Retore, Ana Lúcia R438a Aspects of classical and quantum integrable models / Ana Lúcia Retore. – São Paulo, 2019 127 f. : il. Tese (doutorado) - Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: José Francisco Gomes 1. Física matemática. 2. Teoria de campos (Física). 3. Sólitons. I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). Acknowledgments Primeiramente eu gostaria de agradecer meu orientador José Francisco Gomes pela orientação, tranqui- lidade e paciência durante todo esse tempo, e por tudo que me ensinou. Foi um grande prazer trabalhar com o senhor! Eu queria agradecer também ao Professor Zimerman, por estar sempre presente nas nossas discussões com sua sabedoria e bom humor. I am very grateful to my co-advisor Rafael Nepomechie for all kindness and patience during supervi- sion, for everything he taught me and for his enormous help in my career. It was a great pleasure for me to work with you, Rafael! I would like to thank Rafael, Susana, Matt, Orlando, Nicolas and Zoltan for your kindness during my stay in US. To all my collaborators Frank, Rafael, Nathaly, Alexis, Zimerman, and Rodrigo my sincere thank you for all you taught me during these projects. Gostaria de agradecer aos meus amigos Nathaly, Matheus, Thiago, Rafaela, Leônidas, Geórgia, Ju- liana, Ernane, Vı́ctor, Jogean, Gabriel e Bárbara por todos as brincadeiras, conversas e os inúmeros momentos bons durante estes anos. Eu gostaria de agradecer especialmente ao Luan e a Vivi por estarem sempre comigo mesmo longe. Amo vocês! Aos professores por todo conhecimento que me passaram ao longo desses anos. Aos funcionários do IFT pelas inúmeras vezes que me ajudaram, em especial, à dona Jô. Gostaria de agradecer aos meus pais, Angelo e Rose, minha irmã Lu e minha vó Nena por todo o amor, carinho e apoio durante todos esses anos. Amo vocês! E finalmente eu gostaria de agradecer ao meu amor, Henrique, por todo carinho, amor, apoio e companheirismo. Obrigada por me fazer acreditar mais em mim! Você me inspira a ser melhor todos os dias! Te amo! Eu gostaria de agradecer à CAPES e à FAPESP pelo apoio financeiro na bolsa de doutorado regular no páıs sob Processo número: 2015/00025-9 Fundação de Amparo à Pesquisa do Estado de São Paulo e na bolsa BEPE nos Estados Unidos sob Processo número: 2017/03072-3 Fundação de Amparo à Pesquisa do Estado de São Paulo. O suporte foi imprescind́ıvel para realização desse trabalho. As opiniões, hipóteses e conclusões ou recomendações expressas nesse material são de responsabilidade do autor e não necessariamente refletem a visão da FAPESP e da CAPES. Abstract Aspects of classical and quantum integrability are explored. Gauge transformations play a fundamental role in both cases. Classical integrable hierarchies have an underlying algebraic structure which brings universality for the solutions of all the equations belonging to a hierarchy. Such universality is explored together with the gauge invariance of the zero curvature equation to systematically construct the Bäcklund trans- formations for the mKdV hierarchy, as well as to relate it with the KdV hierarchy. As a consequence the defect-matrix for the KdV hierarchy is obtained and a few explicit Bäcklund transformations are computed for both Type-I and Type-II. The generalization for super mKdV hierarchy is also explored. We studied symmetries and degeneracies of families of integrable quantum open spin chains with finite length associated to affine Lie algebras ĝ = A (2) 2n , A (2) 2n−1, B (1) n , C (1) n , D (1) n whose K-matrices depend on a discrete parameter p (p = 0, ..., n). We show that all these transfer matrices have quantum group symmetry corresponding to removing the pth node of the Dynking diagram of ĝ. We also show that the transfer matrices for C (1) n and D (1) n also have duality symmetry and the ones for A (2) 2n−1, B (1) n and D (1) n have Z2 symmetries that map complex representations into their conjugates. Gauge transformations simplify considerably the proofs by allowing us to work in a way that only unbroken generators appear. The spectrum of the same integrable spin chains with the addition of D (2) n+1 is then determined using analytical Bethe ansatz. We conjecture a generalization for open chains for the Bethe ansatz Reshetikhin’s general formula and propose a formula relating the Dynkin labels of the Bethe states with the number of Bethe roots of each type. Key-words: Integrable Hierarchies, Bäcklund transformations, quantum group symmetries, open spin chains, Bethe ansatz. 2 Resumo Aspectos de integrabilidade clássica e quântica são explorados. Transformações de gauge têm papel fundamental em ambos os casos. Hierarquias integráveis clássicas tem uma estrutura algébrica subjacente que traz uma universa- lidade para as soluções de todas as equações que a compoem. Essa universalidade é explorada jun- tamente com a invariância da equação de curvatura nula por transformações de gauge para construir sistematicamente as transformações de Bäcklund da hierarquia mKdV, assim como para relacioná-la com a hierarquia KdV. Como uma consequência a matriz de defeito para a hierarquia KdV é obtida e alguns exemplos expĺıcitos são calculados tanto para o Tipo-I quanto para o Tipo-II. A generalização para a hierarquia mKdV supersimétrica também é discutida. Nós estudamos simetrias e degenerecências de famı́lias de cadeias de spin quânticas integráveis com comprimento finito associadas a algebras de Lie afins ĝ = A (2) 2n , A (2) 2n−1, B (1) n , C (1) n , D (1) n cujas matrizes K dependem de um parâmetro discreto p (p = 0, ..., n). Nós mostramos que todas essas matrizes de transferências têm simetrias de grupos quânticos correspondente a remover o nodo p do diagrama de Dynkin de ĝ. Também mostramos que as matrizes de transferência para C (1) n e D (1) n têm também simetria de dualidade enquanto A (2) 2n−1, B (1) n e D (1) n têm simetrias Z2 que mapeiam representações complexas em seus conjugados. Transformações de gauge simplificam consideravelmente as provas pois permitem-nos trabalhar com apenas os geradores que não foram quebrados. O espectro dessas matrizes de transferência juntamente com D (2) n+1 é então calculado usando o método do Bethe ansatz anaĺıtico. Nós conjecturamos uma generalização para cadeias de spin abertas para a fórmula de Reshetikhin e propomos uma fórmula relacionando os ı́ndices de Dynkin dos estados de Bethe com o número de ráızes de Bethe de cada tipo. Palavras-chave: Hierarquias Integráveis, Transformações de Bäcklund, simetrias de grupos quânticos, cadeias de spin abertas e Bethe ansatz. 3 4 Preface This PhD thesis is divided in two completely separated parts. The Part 1 is devoted to classical integrable models while the Part 2 is focused in quantum integrable models. Since the models we work in each part are completely different we write separated introductions and conclusions for each part. The thesis is heavily based on the following published papers: • R. I. Nepomechie and A. L. Retore, “The spectrum of quantum-group-invariant transfer matri- ces,” Nucl. Phys. B 938, 266 (2019), https://arxiv.org/pdf/1810.09048.pdf. • R. I. Nepomechie and A. L. Retore, “Surveying the quantum group symmetries of integrable open spin chains,” Nucl. Phys. B 930, 91 (2018), https://arxiv.org/abs/1802.04864 • A. R. Aguirre, A. L. Retore, J. F. Gomes, N. I. Spano and A. H. Zimerman, “Defects in the supersymmetric mKdV hierarchy via Bäcklund transformations,” JHEP 1801, 018 (2018), https://arxiv.org/abs/1709.05568. • J. F. Gomes, A. L. Retore and A. H. Zimerman, “Miura and generalized Bäcklund transformation for KdV hierarchy,” J. Phys. A 49, no. 50, 504003 (2016), https://arxiv.org/abs/1610. 02303. • J. F. Gomes, A. L. Retore and A. H. Zimerman, “Construction of Type-II Backlund Transforma- tion for the mKdV Hierarchy,” J. Phys. A 48, 405203 (2015), https://arxiv.org/abs/1505. 01024. and on the conference proceedings : • J. F. Gomes, A. L. Retore, N. I. Spano and A. H. Zimerman, “Backlund Transformation for Integrable Hierarchies: example - mKdV Hierarchy,” J. Phys. Conf. Ser. 597, no. 1, 012039 (2015), https://arxiv.org/abs/1501.00865. but also includes some results and comments from the following works 6 • R. I. Nepomechie, R. A. Pimenta and A. L. Retore, “The integrable quantum group invariant A (2) 2n−1 and D (2) n+1 open spin chains,” Nucl. Phys. B 924, 86 (2017), https://arxiv.org/abs/ 1707.09260. • A. R. Aguirre, A. L. Retore, N. I. Spano, J. F. Gomes and A. H. Zimerman, “Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy,”, https://arxiv.org/abs/1804. 06463. • N. I. Spano, A. L. Retore, J. F. Gomes, A. R. Aguirre and A. H. Zimerman, “The sinh-Gordon defect matrix generalized for n defects,” https://arxiv.org/abs/1610.01856. • A. R. Aguirre, J. F. Gomes, A. L. Retore, N. I. Spano and A. H. Zimerman, “An alternative construction for the Type-II defect matrix for the sshG,” , https://arxiv.org/abs/1610. 01855. and on the following paper which was already submitted to J.Phys.A • R. I. Nepomechie, R. A. Pimenta and A. L. Retore, “Towards the solution of an integrable D (2) 2 spin chain, https://arxiv.org/abs/1905.11144 Since the two parts of the thesis are very different some notations were used in both parts meaning different things. K-matrix for example has two completely different meaning in Part 1 and Part 2. I ask the reader to keep this in mind when reading this work. Contents I Classical Integrability 13 1 Introduction 15 2 Integrable Hierarchies and Bäcklund transformations 17 2.1 Zero Curvature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Constructing integrable hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Positive sub-hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Negative sub-hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 mKdV hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 AKNS hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.5 KdV hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Gauge-invariance of the zero curvature equation . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Bäcklund transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Gauge transformations and the Bäcklund transformations for mKdV hierarchy . . . . 23 2.4.3 Integrable defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Miura and Generalized Bäcklund transformations for KdV hierarchy 25 3.1 The Algebraic Formalism for KdV Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Bäcklund Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 mKdV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.2 KdV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Fusing and Type-II Bäcklund Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 Examples and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Super mKdV hierarchy and its Bäcklund transformations 35 4.1 The supersymmetric mKdV hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Super-Bäcklund transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2.1 N=3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.2 N=5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Conclusions and further developments 41 II Quantum Integrability 43 6 Introduction 45 7 Surveying the quantum group symmetries of integrable open spin chains 47 7.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.1.1 R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.1.2 K-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.1.3 Transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.2 Quantum group symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.3 Duality symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.3.1 Action of duality on the QG generators . