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## Secret Symmetries in AdS/CFT Proceedings to the Nordita program 'Exact Results in Gauge-String Dualities'

### Citations

5604 | The large N limit of superconformal field theories and supergravity
- Maldacena
- 1998
(Show Context)
Citation Context .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Secret symmetry 6 3.1 Secret symmetry of the R-matrix . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Secret symmetries of the K-matrices . . . . . . . . . . . . . . . . . . . . . 8 3.3 Secret symmetry in Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Secret symmetry in the pure spinor formalism . . . . . . . . . . . . . . . 11 3.5 Quantum deformed secret symmetry . . . . . . . . . . . . . . . . . . . . 12 1 Introduction In recent years there has been a remarkable progress towards a proof of the AdS/CFT conjecture [1]. The problem of calculating anomalous dimensions in N = 4 SYM can be translated into the spectral problem associated to a certain integrable Hamiltonian [2]. There are by now extensive reviews on the subject, see for instance [3]. From the algebraic viewpoint, it has been possible to rephrase the problem in the language of Hopf algebras and their representation theory. The Hopf superalgebra relevant to AdS/CFT is quite unconventional, and, as of today, its properties are only partially understood. It is infinite dimensional, with a structure similar to Yangians [4, 5, 6, 7, 8]. It admits a le... |

1100 |
Quantum groups
- Drinfeld
- 1986
(Show Context)
Citation Context ...f of the AdS/CFT conjecture [1]. The problem of calculating anomalous dimensions in N = 4 SYM can be translated into the spectral problem associated to a certain integrable Hamiltonian [2]. There are by now extensive reviews on the subject, see for instance [3]. From the algebraic viewpoint, it has been possible to rephrase the problem in the language of Hopf algebras and their representation theory. The Hopf superalgebra relevant to AdS/CFT is quite unconventional, and, as of today, its properties are only partially understood. It is infinite dimensional, with a structure similar to Yangians [4, 5, 6, 7, 8]. It admits a level zero given by the centrallyextended psl(2|2) Lie superalgebra, and level one generators giving rise to an infinite dimensional tower1. Nevertheless, the actual algebra sits rather outside the standard theory of Yangians, in that it displays an additional symmetry at level one, which is absent at level zero. Were this symmetry present at level zero, it would extend the Yangian to that of gl(2|2). However, this is not compatible with the central extension. Moreover, if one starts commuting the new generator with the old ones, one obtains a growth in the algebra which is not c... |

1053 |
A Guide to Quantum Groups,
- Chari, ley
- 1994
(Show Context)
Citation Context ...f of the AdS/CFT conjecture [1]. The problem of calculating anomalous dimensions in N = 4 SYM can be translated into the spectral problem associated to a certain integrable Hamiltonian [2]. There are by now extensive reviews on the subject, see for instance [3]. From the algebraic viewpoint, it has been possible to rephrase the problem in the language of Hopf algebras and their representation theory. The Hopf superalgebra relevant to AdS/CFT is quite unconventional, and, as of today, its properties are only partially understood. It is infinite dimensional, with a structure similar to Yangians [4, 5, 6, 7, 8]. It admits a level zero given by the centrallyextended psl(2|2) Lie superalgebra, and level one generators giving rise to an infinite dimensional tower1. Nevertheless, the actual algebra sits rather outside the standard theory of Yangians, in that it displays an additional symmetry at level one, which is absent at level zero. Were this symmetry present at level zero, it would extend the Yangian to that of gl(2|2). However, this is not compatible with the central extension. Moreover, if one starts commuting the new generator with the old ones, one obtains a growth in the algebra which is not c... |

309 |
Super-Poincaré covariant quantization of the superstring
- Berkovits
(Show Context)
Citation Context ...t symmetry in the pure spinor formalism The psl(2|2)c symmetry of the spin chain appears after choosing one of the complex scalars as a vacuum, while the original super Yang-Mills theory has manifest psl(2, 2|4) symmetry. It is natural to ask whether the original super Yang-Mills theory itself also possesses a secret symmetry, embedded in some fashion in the Yangian Y (gl(2, 2|4)). The answer seems to be affirmative [16]. Compared with the anomalous dimensions or the n-point amplitudes, which are derived quantities, the statement is more direct in the pure spinor formalism. It was proposed in [70] that string theory on AdS5 × S5, or its dual, the super Yang-Mills theory, can be formulated in the pure spinor formalism. In [16], interestingly, it was shown that one can find a secret symmetry (only at odd levels) in this formalism. The fact that the super Yang-Mills theory has a symmetry bigger than the Yangian Y (psl(2, 2|4)) but smaller than Y (gl(2, 2|4)) may look bewildering. However, an interesting interpretation was suggested by [16]. Free Yang-Mills theory, which corresponds to the sigma model at zero AdS radius, preserves the whole Y (gl(2, 2|4)), and is conjectured to be dual to ... |

221 | The su (2|2) dynamic S-matrix
- Beisert
- 2008
(Show Context)
Citation Context ...with three-fold central extension. We will call this algebra psl(2|2)c. Such a large central extension is unique among the basic classical simple Lie superalgebras [18]. The even part of psl(2|2)c consists of sl(2)⊕ sl(2) and of the space generated by the central elements, which we will denote4 as H, C and C†. Latin indices refer to the first sl(2), generated by L ba subject to ∑2 a=1 L aa = 0, greek indices to the second sl(2), generated by R βα subject to ∑4 α=3 R αα = 0. The fermionic generators will be denoted by Q aα and G αa . Besides standard sl(2)⊕ sl(2) commutation relations, one has [19]: [L ba , Jc] = δbc Ja − 12δ b a Jc, [R βα , Jγ] = δβγ Jα − 12δ β α Jγ, [L ba , Jc] = −δca Jb + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβabC, {G αa ,G β b } = αβabC†, {Q aα ,G β b } = δab R βα + δβα L ab + 12δ a b δ β αH. where J denotes any odd generator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5.... |

175 |
Giant magnons
- Hofman, Maldacena
(Show Context)
Citation Context ...ical cobrackets8. This is somehow echoed in [16], as we will see later on. Other approaches do not seem to detect such obstacles in principle, which is a sign of how non-trivial a task the complete quantum formulation of this symmetry still remains. It is worth noticing that in the opposite (gauge-theory) regime of small ‘t Hooft coupling, the one-loop R-matrix is a twisted version of the gl(2|2) Yangian R-matrix in the fundamental representation. The presence of the secret symmetry at level zero is in this case a one-loop accident. 3.2 Secret symmetries of the K-matrices In a series of works [59, 60, 61, 62, 63, 64] reflection K-matrices were found for open strings ending on D3, D5 and D7-branes. It was also observed that the secret symmetry manifests itself in some of these reflection matrices [14]. The Y = 0 maximal giant graviton is a D3-brane wrapping a maximal S3 ⊂ S5 of the AdS5 × S5 background and preserves an sl(2|1)L = {L βα , R 11 , R 22 , Q 1α , G α1 , H} subalgebra of the bulk algebra psl(2|2)c. The fundamental reflection matrix, describing the scattering of fundamental magnons from the boundary, is diagonal and the helicity generator B is a symmetry of it, but there is no secret symmetry at ... |

