Bringing an interface to the boundary is a special case of the more general process of fusion, i.e., of juxtaposing and then bringing two interfaces together on the string worldsheet. This is of course only possible when the CFT on the right side of the first interface coincides with the cft on the left side of the second. furthermore two interfaces can only be added when their left and right CFTs are identical. Since fusion and addition cannot be defined for arbitrary elements, the set of all conformal interfaces is neither an algebra nor a group. By abuse of language, I will nevertheless refer to it as the"interface algebra"2 The first thing to note is that the interface"algebra"is non-trivial even if re- stricted only to elements with non-singular fusion. These include all the topological interfaces, for which fusion is the regular product of the corresponding operators, OAOOB=OAOB. The simplest topological defects are those whose internal state is decoupled from the dynamics in the bulk. They correspond to multiples of the identity operator, O=nl with n a natural number. Their action on any D-brane endows this latter with Chan-Paton multiplicity. Less trivial are the topological de- fects which generate symmetries of the CFT, as well as the topological interfaces that generate perturbative T-dualities. These were first studied, for several exam- ples, in two beautiful papers by Frohlich et al. [23, 24]. The fact that all perturbative string symmetries can be realized through the action of local defects is not a pri- ori obvious(and needs still to be generally established). Other interesting examples are the minimal-model topological defects, shown to generate universal boundary flows [22, 28]. a different set of conformal interfaces whose fusion is non-singular are those that preserve at least N=(2, 2) supersymmetry [14, 15]. Some of these descend from supersymmetric gauge theories in higher dimensions [29, 33-35 Such interfaces were, in particular, used to generate the monodromy transformations of supersymmetric D-branes transported around singular points in the Calabi-Yau is very rich even if restricted to interfaces with non-singular fusion. Extending the structure to arbitrary interfaces is, nevertheless, an interesting ly, the algebras(without quotation marks) Tects would provide, if they could be defined, large extensions of the automorphism groups of various CFTs. Furthermore, while topological interfaces are rare-they may only join CFTs that have isomorphic Virasoro representations-the conformal ones are on the contrary common. A useful quantity is the reflection coefficient, ge, [44] which vanishes in the topological case. To see that conformal interfaces are not rare, consider the nth multiple of the identity defect which is mapped, after folding, to n diagonally-embedded middle-dimensional branes in .lx.l[10] A generic Hamiltonian of the form(2), with the tachyon potential T set to zero orresponds to arbitrary geometric and gauge-field perturbations of these diagonal ranes. Any solution of the (non-abelian, corrected) Dirac-Born-Infeld equa- tions for these branes gives therefore rise to a conformal defect [8]. Likewise, any The correct term for the interfaces is"functors " For a more accurate mathematical terminology the reader should consult, for instance, ref. [26
On the Symmetries of Classical String Theory 21 Bringing an interface to the boundary is a special case of the more general process of fusion, i.e., of juxtaposing and then bringing two interfaces together on the string worldsheet. This is of course only possible when the CFT on the right side of the first interface coincides with the CFT on the left side of the second. Furthermore, two interfaces can only be added when their left and right CFTs are identical. Since fusion and addition cannot be defined for arbitrary elements, the set of all conformal interfaces is neither an algebra nor a group. By abuse of language, I will nevertheless refer to it as the “interface algebra.”2 The first thing to note is that the interface “algebra” is non-trivial even if restricted only to elements with non-singular fusion. These include all the topological interfaces, for which fusion is the regular product of the corresponding operators, OA ◦ OB = OAOB. The simplest topological defects are those whose internal state is decoupled from the dynamics in the bulk. They correspond to multiples of the identity operator, O = n1 with n a natural number. Their action on any D-brane endows this latter with Chan–Paton multiplicity. Less trivial are the topological defects which generate symmetries of the CFT, as well as the topological interfaces that