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.3.2 Self-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.4 Z2 symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9 10 CONTENTS 7.4.1 The “right” Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.4.2 The “left” Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.5 Degeneracies of the transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.5.1 A (2) 2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.5.2 A (2) 2n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.5.3 B (1) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.5.4 C (1) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.5.5 D (1) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8 The spectrum of quantum-group-invariant transfer matrices 65 8.1 Review of previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.1.1 R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.1.2 K-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8.1.3 Symmetries of the transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.2 Analytical Bethe ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2.1 Eigenvalues of the transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2.2 Determining yl(u, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.2.3 Bethe equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3 Dynkin labels of the Bethe states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.3.1 Eigenvalues of the Cartan generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.3.2 Formulas for the Dynkin labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.4 Duality and the Bethe ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.4.1 Duality of the Bethe equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.4.2 Duality of the transfer-matrix eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.4.3 Duality of the Dynkin labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.4.4 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9 Conclusions and further developments 87 A Zero Curvature for KdV hierarchy 89 B Equivalence between mKdV and KdV variables 91 C Consistency with Equations of Motion 93 D Representation of the ŝl(2,1) affine Lie superalgebra 95 E N = 5 Lax component 97 F Coefficients of the Bäcklund transformations for N = 5 member 99 G R-matrices 101 H Uq(g (l))⊗ Uq(g(r)) and T̃±(p) 105 H.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 H.2 Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 H.2.1 “Left” generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 H.2.2 “Right” generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 H.3 T̃±(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 I The Hamiltonian 109 I.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 J Proofs of four lemmas 111 J.1 Lemma 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 J.1.1 R(1) and R(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 J.1.2 R(3) and R(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 J.1.3 R(5)(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 J.2 Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 J.3 Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 CONTENTS 11 J.3.1 R(1) and R(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 J.3.2 R(3) and R(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 J.3.3 R(5)(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 J.4 Lemma 1 for d = 2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 J.4.1 Finding R̃12(u, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 J.4.2 Performing the large-u limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 J.4.3 Evaluating the commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 J.5 Lemma 1 for d = 2n+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 K Bonus symmetry and singular solutions 119 L Bethe ansatz solutions for some additional cases 121 12 CONTENTS Part I Classical Integrability 13 Chapter 1 Introduction Classical integrable models are known to have an infinite number of conserved charges that guarantees the stability of their solitons solutions. Along the history they were studied in various approaches such as inverse scattering method, the Lax method and the Zakharov-Shabat formulation [1]-[3]. These methods are systematic providing ways to construct many integrable non-linear differential equations such as the Korteweg-de Vries (KdV) equation, the non-linear Schrödinger (NLS) equation and the Sinh-Gordon (S-G) equation . There is another method, however, which has several interesting qualities. It consists of constructing infinite towers of nonlinear integrable differential equations starting from a zero curvature equation [5, 6] and an underlying graded affine algebra. These models are called integrable hierarchies. The advantage of them is that due to their algebraic construction one can compute the features of an infinite number of differential equations in a universal way. For example, ŝl(2) with a principal gradation generates the so-called mKdV hierarchy. This process turns clear that both mKdV (modified KdV) equation and S-G equation are part of the same integrable hierarchy and therefore have soliton solutions with the same general structure [4]. This method involves two (1+1)-dimensional gauge potentials Ax and AtN called Lax pair. While the Ax is the same for all the equations within a hierarchy, each AtN is related to a different time evolution and therefore generates a different evolution equation. The universality of the Ax in each hierarchy has a fundamental role in the computation of many physical quantities in a general way. The solutions of these models can be obtained through the so-called Dressing method [7]-[9] which makes use of gauge transformations to create multi-soliton solutions by starting with a vacuum solution. The Dressing method provides a way to simultaneously construct the soliton solutions for all the equations of a given hierarchy. Another interesting way to compute solutions of integrable models is called Bäcklund transformations (BT). These Bäcklund transformations, among other applications, generate an infinite sequence of soliton solutions from a non-linear superposition principle (see [10],[1],[11],[12]). Bäcklund transformations have also been employed to describe integrable defects [13]-[17] in the sense that two solutions of an integrable model may be interpolated by a defect at certain spatial position. After the introduction of the defect, the integrability is only preserved if the two field configurations are related by a Bäcklund transformation. Under such formulation the energy and momentum have to be modified to take into account the contribution of the defect [13]. Well known (relativistic) integrable models as the sine (sinh)- Gordon, Tzitzeica [18], Lund-Regge [14] and other (non-relativistic) models as Non-Linear Schroedinger (NLS), mKdV, etc have been studied within such context [16]. Also the N=1 and N=2 supersymmetric sinh-Gordon and the super Liouville were studied using integrable defects. There are two known types of Bäcklund transformations. The first involves only the fields of the theory and is called Type-I. In particular, it may be observed that the space component of the Type-I Bäcklund transformations for the mKdV and sinh-Gordon equations coincides for their corresponding fields [10]. Today we know that this happens because they are part of the same integrable hierarchy (the mKdV hierarchy) and that actually all the equations in this hierarchy have the same spatial part of the Bäcklund transformation. More recently a new type of Bäcklund transformations involving auxiliary fields was shown to be com- patible with the equations of motion for the sine (sinh)-Gordon and Tzitzeica models [18]. These are known as Type-II Bäcklund transformations. They are obtained by introducing two defects instead of only one and taking the limit where both defects are the same point is [21]-[27]. Historically all the methods to compute Bäcklund transformations require an individual computation for each model. When studying integrable defects, for example, for each model we want to compute the Bäcklund transformations we need to construct its Lagrangian. Although this would be in principle feasible and would bring a lot of interesting discussions, it would also be very hard. This is because the Lagrangians 15 16 CHAPTER 1. INTRODUCTION for differential equations with high orders could take very complicated forms. In [19] it was shown that the Bäcklund transformations may be constructed from gauge transformation relating two field configurations of the same equation of motion. Such gauge transformation is encoded in the so-called K-matrix (or defect matrix). The outline of Part 1 is as follows. The chapter 2 is dedicated to explain the construction of integrable hierarchies and to extend the results in [19] to all integrable equations of the mKdV hierarchy[20], [28]. This is again a consequence of the universality of the spacial Lax along the hierarchy. In the chapter 3 we construct the KdV hierarchy starting from the Lax pair of the mKdV hierarchy and using gauge transformations. As a consequence we also obtain the K-matrix of the KdV hierarchy in terms of the one for the mKdV hierarchy. We also introduce the idea that the Type-II K-matrix can be constructed as a product of two Type-I K- matrices and discuss some solutions [29]-[30]. The Chapter 4 is dedicated to the generalization for the super mKdV hierarchy[31]-[33]. In Chapter 5 we present some conclusions and further developments. Chapter 2 Integrable Hierarchies and Bäcklund transformations The outline of this chapter is as follows. The zero curvature equation is introduced in section 2.1. In section 2.2 is introduced the concept of integrable hierarchy. The construction of integrable hierarchies is discussed and quickly exemplified using the mKdV, AKNS and KdV hierarchies. The gauge invariance of the zero curvature equation is presented in section 2.3. The section 2.4 is dedicated to introduce the concept of Bäcklund transformations as well as to discuss a its construction using gauge transformations as well as integrable defects. 2.1 Zero Curvature equation Consider a linear system, with coordinates (x, tN ), (∂x +Ax)ψ = 0, (∂tN +AtN )ψ = 0, (2.1) in which, Ax and AtN are two-dimensional gauge potentials, in our case written as matrices whose elements are functions of the n fields of the theory and their derivatives of any order. They are known as Lax pair. By acting with ∂x in the second equation of (2.1) and with ∂tN on the first equation and then subtracting the results we obtain [∂x +Ax, ∂tN +AtN ] = 0. (2.2) which is called zero curvature equation. In (1+1) dimensions, we say that a model is classically integrable if it can be generated by a zero curvature representation. The zero curvature equation plays a very important role in classical integrability since it enables the construction of the so called integrable hierarchies. An integrable hierarchy is an infinite set of integrable equations constructed from a given algebra. The differential equations belonging to the same integrable hierarchy have several properties in common such as: its equations have soliton solutions in a universal form and also its Bäcklund transformations can be constructed in a systematic way. Such universality due to their algebraic structure makes possible to understand features of an infinite number of equations at once. Usually, for each hierarchy, Ax remains the same for all its equations while each AtN gives rise to a new differential equation. The order of the differential equation is directly related with N . For example, for the mKdV (modified Kortweg-deVries) hierarchy, the At3 generates the mKdV equation 4∂t3v = ∂3 xv − 6v2∂xv, (2.3) while At−1 generates the Sinh-Gordon equation ∂t−1 v = e2 ∫ x 0 v(y,t−1)dy − e−2 ∫ x 0 v(y,t−1)dy. (2.4) So, the At3 generates a non-linear differential equation with order 3, while At−1 generates one with one integral, so we can say, that it has order −1. Notice that the equation (2.4) becomes the usual Sinh-Gordon equation1 1Notice that for Sinh-Gordon equation t−1 = z and x = z̄ are the light-cone coordinates . 