155 | Magnon bound states and the AdS/CFT correspondence,”J. Phys. A39
- Dorey
- 2006
(Show Context)
Citation Context ...I produces the simple Lie superalgebra psl(2|2). Its representations can be understood as that of sl(2|2) at q = 0 [38]. Irreps of gl(2|2) are divided into typical (long), with generic values of j1, j2, q and dimension 16(2j1 + 1)(2j2 + 1), and atypical (short), for which special relations are satisfied by the labels, namely ±q = j1 − j2 and ±q = j1 + j2 + 1. When these relations are satisfied, the dimension of the representation is smaller than 16(2j1 + 1)(2j2 + 1). The 4-dimensional fundamental representation [19] corresponds to j1 = 1 2 , j2 = 0 and q = 1 2 . The symmetric bound-state reps [39, 40, 41, 42, 24, 43] are given by j2 = 0, q = j1, with j1 = 1 2 , 1, .... The antisymmetric bound-state reps are given by j1 = 0, q = 1 + j2, with j2 = 0, 1 2 , .... Symmetric and antisymmetric bound-state reps have dimension 4M , with M = 2j1 for symmetric, M = 2(j2 + 1) for antisymmetric. An sl(2) rotation back of such reps provides an explicit matrix representation of sl(2|2)c: L ba = Eab , ∀ a 6= b , R βα = Eαβ , ∀ α 6= β , Q aα = aEαa + b αβabEbβ , G αa = c abαβEβb + dEaα , (2.11) subject to the constraint ad− bc = 1. (2.12) Diagonal generators are automatically obtained by commuting positive and negativ... |

142 | The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2
- Beisert
(Show Context)
Citation Context ... Jα − 12δ β α Jγ, [L ba , Jc] = −δca Jb + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβabC, {G αa ,G β b } = αβabC†, {Q aα ,G β b } = δab R βα + δβα L ab + 12δ a b δ β αH. where J denotes any odd generator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are the supercharges Q and G. 5This feature is shared by D(2, 1;α), psl(n|n) and osp(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in t... |

133 |
Solutions of the classical Yang–Baxter equation for simple Lie algebras.” Functional Analysis and Its Applications 16
- Belavin, Drinfeld
- 1981
(Show Context)
Citation Context ...rator for the Yangian algebra. The relations (3.7) are also compatible with the coalgebra structure. Shadows of this situation are observed at the level of the classical r-matrix, where the secret generator is needed in order to achieve factorization in terms of a Drinfeld double. In fact, all short representations admit a notion of classical limit in a small parameter ~ [28, 48, 49, 44]. This limit involves a scaling of the eigenvalues of the central elements, since they depend on ~. The R-matrix can be Taylor-expanded, the first order r being a solution of the classical Yang-Baxter equation [50, 51, 52, 53, 54, 55, 56, 57]. The element r displays a single pole at the origin in some appropriate classical spectral variables, with residue the quadratic Casimir of gl(2|2)⊗gl(2|2). There exists an infinitedimensional Lie bialgebra [45], formulated purely in abstract terms, which admits R as coboundary structure in these short representations. Its nature is quite unconventional, and its quantization is a fascinating open problem. This Lie bialgebra accomodates a class of generators of the type B, which appear naturally in the classical limit [58]. The indentation described above is interpreted in [45] as a different... |

127 | Yangian symmetry of scattering amplitudes
- Drummond, Henn, et al.
(Show Context)
Citation Context ...d the presence of a mechanism completely analog to the one we have been describing for the spectral problem. This time, it involves tree-level planar n-particle color-ordered amplitudes An. One describes such amplitudes as functions of spinor-helicity variables (λk, λk, ηk), k = 1, . . . , n, with λk, λk ∈ C2 complex conjugate spinors, and ηk ∈ C0|4 a Grassmann variable encoding flavour. The lightlike momentum of the particle k is given by pk = λkλk. The psu(2, 2|4) superconformal symmetry generators JA act on particles as differential operators JAk , and they annihilate the amplitudes. In [68], an additional set of Yangian generators JA annihilating the amplitudes was found, such that JA = n∑ i=1 JAi , J A = fABC n∑ j<k=1 JBj J C k . (3.9) fABC are the psu(2, 2|4) structure constants. Because of the cyclicity of color-ordered amplitudes, the symmetry generators have to satisfy specific constraints in order to be well-defined. One has for instance [68] JA(2,n+1) − JA(1,n) = fABCJB1 JC + fABCfBCD JD1 = 0 . (3.10) 9 Indeed, the first term vanishes because JA annihilates the amplitudes, whereas the second term vanishes because the dual Coxeter number of psu(2, 2|4) is zero. Curious... |

121 | Yangians and classical Lie algebras
- Molev, Nazarov, et al.
- 1996
(Show Context)
Citation Context ...f of the AdS/CFT conjecture [1]. The problem of calculating anomalous dimensions in N = 4 SYM can be translated into the spectral problem associated to a certain integrable Hamiltonian [2]. There are by now extensive reviews on the subject, see for instance [3]. From the algebraic viewpoint, it has been possible to rephrase the problem in the language of Hopf algebras and their representation theory. The Hopf superalgebra relevant to AdS/CFT is quite unconventional, and, as of today, its properties are only partially understood. It is infinite dimensional, with a structure similar to Yangians [4, 5, 6, 7, 8]. It admits a level zero given by the centrallyextended psl(2|2) Lie superalgebra, and level one generators giving rise to an infinite dimensional tower1. Nevertheless, the actual algebra sits rather outside the standard theory of Yangians, in that it displays an additional symmetry at level one, which is absent at level zero. Were this symmetry present at level zero, it would extend the Yangian to that of gl(2|2). However, this is not compatible with the central extension. Moreover, if one starts commuting the new generator with the old ones, one obtains a growth in the algebra which is not c... |

113 |
A new realization of Yangians and quantum affine algebras
- Drinfeld
- 1988
(Show Context)
Citation Context ...With these conditions, ∆(C) = C⊗ 1 + 1⊗ C + C⊗ C = ∆op(C) , (2.5) and similarly for ∆(C†). The conditions (2.4) imply the equivalence relation CC† + C + C† = 0 which we will always assume. This already represents a departure from the standard theory of Hopf algebras. 2.2 The algebra: level one The symmetry algebra of the R-matrix contains another set of generators, partners to those described in the previous section. Together with the level zero, these new charges generate an infinite-dimensional Hopf algebra (which we will call Y ) similar to a Yangian [30]. Its Drinfeld’s second realization [31] is given in terms of Cartan generators κi,m and fermionic ladder generators ξ±i,m, i = 1, 2, 3, m = 0, 1, 2, . . . , subject to the following relations [32]: [κi,m, κj,n] = 0, [κi,0, ξ + j,m] = aij ξ + j,m, [κi,0, ξ − j,m] = −aij ξ−j,m, {ξ+i,m, ξ−j,n} = δi,j κj,n+m, [κi,m+1, ξ ± j,n]− [κi,m, ξ±j,n+1] = ± 1 2 aij{κi,m, ξ±j,n}, {ξ±i,m+1, ξ±j,n} − {ξ±i,m, ξ±j,n+1} = ± 1 2 aij[ξ ± i,m, ξ ± j,n], (2.6) 3 i 6= j, nij = 1 + |aij|, Sym{k}[ξ±i,k1 , [ξ ± i,k2 , . . . {ξ±i,knij , ξ ± j,l} . . . }} = 0 , except for {ξ+2,n, ξ+3,m} = Cn+m, {ξ−2,n, ξ−3,m} = C † n+m. (2.7) The non zero entries of the (degene... |