generate perturbative T-dualities. These were first studied, for several examples, in two beautiful papers by Fr¨ohlich et al. [23,24]. The fact that all perturbative string symmetries can be realized through the action of local defects is not a priori obvious (and needs still to be generally established). Other interesting examples are the minimal-model topological defects, shown to generate universal boundary flows [22, 28]. A different set of conformal interfaces whose fusion is non-singular are those that preserve at least N = (2,2) supersymmetry [14, 15]. Some of these descend from supersymmetric gauge theories in higher dimensions [29, 33–35]. Such interfaces were, in particular, used to generate the monodromy transformations of supersymmetric D-branes transported around singular points in the Calabi–Yau moduli space [16]. As these and other examples demonstrate, the interface “algebra” is very rich even if restricted to interfaces with non-singular fusion. Extending the structure to arbitrary interfaces is, nevertheless, an interesting problem. Firstly, the algebras (without quotation marks) of non-topological defects would provide, if they could be defined, large extensions of the automorphism groups of various CFTs. Furthermore, while topological interfaces are rare – they may only join CFTs that have isomorphic Virasoro representations – the conformal ones are on the contrary common. A useful quantity is the reflection coefficient, R, [44] which vanishes in the topological case. To see that conformal interfaces are not rare, consider the nth multiple of the identity defect which is mapped, after folding, to n diagonally-embedded middle-dimensional branes in M × M [10]. A generic Hamiltonian of the form (2), with the tachyon potential T set to zero, corresponds to arbitrary geometric and gauge-field perturbations of these diagonal branes. Any solution of the (non-abelian, α corrected) Dirac–Born–Infeld equations for these branes gives therefore rise to a conformal defect [8]. Likewise, any 2 The correct term for the interfaces is “functors.” For a more accurate mathematical terminology the reader should consult, for instance, ref. [26]
non-factorizable D-brane of CFTlCFT2 unfolds to a non-trivial (i.e, not purely reflecting) interface between the two conformal field theories. For most of these interfaces the product of the corresponding operators is singu- lar, so the fusion needs to be appropriately defined. A first step in this direction was taken, in the context of a free-scalar theory, in [7]. The rough idea is to define the fusion product as the renormalization-group fixed point to which the system of the ng this, consistent with the distributive property of fusion, has not yet Bp vay two interfaces flows when their separation, E, goes to zero. A systematic way of do- out for interacting theories. For free fields, on the other hand, the story is simpler. The short-distance singularities are in this case expected to be of the general form O4e-(o+0)O where ea 0 is the separation of the two(circular) interfaces on the cylinder, Lo+Lo the translation operator in the middle CFT, the dAB are(non-universal)constant and the NAB are integer multiplicities. The singular coefficients in the above ex- pression are Boltzmann factors for divergent Casimir energies. The latter must be proportional to 1/E which is the only scale in the problem(other than the inverse temperature normalized to B=2r) By analogy with the operator-product expansion and the verlinde algebra [45] we may extract from expression( 8)the fusion rule OAoB=∑NO The following iterative argument shows that this definition respects the conformal symmetry: first multiply the left-hand-side of (8)with the most singular inverse Boltzmann factor(the one with the largest dB) and take the limit e-0 so as to extract the leading term of the product. Since [ Lv-L-N, e-e(Lo+o)]co(e)the result commutes with the diagonal virasoro algebra. Next subtract the leading term from the left-hand-side of (8), and mutliply by the inverse Boltzmann factor with the second-largest dAR. This picks up the subleading term which, thanks to the above argument and the conformal symmetry of the leading term, commutes also with the diagonal Virasoro algebra. Continuing this iterative reasoning proves that the right-hand-side of (9)is conformal as claimed. 3 The c= l CFT and a black Hole Analogy A simple context in which to illustrate the above ideas is the c= l conformal neory of a periodically-identified free scalar field, o =o+2TR. Consider the interfaces that preserve a U(1)xU(1)symmetry, i.e., those described by linear 3 I thank Maxim Kontsevich for stressing this point