17 18 CHAPTER 2. INTEGRABLE HIERARCHIES AND BÄCKLUND TRANSFORMATIONS ∂t−1 ∂xφ = e2φ − e−2φ (2.5) if we change variable v = ∂xφ. Let us now see more in detail how such hierarchies are constructed. 2.2 Constructing integrable hierarchies In order to construct an integrable hierarchy we need: • choose an algebra; • choose a gradation of this algebra; • choose a semi simple element E(1). Let us consider the ŝl(2) centerless Kac-Moody algebra. The gradation operator Q decomposes the algebra in graded subspaces in the form Ĝ = ⊕Gm, (2.6) where [Q,Gn] = nGn, (2.7) with n ∈ Z and it is called degree. In addition, due to Jacobi identity we have [Gn,Gm] ⊂ Gm+n. (2.8) As we mentioned, in this and on the next chapter we are dealing with hierarchies related to a ŝl(2) centerless Kac-Moody algebra. It is generated by 2 h(m) = λmh, E (m) ±α = λmE±α, λ ∈ C, m ∈ Z satisfying [h(m), E (n) ±α] = ±2E (m+n) ±α , [E(m) α , E (n) −α] = h(m+n). (2.9) For this algebra two different gradations will be considered: the principal gradation and the homoge- neous gradation. The principal gradation is defined by Qp = 2λ d dλ + 1 2h and generates the so called mKdV hierarchy (modified Kortweg-de-Vries hierarchy). Qp decomposes the affine ŝl(2) algebra into graded subspaces accord- ing to powers of the spectral parameter λ (2.6)-(2.8), where the subspaces G2m, G2m+1 and G2m−1 contain the following generators G2m = {h(m) = λmh}, G2m+1 = {E(m) α = λmEα}, G2m−1 = {E(m) −α = λmE−α} (2.10) i.e. [Qp,G2n+1] = (2n+ 1)G2m+1, [Qp,G2n−1] = (2n− 1)G2m+1 and [Qp,G2n] = (2n)G2n. (2.11) The homogeneous gradation is defined by Qh = ζ d dζ and it generates the AKNS hierarchy (Ablowitz- Kaup-Newel-Segur hierarchy) and the KdV hierarchy (Kortweg-de-Vries hierarchy). Qh decomposes the affine ŝl(2) algebra into graded subspaces according to powers of the spectral parameter ζ3 (2.6)-(2.8) where the subspace Gn contains the following generators Gn = { h(n) = ζnh,E(n) α = ζnEα, E (n) −α = ζnE−α } , (2.12) i.e. [Qh,Gn] = nGn. (2.13) 2The representation used is: h = ( 1 0 0 −1 ) , Eα = ( 0 1 0 0 ) and E−α = ( 0 0 1 0 ) 3Notice that for the principal gradation we are using λ as spectral parameter while in the homogeneous gradation we are using ζ as spectral parameter. This is to avoid some confusion. 2.2. CONSTRUCTING INTEGRABLE HIERARCHIES 19 After choosing a gradation, the next step is to construct the Lax pair (Ax, AtN ) of the model. Let us define Ax = A0 + E(1) where E(1) is a semi simple element and has degree 1, while A0 contains the fields of the theory and has degree 0. But how to construct E(1) and A0? First of all, by choosing a gradation we automatically know which generators have degree 0 and 1, according to (2.10) and (2.12). We do now a new decomposition in the algebra: Ĝ = K ⊕M where K stands for Kernel and M is the image. The kernel is defined as K = { f ∈ Ĝ/ [ f,E(1) ] = 0 } (2.14) and M is its complement. Notice that to choose a semi simple E(1) means that K and M have to satisfy [K,K] ⊂ K, [K,M] ⊂M and [M,M] ⊂ K. (2.15) With these informations we are able to construct the spatial Lax operator Ax. And how about the AtN ? In order to find AtN we separate the hierarchy into two sub-hierarchies: the positive sub-hierarchy and the negative sub-hierarchy. 2.2.1 Positive sub-hierarchy The positive integrable sub-hierarchy is obtained by writing AtN = D(N) +D(N−1) + ...+D(0) in such a way that the zero curvature equation decomposes (2.2) according to the graded structure as [E(1), D(N)] = 0 [E(1), D(N−1)] + [A0, D (N)] + ∂xD (N) = 0 ... = ... [A0, D (0)] + ∂xD (0) − ∂tNA0 = 0, (2.16) Here D(i) ∈ Gi. The equations in (2.16) allows solving for D(i), i = 0, · · ·N and the last equation in (2.16) yields the time evolution for fields in A0. We should point out that D(i) are constructed systematically for each value of N and so is AtN . 2.2.2 Negative sub-hierarchy The negative integrable sub-hierarchy is obtained by writing At−N = D(−N) + D(−N+1) + ... + D(−1). The zero curvature equation then is decomposed in ∂xD (−N) + [A0, D (−N)] = 0 ∂xD (−N+1) + [A0, D (−N+1)] + [E(1), D(−N)] = 0 ... = ... ∂xD (−1) + [A0, D (−1)] + [E(1), D(−2)] = 0 [E(1), D(−1)]− ∂t−NA0 = 0, (2.17) Notice that just as in the positive case, the equation with degree zero is the only one which depends on the t−N . So, we start by solving the equation with degree equal to −N and recursively solve until the one with degree 0. We discuss in the following the mKdV hierarchy, the AKNS hierarchy and the KdV hierarchy. 2.2.3 mKdV hierarchy The informations about E(1) and A0 for all the hierarchies we work on this and on the next chapter can be find in the Table 2.1 Gradation Gradation Operator E(1) A0 Hierarchy principal Qp = 2λ d dλ + 1 2h E (0) α + E (1) −α v(x, tN )h mKdV homogeneous Qh = ζ d dζ h(1) q(x, tN )Eα + r(x, tN )E−α AKNS homogeneous Qh = ζ d dζ h(1) −Eα + J(x, tN )E−α KdV Table 2.1: In this table we put the explicit forms of E(1) and A0 for the hierarchies considered. 20 CHAPTER 2. INTEGRABLE HIERARCHIES AND BÄCKLUND TRANSFORMATIONS Let us start with the mKdV hierarchy and therefore with the principal gradation. The semi simple element E(1) is defined as E(1) = E (0) α +E (1) −α while A0 = v(x, tN )h. Notice that there is not much freedom to choose E(1) and A0 in this case, since the only objects in G1 are E (0) α and E (1) −α and the only object in G0 is h. Positive sub-hierarchy Let us do an example in order to make things clearer. Consider N = 3 and therefore At3 = D(3) + D(2) + D(1) +D(0) with D(i)’s as linear combinations of the generators in Gi. For N = 3, (2.10) becomes G3 = {E(1) α , E (2) −α}, G2 = {h(1)}, G1 = {E(0) α , E (1) −α}, G0 = {h}. (2.18) Consequently, D(3) = a3E (1) α + b3E (2) −α, D(2) = c2h (1), D(1) = a1E (0) α + b1E (1) −α, D(1) = c0h. (2.19) Until this moment we do not know what are the ai’s, bi’s and ci’s. In order to compute them we substitute the D(3) in the first equation in(2.16) for N = 3. From this equation we conclude that D(3) needs to be on the Kernel K. So, b3 = a3. Now we substitute this new D(3) and the D(2) (from (2.19)) in the second equation in (2.16) and solving the resulting equations we obtain a3 = constant ≡ 1 and c2 = v. By continuing this process we find that the complete solution is a3 = b3 ≡ 1, b2 = v, a1 = −1 2 v2 + 1 2 ∂xv, b1 = −1 2 v2 − 1 2 ∂xv and c0 = 1 4 ( ∂2 xv − 2v3 ) , (2.20) while the last equation in (2.16) is the one which has the time-derivative and therefore it gives the time- evolution. By substituting on it D(0) with c0 given by (2.20) we obtain the mKdV equation 4∂t3v − ∂x ( ∂2 xv − 2v3 ) = 0 mKdV (2.21) which is the equation that names the hierarchy. The first next few explicit equations are 16∂t5v − ∂x ( ∂4 xv − 10v2(∂2 xv)− 10v(∂xv)2 + 6v5 ) = 0 (2.22) 64∂t7v − ∂x ( ∂6 xv − 70(∂xv)2(∂2 xv)− 42v(∂2 xv)2 − 56v(∂xv)(∂3 xv) ) + ∂x ( 14v2∂4 xv − 140v3(∂xv)2 − 70v4(∂2 xv) + 20v7 ) = 0 · · · etc. (2.23) constructed from N = 5 and N = 7, respectively. Notice that, for positive mKdV sub-hierarchy we did not mentioned any differential equation for even N . This has a very fundamental reason. If we start with an even N , let us say N = 2n, D(N) has to be of the form D(2n) = h(n) (due to (2.10)) which belongs to the image M. But the first equation in (2.16) requires D(N) being in the Kernel. Therefore if we try to construct a differential equation with even order in the mKdV hierarchy we just find that all the coefficients are equal to zero. Notice that the equations (2.21)-(2.23) are invariant under v → −v. 2.2. CONSTRUCTING INTEGRABLE HIERARCHIES 21 Negative sub-hierarchy An interesting fact is that if we do the same procedure for the negative sub-hierarchy, for v = ∂xφ with N = 1 we will obtain the Sinh-Gordon equation ∂t−1 ∂xφ = 2 sinh(2φ) (2.24) and for N = 3 and N = 5 we have ∂t−3∂xφ = 4 e−2φd−1 ( e2φd−1 (sinh 2φ) ) + 4 e2φd−1 ( e−2φd−1 (sinh 2φ) ) ∂t−5 ∂xφ = 8 e−2φd−1 ( e2φd−1 ( e−2φd−1 ( e2φd−1 (sinh 2φ) ) + e2φd−1 ( e−2φd−1 (sinh 2φ) ))) + 8 e+2φd−1 ( e−2φd−1 ( e−2φd−1 ( e2φd−1 (sinh 2φ) ) + e2φd−1 ( e−2φd−1 (sinh 2φ) ))) .... (2.25) where d−1f = ∫ x f(y)dy. And for N = 2 and N = 4 we have ∂t−2∂xφ = 4e2φd−1 ( e−2φ ) + 4e−2φd−1 ( e2φ ) , ∂t−4 ∂xφ = 4e2φd−1 ( e−2φ ( e2φd−1 ( e−2φ ) + e−2φd−1 ( e2φ ))) + 4e−2φd−1 ( e+2φ ( e2φd−1 ( e−2φ ) + e−2φd−1 ( e2φ ))) (2.26) Notice, that opposite to what happens in the positive sub-hierarchy we do not have any restriction for even N . For the negative sub-hierarchy decomposition (2.17) the first equation does not require that D(N) commutes with the E(1) and therefore does not create any difficulty. However, the equations constructed from even N do have some properties that are different from the ones with odd N . The first property is that while the equations for odd N are invariant under φ→ −φ (by inspection on equations (2.24)-(2.25))the ones for even N are not (see (2.26)). The second property is that on contrary of odd N (2.24)-(2.25), the equations for even N (2.26) do not have the vacuum φ = 0 as a solution. 2.2.4 AKNS hierarchy Following the same procedure, but now considering the homogeneous gradation and E(1) and A0 as in the Table 2.1 we obtain for N = 2 ∂t2q = −α 2 ∂2 xq + γ∂xq + αq2r ∂t2r = α 2 ∂2 xr + γ∂xr − αr2q (2.27) where α and γ are constants. Notice that for q = ψ and r = ψ∗ we obtain the nonlinear Schrödinger equation. Now for N = 3 we obtain ∂t3q = α 4 ∂3 xq − 3 2 αrq∂xq − 1 2 γ∂2 xq + γq2r ∂t3r = α 4 ∂3 xr − 3 2 αqr∂xr + 1 2 γ∂2 xr − γr2q. (2.28) The above equations are just examples. One could continue obtaining higher and higher degree differential equations by assuming larger values of N . 2.2.5 KdV hierarchy By making q = −1 and r = J in the AKNS Lax and doing again the computations for N = 3 we obtain the KdV equation 4∂t3J = ∂3 xJ + 6J∂xJ, (2.29) Notice that by doing γ = 0, q = −1 and r = J directly on equation (2.28) we obtain the equation (2.29). 22 CHAPTER 2. INTEGRABLE HIERARCHIES AND BÄCKLUND TRANSFORMATIONS Similarly for At5 and for At7 , (details are given in the appendix A), we find the Sawada-Kotera equation [12] 16∂t5J = ∂5 xJ + 20∂xJ∂ 2 xJ + 10J∂3 xJ + 30J2∂xJ, (2.30) and 64∂t7J = ∂7 xJ − 70∂2 xJ∂ 3 xJ + 42∂xJ∂ 4 xJ + 70(∂xJ)3 + +14J∂5 xJ +280J∂xJ∂ 2 xJ + 70J2∂3 xJ + 140J3∂xJ, (2.31) respectively. Higher flows (time evolutions) can be systematically constructed for generic N from the same formalism. 2.3 Gauge-invariance of the zero curvature equation If we do a gauge transformation on the Lax pair as follows A′x = KAxK −1 − ∂xKK−1, (2.32) A′tN = KAtNK −1 − ∂tNKK−1 (2.33) the zero curvature equation (2.2) remains invariant. The gauge invariance of the zero curvature equation plays a very important in finding universal features of integrable hierarchies. It allows the systematic construction of universal soliton solutions for the hierarchies through the so called Dressing Method [7]-[9]. Also, it allows the generation of Bäcklund transformations for entire hierarchies in a systematic way as we will see in next section and next two chapters. 