59 |
Triangle equation for simple Lie algebras
- Belavin, Drinfeld
- 1984
(Show Context)
Citation Context ...rator for the Yangian algebra. The relations (3.7) are also compatible with the coalgebra structure. Shadows of this situation are observed at the level of the classical r-matrix, where the secret generator is needed in order to achieve factorization in terms of a Drinfeld double. In fact, all short representations admit a notion of classical limit in a small parameter ~ [28, 48, 49, 44]. This limit involves a scaling of the eigenvalues of the central elements, since they depend on ~. The R-matrix can be Taylor-expanded, the first order r being a solution of the classical Yang-Baxter equation [50, 51, 52, 53, 54, 55, 56, 57]. The element r displays a single pole at the origin in some appropriate classical spectral variables, with residue the quadratic Casimir of gl(2|2)⊗gl(2|2). There exists an infinitedimensional Lie bialgebra [45], formulated purely in abstract terms, which admits R as coboundary structure in these short representations. Its nature is quite unconventional, and its quantization is a fascinating open problem. This Lie bialgebra accomodates a class of generators of the type B, which appear naturally in the classical limit [58]. The indentation described above is interpreted in [45] as a different... |

50 | Introduction to Yangian symmetry in integrable field theory
- MacKay
(Show Context)
Citation Context |

49 | Magnon Bound-state Scattering in Gauge and String Theory
- Roiban
- 2007
(Show Context)
Citation Context ...I produces the simple Lie superalgebra psl(2|2). Its representations can be understood as that of sl(2|2) at q = 0 [38]. Irreps of gl(2|2) are divided into typical (long), with generic values of j1, j2, q and dimension 16(2j1 + 1)(2j2 + 1), and atypical (short), for which special relations are satisfied by the labels, namely ±q = j1 − j2 and ±q = j1 + j2 + 1. When these relations are satisfied, the dimension of the representation is smaller than 16(2j1 + 1)(2j2 + 1). The 4-dimensional fundamental representation [19] corresponds to j1 = 1 2 , j2 = 0 and q = 1 2 . The symmetric bound-state reps [39, 40, 41, 42, 24, 43] are given by j2 = 0, q = j1, with j1 = 1 2 , 1, .... The antisymmetric bound-state reps are given by j1 = 0, q = 1 + j2, with j2 = 0, 1 2 , .... Symmetric and antisymmetric bound-state reps have dimension 4M , with M = 2j1 for symmetric, M = 2(j2 + 1) for antisymmetric. An sl(2) rotation back of such reps provides an explicit matrix representation of sl(2|2)c: L ba = Eab , ∀ a 6= b , R βα = Eαβ , ∀ α 6= β , Q aα = aEαa + b αβabEbβ , G αa = c abαβEβb + dEaα , (2.11) subject to the constraint ad− bc = 1. (2.12) Diagonal generators are automatically obtained by commuting positive and negativ... |

40 |
On the Scattering of Magnon Boundstates
- Chen, Dorey, et al.
- 611
(Show Context)
Citation Context ...I produces the simple Lie superalgebra psl(2|2). Its representations can be understood as that of sl(2|2) at q = 0 [38]. Irreps of gl(2|2) are divided into typical (long), with generic values of j1, j2, q and dimension 16(2j1 + 1)(2j2 + 1), and atypical (short), for which special relations are satisfied by the labels, namely ±q = j1 − j2 and ±q = j1 + j2 + 1. When these relations are satisfied, the dimension of the representation is smaller than 16(2j1 + 1)(2j2 + 1). The 4-dimensional fundamental representation [19] corresponds to j1 = 1 2 , j2 = 0 and q = 1 2 . The symmetric bound-state reps [39, 40, 41, 42, 24, 43] are given by j2 = 0, q = j1, with j1 = 1 2 , 1, .... The antisymmetric bound-state reps are given by j1 = 0, q = 1 + j2, with j2 = 0, 1 2 , .... Symmetric and antisymmetric bound-state reps have dimension 4M , with M = 2j1 for symmetric, M = 2(j2 + 1) for antisymmetric. An sl(2) rotation back of such reps provides an explicit matrix representation of sl(2|2)c: L ba = Eab , ∀ a 6= b , R βα = Eαβ , ∀ α 6= β , Q aα = aEαa + b αβabEbβ , G αa = c abαβEβb + dEaα , (2.11) subject to the constraint ad− bc = 1. (2.12) Diagonal generators are automatically obtained by commuting positive and negativ... |

39 | Tseytlin, “Pohlmeyer reduction of AdS5×S5 superstring sigma model,” Nucl. Phys. B800 - Grigoriev, A - 2008 |

37 |
Central extension of Lie superalgebra
- Iohara, Koga
- 2001
(Show Context)
Citation Context ...n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we will humbly attempt at reproducing in its salient features. 2 The algebra 2.1 The algebra: level zero We will start by discussing the Hopf algebra based on the Lie superalgebra A(1, 1) = psl(2|2) with three-fold central extension. We will call this algebra psl(2|2)c. Such a large central extension is unique among the basic classical simple Lie superalgebras [18]. The even part of psl(2|2)c consists of sl(2)⊕ sl(2) and of the space generated by the central elements, which we will denote4 as H, C and C†. Latin indices refer to the first sl(2), generated by L ba subject to ∑2 a=1 L aa = 0, greek indices to the second sl(2), generated by R βα subject to ∑4 α=3 R αα = 0. The fermionic generators will be denoted by Q aα and G αa . Besides standard sl(2)⊕ sl(2) commutation relations, one has [19]: [L ba , Jc] = δbc Ja − 12δ b a Jc, [R βα , Jγ] = δβγ Jα − 12δ β α Jγ, [L ba , Jc] = −δca Jb + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβ... |

29 |
Automorphisms of simple Lie superalgebras
- Serganova
- 1984
(Show Context)
Citation Context ...b + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβabC, {G αa ,G β b } = αβabC†, {Q aα ,G β b } = δab R βα + δβα L ab + 12δ a b δ β αH. where J denotes any odd generator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are the supercharges Q and G. 5This feature is shared by D(2, 1;α), psl(n|n) and osp(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in the associated string sigma models. 2 J... |

26 |
The magnon kinematics of the AdS/CFT correspondence,
- Gomez, Hernandez
- 2006
(Show Context)
Citation Context ...enerator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are the supercharges Q and G. 5This feature is shared by D(2, 1;α), psl(n|n) and osp(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in the associated string sigma models. 2 JA ∈ psl(2|2)c ∆(JA) = JA ⊗ 1 + ei[[A]]p ⊗ JA, ∆(eip) = eip ⊗ eip, (2.1) where p is central. The additive quantum number [[A]] equals 0 for generators in sl(2)⊕ sl(2) and for H, 1 2 for Q a... |

