.3.2 Reflected algebra . 149 4.4 q-deformed Y=0 giant graviton . 151 4.4.1 Coideal subalgebra . 152 4.4.2 Boundary scattering . 154 4.5 q-deformed Z=0 giant graviton . 159 4.5.1 Coideal subalgebra . 159 4.5.2 Boundary scattering . 163 4.6 q-deformed Z=0 D7-brane . 163 4.6.1 Coideal subalgebra . 164 4.6.2 Boundary scattering . 165 C Elements of the S-matrix . 166 C.1 Subspace II . 166 C.2 Subspace III . 169 D Elements of the special cases of the S-matrix . 172 D.1 Elements of the fundamental S-matrix . 172 D.2 Elements of the S-matrix SQ1 ...................... 173 E Yang-Baxter equation . 176 F q-deformed reflection matrices . 177 5 Secret symmetries of the AdS/CFT scattering matrices 183 5.1 Secret symmetries of the S-matrix . 184 5.2 Secret symmetries of the K-matrices . 186 5.2.1 Y = 0 giant graviton . 186 5.2.2 D5-brane . 187 5.3 Secret symmetries of the q-deformed S-matrix . 190 5.3.1 The quantum affine superalgebra Uq(glb(1j1)) ............. 190 5.3.1.1 Chevalley-Serre realization . 190 5.3.1.2 Drinfeld’s second realization . 191 5.3.1.3 Fundamental representation . 192 5.3.2 The quantum affine superalgebra Uq(glb(2j2)) ............. 193 5.3.2.1 Drinfeld’s second realization . 193 5.3.2.2 Fundamental representation . 195 5.3.3 Deformed quantum affine algebra Qb .................. 199 5.3.3.1 Conventional affine limit . 200 5.3.3.2 q-deformed AdS/CFT: the Secret symmetry . 203 Conclusions and Outlook 207 Bibliography 209 vi Preface This manuscript will take the reader to a magical world of symmetries, where Yangian and quantum affine algebras converge. It will lead on a magnificent journey following a yellow-brick road of a superstring of infinite length and light-cone momentum through- 5 out the AdS5 × S spacetime. We will encounter various D-branes blocking our way and will find elegant integrable solutions giving a safe bypass from these obstacles. The experience gained will allow the reader a glimpse into an even more extraor- dinary world of quantum deformations, where the previously encountered structures emerge in completely new prospects and require more elaborate methods for being con- quered. The final part of the manuscript will lead to a quest for the origins of the secret sym- metry. This quest will require to scout through many strange worlds of quantum sym- metries that are fearlessly protecting secrets of the AdS/CFT. vii Acknowledgements I am very thankful to my PhD supervisor Niall MacKay for his support and inspiration over the past four years, and especially for bringing me to this fascinating field of re- search. I especially thank my collaborators, D. Correa, M. de Leeuw, T. Matsumoto, S. Mori- yama, A. Torrielli and C. Young, for the work we did together in the quest of integrability in the AdS/CFT duality, and without whom I would have not succeeded in having such a fruitful experience of research. I also want to thank all members of the AdS/CFT community, including Ch. Ahn, N. Beisert, A. Dekkel, N. Drukker, N. Gromov, B. Hoare, T. Lukowski, R. Nepomechie, B. Schwab, P. Vieira, K. Zoubos and many others for innumerable discussions, for inspir- ing talks I was honoured to listen to in many meetings that I was pleased to attend, even though sometimes I had to travel half of the globe, and for the many comments I have received upon my work. There are also a number of people outside the AdS/CFT community whom I want to thank. In particular I want to thank S. Kolb and P. Sutcliffe, whom I have benefited a lot from communicating with. I also what to thank all remembers of the Department of Mathematics of the Uni- versity of York, especially members of the mathematical physics group, H. Bostelmann, P. Bush, E. Corrigan, G. Delius, Ch. Fewster, E. Hawkins, A. Higuchi, D. Hunt, B. Kay, L. Loveridge, E. Sklyanin and B. Vicedo. I also want to take a change to thank S. Nor- vaisas for supervising me during my studies at Vilnius University, and also everyone with whom I had pleasure to have challenging discussions in fundamental physics, – A. Acus, Th. Gajdosik, A.