2.4 Bäcklund transformations 2.4.1 Basics Bäcklund transformations (BT) [10, 2, 1, 3] consists of a system of equations which relates two solutions of the same nonlinear differential equation 4. Bäcklund transformations have at least one order in x less than the original equation which makes them, in most of the cases, a lot easier to solve. Also, they depend on at least one new parameter (which is not present on the original equation) and is called Bäcklund parameter. Usually nonlinear differential equations can be very hard to solve. What makes BT so important is that they work as generating functions of new solutions given that you already know some solutions (at least one). One simple example is that if you know that the vacuum is a solution, you can substitute it in the BT and then solve the system to find a 1-soliton solution. Let us see the Bäcklund transformations of the Sinh-Gordon equation as an example. The equations ∂t−1 (φ1 + φ2) = − 4 β sinh (φ1 − φ2) ∂x (φ1 − φ2) = −β sinh (φ1 + φ2) (2.34) are the BT for the Sinh-Gordon equation (2.24). The β is the Bäcklund parameter, while φ1 and φ2 are two solutions of the S-G equation. Notice that while the S-G equation has two derivatives, one in x and one in t−1, its Bäcklund transformations have or one derivative in t−1 or one derivative in x. If we act with ∂x on the first equation on (2.34) and use the second one, the β disappears and we obtain two S-G equations, one for φ1 and one for φ2. 4There are also BT which can relate a solution of a differential equation with a solution of another differential equation. Those, however, will not be considered in this thesis. 2.4. BÄCKLUND TRANSFORMATIONS 23 The solutions of the S-G equation for vacuum, 1-soliton and 2-soliton solutions are given by 56. φ0−sol = 0 φ1−sol = ln ( 1 +R1ρ1 1−R1ρ1 ) φ2−sol = ln ( 1 + δ (ρ1 − ρ2) + ρ1ρ2 1− δ (ρ1 − ρ2) + ρ1ρ2 ) (2.35) where ρi = Exp ( 2kix+ 2k−1 i t−1 ) , R1 is a constant and δ = k1+k2 k1−k2 . Φ 1 (k ) 1 Φ1 (k )2 Φ0 =0 Φ2 (k ,k )1 2 k 1 k 2 k 1 k 2 Figure 2.1: Permutability theorem for solutions of vacuum, one-soliton and two-solitons. The solutions were introduced now in order to present a very important theorem: the permutabil- ity theorem. It says that if we start with a vac- uum solution φ0−sol and using the BT go to 1-soliton solution φ1−sol(k1) with β = f(k1) and then to a φ2−sol(k1, k2) with β = f(k2) we will obtain a result which is completely equivalent to the one obtained by going from φ0−sol to 1-soliton solution φ1−sol(k2) with β = f(k2) and then to a φ2−sol(k1, k2) with β = f(k1) as is represented in the figure 2.1. Another possible solution is to start with a 1-soliton and obtain another 1-soliton with a phase difference. If we start with a φ(1−sol) = ln ( 1+R1ρ1 1−R1ρ1 ) we could obtain φ(1−sol) = ln ( 1+R2ρ1 1−R2ρ1 ) with R2 = ( 2k1+β 2k1−β ) R1. The Bäcklund transformations presented in this section are called Type-I Bäcklund transformations. They depend only on the fields of the theory φ1 and φ2 and have one Bäcklund parameter only (called β). There is another type which was introduced by Corrigan et al [18] in the context of defects which depends on the fields of the theory but also on an auxiliary field which they called Λ and depends on two Bäcklund parameters. 2.4.2 Gauge transformations and the Bäcklund transformations for mKdV hi- erarchy Consider a gauge transformation as in (2.33) but now make ( A′x, A ′ tN ) ≡ (Ax(φ2), AtN (φ2)) and (Ax, AtN ) ≡ (Ax(φ1), AtN (φ1)) in such a way that (2.33) becomes ∂xK = KAx(φ1)−Ax(φ2)K (2.36) ∂tNK = KAtN (φ1)−AtN (φ2)K (2.37) i.e. the K matrix connects two field configurations φ1 and φ2. But how to obtain the K? Remember that Ax(φ) is the same for the whole hierarchy. So if we can find K using only the spacial part of the Lax we automatically have a K which is valid for the entire hierarchy. Notice that both Type-I and Type-II Bäcklund transformations for S-G equation were well known. What we did was to write K as a polynomial function on the spectral parameter, substitute the Ax(φi) (which we had from the hierarchy construction) on the first equation of (2.37) and solve for the coefficients7 in order to find the spacial part of the Type-I BT and the spacial part of Type-II BT. By doing this we found KType−I = [ 1 −β 2λ e −p −β2 ep 1 ] and KType−II = [ 1− 1 σ2λe q −β 2σλe Λ−p (eq + e−q + η) − 2 σ e p−Λ 1− 1 σ2λe −q. ] (2.38) Using those transformations, we explicitly constructed the Bäcklund transformations for several equations of the mKdV hierarchy, including many in the negative sub-hierarchy (including with even N). In order to 5The solution of any other equation with odd order in the mKdV hierarchy is the same as (2.35) by just making −1→ N in the definition of ρi. 6The equations for even N will be very similar, except that vacuum is φ = constant nonzero and that we have to add a constant in the 1-soliton and 2-soliton solutions. 7In order to see the complete calculation see the appendix B of [my first proceedings] 24 CHAPTER 2. INTEGRABLE HIERARCHIES AND BÄCKLUND TRANSFORMATIONS see some examples please see [20, 28]. But since the K matrix is general we could construct the Bäcklund transformations for any equation in the mKdV hierarchy. We would like to highlight that the K matrix for Type-II was already developed before by [19]. Our contribution in this part was to notice that since the K could be constructed using only the spacial part of the Lax pair we could use this K to obtain any Bäcklund transformation in the hierarchy. 2.4.3 Integrable defects There are also applications of Bäcklund transformations to describe integrable defects [13, 16, 18] in the sense that one solution when hits a defect becomes another solution through Bäcklund transformations. So, it is possible to compute Bäcklund transformations using Lagrangian defects. It is considered a line and introduced a defect in x = 0. In each side of the defect we have a theory described by φ1 and φ2 respectively. In order to preserve the integrability after the introduction of the defect the φ1 and φ2 have to satisfy a Type-I Bäcklund transformation at the defect point. Historically the way the Type-II defect was introduced was by considering two defects, one at a point x1 and one at a point x2, and considering a field φ1 in (−∞, x1), a field Λ in (x1, x2) and a field φ2 in (x2,∞). Then they took the limit of x2 → x1 and obtained a Bäcklund transformation depending on two parameters and on the fields φ1, φ2 and Λ. We thought that maybe something similar could be done using K matrices (defect matrices). In chapter 3 one of the things we show is that the Type-II K matrix can be obtained by the product of two Type-I K matrices, i.e. KType−II(φ1, φ2) = KType−I(φ1, φ0)KType−I(φ0, φ2) [29]. In the proceedings [30] we discuss the generalization for n-defects. And in [31] we use this idea to reconstruct the K matrix for the super Sinh-Gordon equation. Another thing we show in the next chapter is related with Miura transformations. It is well known that one can relate the mKdV and the KdV equations through Miura transformations. What we show is that this can be generalized in order to connect the whole mKdV hierarchy with the whole KdV hierarchy [29], by using a sequence of two gauge transformations. This allows us also to construct the K matrix for the KdV hierarchy using the known one for the mKdV hierarchy, and therefore to have the Bäcklund transformations of all the equations of the KdV hierarchy. The chapter 4 will be dedicated to construct explicitly the Bäcklund transformations for the super mKdV hierarchy. Chapter 3 Miura and Generalized Bäcklund transformations for KdV hierarchy This chapter is divided in three main parts. The section 3.1 is dedicated to construct the KdV hierarchy through gauge transformations applied on the Lax pair of the mKdV hierarchy. In the section 3.2 we present the construction of the K matrix for the KdV hierarchy using the one for the mKdV hierarchy and the results obtained in section 3.1. In this same section, we show some solutions of the Type-I Bäcklund transformations constructed in this section. The section 3.3 is then dedicated to the construction of the Type-II K-matrix for the KdV hierarchy. Actually, we start by showing that the mKdV Type-II K-matrix can be obtained by the product of two Type-I K-matrices. And then we proceed to construct the one for the KdV hierarchy. Again, some solutions are discussed. 3.1 The Algebraic Formalism for KdV Hierarchy Following the algebraic formalism described in the chapter 2 we recall that the nonlinear equations of the mKdV hierarchy can be derived from the zero curvature representation(2.2) underlined by an affine ŝl(2) centerless Kac-Moody algebra and using a principal gradation. With this in mind we discuss in this section the construction of the KdV using a different approach from the one discussed in the section 2.2.5. The gauge transformations will be again a powerful tool for the process. The procedure will consist in construct the KdV hierarchy starting from a mKdV hierarchy. Consider now the global gauge transformation generated by g1 = ( ζ 1 ζ −1 ) , ζ2 = λ (3.1) which transforms Aprincx,mKdV = E(1) + v(x, tN )h = ( v 1 λ −v ) , into Ahomx,mKdV = g1 ( Aprincx,mKdV ) g−1 1 = g1 ( E(1) + v(x, tN )h ) g−1 1 = ( ζ v v −ζ ) , (3.2) i.e., transforms the principal into homogeneous gradation, Qh = ζ d dζ . A subsequent local Miura-gauge transformation [39], [36] g2(v, ε) = ( 1 ε −εv −v + 2εζ ) , (3.3) transforms Ax,mKdV → Ax,KdV . i.e., Ax,KdV = g2(v, ε)Ahomx,mKdV g −1 2 (v, ε)− ∂xg2(v, ε)g−1 2 (v, ε) = ( ζ −1 J −ζ ) (3.4) and realizes the Miura transformation, J = ε∂xv − v2, ε2 = 1. (3.5) 25 26CHAPTER 3. MIURA ANDGENERALIZED BÄCKLUND TRANSFORMATIONS FORKDVHIERARCHY We should emphasize that for each solution v(x, tN ) of the evolution equations for the mKdV hierarchy, the Miura transformation (3.5) generates two towers of solutions, Jε(x, tN ), ε = ±1, of the KdV hierarchy [36]. The zero curvature under the homogeneous gradation [∂x +Ax,KdV , ∂tN +AtN ,KdV ] = 0, (3.6) with AtN ,KdV = D̃(N) + D̃(N−1) + · · ·+ D̃(0), D̃(j) ∈ G̃j yields the KdV hierarchy equations of motion. For instance At3,KdV = [ ζ3 + 1 2ζJ + 1 4∂xJ −ζ2 − 1 2J ζ2J + 1 2ζ∂xJ + 1 4∂ 2 xJ + 1 2J 2 −ζ3 − 1 2ζJ − 1 4∂xJ ] (3.7) yields the KdV equation 4∂t3J − ∂3 xJ − 6J∂xJ = (ε∂x − 2v) [4∂t3v − ∂x(∂2 xv − 2v3)] = 0, (3.8) Similarly from At5 and for At7 , given in the appendix, we find the Sawada-Kotera equation [12] 16∂t5J − ∂5 xJ − 20∂xJ∂ 2 xJ − 10J∂3 xJ − 30J2∂xJ = (ε∂x − 2v) [16∂t5v − ∂x(∂4 xv − 10v2∂2 xv − 10v(∂xv)2 + 6v5)] = 0, (3.9) and 1 64∂t7J − ∂7 xJ − 70∂2 xJ∂ 3 xJ − 42∂xJ∂ 4 xJ − 70(∂xJ)3 − 14J∂5 xJ − 280J∂xJ∂ 2 xJ − 70J2∂3 xJ − 140J3∂xJ = (ε∂x − 2v) (64∂t7v − ∂x(∂6 xv − 70(∂xv)2∂2 xv − 42v(∂2 xv)2 − 56v∂xv∂ 3 xv − 14v2∂4 xv + 140v3(∂xv)2 + 70v4∂2 xv − 20v7)) = 0 (3.10) respectively. Eqns.( 3.8-3.10) are displayed as explicit examples as illustration of the formalism. Higher flows (time evolutions) can be systematically constructed for generic N from the same formalism. 3.2 Bäcklund Transformation 3.2.1 mKdV In this section we start by noticing that the zero curvature representation (2.2) and (3.6) are invariant under gauge transformations of the type Aµ(φ, ∂xφ, · · · )→ õ = K−1AµK +K−1∂µK, (3.11) where Aµ stands for either AtN or Ax. The key ingredient of this section is to consider two field configurations φ1 and φ2 embedded in Aµ(φ1) and Aµ(φ2) satisfying the zero curvature representation and assume that they are related by a Bäcklund- gauge transformation generated by K(φ1, φ2) preserving the equations of motion (e.g,, zero curvature (2.2) or (3.6) ) , i.e., K(φ1, φ2)Aµ(φ1) = Aµ(φ2)K(φ1, φ2) + ∂µK(φ1, φ2). (3.12) If we now consider the Lax operator L = ∂x +Ax for mKdV case within the principal gradation, Ax,mKdV = E(1) +A0 = [ ∂xφ(x, tN ) 1 λ −∂xφ(x, tN ) ] (3.13) is common to all members of the hierarchy defined by (2.2). We find that K(φ1, φ2)Ax,mKdV (φ1) = Ax,mKdV (φ2)K(φ1, φ2) + ∂xK(φ1, φ2), (3.14) where the Bäcklund-gauge generator K(φ1, φ2) is given by [20], [28] K(φ1, φ2) = [ 1 − β 2λe −(φ1+φ2) −β2 e(φ1+φ2) 1 ] (3.15) 1 In general, we find KdV (J) = (ε∂x − 2v)mKdV (v). 