25 |
Perturbative study of the transfer matrix on the string worldsheet in
- Mikhailov, Schafer-Nameki
- 2007
(Show Context)
Citation Context ...AdS radius, preserves the whole Y (gl(2, 2|4)), and is conjectured to be dual to a topological string. After we turn on the vertex operator that changes the radius, we break this secret gl(1) symmetry spontaneously, with some of its Yangian cousins remaining. The resulting indentation is very reminiscent of the one we have been previously discussing. In the pure-spinor sigma model, one works with a group variable g ∈ PSU(2, 2|4), and the action is given in terms of the right-invariant current J = −dg g−1 . (3.15) Integrability is guaranteed by the existence of a Lax connection J±(z) such that [71] [∂+ + J+(z) , ∂− + J−(z)] = 0 . (3.16) Non-local conserved charges are generated by the transfer matrix T (z) = g(+∞)−1 ( P exp ∫ C ( −J+(z)dτ+ − J−(z)dτ− )) g(−∞) (3.17) upon suitable expansion in the spectral parameter, for some contour C. The observation of [16] is that, although T (z) takes values in PSU(2, 2|4), one can lift it to SU(2, 2|4) by lifting the group element g. At this point, one singles out the central component of the transfer matrix by tracing with the hypercharge (which is indeed conjugated to the identity w.r.t. the Killing form of u(2, 2|4)). Namely, one takes Str(s log... |

24 | Towards the quantum S-matrix of the Pohlmeyer reduced version of - Hoare, Tseytlin - 2011 |

19 | Central Extension of the Yangian Double
- Khoroshkin
- 1995
(Show Context)
Citation Context |

17 |
Drinfeld second realization of the quantum affine superalgebras
- Heckenberger, Spill, et al.
- 2008
(Show Context)
Citation Context ...c generators will be denoted by Q aα and G αa . Besides standard sl(2)⊕ sl(2) commutation relations, one has [19]: [L ba , Jc] = δbc Ja − 12δ b a Jc, [R βα , Jγ] = δβγ Jα − 12δ β α Jγ, [L ba , Jc] = −δca Jb + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβabC, {G αa ,G β b } = αβabC†, {Q aα ,G β b } = δab R βα + δβα L ab + 12δ a b δ β αH. where J denotes any odd generator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are... |

17 |
A Quantum Affine Algebra for the Deformed Hubbard Chain
- Beisert, Galleas, et al.
(Show Context)
Citation Context ...+ 12εac ε αγ e−ipQ cγ ⊗ C† − 12εac e i 2 p εαγ C† ⊗Q cγ . (3.5) The new supercharges display different spectral parameters in the fermion-boson block vs. the boson-fermion block, and when combined together produce the usual Yangian generators, Q aα = Q aα,+1 + Q aα,−1 , G αa = G αa,+1 + G αa,−1 . (3.6) The growth of the algebra depends on how many new independent charges are generated by subsequent commutation, and in the present case it is still unclear how to control this growth. However, by allowing non-linearity, one can recast some of the commutation relations as follows [46] (see also [47, 45]): Q aα = +[B,Q aα ] + i(1 + e−ip) abαβ G β b , G αa = −[B,G αa ]− i(1 + eip) αβabQ bβ . (3.7) This means that the secret symmetry works as a level raising operator for the Yangian algebra. The relations (3.7) are also compatible with the coalgebra structure. Shadows of this situation are observed at the level of the classical r-matrix, where the secret generator is needed in order to achieve factorization in terms of a Drinfeld double. In fact, all short representations admit a notion of classical limit in a small parameter ~ [28, 48, 49, 44]. This limit involves a scaling of the eige... |

17 | Tseytlin, “Tree-level S-matrix of Pohlmeyer reduced form - Hoare, A - 2010 |

16 |
Strings on Semisymmetric Superspaces”,
- Zarembo
- 2010
(Show Context)
Citation Context ...ly5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are the supercharges Q and G. 5This feature is shared by D(2, 1;α), psl(n|n) and osp(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in the associated string sigma models. 2 JA ∈ psl(2|2)c ∆(JA) = JA ⊗ 1 + ei[[A]]p ⊗ JA, ∆(eip) = eip ⊗ eip, (2.1) where p is central. The additive quantum number [[A]] equals 0 for generators in sl(2)⊕ sl(2) and for H, 1 2 for Q aα , −12 for G α a , 1 for C and −1 for C†. The above coproduct can be easily shown to be a Lie algebra homomorphism. The corresponding counit and antipode are straightforwardly derived from the Hopf algebra axioms [27]. The R-matrix R of [19] renders this Hopf algebra quasi-cocommutative (see also [28, 29]). One ha... |

16 |
On Drinfeld’s second realization of the AdS/CFT su(2|2
- Spill, Torrielli
(Show Context)
Citation Context ... 0 which we will always assume. This already represents a departure from the standard theory of Hopf algebras. 2.2 The algebra: level one The symmetry algebra of the R-matrix contains another set of generators, partners to those described in the previous section. Together with the level zero, these new charges generate an infinite-dimensional Hopf algebra (which we will call Y ) similar to a Yangian [30]. Its Drinfeld’s second realization [31] is given in terms of Cartan generators κi,m and fermionic ladder generators ξ±i,m, i = 1, 2, 3, m = 0, 1, 2, . . . , subject to the following relations [32]: [κi,m, κj,n] = 0, [κi,0, ξ + j,m] = aij ξ + j,m, [κi,0, ξ − j,m] = −aij ξ−j,m, {ξ+i,m, ξ−j,n} = δi,j κj,n+m, [κi,m+1, ξ ± j,n]− [κi,m, ξ±j,n+1] = ± 1 2 aij{κi,m, ξ±j,n}, {ξ±i,m+1, ξ±j,n} − {ξ±i,m, ξ±j,n+1} = ± 1 2 aij[ξ ± i,m, ξ ± j,n], (2.6) 3 i 6= j, nij = 1 + |aij|, Sym{k}[ξ±i,k1 , [ξ ± i,k2 , . . . {ξ±i,knij , ξ ± j,l} . . . }} = 0 , except for {ξ+2,n, ξ+3,m} = Cn+m, {ξ−2,n, ξ−3,m} = C † n+m. (2.7) The non zero entries of the (degenerate) symmetric Cartan matrix aij are a12 = a21 = 1 and a13 = a31 = −1. The second subscript n in ξ±i,n, κi,n denotes the level, with n = 0 providing a subal... |

16 |
The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure,
- Beisert, Spill
- 2009
(Show Context)
Citation Context ...se representations. Evaluation here means that the level n generators are obtained by multiplying the corresponding level zero matrices by certain polynomials of degree n in a spectral parameter u [32]. The hatted constraints fix u to be a function of the eigenvalues of the level zero central charges (basically, a function of the momentum p) [30]. 3 Secret symmetry 3.1 Secret symmetry of the R-matrix After computing the element R for all short representations and verifying that it satisfies the Yang-Baxter equation [19, 43, 44], one notices that it automatically solves the additional equation [13, 45, 46] ∆op(B)R = R∆(B), (3.1) with ∆(B) = B⊗ 1 + 1⊗ B + ( e i 2 pG αa ⊗Q aα + e− i 2 pQ aα ⊗ G αa ) , B = B0 diag(1, 1, ..., 1,−1,−1...,−1). (3.2) The “1”s run over the bosonic subspace of the representation module, the “−1”s over the fermionic subspace, and B0 is a certain function of p. One notices that B is not supertraceless, unlike all the generators of Y . One is tempted to postulate an extension of the symmetry algebra to the Yangian of gl(2|2). However, this is not possible. The coproduct which one would normally attribute to the level zero partner of B, namely ∆(B) = B⊗ 1 + 1⊗ B, (3.... |