22 C.P. Bachas non-factorizable D-brane of CFT1⊗CFT2 unfolds to a non-trivial (i.e., not purely reflecting) interface between the two conformal field theories. For most of these interfaces the product of the corresponding operators is singular, so the fusion needs to be appropriately defined. A first step in this direction was taken, in the context of a free-scalar theory, in [7]. The rough idea is to define the fusion product as the renormalization-group fixed point to which the system of the two interfaces flows when their separation, ε, goes to zero. A systematic way of doing this, consistent with the distributive property of fusion,3 has not yet been worked out for interacting theories. For free fields, on the other hand, the story is simpler. The short-distance singularities are in this case expected to be of the general form OA e−ε(L0+L0) OB ∑ C (e2π/ε ) d C AB N C AB OC, (8) where ε 0 is the separation of the two (circular) interfaces on the cylinder, L0 +L0 is the translation operator in the middle CFT, the d C AB are (non-universal) constants, and the N C AB are integer multiplicities. The singular coefficients in the above expression are Boltzmann factors for divergent Casimir energies. The latter must be proportional to 1/ε which is the only scale in the problem (other than the inverse temperature normalized to β = 2π). By analogy with the operator-product expansion and the Verlinde algebra [45] we may extract from expression (8) the fusion rule OA ◦ OB = ∑ C N C AB OC. (9) The following iterative argument shows that this definition respects the conformal symmetry: first multiply the left-hand-side of (8) with the most singular inverse Boltzmann factor (the one with the largest d C AB) and take the limit ε → 0 so as to extract the leading term of the product. Since [LN − L−N,e−ε(L0+L0) ] o(ε) the result commutes with the diagonal Virasoro algebra. Next subtract the leading term from the left-hand-side of (8), and mutliply by the inverse Boltzmann factor with the second-largest d C AB. This picks up the subleading term which, thanks to the above argument and the conformal symmetry of the leading term, commutes also with the diagonal Virasoro algebra. Continuing this iterative reasoning proves that the right-hand-side of (9) is conformal as claimed. 3 The c = 1 CFT and a Black Hole Analogy A simple context in which to illustrate the above ideas is the c = 1 conformal theory of a periodically-identified free scalar field, φ = φ + 2πR. Consider the interfaces that preserve a U(1) ×U(1) symmetry, i.e., those described by linear 3 I thank Maxim Kontsevich for stressing this point
Symmetries of Cl gluing conditions for the field o. They correspond, after folding, to combinations of Dl-branes and of magnetized D2-branes on the orthogonal two-torus whose radii, RI and R, are the radii on either side of the interface. The Dl-branes are characterized by their winding numbers, kI and k2, and by the wilson line and periodic position moduli a and B. The magnetized D2-branes are obtained from the DI-branes by T-dualizing one of the two directions of the torus-they have therefore the same number of discrete and of continuous moduli Let us focus here on the DI-branes The corresponding boundary states read )|-k1N,k2M),(1 where ai and ai are the left -and right-moving annihilation operators of the field o (for j=1, 2)and the dagger denotes hermitean conjugation. The ground states Im, m) of the scalar fields are characterized by a momentum(m)and a winding num ber(m). The states in the above tensor product correspond to o1 and o2. Furthermore s 20-sin 21 where (a)is a rotation matrix and 1= arctan(k2R2/ki Ri) is the angle between the D1-brane and the oI direction. Finally, the normalization constant is the g-factor [2]of the boundary state. It is given by √2V 2RIR V sin20 where t is the length of the D1-brane v the volume of the two-torus. and the last rewriting follows from straightforward trigonometry. The logarithm of the g factor he invariant entropy of the interface. Inspection of the expression(10) shows that the non-zero modes of the fields oj are only sensitive to the angle v, which also determines the reflection coeffi cient of the interface. For fixed kI and k2 the g factor is minimal when 1=+r/4, in which case the reflection 9=0 and the interface is topological. Note that this requirement fixes the ratio of the two bulk moduli: R1/R2=k2/k1. When kIl kk2l= l the two radii are equal and the invariant entropy is zero. The correspond ing topological defects generate the automorphisms of the CFT, i.e., sign flip of the field o and separate translations of its left-and right-moving pieces. The iden tity defect corresponds to the diagonal Dl-brane, with ki=k2= 1 and a=B=0 A T-duality along I maps this topological defect to a D2-brane with one unit of magnetic flux. The corresponding interface operator is the generator of the radius inverting T-duality transformation. All other topological interfaces have positive entropy, logg=logykjk2l>0. One may conjecture that the following statement