3.2. BÄCKLUND TRANSFORMATION 27 and β is the Bäcklund parameter. Eqn. (3.14) is satisfied provided ∂x (φ1 − φ2) = −β sinh (φ1 + φ2) . (3.16) For the sinh-Gordon (s-g) model, the equations of motion ∂t∂xφa = 2 sinh 2φa, a = 1, 2 are satisfied if we further introduce the time component of the Bäcklund transformation, ∂t (φ1 + φ2) = 4 β sinh (φ2 − φ1) . (3.17) Eqn. (3.17) is compatible with (3.12) for Aµ = AtN with At,s−g = [ 0 λ−1e−2φ e2φ 0 ] . (3.18) For higher graded time evolutions the time component of the Bäcklund transformation can be derived from the appropriated time component of the two dimensional gauge potential. Several explicit examples within the positive and negative graded mKdV sub-hierarchies were discussed in [20]. We now give a general argument that the Bäcklund Transformation derived from the gauge transformation (3.14) for arbitrary N provides equations compatible with the eqn. of motion. Consider the zero curvature representation for certain field configuration, namely φ1, i.e., [∂x +Ax(φ1), ∂tN +AtN (φ1)] = 0. (3.19) Under the gauge transformation, K(φ1, φ2)[∂x +Ax(φ1), ∂tN +AtN (φ1)]K(φ1, φ2)−1 = [K(∂x +Ax(φ1))K−1,K(∂tN +AtN (φ1))K−1] = [∂x +Ax(φ2), ∂tN +AtN (φ2)] = 0. (3.20) where the last equality comes from our assumption (3.14). The gauge transformation of the first entry in the zero curvature representation implies the x-component of the Bäcklund transformation (3.16). Since the zero curvature (3.19) and (3.20) implies that both φ1 and φ2 satisfy the same equation of motion, the gauge transformation (3.14) for Aµ = AtN of the second entry in (3.20) generates the time component of BT which, by construction has to be consistent with the equations of motion with respect to time tN . 3.2.2 KdV In order to extend the same philosophy to the KdV hierarchy recall the fact that the two dimensional gauge potential Ax,KdV can be obtained by Miura-gauge transformation from the homogeneous mKdV gauge potentials AhommKdV as in (3.4), i.e., Ax,KdV (J) = g2(v, ε)g1 (Ax,mKdV (v)) g−1 1 g−1 2 (v, ε)− ∂xg2(v, ε)g−1 2 (v, ε), (3.21) where v = ∂xφ(x, tN ) By assuming (3.12) for the KdV hierarchy, i.e., K̃(J1, J2)Ax,KdV (J1) = Ax,KdV (J2)K̃(J1, J2) + ∂xK̃(J1, J2). (3.22) the Bäcklund-gauge transformation for the KdV hierarchy K̃(J1, J2) constructed in terms of K(φ1, φ2) can be written as K̃ = g2(v2, ε2) ( g1K(φ1, φ2)g−1 1 ) g2(v1, ε1)−1. (3.23) At this stage we should recall that for each solution of the mKdV hierarchy v, the Miura transformation (3.5) generates two solutions, Jεi = εi∂xvi − v2 i , ε = ±1 satisfying the associated equation of motion of the KdV hierarchy. This is precisely why we assume ε1 and ε2 in eqn. (3.23) independent. In terms of mKdV variables 28CHAPTER 3. MIURA ANDGENERALIZED BÄCKLUND TRANSFORMATIONS FORKDVHIERARCHY vi = ∂xφi, K̃ is given for the particular case where ε1 = −ε2 ≡ ε and denote K̃(J1, J2) = K̃(ε1 = −ε2 ≡ ε) 2, K̃(J1, J2)11 = 1− v1ε ζ − β 4ζ (1− ε)e−p − β 4ζ (1 + ε)ep K̃(J1, J2)12 = −1 ζ K̃(J1, J2)21 = β 4ζ (−v1(1 + ε) + v2(1− ε)) e−p + β 4ζ (v1(1− ε)− v2(1 + ε)) ep − v1v2 ζ K̃(J1, J2)22 = −1− v2ε ζ − β 4ζ (1 + ε)e−p − β 4ζ (1− ε)ep (3.24) where p = φ1 + φ2. Substituting in eqn. (3.22) we find the following equations: • Matrix element 11: ζ−1 : J1 − ε∂xv1 − 1 2 βv1(ep − e−p) + v1v2 = 0 (3.25) • Matrix element 12: ζ−1 : v1 − v2 + β 2 (ep − e−p) = 0 (3.26) • Matrix element 21: ζ0 : J1 + J2 + βv1 2 (1 + ε)e−p − βv1 2 (1− ε)ep − βv2 2 (1− ε)e−p + βv2 2 (1 + ε)ep + 2v1v2 = 0 (3.27) ζ−1 : ε(J1v2 − J2v1)− v1∂xv2 − v2∂xv1 − εβ 2 v1v2(ep − e−p) = 0 (3.28) • Matrix element 22: ζ−1 − J2 − ε∂xv2 − βv2 2 (ep − e−p) = 0. (3.29) Using the mixed Miura transformation, i.e., ε2 = −ε1 ≡ ε, J1 = ε∂xv1 − v2 1 , J2 = −ε∂xv2 − v2 2 (3.30) together with the mKdV Bäcklund transformation (3.16) v1 − v2 = −β 2 (ep − e−p), (3.31) we find that eqns. (3.25), (3.26), (3.28) and (3.29) are identically satisfied. Defining the new variable Q and taking into account the Bäcklund eqn. (3.31) we find the following equality 1 2 Q = εv1 + β 4 (1− ε)e−p + β 4 (1 + ε)ep (3.32) = εv2 + β 4 (1 + ε)e−p + β 4 (1− ε)ep. (3.33) 2Notice that K̃ is given in terms of mKdV variables v1, v2 and we need to rewrite it in terms of KdV variables J1, J2. This requires solving Riccati eqn. v = v(J) (3.5) 3.2. BÄCKLUND TRANSFORMATION 29 Eliminating v1 and v2 from eqns (3.32) and (3.33) we find v1 = ε 2 Q− β 4 (ε− 1)e−p − β 4 (ε+ 1)ep (3.34) v2 = ε 2 Q− β 4 (ε+ 1)e−p − β 4 (ε− 1)ep (3.35) and henceforth β 4 (−v1(1 + ε) + v2(1− ε)) e−p + β 4 (v1(1− ε)− v2(1 + ε)) ep − v1v2 = β2 4 − Q2 4 . (3.36) Eqn. (3.27) then becomes J1 + J2 = β2 2 − Q2 2 . (3.37) From (3.32) and (3.33) we find that Q = ε(v1 + v2) + β 2 (ep + e−p) (3.38) Acting with ∂x in (3.38) and using (3.30) and (3.31), ∂xQ = ε∂x(v1 + v2) + β 2 (v1 + v2)(ep − e−p) = ε∂x(v1 + v2)− (v1 − v2)(v1 + v2) = J1 − J2 = ∂x (ω1 − ω2) (3.39) where we have used Ji ≡ ∂xwi, i = 1, 2. It therefore follows that Q = w1 − w2 (3.40) and the Bäcklund transformation for the spatial component of the KdV equation becomes, J1 + J2 = ∂xP = β2 2 − (w1 − w2)2 2 , P = w1 + w2. (3.41) which is in agreement with the Bäcklund transformation proposed in [11] and with [40]. In the new variable Q defined in (3.32) and (3.33) we rewrite the gauge-Bäcklund transformation K̃(J1, J2) in (3.24) as K̃(J1, J2, β) = −1 ζ ( −ζ + 1 2Q 1 −β2 4 + 1 4Q 2 ζ + 1 2Q ) , (3.42) Other cases with ε1 = ε2 = ±1 lead to trivial Bäcklund transformations in the sense that (3.22) for K̃(±1,±1) is trivially satisfied for mKdV Bäcklund and Miura transformations (3.16) and (3.5). There is no new equation relating the two KdV fields J1 and J2. From now on we shall only consider K̃(+1,−1) ≡ K̃ given in (3.42) and Miura transformation given by (3.30) We now discuss the extension of the Bäcklund transformation to the time component of the KdV hierarchy. Notice that in the zero curvature representation the spatial component of the two dimensional gauge potential Ax is the same for all flows and therefore universal among the different evolution equations. They differ from the time component AtN written according to the algebraic graded structure and parametrized by the integer N . AtN ,KdV = D̃(N) + D̃(N−1) + · · ·+ D̃(0), D̃(j) ∈ G̃j . (3.43) The Bäcklund-gauge transformation (3.42) acting on the potentials At3,KdV , At5,KdV and At7,KdV given by 30CHAPTER 3. MIURA ANDGENERALIZED BÄCKLUND TRANSFORMATIONS FORKDVHIERARCHY eqns. (A.2)- (A.4) of the appendix leads to the following Bäcklund equations respectively 4∂t3P = −Q∂2 xQ+ 1 2 ( (∂xQ)2 + 3(∂xP )2 ) (3.44) 16∂t5P = −Q∂4 xQ+ ∂xQ∂ 3 xQ+ 5∂xP∂ 3 xP + 1 2 ( 5(∂2 xP )2 − (∂2 xQ)2 ) + 5 2 ∂xP ( (∂xP )2 + 3(∂xQ)2 ) (3.45) 64∂t7P = −Q∂6 xQ+ ∂xQ∂ 5 xQ+ 7∂xP∂ 5 xP − ∂2 xQ∂ 4 xQ+ 14∂2 xP∂ 4 xP + 35∂xQ∂ 2 xQ∂ 2 xP + 35∂xP∂xQ∂ 3 xQ+ 21 2 (∂3 xP )2 + 1 2 (∂3 xQ)2 + 35 2 ∂xP ( (∂2 xP )2 + (∂2 xQ)2 ) + 35 2 ∂3 xP ( (∂xP )2 + (∂xQ)2 ) + 105 4 (∂xP )2(∂xQ)2 + 35 8 (∂xP )4 + 35 8 (∂xQ)2, (3.46) where ∂xP = J1 + J2. Equations (3.41) and (3.44) coincide with the Bäcklund transformation proposed in [11] for the KdV equation. Equations (3.41) and (3.45) correspond to those derived for the Sawada-Kotera equation in [40] 3. In the appendix we have checked the consistency between the spatial, (3.41) and time components (3.44) -(3.46) of the Bäcklund transformations for N = 3, 5 and 7. By direct calculation, using software Mathematica, we indeed recover the evolution equations (3.8)-(3.10). We would like to point out that our method is systematic and provides the Bäcklund transformations for arbitrary time evolution in terms of its time component 2-d gauge potential AtN ,KdV in terms of graded subspaces D̃(i), i = 0, · · ·N . The examples given above for t3, t5 and t7 just illustrate the potential of the formalism. 3.2.3 Examples • Vacuum - One soliton solution Consider φ1 = 0 and φ2 = ln( 1+ρ 1−ρ ), ρ = e2kx+2kN tN , N = 3, 5, 7 two solutions of the mKdV hierarchy. The mixed Miura transformation yields J1 + = ∂2 xφ1 − (∂xφ1)2 = 0, J2 − = −∂2 xφ2 − (∂xφ2)2 (3.47) Integrating to obtain J = ∂xw we find w1 = 0, w2 = − 4k 1 + ρ + 2k (3.48) Type-I Bäcklund transformation ∂x(w1 + w2) = β2 2 − 1 2 (w1 − w2)2 is satisfied by (3.48) for β = ±2k • Scattering of two One-soliton Solutions Consider the one-soliton of the mKdV hierarchy given by φi = ln ( 1 +Riρ 1−Riρ ) , i = 1, 2 ρ = e2kx+2kN tN , N = 3, 5, 7 · · · (3.49) Miura transformation generates two one-soliton solutions of the KdV hierarchy, namely J1 + = ∂2 xφ1 − (∂xφ1)2; (3.50) J2 − = −∂2 xφ2 − (∂xφ2)2; (3.51) leading to w1 = − 4k 1 +R1ρ + 2k, w2 = − 4k 1−R2ρ + 2k (3.52) The Type-I Bäcklund transformation is satisfied for R1 = R2. Notice that although R1 = R2, J1 and J2 correspond to different solutions due to opposite ε-sings in the Miura transformation. 3 Notice that there are typos in eqn. (45.11) of ref. [40] 3.3. FUSING AND TYPE-II BÄCKLUND TRANSFORMATION 31 • One-Soliton into Two-Soliton Solution Taking φ1 given by the one-soliton solution (3.49) and φ2 by φ2 = ln ( 1 + δ(ρ1 − ρ2)− ρ1ρ2 1− δ(ρ1 − ρ2)− ρ1ρ2 ) , δ = k1 + k2 k1 − k2 (3.53) leading to w2 = − 2(k2 1 − k2 2)(1 + ρ1)(1 + ρ2) k1 − k2 − (k1 + k2)(ρ1 − ρ2)− (k1 − k2)ρ1ρ2 (3.54) where ρi = e2kix+2kNi tN , i = 1, 2 satisfy the Type-I Bäcklund transformation for β = ±2k2. All these verifications were made in the software Mathematica. 3.3 Fusing and Type-II Bäcklund Transformation In this section we shall consider the composition of two gauge-Bäcklund transformations leading to the Type-II Bäcklund transformation. Let us consider a situation in which we start with a Bäcklund relation transforming solution v1 into another solution v0. A second subsequent Bäcklund relation transforms v0 into v2. Such algebraic relation for the mKdV hierarchy is described by KII(v1, v0, v2) = K(v2, v0)K(v0, v1) (3.55) where K(vi, vj) is given in (3.15) with β = βij . It leads to KII(v1, v0, v2) = [ 1 + β10β02 4λ eq e−φ0 2λ (β01e −φ1 + β02e −φ2) − 1 2e φ0(β01e φ1 + β02e φ2) 1 + β10β02 4λ e−q ] (3.56) where q = φ1 − φ2 and σ2 = − 4 β10β02 . Inserting the following identity (β01e φ1 + β02e φ2)(β01e −φ1 + β02e −φ2) = β01β02(η + eq + e−q) (3.57) where η = β2 10+β2 02 β10β02 . Defining Λ = −φ0 − ln(β02e −φ1 + β01e −φ2)− ln σ 4 , eqn. (3.56) becomes KII(v1, v0, v2) = [ 1− 1 σ2λe q eΛ−p 2λσ (eq + e−q + η) − 2 σ e p−Λ 1− 1 λσ2 e −q ] . (3.58) Eqn. (3.14) with KII(v1, v0, v2) leads to the following Bäcklund equations ∂xq = − 1 2σ eΛ−p(eq + e−q + η)− 2 σ ep−Λ (3.59) ∂xΛ = 1 2σ eΛ−p(eq − e−q). (3.60) Eqns. (3.59) and (3.60) coincide with the x-component of the Type-II Bäcklund transformation proposed for the sine-gordon model in [18]. Considering now the time component of the 2-d gauge potential for t = t3 , (i.e., for the mKdV equation), At3 = λEα + λ2E−α + vλh+ 1 2 (∂xv − v2)Eα − 1 2 (∂xv + v2)λE−α + 1 4 (∂2 xv − 2v3)h, (3.61) we find from eqn. (3.12), 16σ3∂t3q = eΛ−p (eq + e−q + η ) [ 2σ2(∂2 xp+ ∂2 xq) + σ2 (∂xp+ ∂xq) 2 − 8eq ] + + 4ep−Λ [ −2σ2(∂2 xp+ ∂2 xq) + σ2 (∂xp+ ∂xq) 2 − 8e−q ] + + 16σ∂xp ( eq + e−q + η ) (3.62) 32CHAPTER 3. MIURA ANDGENERALIZED BÄCKLUND TRANSFORMATIONS FORKDVHIERARCHY together with 4σ∂t3Λ = (v2 1 + ∂xv1)eΛ−q−p − (v2 2 + ∂xv2)eΛ+q−p, (3.63) which is compatible with equations of motion for the mKdV model. These Type-II Bäcklund equations (3.59)-(3.63) coincide with those derived in detail in ref. [20] where x→ x+, t→ x− and was extended to all positive higher graded equation within the mKdV hierarchy 4. In the case of the KdV hierarchy K̃typeII(J1, J0, J2) = K̃(J2, J0, β02)K̃(J0, J1, β01) (3.64) where K̃(Jj , Ji, βij) = −1 ζ [ −ζ + 1 2Qij 1 −β 2 ij 4 + 1 4Q 2 ij ζ + 1 2Qij ] . Such transformation can be interpreted as an extended Bäcklund transformation dubbed Type-II Bäcklund transformation (see [18]). Explicitly we find directly from (3.64) [K̃II(J1, J0, J2)]11 = 1− 1 2ζ Q− (β+ + β−) 2ζ2 + Q 8ζ2 (Q+ P − 2Ω) [K̃II(J1, J0, J2)]12 = 1 2ζ2 Q [K̃II(J1, J0, J2)]22 = 1 + 1 2ζ Q− (β+ − β−) 2ζ2 + Q 8ζ2 (Q− P + 2Ω) [K̃II(J1, J0, J2)]21 = − β+ 4ζ2 Q+ β− 4ζ2 (P − 2Ω) + Q 8ζ2 (−Ω2 + ΩP − P 2 4 + Q2 4 ) − β− ζ + Q 4ζ (P − 2Ω) (3.65) where Q = Q10 + Q02 = w1 − w2, P = Q10 − Q02 + 2Ω = w1 + w2, Ω = w0, 4β± = β2 01 ± β2 02 and Qij = wi − wj . Acting with K̃typeII(J1, J0, J2) in (3.22) we find the Bäcklund transformation for the KdV equation, i.e., ∂xQ = 2β− − 1 2 PQ+ ΩQ, ∂x(2Ω + P ) = 2β+ − 1 4 P 2 − 1 4 Q2 − Ω2 + ΩP. (3.66) Similarly for the time component gauge potential (A.2) we find ∂t3Q = 1 2 ∂xP∂xQ+ 1 2 ∂xΩ∂xQ+ 1 4 Q∂2 xΩ + 1 4 Ω∂2 xQ− P 8 ∂2 xQ− Q 8 ∂2 xP ∂t3(2Ω + P ) = 1 4 (∂xP )2 + 1 4 (∂xQ)2 + 1 2 ∂xP∂xΩ + (∂xΩ)2 − P 8 ∂2 xP − Q 8 ∂2 xQ + 1 4 P∂2 xΩ + 1 4 Ω∂2 xP − 1 2 Ω∂2 xΩ. (3.67) Equations (3.66) and (3.67) are compatible and lead to the eqns. of motion (3.8). Alternatively in terms of the mKdV Bäcklund transformation (3.23), eqn. (3.64) can be obtained by gauge-Miura transformation, i.e., K̃TypeII(J1, J0, J2) = g2(v2, ε2)g1 (K(φ2, φ0)IK(φ0, φ1)) g−1 1 g2(v1, ε1)−1 where we may introduce the identity element, I = g−1 1 g2(v0, ε0)−1g2(v0, ε0)g1 depending upon an arbitrary ε-sign, say, ε0. As argued when establishing (3.23), we are considering transitions with opposite ε-signs such that ε1 = −ε0 = ε and ε0 = −ε2 = −ε. It therefore follows that K̃TypeII(J1, J0, J2) = g2(v2, ε)g1 [K(φ2, φ0)K(φ0, φ1)] g−1 1 g2(v1, ε) −1 = g2(v2, ε)g1K II(φ2, φ1)g−1 1 g2(v1, ε) −1 (3.68) 4 Observe that the Type-II Bäcklund transformation via gauge transformation was constructed in [19] where a solution presented there was chosen to reproduce the Bäcklund transformation proposed in [18]. Here we choose a gauge transformation solution of [19] that reproduces [28]. 