15 | Tensor products of psl(2|2) representations
- Götz, Quella, et al.
(Show Context)
Citation Context ... Eii|ω〉 = 2q|ω〉, N |ω〉 = 4∑ i=1 (−)[i]Eii|ω〉 = 2y|ω〉, and Eij|ω〉 = 0, ∀ i < j . (2.10) The generator N (to be identified with the level zero generators B or B of the following sections) never appears on the right hand side of the commutation relations, and is defined up to the addition of a central element βI, with β a constant (which we will drop). We can consistently mod N out, and obtain sl(2|2), the algebra of supertraceless 2|2×2|2 matrices. Further modding out of the center I produces the simple Lie superalgebra psl(2|2). Its representations can be understood as that of sl(2|2) at q = 0 [38]. Irreps of gl(2|2) are divided into typical (long), with generic values of j1, j2, q and dimension 16(2j1 + 1)(2j2 + 1), and atypical (short), for which special relations are satisfied by the labels, namely ±q = j1 − j2 and ±q = j1 + j2 + 1. When these relations are satisfied, the dimension of the representation is smaller than 16(2j1 + 1)(2j2 + 1). The 4-dimensional fundamental representation [19] corresponds to j1 = 1 2 , j2 = 0 and q = 1 2 . The symmetric bound-state reps [39, 40, 41, 42, 24, 43] are given by j2 = 0, q = j1, with j1 = 1 2 , 1, .... The antisymmetric bound-state reps are gi... |

14 |
Yangian symmetry of the Y=0 maximal giant graviton, JHEP 1012
- MacKay, Regelskis
- 2010
(Show Context)
Citation Context ...but there is no secret symmetry at level one. Higher order reflection matrices are of non-diagonal form and respect neither B nor B. However, the secret symmetry was found to emerge through additional twisted secret charges, namely Q 2α,+1 = Q 2α,+1 + 12Q 2 α R 22 − 12R 2 1 Q 1α + 12Q 2 γ L γα + 14Q 2 α H , Q 2α,−1 = Q 2α,−1 − 12εαγ CG γ 1 , G α2,+1 = G α2,+1 − 12G α 2 R 22 + 12R 1 2 G α1 − 12G γ 2 L αγ − 14G α 2 H , G α2,−1 = G α2,−1 + 12ε αγ C†Q 1γ , (3.8) corresponding to (3.5) and constructed using the twisted Yangian algebra [14]. The mirror model of the Y = 0 maximal giant graviton [65, 66] preserves the subalgebra sl(2|1)R = {R ba , L 33 , L 44 , Q a3 , G 3a , H}. The reflection matrices are diagonal for all bound-state numbers; thus B is a symmetry for all bound-states. This configuration also possesses additional twisted secret charges, but no level one secret symmetry B itself. The D5-brane wraps an AdS4 ⊂ AdS5 and a maximal S2 ⊂ S5 of the AdS5 × S5. The AdS4 part of the brane defines a 2 + 1 dimensional defect hypersurface of the 3 + 1 dimensional conformal boundary. This brane preserves a diagonal psl(2|2)c subalgebra of the complete bulk algebra psl(2|2)L × psl(2|2)R n R... |

11 | Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills
- Beisert, Schwab
(Show Context)
Citation Context ...n the light of these new observations, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we will humbly attempt at reproducing in its salient features. 2 The algebra 2.1 The algebra: level zero We will start by discussing the Hopf algebra based on the Lie superalgebra A(1, 1) = psl(2|2) with three-fold central extension. We will call this algebra psl(2|2)c. Such a large central extension is unique among the basic classical simple Lie superalgebras [18]. The even part of ... |

11 |
Solutions of the Classical Yang-Baxter Equation for Simple Superalgebras, Theor
- Leites, Serganova
- 1984
(Show Context)
Citation Context ...rator for the Yangian algebra. The relations (3.7) are also compatible with the coalgebra structure. Shadows of this situation are observed at the level of the classical r-matrix, where the secret generator is needed in order to achieve factorization in terms of a Drinfeld double. In fact, all short representations admit a notion of classical limit in a small parameter ~ [28, 48, 49, 44]. This limit involves a scaling of the eigenvalues of the central elements, since they depend on ~. The R-matrix can be Taylor-expanded, the first order r being a solution of the classical Yang-Baxter equation [50, 51, 52, 53, 54, 55, 56, 57]. The element r displays a single pole at the origin in some appropriate classical spectral variables, with residue the quadratic Casimir of gl(2|2)⊗gl(2|2). There exists an infinitedimensional Lie bialgebra [45], formulated purely in abstract terms, which admits R as coboundary structure in these short representations. Its nature is quite unconventional, and its quantization is a fascinating open problem. This Lie bialgebra accomodates a class of generators of the type B, which appear naturally in the classical limit [58]. The indentation described above is interpreted in [45] as a different... |

11 |
Integrable achiral D5-brane reflections and asymptotic Bethe equations
- Correa, Regelskis, et al.
(Show Context)
Citation Context ...ical cobrackets8. This is somehow echoed in [16], as we will see later on. Other approaches do not seem to detect such obstacles in principle, which is a sign of how non-trivial a task the complete quantum formulation of this symmetry still remains. It is worth noticing that in the opposite (gauge-theory) regime of small ‘t Hooft coupling, the one-loop R-matrix is a twisted version of the gl(2|2) Yangian R-matrix in the fundamental representation. The presence of the secret symmetry at level zero is in this case a one-loop accident. 3.2 Secret symmetries of the K-matrices In a series of works [59, 60, 61, 62, 63, 64] reflection K-matrices were found for open strings ending on D3, D5 and D7-branes. It was also observed that the secret symmetry manifests itself in some of these reflection matrices [14]. The Y = 0 maximal giant graviton is a D3-brane wrapping a maximal S3 ⊂ S5 of the AdS5 × S5 background and preserves an sl(2|1)L = {L βα , R 11 , R 22 , Q 1α , G α1 , H} subalgebra of the bulk algebra psl(2|2)c. The fundamental reflection matrix, describing the scattering of fundamental magnons from the boundary, is diagonal and the helicity generator B is a symmetry of it, but there is no secret symmetry at ... |

11 |
Yangian symmetry of boundary scattering in AdS/CFT and the explicit form of bound state reflection matrices, JHEP 1103
- Palla
- 2011
(Show Context)
Citation Context ...but there is no secret symmetry at level one. Higher order reflection matrices are of non-diagonal form and respect neither B nor B. However, the secret symmetry was found to emerge through additional twisted secret charges, namely Q 2α,+1 = Q 2α,+1 + 12Q 2 α R 22 − 12R 2 1 Q 1α + 12Q 2 γ L γα + 14Q 2 α H , Q 2α,−1 = Q 2α,−1 − 12εαγ CG γ 1 , G α2,+1 = G α2,+1 − 12G α 2 R 22 + 12R 1 2 G α1 − 12G γ 2 L αγ − 14G α 2 H , G α2,−1 = G α2,−1 + 12ε αγ C†Q 1γ , (3.8) corresponding to (3.5) and constructed using the twisted Yangian algebra [14]. The mirror model of the Y = 0 maximal giant graviton [65, 66] preserves the subalgebra sl(2|1)R = {R ba , L 33 , L 44 , Q a3 , G 3a , H}. The reflection matrices are diagonal for all bound-state numbers; thus B is a symmetry for all bound-states. This configuration also possesses additional twisted secret charges, but no level one secret symmetry B itself. The D5-brane wraps an AdS4 ⊂ AdS5 and a maximal S2 ⊂ S5 of the AdS5 × S5. The AdS4 part of the brane defines a 2 + 1 dimensional defect hypersurface of the 3 + 1 dimensional conformal boundary. This brane preserves a diagonal psl(2|2)c subalgebra of the complete bulk algebra psl(2|2)L × psl(2|2)R n R... |