On the Symmetries of Classical String Theory 23 gluing conditions for the field φ. They correspond, after folding, to combinations of D1-branes and of magnetized D2-branes on the orthogonal two-torus whose radii, R1 and R2, are the radii on either side of the interface. The D1-branes are characterized by their winding numbers, k1 and k2, and by the Wilson line and periodic position moduli α and β. The magnetized D2-branes are obtained from the D1-branes by T-dualizing one of the two directions of the torus – they have therefore the same number of discrete and of continuous moduli. Let us focus here on the D1-branes. The corresponding boundary states read ||D1,ϑ = g(+) ∞ ∏n=1 (e S (+) i j ai naj n ) † ∞ ∑ N,M=−∞ eiNα−iMβ |k2N,k1M ⊗|−k1N,k2M , (10) where aj n and ˜aj n are the left- and right-moving annihilation operators of the field φj (for j = 1,2) and the dagger denotes hermitean conjugation. The ground states |m,m˜ of the scalar fields are characterized by a momentum (m) and a winding number ( ˜m). The states in the above tensor product correspond to φ1 and φ2. Furthermore S(+) = U T (ϑ) −1 0 0 1 U (ϑ) = −cos2ϑ −sin2ϑ −sin2ϑ cos2ϑ , (11) where U (ϑ) is a rotation matrix and ϑ = arctan(k2R2/k1R1) is the angle between the D1-brane and the φ1 direction. Finally, the normalization constant is the g-factor [2] of the boundary state. It is given by g(+) = √2V = k2 1R2 1 +k2 2R2 2 2R1R2 = k1k2 sin2ϑ, (12) where is the length of the D1-brane, V the volume of the two-torus, and the last rewriting follows from straightforward trigonometry. The logarithm of the g factor is the invariant entropy of the interface. Inspection of the expression (10) shows that the non-zero modes of the fields φj are only sensitive to the angle ϑ, which also determines the reflection coeffi- cient of the interface. For fixed k1 and k2 the g factor is minimal when ϑ = ±π/4, in which case the reflection R = 0 and the interface is topological. Note that this requirement fixes the ratio of the two bulk moduli: R1/R2 = |k2/k1|. When |k1| = |k2| = 1 the two radii are equal and the invariant entropy is zero. The corresponding topological defects generate the automorphisms of the CFT, i.e., sign flip of the field φ and separate translations of its left- and right-moving pieces. The identity defect corresponds to the diagonal D1-brane, with k1 = k2 = 1 and α = β = 0. A T-duality along φ1 maps this topological defect to a D2-brane with one unit of magnetic flux. The corresponding interface operator is the generator of the radiusinverting T-duality transformation. All other topological interfaces have positive entropy, logg = log|k1k2| > 0. One may conjecture that the following statement
is more generally true: the entropy of all topological interfaces is non-negative, and it vanishes only for T-duality transformations and for CFT automorphisms The interfaces given by (10)(12)exist for all values of the bulk radii Ri and r2 By choosing the radii to be equal we obtain a large set of conformal defects whose algebra is an extension of the automorphism group of the CFT. For a detailed deriva tion of this algebra see [7]. The fusion rule for the discrete defect moduli turns out to be multiplicative, kk1, k2; s]ok1, k2: s=kIki, k2k2; ss, where k1, k2; s denotes a defect with integer moduli k1, k2, s, according to whether the folded defect is a D1-brane or a magr inediteD D2-brane The above fusion rule continues to hold for general interfaces, i c the radii on either side are not the same. Let me also give the composition rule for the angle 1 in this general case(assuming s=s=+) tan(9o1)=tan tanu (14 where vo n denotes the angle of the fusion product. The composition rule(13)was first derived, for the topological interfaces, in [25]. In this case the tangents in the last equation are +l and all operator products are non-singular There exist some intriguing similarities [7 between the above conformal inter- faces and supergravity black holes. The counterpart of BPs black holes are the topological interfaces, which(a) minimize the free energy for fixed values of the discrete charges, (b) fix through anattractor mechanism"[21] a combination of the bulk moduli, and(c) are marginally stable against dissociation-the inverse process of fusion. The interface"algebra"is, in this sense, reminiscent