3.3. FUSING AND TYPE-II BÄCKLUND TRANSFORMATION 33 The equation (3.68) yields[ K̃TypeII (J1, J0, J2) ] 11 = 1 + (1 + ε) 4σζ eΛ−p (eq + e−q + η ) − (1− ε) σζ ep−Λ − (1− ε) 2σ2ζ2 e−q − (1 + ε) 2σ2ζ2 eq − (1− ε) σζ2 v1e p−Λ − (1 + ε) 4σζ2 v1e Λ−p (eq + e−q + η ) ,[ K̃TypeII (J1, J0, J2) ] 12 = (1− ε) σζ2 ep−Λ − (1 + ε) 4σζ2 eΛ−p (eq + e−q + η ) , [ K̃TypeII (J1, J0, J2) ] 21 = − ε σ2ζ ( eq − e−q ) − (1− ε) σζ (v1 + v2)ep−Λ − (1 + ε) 4σζ (v1 + v2) eΛ−p (eq + e−q + η ) + (v1 + v2) 2σ2ζ2 ( eq − e−q ) − ε(v1 − v2) 2σ2ζ2 ( eq + e−q ) + (1 + ε) 4σζ2 v1v2e Λ−p (eq + e−q + η ) − (1− ε) σζ2 v1v2e p−Λ,[ K̃TypeII (J1, J0, J2) ] 22 = 1− (1 + ε) 4σζ eΛ−p (eq + e−q + η ) + (1− ε) σζ ep−Λ − (1 + ε) 2σ2ζ2 e−q − (1− ε) 2σ2ζ2 eq + (1− ε) σζ2 v2e p−Λ + (1 + ε) 4σζ2 v2e Λ−p (eq + e−q + η ) . (3.69) Comparing the matrix elements of (3.64) with (3.68) we find the following relations between the mKdV and KdV variables: • matrix element 11 ζ−1 : −1 2 Q = (1 + ε) 4σ eΛ−p (eq + e−q + η ) − (1− ε) σ ep−Λ (3.70) ζ−2 : − (β+ + β−) 2 + Q 8 (Q+ P − 2Ω) = − (1− ε) 2σ2 e−q − (1 + ε) 2σ2ζ2 eq − (1− ε) σζ2 v1e p−Λ − (1 + ε) 4σζ2 v1e Λ−p (eq + e−q + η ) (3.71) • matrix element 21: ζ−1 : −β− + Q 4 (P − 2Ω) = − ε σ2 ( eq − e−q ) − (1− ε) σ (v1 + v2)ep−Λ − (1 + ε) 4σζ (v1 + v2) eΛ−p (eq + e−q + η ) (3.72) ζ−2 : −β+ 4 Q+ β− 4 (P − 2Ω) + Q 8 (−Ω2 + ΩP − P 2 4 + Q2 4 ) = + (v1 + v2) 2σ2 ( eq − e−q ) − ε(v1 − v2) 2σ2 ( eq + e−q ) + (1 + ε) 4σ v1v2e Λ−p (eq + e−q + η ) − (1− ε) σ v1v2e p−Λ (3.73) • matrix element 22: ζ−2 : − (β+ − β−) 2 + Q 8 (Q− P + 2Ω) = − (1 + ε) 2σ2 e−q − (1− ε) 2σ2 eq + (1− ε) σ v2e p−Λ + (1 + ε) 4σ v2e Λ−p (eq + e−q + η ) . (3.74) 34CHAPTER 3. MIURA ANDGENERALIZED BÄCKLUND TRANSFORMATIONS FORKDVHIERARCHY The element 12 and the element 22 with ζ−1 gives us the same result of (3.70). Eliminating the mKdV variables p, q and Λ we recover the Type-II Bäcklund transformation for the KdV hierarchy (3.66) as shown in the Appendix B. 3.3.1 Examples and Solutions • Vacuum - 1-soliton - Vacuum The first example is to consider vacuum to 1-soliton and back to vacuum again given by the following configuration, w1 = 0, w0 = Ω = − 4k 1 + ρ(x, tN ) + 2k, w2 = 0 (3.75) with ρ(x, tN ) = e2kx+2kN tN . It is straightforward to check that eqns. (3.66) and (3.67) are satisfied for β− = 0 and β+ = 2k2. • 1-soliton - 2soliton - 1-soliton Consider now a configuration of 1-soliton transforming into a 2-solitons solution and back to 1-soliton. It is described by wi = − 4ki 1 + ρi(x, tN ) + 2ki, i = 1, 2 ρi = e2kix+2kNi tN , (3.76) Ω = w0 = − 2(k2 1 − k2 2)(1 + ρ1)(1 + ρ2) k1 − k2 − (k1 + k2)(ρ1 − ρ2)− (k1 − k2)ρ1ρ2 (3.77) Eqns. (3.66) and (3.67) are satisfied for β− = k2 2 − k2 1 and β+ = k2 1 + k2 2. • Vacuum - 1-soliton - 2-soliton Consider the solution of eqn. (3.66) and (3.67) w1 = 0, w0 = Ω = − 4k1 1 + ρ1(x, tN ) + 2k1, (3.78) and w2 = − 2(k2 1 − k2 2)(1 + ρ1)(1 + ρ2) k1 − k2 − (k1 + k2)(ρ1 − ρ2)− (k1 − k2)ρ1ρ2 , (3.79) where ρi = e2kix+2kNi tN Eqns. (3.66) and (3.67) are satisfied for β− = k2 1 − k2 2 and β+ = k2 1 + k2 2. Chapter 4 Super mKdV hierarchy and its Bäcklund transformations This chapter is divided in two sections. The section 4.1 is dedicated to a review of the construction of the super mKdV hierarchy. In section 4.2 we obtain the Bäcklund transformations for the super mKdV hierarchy as a generalization of the discussion presented in the previous chapters. The explicit examples of N = 3 and N = 5 are computed. 4.1 The supersymmetric mKdV hierarchy In this section we present a brief review of the systematic construction of the supersymmetric mKdV hierarchy based on the affine Kac-Moody superalgebra Ĝ = ŝl(2,1) [35]. The structure explained in the first chapter of the thesis and used in the second one will be again crucial here (with some adaptations to include the supersymmetric generators). Let us start by considering the super Lie algebra sl(2,1), which has four bosonic generators { h1, h2, E±α1 } , and four fermionic generators { E±α2 , E±(α1+α2) } , where α1 is bosonic simple root and α2, α1 + α2 are fermionic simple roots. The affine ŝl(2,1) structure is introduced by extending each generator Ta ∈ sl(2, 1) to T (n) a , where d is defined by [ d, T (n) a ] = nT (n) a . The hierarchy is further specified by introducing a decom- position of the ŝl(2,1) superalgebra through the definition of a constant grade one element E(1), where E(2n+1) = h (n+1/2) 1 + 2h (n+1/2) 2 − E(n) α1 − E(n+1) −α1 , (4.1) and the so called principal grading operator Qp = 2d+ 1 2 h (0) 1 . (4.2) The grading operator Qp and the constant element E(1) decompose the affine superalgebra Ĝ = ⊕ Ĝm = K⊕M, where m is the degree of the subspace Ĝm according to Qp, K = { x ∈ Ĝ /[x,E(1)] = 0 } is the kernel of E(1), and M its complement, in the following way Ĝ2n+1 = { K (2n+1) 1 , K (2n+1) 2 , M (2n+1) 1 } , Ĝ2n = { M (2n) 2 } , Ĝ2n+ 1 2 = { F (2n+ 1 2 ) 2 , G (2n+ 1 2 ) 1 } , Ĝ2n+ 3 2 = { F (2n+ 3 2 ) 1 , G (2n+ 3 2 ) 2 } , (4.3) where the generators Fi, Gi,Ki, and Mi are defined as linear combinations of the ŝl(2,1) generators [35]. The representation of these generators is given in Appendix D. Now the construction of the integrable hierarchy is based on the zero curvature condition (2.2). In general, Ax is defined as Ax = E(1) +A0 +A1/2 where A0 +A1/2 ∈M, i.e., A0 = uM (0) 2 , A1/2 = √ iψ̄ G (1/2) 1 . (4.4) 35 36 CHAPTER 4. SUPER MKDV HIERARCHY AND ITS BÄCKLUND TRANSFORMATIONS Here, u and ψ̄ are the corresponding fields of the integrable hierarchy. Now, we will assume that AtN = D(N) +D(N−1/2) + ...+D(1/2) +D(0) for the positive hierarchy, where D(m) has grade m. Then, the equation to be solved reads[ ∂x + E(1) +A0 +A1/2, ∂tN +D(N) +D(N−1/2) + ...+D(1/2) +D(0) ] = 0. (4.5) The solving method consists on splitting the above equation grade by grade, which leads us to the following set of relations, (N + 1) : [ E(1), D(N) ] = 0, (N + 1/2) : [ E(1), D(N−1/2) ] + [ A1/2, D (N) ] = 0, (N) : ∂xD N + [ A0, D (N) ] + [ E(1), D(N−1) ] + [ A1/2, D (N−1/2) ] = 0, ... (1) : ∂xD (1) + [ A0, D (1) ] + [ E(1), D(0) ] + [ A1/2, D (1/2) ] = 0, (1/2) : ∂xD (1/2) + [ A0, D (1/2) ] + [ A1/2, D (0) ] − ∂tNA1/2 = 0, (0) : ∂xD (0) + [ A0, D (0) ] − ∂tNA0 = 0. (4.6) Note that, the image part of the zero and the one-half grade components of (4.6) yields the time evolution for the fields introduced in eq. (4.4). Now, it is possible to expand each term D(m) by using the generators in eq. (4.3), as follows D(2n+1) = ã2n+1K (2n+1) 1 + b̃2n+1K (2n+1) 2 + c̃2n+1M (2n+1) 1 , D(2n) = ã2nM (2n) 2 , D(2n+ 1 2 ) = ã2n+ 1 2 F (2n+ 1 2 ) 2 + b̃2n+ 1 2 G (2n+ 1 2 ) 1 , D(2n+ 3 2 ) = ã2n+ 3 2 F (2n+ 3 2 ) 1 + b̃2n+ 3 2 G (2n+ 3 2 ) 2 , (4.7) where the ãm, b̃m, and c̃m are functionals of the fields u and ψ̄. Substituting this parametrization in eq. (4.6), one can solve recursively for all D(m),m = 0, · · ·N . Notice that the Lax component Ax does not depend on the index N and will be the same for the entire hierarchy. It takes the following form (see for instance [26, 34]) Ax =  λ1/2 − ∂xφ −1 √ i ψ̄ −λ λ1/2 + ∂xφ √ i λ1/2 ψ̄ √ i λ1/2 ψ̄ √ i ψ̄ 2λ1/2  , (4.8) where we have redefined u = −∂xφ. This parametrization establishes the explicit relationship between the relativistic (sinh-Gordon) and non-relativistic (mKdV) field variables. In what follows we will apply the procedure and consider explicit solutions of the integrable hierarchy equations (4.6) for the simplest members. For the N = 3 member, we find that the solution for the Lax component At3 = D(3) +D(5/2) +D(2) +D(3/2) +D(1) +D(1/2) +D(0), is given by At3 =  a0 + λ1/2a1/2 − λ∂xφ+ λ3/2 a+ − λ µ+ + λ1/2ν+ + λ √ iψ̄ −λa− − λ2 −a0 + λ1/2a1/2 + λ∂xφ+ λ3/2 λ1/2µ− + λν− + λ3/2 √ iψ̄ λ1/2µ− − λν− + λ3/2 √ iψ̄ µ+ − λ1/2ν+ + λ √ iψ̄ 2λ1/2a1/2 + 2λ3/2  , (4.9) where a0 = −1 4 ( ∂3 xφ− 2(∂xφ)3 + 3i∂xφψ̄∂xψ̄ ) , a1/2 = − i 2 ψ̄∂xψ̄, a± = 1 2 ( ∂2 xφ± (∂xφ)2 ∓ iψ̄∂xψ̄ ) , µ± = √ i 4 ( ∂2 xψ̄ ± ∂xφ∂xψ̄ ∓ ψ̄∂2 xφ− 2ψ̄(∂xφ)2 ) , ν± = √ i 2 ( ∂xψ̄ ± ψ̄∂xφ ) . (4.10) 4.2. SUPER-BÄCKLUND TRANSFORMATIONS 37 The equations of motion, which correspond to the zero and one-half grade components of (4.6), are in this case the N = 1 supersymmetric mKdV equations, namely 4∂t3u = ∂3 xu− 6u2∂xu+ 3iψ̄∂x ( u∂xψ̄ ) , (4.11) 4∂t3 ψ̄ = ∂3 xψ̄ − 3u∂x ( uψ̄ ) . (4.12) Now, for the N = 5 member, the solution for the Lax component At5 = D(5) + D(9/2) + · · · + D(0) is given explicitly in appendix E. In this case, we find the following equations of motion, 16∂t5u = ∂5 xu− 10(∂xu)3 − 40u(∂xu)(∂2 xu)− 10u2(∂3 xu) + 30u4(∂xu) + 5i∂xψ̄∂x(u∂2 xψ̄) + 5iψ̄∂x(u∂3 xψ̄ − 4u3∂xψ̄ + ∂xu∂ 2 xψ̄ + ∂2 xu∂xψ̄), (4.13) 16∂t5 ψ̄ = ∂5 xψ̄ − 5u∂x(u∂2 xψ̄ + 2∂xu∂xψ̄ + ∂2 xuψ̄) + 10u2∂x(u2ψ̄)− 10(∂xu)∂x(∂xuψ̄). (4.14) It is worth pointing out that the negative integrable hierarchy can be also constructed by considering the following zero curvature condition,[ ∂x + E(1) +A0 +A1/2, ∂t−M +D(−M) +D(−M+1/2) + · · ·+D(−1) +D(−1/2) ] = 0. (4.15) The solutions are in general non-local, however, for the simplest case of N = −M = −1, we find that the Lax component At−1 = D(−1) +D(−1/2), corresponds to the N = 1 sshG equation [27, 35], i.e At−1 =  λ−1/2 −λ−1 e2φ −λ1/2 √ i ψeφ −e−2φ λ−1/2 − √ i ψe−φ √ i ψe−φ λ1/2 √ i ψeφ 2λ−1/2  . (4.16) In this case, the fields φ, ψ̄ and ψ satisfy ∂t−1 ∂xφ = 2 sinh 2φ+ 2ψ̄ψ sinhφ, ∂t−1 ψ̄ = 2ψ coshφ, (4.17) ∂xψ = 2ψ̄ coshφ, the equations of motion of the N = 1 sshG model in the light-cone coordinates (x, t−1). We note that the equations of motion for all members of the hierarchy are invariant under the following supersymmetric transformations, δφ = √ iε̄ ψ̄, δψ̄ = 1√ i ε̄ ∂xφ, (4.18) where ε̄ is a Grassmannian parameter. 