11 |
The grassmannian origin Of dual superconformal invariance,”
- Arkani-Hamed, Cachazo, et al.
- 2010
(Show Context)
Citation Context ...mplitudes being superconformal symmetric. The other term is now proportional to the dual Coxeter number of u(2, 2|4), which is non-zero. However, this surviving contribution is multiplied by the central charge C1, which vanishes for all individual particles. Not too dissimilarly from the case of psl(2|2)c, the bonus symmetry is an automorphism of u(2, 2|4) and plays the role of a level raising operator: [B,Qαb] = +Qαb , [B,Sαb] = −Sαb . (3.12) The barred supercharges also satisfy similar relations. The authors of [15] also prove the invariance under B of the Grassmannian integral formula [69] for the leading singularities in tree amplitudes. Moreover, they also show that the secret generator can be consistently corrected to ensure that the conformal anomaly is properly taken into account. In fact, certain distributional contributions arise when the differential operators act on poles of the amplitudes. This violates manifest superconformal invariance, which can be restored by adding suitable length-changing operators. One has for the secret symmetry BAn + B+An−1 = 0 , (3.13) with B+ = n−1∑ k=1 n∑ j=k+1 ( Qαbk S + j,αb − Q α k,bS +,b j,α −Qαbj S+k−1,αb + Q α j,bS +,b k−1,... |

10 | Constructing r-matrices on simple Lie superalgebras
- Karaali
(Show Context)
Citation Context |

10 |
A Yangian Double for the AdS/CFT Classical r-matrix,
- Moriyama, Torrielli
- 2007
(Show Context)
Citation Context ...tion of the classical Yang-Baxter equation [50, 51, 52, 53, 54, 55, 56, 57]. The element r displays a single pole at the origin in some appropriate classical spectral variables, with residue the quadratic Casimir of gl(2|2)⊗gl(2|2). There exists an infinitedimensional Lie bialgebra [45], formulated purely in abstract terms, which admits R as coboundary structure in these short representations. Its nature is quite unconventional, and its quantization is a fascinating open problem. This Lie bialgebra accomodates a class of generators of the type B, which appear naturally in the classical limit [58]. The indentation described above is interpreted in [45] as a different redistribution of the classical generators in the two copies of the double. Similarly, the new supercharges 7The additional index ±1 in (3.5) and (3.6) does not denote levels of any sort, rather the type of linear combination one needs in order to achieve these coproducts. It is mostly kept here for historical reasons. 7 and relations we have described above may be interpreted in the classical framework of [45]. The difficulty in quantizing the classical Lie bialgebra, however, still prevents from settling the question of ... |

9 | A unified and complete construction of all finite dimensional irreducible representations of gl(2|2
- Zhang, Gould
(Show Context)
Citation Context ...al in U(Y )⊗ U(Y ). Following the same argument as in the previous section, quasi-cocommutativity implies ∆(C1) = ∆op(C1), and the same for ∆(C†1), hence extra constraints to be added to (2.4). We will from now on always assume (2.4) and these new level one constraints (which we call hatted constraints), departing even further from the standard theory of Yangians. 2.3 Representations We can use an sl(2) outer automorphism to turn (H,C,C†) into (H′, 0, 0), corresponding to the Lie superalgebra sl(2|2). In turn, sl(2|2) is strictly related to gl(2|2), whose reps we will now describe6. The paper [35] (see also [36] and [37]) explicitly constructs all finite-dimensional irreps of gl(2|2). Generators of gl(2|2) are denoted by Eij, satisfying [Eij, Ekl] = δjkEil − (−)(d[i]+d[j])(d[k]+d[l])δilEkj . (2.9) Indices i, j, k, l run from 1 to 4, and the fermionic grading is assigned as d[1] = d[2] = 0, d[3] = d[4] = 1. The quadratic Casimir of this algebra is C2 = ∑4 i,j=1(−)d[j]EijEji. Finite dimensional irreps are labelled by two half-integers j1, j2 = 0, 1 2 , ..., and two complex numbers q and y. These numbers correspond to the eigenvalues of the Cartan elements 6We will follow [34] in this sec... |

9 | A New Lie Bialgebra Structure on sl(2,1
- Karaali
(Show Context)
Citation Context |

8 |
The S-Matrix of AdS/CFT and Yangian Symmetry. PoS,
- Beisert
- 2006
(Show Context)
Citation Context ...]: eip = C + 1 and e−ip = C† + 1. (2.4) With these conditions, ∆(C) = C⊗ 1 + 1⊗ C + C⊗ C = ∆op(C) , (2.5) and similarly for ∆(C†). The conditions (2.4) imply the equivalence relation CC† + C + C† = 0 which we will always assume. This already represents a departure from the standard theory of Hopf algebras. 2.2 The algebra: level one The symmetry algebra of the R-matrix contains another set of generators, partners to those described in the previous section. Together with the level zero, these new charges generate an infinite-dimensional Hopf algebra (which we will call Y ) similar to a Yangian [30]. Its Drinfeld’s second realization [31] is given in terms of Cartan generators κi,m and fermionic ladder generators ξ±i,m, i = 1, 2, 3, m = 0, 1, 2, . . . , subject to the following relations [32]: [κi,m, κj,n] = 0, [κi,0, ξ + j,m] = aij ξ + j,m, [κi,0, ξ − j,m] = −aij ξ−j,m, {ξ+i,m, ξ−j,n} = δi,j κj,n+m, [κi,m+1, ξ ± j,n]− [κi,m, ξ±j,n+1] = ± 1 2 aij{κi,m, ξ±j,n}, {ξ±i,m+1, ξ±j,n} − {ξ±i,m, ξ±j,n+1} = ± 1 2 aij[ξ ± i,m, ξ ± j,n], (2.6) 3 i 6= j, nij = 1 + |aij|, Sym{k}[ξ±i,k1 , [ξ ± i,k2 , . . . {ξ±i,knij , ξ ± j,l} . . . }} = 0 , except for {ξ+2,n, ξ+3,m} = Cn+m, {ξ−2,n, ξ−3,m} = C † n+m. (... |

7 |
Yangians of classical Lie superalgebras: Basic constructions, quantum double and universal
- Stukopin
- 2004
(Show Context)
Citation Context ...he choice of gauge for the string sigma model. This is because the centrally-extended psl(2|2) algebra is intimately linked to those specific choices. More recently, however, there have been observations of the very same mechanism in several (a priori unrelated) sectors of AdS/CFT. On one hand, this is reassuring that we are not dealing with an accidental problem. On the other hand, even in the light of these new observations, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we wi... |