of an ear lier effort by Harvey and Moore [30] to define an extended symmetry algebra for string theory. Their symmetry generators were vertex operators for supersymn states of the compactified string. One noteworthy difference is that the additively conserved charges in our case are logarithms of natural numbers, rather than taking values in a charge lattice as in [30]. Whether these observations have any deeper meaning remains to be seen. Another direction worth exploring is a possible relation of the above ideas with efforts to formulate string theory in a "doubled geometry, see for instance [ 31]. The doubling of spacetime after folding suggests that this may provide the natural setting in which to formulate the defect algebras Time to conclude: conformal interfaces and defects are examples of extended perators, which are a rich and still only partially-explored chapter of quantum field theory. They describe a variety of critical phenomena in low-dimensional condensed-matter systems which, for lack of time, I did not discuss. They can be furthermore, both added and juxtaposed or fused. When this latter operation can be defined, the conformal interfaces form interesting algebraic structures which could shed new light on the symmetries of string theory. For all these reasons they deserve to be studied more
24 C.P. Bachas is more generally true: the entropy of all topological interfaces is non-negative, and it vanishes only for T-duality transformations and for CFT automorphisms. The interfaces given by (10)–(12) exist for all values of the bulk radii R1 and R2. By choosing the radii to be equal we obtain a large set of conformal defects whose algebra is an extension of the automorphism group of the CFT. For a detailed derivation of this algebra see [7]. The fusion rule for the discrete defect moduli turns out to be multiplicative, [k1,k2;s] ◦ [k 1,k 2;s ]=[k1k 1,k2k 2;ss ], (13) where [k1,k2;s] denotes a defect with integer moduli k1, k2, s, where s = +,− according to whether the folded defect is a D1-brane or a magnetized D2-brane. The above fusion rule continues to hold for general interfaces, i.e., when the radii on either side are not the same. Let me also give the composition rule for the angle ϑ in this general case (assuming s = s = +): tan(ϑ ◦ϑ ) = tanϑ tanϑ , (14) where ϑ ◦ϑ denotes the angle of the fusion product. The composition rule (13) was first derived, for the topological interfaces, in [25]. In this case the tangents in the last equation are ±1 and all operator products are non-singular. There exist some intriguing similarities [7] between the above conformal interfaces and supergravity black holes. The counterpart of BPS black holes are the topological interfaces, which (a) minimize the free energy for fixed values of the discrete charges, (b) fix through an “attractor mechanism” [21] a combination of the bulk moduli, and (c) are marginally stable against dissociation – the inverse process of fusion. The interface “algebra” is, in this sense, reminiscent of an earlier effort by Harvey and Moore [30] to define an extended symmetry algebra for string theory. Their symmetry generators were vertex operators for supersymmetric states of the compactified string. One noteworthy difference is that the additivelyconserved charges in our case are logarithms of natural numbers, rather than taking values in a charge lattice as in [30]. Whether these observations have any deeper meaning remains to be seen. Another direction worth exploring is a possible relation of the above ideas with efforts to formulate string theory in a “doubled geometry,” see for instance [31]. The doubling of spacetime after folding suggests that this may provide the natural setting in which to formulate the defect algebras. Time to conclude: conformal interfaces and defects are examples of extended operators, which are a rich and still only partially-explored chapter of quantum field theory. They describe a variety of critical phenomena in low-dimensional condensed-matter systems which, for lack of time, I did not discuss. They can be, furthermore, both added and juxtaposed or fused. When this latter operation can be defined, the conformal interfaces form interesting algebraic structures which could shed new light on the symmetries of string theory. For all these reasons they deserve to be studied more