4.2 Super-Bäcklund transformations In this section we derive a general method to generate the super-Bäcklund transformations (sBT) for all members of the hierarchy. We will use the defect matrix associated to the hierarchy in order to derive the sBT in components. The key ingredient is the gauge invariance of the zero curvature representation generated by the defect matrix which, in turn is again assumed to relate two field configurations. As explicit examples we construct the super-Bäcklund transformation for the first two flows, namely, N = 3 (smKdV) equation, and for the N = 5 super-equation. One nice check will be to put the fermioninc fields to zero and recover the “classical” Bäcklund transformations. Based upon the fact that the spatial Lax operator is common to all members of the mKdV hierarchy, it has been shown recently in the previous chapters that the spatial component of the Bäcklund transformation, and consequently the associated defect matrix, are also common and henceforth universal within the entire hierarchy. Here, we will extend these results to the supersymmetric mKdV hierarchy starting from the defect matrix already derived for the (N = −1 member), the super sinh-Gordon equation. The so-called type-I 38 CHAPTER 4. SUPER MKDV HIERARCHY AND ITS BÄCKLUND TRANSFORMATIONS defect matrix can be written as follows [26], K =  λ1/2 − 2 ω2 e φ+λ−1/2 − 2 √ i ω e φ+ 2 f1 − 2 ω2 e −φ+λ1/2 λ1/2 − 2 √ i ω e− φ+ 2 f1λ 1/2 2 √ i ω e− φ+ 2 f1λ 1/2 2 √ i ω e φ+ 2 f1 2 ω2 + λ1/2  (4.19) where φ± = φ1±φ2, ω represents the Bäcklund parameter, and f1 is an auxiliary fermionic field. The defect matrix K, connecting two different configurations φ1 and φ2, satisfies the following gauge equation, ∂xK = KAx(φ1, ψ̄1)−Ax(φ2, ψ̄2)K. (4.20) Now, by substituting (4.8) and (4.19) in the (4.20), we get ∂xφ− = 4 ω2 sinh(φ+)− 2i ω sinh( φ+ 2 )f1ψ̄+, (4.21) ψ̄− = 4 ω cosh ( φ+ 2 ) f1, (4.22) ∂xf1 = 1 ω cosh( φ+ 2 )ψ̄+. (4.23) the spatial part of the Bäcklund transformations, where we have denoted ψ̄± = ψ̄1 ± ψ̄2. To derive the time component of the transformation, we consider the corresponding temporal part of the Lax pair Atn . 4.2.1 N=3 For the smKdV equation (N = 3), the second gauge condition reads, ∂t3K = KAt3(φ1, ψ̄1)−At3(φ2, ψ̄2)K. (4.24) By substituting (4.9) and (4.19) in the above equation, we obtain 4∂t3φ− = i ω [ ∂2 xφ+ cosh ( φ+ 2 ) − (∂xφ+) 2 sinh (φ+ 2 )] ψ̄+f1 − i ω [ ∂xφ+ cosh (φ+ 2 ) ∂xψ̄+ − 2 sinh (φ+ 2 ) ∂2 xψ̄+ ] f1 + 2 ω2 [ 2(∂2 xφ+) coshφ+ − (∂xφ+) 2 sinhφ+ + iψ̄+(∂xψ̄+) sinhφ+ ] − 96i ω5 [ sinh (φ+ 2 ) + 4 sinh3 (φ+ 2 ) + 3 sinh5 (φ+ 2 )] ψ̄+f1 − 32 ω6 sinh3 φ+, (4.25) and 4∂t3f1 = 1 2ω cosh (φ+ 2 ) [ 2∂2 xψ̄+ − ψ̄+(∂xφ+)2 ] + 1 2ω sinh (φ+ 2 ) [ ψ̄+∂ 2 xφ+ − (∂xφ+)(∂xψ̄+) ] − 12 ω4 sinhφ+ cosh2 (φ+ 2 ) (∂xφ+)f1 + 12 ω5 sinh2 φ+ cosh (φ+ 2 ) ψ̄+. (4.26) Equations (4.21)–(4.23), and (4.25) and (4.26) correspond to the super-Bäcklund transformations for the smKdV in components. It can easily verified that they are consistent by cross-differentiating any of them. Limit to bosonic case Notice also that by setting all the fermions to zero we recover the bosonic case, i.e., the Bäcklund transfor- mation of the mKdV [28], ∂xφ− = 4 ω2 sinhφ+, (4.27) 4∂t3φ− = 4 ω2 ∂2 xφ+ coshφ+ − 2 ω2 (∂xφ+) 2 sinhφ+ − 32 ω6 sinh3 φ+. (4.28) 4.2. SUPER-BÄCKLUND TRANSFORMATIONS 39 4.2.2 N=5 Now, to derive the temporal part of the super-Bäcklund transformation for the N = 5 member of the hierarchy we consider the corresponding Lax operator At5 . The gauge condition reads, ∂t5K = KAt5(φ1, ψ̄1)−At5(φ2, ψ̄2)K. (4.29) By solving this condition for At5 given in Appendix E, we obtain 16∂t5φ− = − i ω [ c0 ψ̄+ + c1 ∂xψ̄+ + c2 ∂ 2 xψ̄+ + c3 ∂ 3 xψ̄+ + c4 ∂ 4 xψ̄+ ] f1 + 1 ω2 [ c5 + ic6 ψ̄+∂xψ̄+ + ic7 ψ̄+∂ 2 xψ̄+ + ic8 ( ψ̄+∂ 3 xψ̄+ − (∂xψ̄+)(∂2 xψ̄+) )] − i ω5 [ c9 ψ̄+ + c10 ∂xψ̄+ + c11 ∂ 2 xψ̄+ ] f1 + 1 ω6 [ c12 + i c13 ψ̄+∂xψ̄+ ] + i ω9 c14f1ψ̄+ + c15 ω10 , (4.30) 16∂t5f1 = 1 ω [ d0 ψ̄+ + d1 ∂xψ̄+ + d2 ∂ 2 xψ̄+ + d3 ∂ 3 xψ̄+ + d4 ∂ 4 xψ̄+ ] + 1 ω4 [ d6 + i d5 ψ̄+∂xψ̄+ ] f1 + 1 ω5 [ d7 ψ̄+ + d8 ∂xψ̄+ + d9 ∂ 2 xψ̄+ ] + d10 ω8 f1 + d11 ω9 ψ̄+, (4.31) where ci, i = 0, .., 15 and dj , j = 0, .., 11 are functions depending on φ+ and its derivatives, and their explicit forms are given by (F.1)-(F.16) and (F.17)-(F.28) from Appendix F, respectively. The equations (4.21)–(4.23), and (4.30) and (4.31) correspond to the super-Bäcklund transformations for the N = 5 super equation. Cross differentiating (4.30) and (4.31) with respect of x we recover the equations of motion (4.13) and (4.14) after using eqns. (4.21)–(4.23). In this chapter we reviewed the construction of the super mKdV hierarchy using the graded super algebra ŝl(2, 1) and the principal gradation. We then used the universality of the K-matrix construction along the hierarchy to explicitly construct the Bäcklund transformations for the super N = 3 and super N = 5 equations. Although we are presenting it here,we also computed the Bäcklund transformations for those equations using the super fields formalism. The super charges were also computed, giving particular attention to the changes on the conserved charges due to the introduction of the defect [31, 33]. 40 CHAPTER 4. SUPER MKDV HIERARCHY AND ITS BÄCKLUND TRANSFORMATIONS Chapter 5 Conclusions and further developments In the Part I of the thesis we explored the algebraic structure underlying integrable hierarchies in order to find universal features of their equations. The gauge invariance of the zero curvature equation allowed the construction of a defect-gauge matrix connecting two different field configurations of the same integrable model and hence generating its Bäcklund transformations. In Chapter 2 the main result is the fact that the construction of the defect matrix depends only on the Ax and therefore is universal within the hierarchy. This means that one can systematically construct the Bäcklund transformations for any equation in the mKdV hierarchy [20], [28]. The main result of the Chapter 3 is the extension of such construction to the KdV hierarchy by proposing a Miura-gauge transformation denoted by the product g2g1 given in (3.1) and (3.3) mapping the mKdV into the KdV hierarchy (see (3.4)) [29]. A subtle point is that such Miura mapping allows a sign ambiguity such that each solution of the mKdV hierarchy defines two solutions for its KdV counterpart. The Bäcklund-gauge transformation for the KdV hierarchy is constructed by Miura-gauge transforming the Bäcklund transforma- tion of the mKdV system as shown in (3.23). An interesting fact is that the Bäcklund transformation for the KdV hierarchy is solved by mixed Miura solutions generated by the mKdV Bäcklund solutions. A few simple explicit examples illustrate our conjecture. A more general evidence of the mixed Miura solutions is shown to agree with the Bäcklund transformation proposed in [11] for the first two KdV flows. The composition law of two subsequent Bäcklund-gauge transformations leading to Type-II Bäcklund transformation (see (3.55)) introduced in [18] in the context of sine-Gordon and Tzitzeica models was extended to the KdV hierarchy. We have showed that the ideia of fusing to defects in the Lagrangian formalism can be translated within our construction, to the direct fusion of two KdV Bäcklund-gauge transformations in (3.64) and alternatively, the Miura transformation of mKdV Type-II Bäcklund transformation as shown in (3.68). These two approaches generate relations between the mKdV and KdV variables which were shown in the Appendix B, to be consistent. In [30] we discuss the generalization of this idea for n defects for the Sinh-Gordon equation. Another interesting point is that we explored only the positive part of the mKdV hierarchy, by using it to construct the positive part of KdV hierarchy and corresponding Bäcklund transformations. It would be very interesting to understand which kind of result it would be obtained for the negative part of the KdV hierarchy by starting with the negative part of the mKdV hierarchy, both in a the sense of hierarchy itself as well as for its Bäcklund transformations. In the Chapter 4 we have studied the presence of a Type-I integrable defects in the ŝl(2, 1) supersymmetric integrable hierarchy through super Bäcklund transformations. What we computed in principle would be called Type-I defect in the literature. However, let us call the attention to an important property appearing in the corresponding supersymmetric extensions. It turns to be that what we are calling type-I integrable defect for the supersymmetric mKdV hierarchy contains intrinsically an auxiliary fermionic field necessary to describe defect conditions for the fermionic fields. In that sense, this kind of defect should be treated as a “partial” Type-II defect, i.e. there is only one auxiliary fermionic time dependent quantity defined on the defect, but not a bosonic auxiliary field. A genuine Type-II defect will then contain one bosonic and two fermionic auxiliary fields as it is the case of the super sinh-Gordon model. Then, by using the universality argument, the type-II super Bäcklund transformation for the smKdV can be obtained either directly from the type-II defect matrix for the super sinh-Gordon previously obtained in [27], or by applying the fusing procedure of two partial type I defect matrices. We discuss this in [31]. The latter procedure can be achieved by performing two type-I Bäcklund transformations frozen at different points and then taking the limit when both points coincide. The auxiliary fields will then appear after an appropriate reparameterization of the “squeezed” fields valued only at the defect point. We expect that solutions for the auxiliary fields will be same as those for the super Sinh-Gordon equation due to the universality of the spatial component of the 41 42 CHAPTER 5. CONCLUSIONS AND FURTHER DEVELOPMENTS Lax within the hierarchy, as it is in the bosonic case [20, 28]. We expect to return to these issues in future investigations. One natural continuation for this work would be to construct the super KdV hierarchy and its K-matrix through gauge transformations starting with the super mKdV hierarchy as a generalization for what we did in the bosonic case. An interesting result we did not discuss here but it is worth to mention is that in [32] we also showed the construction of the super integrable hierarchy and of the Bäcklund transformations by recursion relations as a generalization of [36]. Finally we should mention that the idea of an universality of the Bäcklund-gauge transformation is most probably valid for other hierarchies such as the AKNS and higher rank Toda theories. It would be interesting to see how such examples can be worked out technically. Also, for an ŝl(3) for example we could construct a Toda hierarchy if we assume the principal gradation. A good question is if there exist something analog to the Miura-gauge transformations which would lead to the correspondent hierarchy for the homogeneous gradation. One extra question would be how to translate the discussion present in this Part I to the classical r-matrix formalism. It should be interesting to develop the concept of integrable hierarchies for discrete cases and investigate whether the arguments involving Bäcklund-gauge transformation employed in this thesis can be extended. The relation between the integrable discrete mKdV [37],[38] and its Miura transformation to discrete KdV equations should be understood under the algebraic formalism. Part II Quantum Integrability 43 Chapter 6 Introduction The Part II of this thesis is focused in quantum integrable spin chains. More specifically we are interested in compute the spectrum of a certain class of finite length spin chains and explain the high degeneracies present in their spectra. Quantum spin chains have many interesting applications in several different areas of physics including but not restricted to condensed matter [41], statistical mechanics [42, 43] and AdS/CFT [44] with possible applications in black-holes [45]. Recently, relations between integrable spin chains and a four dimensional Chern-Simons theory were also found [46]-[48]. To have good ways to compute the spectra of such systems is therefore very important. However, this is not usually easy since they are