7 |
The secret symmetries of the AdS/CFT reflection matrices
- Regelskis
- 2011
(Show Context)
Citation Context ...On the other hand, even in the light of these new observations, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we will humbly attempt at reproducing in its salient features. 2 The algebra 2.1 The algebra: level zero We will start by discussing the Hopf algebra based on the Lie superalgebra A(1, 1) = psl(2|2) with three-fold central extension. We will call this algebra psl(2|2)c. Such a large central extension is unique among the basic classical simple Lie superalgebra... |

7 |
Sergey Frolov, and Marija Zamaklar. The Zamolodchikov-Faddeev algebra for
- Arutyunov
- 2007
(Show Context)
Citation Context ...p(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in the associated string sigma models. 2 JA ∈ psl(2|2)c ∆(JA) = JA ⊗ 1 + ei[[A]]p ⊗ JA, ∆(eip) = eip ⊗ eip, (2.1) where p is central. The additive quantum number [[A]] equals 0 for generators in sl(2)⊕ sl(2) and for H, 1 2 for Q aα , −12 for G α a , 1 for C and −1 for C†. The above coproduct can be easily shown to be a Lie algebra homomorphism. The corresponding counit and antipode are straightforwardly derived from the Hopf algebra axioms [27]. The R-matrix R of [19] renders this Hopf algebra quasi-cocommutative (see also [28, 29]). One has R ∈ U(psl(2|2)c) ⊗ U(psl(2|2)c), with U(psl(2|2)c) the universal enveloping algebra of psl(2|2)c, such that ∆op(JA)R = R∆(JA) . (2.2) Since ∆(C) is central in U(psl(2|2)c)⊗ U(psl(2|2)c), one must have ∆op(C)R = R∆(C) = ∆(C)R =⇒ ∆op(C) = ∆(C) (2.3) (analogously for C†). This is guaranteed by the conservation of total momentum in the scattering [19]: eip = C + 1 and e−ip = C† + 1. (2.4) With these conditions, ∆(C) = C⊗ 1 + 1⊗ C + C⊗ C = ∆op(C) , (2.5) and similarly for ∆(C†). The conditions (2.4) imply the equivalence relation CC† + C + C† = 0 which we will always assume. This already... |

6 | Gauss Decomposition of the Yangian Y (gl(m|n)),
- Gow
- 2007
(Show Context)
Citation Context ...he choice of gauge for the string sigma model. This is because the centrally-extended psl(2|2) algebra is intimately linked to those specific choices. More recently, however, there have been observations of the very same mechanism in several (a priori unrelated) sectors of AdS/CFT. On one hand, this is reassuring that we are not dealing with an accidental problem. On the other hand, even in the light of these new observations, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we wi... |

5 |
Super-Yangian double and its central extension.
- Zhang
- 1997
(Show Context)
Citation Context ...he choice of gauge for the string sigma model. This is because the centrally-extended psl(2|2) algebra is intimately linked to those specific choices. More recently, however, there have been observations of the very same mechanism in several (a priori unrelated) sectors of AdS/CFT. On one hand, this is reassuring that we are not dealing with an accidental problem. On the other hand, even in the light of these new observations, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we wi... |

5 |
Finite-dimensional representations of the Lie superalgebra gl(2|2) in a gl(2)⊕gl(2) basis
- Kamupingene, Ky, et al.
- 1989
(Show Context)
Citation Context ...owing the same argument as in the previous section, quasi-cocommutativity implies ∆(C1) = ∆op(C1), and the same for ∆(C†1), hence extra constraints to be added to (2.4). We will from now on always assume (2.4) and these new level one constraints (which we call hatted constraints), departing even further from the standard theory of Yangians. 2.3 Representations We can use an sl(2) outer automorphism to turn (H,C,C†) into (H′, 0, 0), corresponding to the Lie superalgebra sl(2|2). In turn, sl(2|2) is strictly related to gl(2|2), whose reps we will now describe6. The paper [35] (see also [36] and [37]) explicitly constructs all finite-dimensional irreps of gl(2|2). Generators of gl(2|2) are denoted by Eij, satisfying [Eij, Ekl] = δjkEil − (−)(d[i]+d[j])(d[k]+d[l])δilEkj . (2.9) Indices i, j, k, l run from 1 to 4, and the fermionic grading is assigned as d[1] = d[2] = 0, d[3] = d[4] = 1. The quadratic Casimir of this algebra is C2 = ∑4 i,j=1(−)d[j]EijEji. Finite dimensional irreps are labelled by two half-integers j1, j2 = 0, 1 2 , ..., and two complex numbers q and y. These numbers correspond to the eigenvalues of the Cartan elements 6We will follow [34] in this section. 4 on highest weigh... |

5 | On the reflection of magnon bound states
- MacKay, Regelskis
(Show Context)
Citation Context ...ical cobrackets8. This is somehow echoed in [16], as we will see later on. Other approaches do not seem to detect such obstacles in principle, which is a sign of how non-trivial a task the complete quantum formulation of this symmetry still remains. It is worth noticing that in the opposite (gauge-theory) regime of small ‘t Hooft coupling, the one-loop R-matrix is a twisted version of the gl(2|2) Yangian R-matrix in the fundamental representation. The presence of the secret symmetry at level zero is in this case a one-loop accident. 3.2 Secret symmetries of the K-matrices In a series of works [59, 60, 61, 62, 63, 64] reflection K-matrices were found for open strings ending on D3, D5 and D7-branes. It was also observed that the secret symmetry manifests itself in some of these reflection matrices [14]. The Y = 0 maximal giant graviton is a D3-brane wrapping a maximal S3 ⊂ S5 of the AdS5 × S5 background and preserves an sl(2|1)L = {L βα , R 11 , R 22 , Q 1α , G α1 , H} subalgebra of the bulk algebra psl(2|2)c. The fundamental reflection matrix, describing the scattering of fundamental magnons from the boundary, is diagonal and the helicity generator B is a symmetry of it, but there is no secret symmetry at ... |

4 |
Solutions of the graded classical Yang-Baxter equation and integrable models
- Zhang, Gould, et al.
- 1991
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Weakly coupled
- Spill
- 2009
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Reflection algebra, Yangian symmetry and bound-states in AdS/CFT.
- MacKay, Regelskis
- 2012
(Show Context)
Citation Context ...er cumbersome and we refer the reader to [14]. These results show how the twisted Yangians inherit most of the properties of the original Yangians. However, while the reflection from the D5-brane is tightly related to the scattering in the bulk (thus the appearance of the secret symmetry follows naturally), the role of the secret symmetry in the reflection from the Y = 0 maximal giant graviton is not yet understood. An even more complicated question is related to the reflection from the Z = 0 maximal giant graviton and the D7-brane. Such scattering is governed by the level two twisted Yangian [67]. One could expect secret charges to be present in this case as well. However, addressing this issue would require knowledge of the level two twisted secret symmetry, which has not been explored so far. This is related to the open question whether the secret symmetry is present at odd levels only, or instead at all higher levels starting from level one. 3.3 Secret symmetry in Amplitudes In this section, we summarize the findings of [15], who observed the presence of a mechanism completely analog to the one we have been describing for the spectral problem. This time, it involves tree-level plan... |