interacting systems with many particles. For those quantum spin chains which are integrable, several techniques have been developed, such as quantum inverse scattering method (QISM), algebraic Bethe ansatz, analytical Bethe ansatz, nested algebraic Bethe ansatz, separation of variables, etc [49]-[53]. In the context of open integrable quantum spin chains the most important objects are the R-matrix and the K-matrices. The R-matrix encodes the bulk information and satisfies the so-called Yang-Baxter equation (YBE)[50, 51]. The K-matrices, KL and KR, on the other hand contain the information about the left and right boundaries, respectively. The K-matrices satisfy an equation called Boundary Yang-Baxter equation (BYBE) or reflection equation [52, 53]. By having R, KL and KR we are able to construct the so-called transfer matrix which is the generating function of an infinite number of conserved quantities. For a review see [54, 55]. Three important types of R-matrices are: rational, trigonometric and elliptic. Spin chains constructed from rational R-matrices have classical (Lie group) symmetries which are in fact, the same symmetries of the R-matrix. This is not the case for trigonometric and elliptic R-matrices, whose transfer matrices do not have the same symmetries of the R-matrices. Several explicit R-matrices were computed by [56]-[58]. For each R-matrix the BYBE has to be solved in order to find the corresponding K-matrices. Several solutions for the reflection equation were find by [59]-[62]. In this work we focus on anisotropic spin chains. They are constructed from trigonometric R-matrices, which are themselves associated to affine Lie algebras. Some of these spin chains have quantum group (QG) symmetries that help to explain the degeneracies and multiplicities of their spectrum. The first example to be solved was the XXZ integrable spin chain which was proved to have QG symmetry Uq(sl(2)) [63, 64]. Since then many other examples with higher ranks have been investigated, see e.g. [65]-[87]. In this work we construct finite length integrable quantum spin chains using the R-matrices for the affine Lie algebras algebras ĝ = {A(2) 2n , A (2) 2n−1, B (1) n , C (1) n , D (1) n } [56]-[58]. Since we want to describe the spin chains which have more symmetry we use diagonal K-matrices [59]-[62]. Those K-matrices depend on a discrete parameter p which runs from 0 to n. As we prove [88] in chapter 7 those spin chains have quantum group symmetry corresponding to removing the pth node from the Dynkin diagram of ĝ. In the work of Nepomechie and Mezincescu [67] for R-matrices associated with A (1) 1 and A (2) 2 they noticed that the asymptotic monodromy matrix could be expressed in terms of only the coproducts of the generators. This is not immediately true for the ĝ models above. A fundamental step is to do a gauge transformation on the R-matrix and K-matrices in such a way that they still satisfy YBE and BYBE but the new asymptotic monodromy would depend only on the unbroken generators, i.e. the ones corresponding to the nodes on lhs and rhs of the pth node. In the Table 6.1 are summarized the QG symmetries each spin chain has. It is important to notice that some cases were already well understood. Our main contribution is for the cases where 0 < p < n which from our knowledge are new. The cases for p = 0 were know already from a long 45 46 CHAPTER 6. INTRODUCTION ĝ QG symmetry Representation at each site A (2) 2n Uq(Bn−p)⊗ Uq(Cp) (2(n− p) + 1, 1)⊕ (1, 2p) A (2) 2n−1 Uq(Cn−p)⊗ Uq(Dp) (p 6= 1) (2(n− p), 1)⊕ (1, 2p) B (1) n Uq(Bn−p)⊗ Uq(Dp) (n > 1, p 6= 1) (2(n− p) + 1, 1)⊕ (1, 2p) C (1) n Uq(Cn−p)⊗ Uq(Cp) (2(n− p), 1)⊕ (1, 2p) D (1) n Uq(Dn−p)⊗ Uq(Dp) (n > 1, p 6= 1, n− 1) (2(n− p), 1)⊕ (1, 2p) Table 6.1: QG symmetries of the open-chain transfer matrix, where p = 0, 1, . . . , n. time [65]-[69]. The case of p = n and ĝ = A (2) 2n was computed more recently in [78, 86] while for ĝ = A (2) 2n−1 was computed by us in [87]. In addition to the symmetries described in Table 6.1, the cases C (1) n and D (1) n have also a duality symmetry p→ n−p. Such symmetry is a consequence of the Dynkin diagram be invariant under reflection. When p = n 2 we also have a self-duality symmetry. In addition to that, when p = n 2 and the parameter of the K-matrix γ0 = −1 we also have what we called bonus symmetry. The bonus symmetry makes representations as 2(a, a) degenerate to 2a2. The cases for A (2) 2n−1, B (1) n and D(1) have also Z2 symmetries transforming complex representations into their conjugates. The duality symmetry, self-duality symmetries, bonus symmetry and the Z2 symmetries are all explicitly constructed in the Chapter 7 and are used to explain the degeneracies and multiplicities of the spectrum. Recently the case of D (2) n+1 was also studied [89] and showed to have QG symmetry Uq(Bn−p)⊗ Uq(Bp), duality p→ n− p, self duality and bonus symmetry in the same situations as the ones described above. The process of directly diagonalize the transfer matrix can be computationally very difficult. The di- mension of the matrices increases very fast with the number of sites in such a way that this process quickly becomes impossible to continue. An alternative is to use the method of analytical Bethe ansatz to obtain the eigenvalues. Again the cases for p = 0 were already been considered several years ago [68, 69, 84, 85, 87]. And some of the cases for p = n [84, 86, 87]. When talking about Bethe ansatz, on Chapter 8, in addition to the cases described in Table 6.1, we also consider D (2) n+1 . For closed spin chains there is a general formula by Reshetikhin for Bethe ansatz for all algebras. In this work we conjecture a generalization of such formula for open spin chains. We also construct formulas for the Dynkin labels of the Bethe states in terms of the number of Bethe roots of each type. In chapter 7 we prove that the spin chains have the QG symmetries presented in Table 6.1, the dualities and Z2 are explicitly constructed and proved. In the process to prove these symmetries several interesting properties for the R-matrices were found and proved. Several examples are given in order to illustrate the symmetries. In Chapter 8 we compute the Bethe ansatz and give explicit formulas for all the algebras mentioned, as well as a conjecture for a general formula. The relation between Dynkin labels of the Bethe states and the number of Bethe roots of each type is also presented. A more detailed outline of each chapter is presented at the beginning of them. Chapter 7 Surveying the quantum group symmetries of integrable open spin chains The outline of this chapter is as follows. The transfer matrix is introduced in Sec. 7.1. The QG symmetry of the transfer matrix is proved in Sec. 7.2. The duality symmetry of the transfer matrix (for the cases C (1) n and D (1) n ), and the action of duality on the QG generators, are worked out in Sec. 7.3. The additional Z2 symmetries of the transfer matrix (for the cases A (2) 2n−1, B (1) n and D (1) n ), and the action of these symmetries on the QG generators, are worked out in Sec. 7.4. These symmetries are used in Sec. 7.5 to explain the degeneracies in the spectrum of the transfer matrix for generic values of the anisotropy parameter η. The R-matrices are recalled in Appendix G, details about the QG generators are presented in Appendix H, and the Hamiltonian is noted in Appendix I. Proofs of several lemmas are outlined in Appendix J. 7.1 Basics We consider an integrable open quantum spin chain with a vector space V = Cd at each of its N sites, where d = { 2n+ 1 for A (2) 2n , B (1) n 2n for A (2) 2n−1 , C (1) n , D (1) n , n = 1, 2, . . . . (7.1) The Hilbert space (“quantum” space) of the spin chain is therefore V⊗N . 7.1.1 R-matrix The bulk interactions of the spin chain are encoded in the R-matrix R(u), which maps V ⊗ V to itself, and satisfies the Yang-Baxter equation (YBE) on V ⊗ V ⊗ V R12(u− v)R13(u)R23(v) = R23(v)R13(u)R12(u− v) . (7.2) We use the standard notations R12 = R ⊗ I , R23 = I ⊗ R ,R13 = P23R12P23 = P12R23P12, where I is the identity matrix on V, and P is the permutation matrix on V ⊗ V P = d∑ i,j=1 eij ⊗ eji , (7.3) where eij are the d× d elementary matrices with elements (eij)αβ = δi,αδj,β . We consider here the anisotropic R-matrices (with anisotropy parameter η) corresponding to the following affine Lie algebras 1 ĝ = {A(2) 2n , A (2) 2n−1 , B (1) n , C(1) n , D(1) n } . (7.4) 1We do not consider here the case A (1) n , which does not have crossing symmetry; it has been studied in a similar context in [71, 72, 77]. 47 48CHAPTER 7. SURVEYING THEQUANTUMGROUP SYMMETRIES OF INTEGRABLE OPEN SPIN CHAINS These R-matrices, which are given by Jimbo [56] (except for A (2) 2n−1, in which case we consider instead Kuniba’s R-matrix [58]), are in the homogeneous picture (gauge).2 These R-matrices, which can be found in Appendix G, all have the following additional properties: PT symmetry R21(u) ≡ P12R12(u)P12 = Rt1t212 (u) , (7.5) unitarity R12(u) R21(−u) = ζ(u) I⊗ I , (7.6) where ζ(u) is given by ζ(u) = ξ(u) ξ(−u) , ξ(u) = −2 δ1 sinh( 1 2 (u+ 4η)) sinh( 1 2 (u+ ρ)) , (7.7) where δ1 is given by δ1 = { i for A (2) 2n , A (2) 2n−1 1 for B (1) n , C (1) n , D (1) n , (7.8) and crossing symmetry R12(u) = V1R t2 12(−u− ρ)V1 = V t22 Rt112(−u− ρ)V t22 , (7.9) where the crossing parameter ρ is given by ρ = { −2κη − iπ for A (2) 2n , A (2) 2n−1 −2κη for B (1) n , C (1) n , D (1) n , (7.10) with κ defined in (G.4). The crossing matrix V is an antidiagonal matrix given by V = δ2 d∑ α=1 εαe (ᾱ−ᾱ′)ηeαα′ , V 2 = I , (7.11) where δ2 is given by δ2 = { 1 for A (2) 2n , B (1) n , D (1) n i for A (2) 2n−1 , C (1) n , and the other notations are defined in (G.5)-(G.7). The corresponding matrix M is defined by M = V t V , (7.12) and it is given by the diagonal matrix M = δ2 2 d∑ α=1 e4( d+1 2 −ᾱ)ηeαα . (7.13) 7.1.2 K-matrices The boundary interactions are encoded in the right and left K-matrices, denoted here by KR(u) and KL(u), respectively, which map V to itself.3 We choose KR(u) to be the diagonal d× d matrix KR(u) = KR(u, p) = diag ( e−u , . . . , e−u︸ ︷︷ ︸ p , γeu + 1 γ + eu , . . . , γeu + 1 γ + eu︸ ︷︷ ︸ d−2p , eu , . . . , eu︸ ︷︷ ︸ p ) , (7.14) where p = 0, 1, . . . , n, and γ =  γ0e (4p−2)η+ 1 2ρ for A (2) 2n−1 , B (1) n , D (1) n γ0e (4p+2)η+ 1 2ρ for A (2) 2n , C (1) n , γ0 = ±1 , (7.15) 2Bazhanov’s R-matrices [57] are equivalent, but are instead in the principal picture. 3Following Sklyanin [52], the right and left K-matrices are usually denoted instead by K−(u) and K+(u), respectively. However, we adopt a different notation here in order to avoid confusion with the ± used in subsequent sections to denote the limits u→ ±∞. 7.2. QUANTUM GROUP SYMMETRY 49 where ρ is the crossing parameter (7.10). Unless otherwise noted, all the results in this chapter hold for both values (±1) of the parameter γ0. As observed in [62] (see also [59, 60, 61]), the matrices (7.14) are solutions of the boundary Yang-Baxter equation (BYBE) on V ⊗ V [52, 92, 93] R12(u− v)KR 1 (u) R21(u+ v)KR 2 (v) = KR 2 (v)R12(u+ v)KR 1 (u)R21(u− v) . (7.16) For p = 0, we see that KR(u, p) in (7.14) is proportional to the identity matrix, KR(u, 0) ∝ I , (7.17) which is the solution noted in [65]. We emphasize that the solution (7.14) depends on the bulk anisotropy parameter η and the discrete boundary parameters p and γ0, but does not have any continuous boundary parameters. For the left K-matrix, we take KL(u) = KL(u, p) = KR(−u− ρ, p)M , (7.18) where M is given by (7.12), which is a solution of the corresponding BYBE [52, 53] R12(−u+ v)KL t1 1 (u)M−1 1 R21(−u− v − 2ρ)M1K L t2 2 (v) = KL t2 2 (v)M1R12(−u− v − 2ρ)M−1 1 KL t1 1 (u)R21(−u+ v) . (7.19) 7.1.3 Transfer matrix The open-chain transfer matrix, which maps the quantum space V⊗N