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Nonlocal Charges for Bonus Yangian Symmetries of Super-Yang-Mills.
- Berkovits, Mikhailov
- 2011
(Show Context)
Citation Context ...ons, the fundamental nature of the secret symmetry remains unclear, and it is still not known how to consistently embed it 1For Yangians based on Lie superalgebras, see e.g. [9, 10, 11, 12]. 1 into a satisfactory mathematical framework. After all, we might simply have in front of us a new type of quantum group2. In this proceedings3, we will try and give a survey of the places where the secret symmetry manifests itself. We will start with the spectral problem, where it was originally observed. We will then move to the boundary problem [14], n-point amplitudes [15], the pure-spinor formulation [16] and, finally, to quantum affine deformations [17]. We will not try and be exhaustive, also due to the fact that some of the above examples were found in the work of others, which we will humbly attempt at reproducing in its salient features. 2 The algebra 2.1 The algebra: level zero We will start by discussing the Hopf algebra based on the Lie superalgebra A(1, 1) = psl(2|2) with three-fold central extension. We will call this algebra psl(2|2)c. Such a large central extension is unique among the basic classical simple Lie superalgebras [18]. The even part of psl(2|2)c consists of sl(2)⊕ sl(2)... |

2 | An Exceptional Algebraic Origin of the AdS/CFT Yangian Symmetry.
- Matsumoto, Moriyama
- 2008
(Show Context)
Citation Context ...c generators will be denoted by Q aα and G αa . Besides standard sl(2)⊕ sl(2) commutation relations, one has [19]: [L ba , Jc] = δbc Ja − 12δ b a Jc, [R βα , Jγ] = δβγ Jα − 12δ β α Jγ, [L ba , Jc] = −δca Jb + 12δ b a Jc, [R βα , Jγ] = −δγα Jβ + 12δ β α Jγ, {Q aα ,Q bβ } = αβabC, {G αa ,G β b } = αβabC†, {Q aα ,G β b } = δab R βα + δβα L ab + 12δ a b δ β αH. where J denotes any odd generator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are... |

2 | Note on Centrally Extended su(2|2) and Serre Relations,
- Dobrev
- 2009
(Show Context)
Citation Context ...i,m+1, ξ ± j,n]− [κi,m, ξ±j,n+1] = ± 1 2 aij{κi,m, ξ±j,n}, {ξ±i,m+1, ξ±j,n} − {ξ±i,m, ξ±j,n+1} = ± 1 2 aij[ξ ± i,m, ξ ± j,n], (2.6) 3 i 6= j, nij = 1 + |aij|, Sym{k}[ξ±i,k1 , [ξ ± i,k2 , . . . {ξ±i,knij , ξ ± j,l} . . . }} = 0 , except for {ξ+2,n, ξ+3,m} = Cn+m, {ξ−2,n, ξ−3,m} = C † n+m. (2.7) The non zero entries of the (degenerate) symmetric Cartan matrix aij are a12 = a21 = 1 and a13 = a31 = −1. The second subscript n in ξ±i,n, κi,n denotes the level, with n = 0 providing a subalgebra isomorphic to psl(2|2)c. Our choice corresponds to the following Chevalley-Serre presentation of psl(2|2)c [33]: ξ+1,0 = G 42 , ξ−1,0 = Q 24 , κ1,0 = −L 11 − R 33 + 12H , ξ+2,0 = iQ 14 , ξ−2,0 = iG 41 , κ2,0 = −L 11 + R 33 − 12H , ξ+3,0 = iQ 23 , ξ−3,0 = iG 32 , κ3,0 = L 11 − R 33 − 12H . (2.8) The generators Cn and C†n are central in Y for all n. The level one coproduct compatible with (2.1) is quite cumbersome and can be found in the literature. As for ordinary Yangians, it is enough to specify the coproduct at level 0 and 1. Recursive use of the defining relations and of the algebra-homomorphism property fixes the coproduct for all other levels. One now understands the R-matrix and all generators as... |

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Finite dimensional representations of the Lie superalgebra gl(2|2) in a gl(2)×gl(2) basis
- Palev, Soilova
- 1990
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Citation Context ...Y ). Following the same argument as in the previous section, quasi-cocommutativity implies ∆(C1) = ∆op(C1), and the same for ∆(C†1), hence extra constraints to be added to (2.4). We will from now on always assume (2.4) and these new level one constraints (which we call hatted constraints), departing even further from the standard theory of Yangians. 2.3 Representations We can use an sl(2) outer automorphism to turn (H,C,C†) into (H′, 0, 0), corresponding to the Lie superalgebra sl(2|2). In turn, sl(2|2) is strictly related to gl(2|2), whose reps we will now describe6. The paper [35] (see also [36] and [37]) explicitly constructs all finite-dimensional irreps of gl(2|2). Generators of gl(2|2) are denoted by Eij, satisfying [Eij, Ekl] = δjkEil − (−)(d[i]+d[j])(d[k]+d[l])δilEkj . (2.9) Indices i, j, k, l run from 1 to 4, and the fermionic grading is assigned as d[1] = d[2] = 0, d[3] = d[4] = 1. The quadratic Casimir of this algebra is C2 = ∑4 i,j=1(−)d[j]EijEji. Finite dimensional irreps are labelled by two half-integers j1, j2 = 0, 1 2 , ..., and two complex numbers q and y. These numbers correspond to the eigenvalues of the Cartan elements 6We will follow [34] in this section. 4 on high... |

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Yangian symmetry and bound states in AdS/CFT boundary scattering,
- Ahn, Nepomechie
- 2010
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On the Hopf algebra
- Plefka, Spill, et al.
- 2006
(Show Context)
Citation Context ...enerator with the appropriate index. The elements H, C and C† commute with all the generators. The algebra psl(2|2)c can be obtained as a contraction from the simple Lie superalgebra D(2, 1;α) (see for instance [19, 20, 21, 22]), and for this reason it is sometimes called D(2, 1;−1). The Killing form vanishes identically5. The algebra admits an sl(2) outer automorphism group [24], which is inherited from A(1, 1) [25]. This automorphism rotates for instance the three-vector of central charges (H,C,C†) preserving the “norm” H2 − CC†. One can put a non-trivial Hopf algebra structure on psl(2|2)c [26, 27]. For any 2P. Etingof, private communication. 3Based on the talk presented by A.T., Nordita, 15 February 2012. 4In unitary representations, the central elements C and C† are hermitean conjugate to each other, and so are the supercharges Q and G. 5This feature is shared by D(2, 1;α), psl(n|n) and osp(2n + 2|2n) [23], and it is crucial for the cancellation of anomalies in the associated string sigma models. 2 JA ∈ psl(2|2)c ∆(JA) = JA ⊗ 1 + ei[[A]]p ⊗ JA, ∆(eip) = eip ⊗ eip, (2.1) where p is central. The additive quantum number [[A]] equals 0 for generators in sl(2)⊕ sl(2) and for H, 1 2 for Q a... |

1 | Takuya Matsumoto, and Vidas Regelskis. The Bound State Smatrix of the Deformed Hubbard Chain. - Leeuw - 1109 |

1 | Hollowood and J.Luis Miramontes. Magnons, their Solitonic Avatars and the Pohlmeyer Reduction. - Timothy - 2009 |