On the Symmetries of Classical String Acknowledgments I thank Ilka Brunner, Jurg Frohlich and Samuel Monnier for very pleasant collaborations during the last couple of years, on different aspects of this talk. Many thanks also to Eric D Hoker, Mike douglas, Sergei Gukov, Chris Hull and Maxim Kontsevich for useful con ersations and comments. Claudio Bunster supervised my senior thesis and helped me publish m first scientific article -always a source of considerable pride for a student. My gratitude, after all these years, remains intact. References I.I. Affleck, Acta Phys. Polon. B 26(1995)1869 arXiv: cond-mat/95120991 2. I. Affleck and A. w. w. Ludwig, Phys. Rev. Lett. 67(1991)161 3. A Alekseev and s Monnier, JHEP 0708(2007)039 [arXiv: hep-th/07021741 4. A Alekseev and S L. Shatashvili, Commun. Math. Phys. 133(1990)353 5. N. Andrei, Phys. Rev. Lett. 45(1980)379 6. 0. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Syst (Cambridge University Press, Cambridge, 2003). 7. C. Bachas and I Brunner, JHEP 0802(2008)085 [arXiv: 0712.0076 [hep-thIl 8. C. Bachas and M. Gaberdiel, JHEP 0411(2004)065 [arXiv: hep-th/04110671 9. C. Bachas and M. Petropoulos, JHEP 0102 (2001)025 [arXiv: hep-th/0012234 10. C. Bachas, J de boer, R. Dijkgraaf and H. Ooguri, JHEP 0206(2002)027 [arXiv hep-th/ I1. V. V Bazhanov, S L Lukyanov and A B Zamolodchikov, Commun. Math. Phys. 177(1996) 381 arXiv: hep-th/94122291 12. VV Bazhanov, S L. Lukyanov and A B Zamolodchikov, Commun. Math. Phys. 190(1997) 47 [arXiv: hep-th/96040441 13. V.V. Bazhanov. s L Lukyanov and A. B Zamolodchikov, Commun. Math. Phys. 200(1999) 97 [arXiv: hep-th/9805008]- 14. I Brunner and D Roggenkamp, JHEP 0708(2007)093 [arXiv: 0707.0922 [hep-thlI 15. I Brunner and D Roggenkamp, JHEP 0804 (2008)001 [arXiv: 0712.0188 [hep-thlI 16. 1. Brunner, H Jockers and D. Roggenkamp, arXiv: 0806.4734 [hep-th](unpublished) 17. C. G. Callan, C. Lovelace, C R Nappi and S. A. Yost, Nucl. Phys. B 293(1987)83 18 0. DeWolfe, D. Z Freedman and H Ooguri, Phys. Rev. D 66(2002)025009 [arXiv: hep-th/ 01111351 19. J Ehlers, Konstruktionen und Charakterisierung von Lsungen der Einsteinschen gravitations feldgleichungen, Dissertation, Hamburg University, Hamburg, Germany(1957) 20. J. Erdmenger, Z. Guralnik and I. Kirsch, Phys. Rev. D 66(2002)025020 [arXiv: hep-th/ 21.S. Ferrara, R. Kallosh and A. Strominger, Phys. Rev. D 52(1995)5412 [arXiv: hep-th/ 9508072 22. S Fredenhagen, Nucl. Phys. B 660(2003)436 [arXiv: hep-th/03012291 23. J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Phys. Rev. Lett. 93(2004)070601 [ar Xiv: cond-mat/0404051] 4. J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Nucl. Phys. B 763(2007)354 [ar Xiv: h/O60724 25. J. Fuchs, M.R. Gaberdiel, I. Runkel and C. Schweigert, J. Phys. A 40(2007)11403 [arXiv: 0705.3129 [hep-thIl 26. J. Fuchs, I Runkel and C Schweigert, ar Xiv math/0701223(unpublished) 27.R. Geroch, J. Math. Phys. 12(1971)918 8. K Graham and G M. T Watts, JHEP 0404 (2004)019 [ar Xiv hep-th/03061671 9. S. Gukov and E. Witten, arXiv: hep-th/0612073(unpublished) 0. J. A Harvey and G w. Moore, Commun. Math. Phys. 197(1998)489 [ar Xiv: hep-th/96090171
On the Symmetries of Classical String Theory 25 Acknowledgments I thank Ilka Brunner, J¨urg Fr¨ohlich and Samuel Monnier for very pleasant collaborations during the last couple of years, on different aspects of this talk. Many thanks also to Eric D’Hoker, Mike Douglas, Sergei Gukov, Chris Hull and Maxim Kontsevich for useful conversations and comments. Claudio Bunster supervised my senior thesis and helped me publish my first scientific article – always a source of considerable pride for a student. My gratitude, after all these years, remains intact. References 1. I. Affleck, Acta Phys. Polon. B 26 (1995) 1869 [arXiv:cond-mat/9512099]. 2. I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 67 (1991) 161. 3. A. Alekseev and S. Monnier, JHEP 0708 (2007) 039 [arXiv:hep-th/0702174]. 4. A. Alekseev and S. L. Shatashvili, Commun. Math. Phys. 133 (1990) 353. 5. N. Andrei, Phys. Rev. Lett. 45 (1980) 379. 6. O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems (Cambridge University Press, Cambridge, 2003). 7. C. Bachas and I. Brunner, JHEP 0802 (2008) 085 [arXiv:0712.0076 [hep-th]]. 8. C. Bachas and M. Gaberdiel, JHEP 0411 (2004) 065 [arXiv:hep-th/0411067]. 9. C. Bachas and M. Petropoulos, JHEP 0102 (2001) 025 [arXiv:hep-th/0012234]. 10. C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, JHEP 0206 (2002) 027 [arXiv:hep-th/ 0111210]. 11. V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, Commun. Math. Phys. 177 (1996) 381 [arXiv:hep-th/9412229]. 12. V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, Commun. Math. Phys. 190 (1997) 247 [arXiv:hep-th/9604044]. 13. V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, Commun. Math. Phys. 200 (1999) 297 [arXiv:hep-th/9805008]. 14. I. Brunner and D. Roggenkamp, JHEP 0708 (2007) 093 [arXiv:0707.0922 [hep-th]]. 15. I. Brunner and D. Roggenkamp, JHEP 0804 (2008) 001 [arXiv:0712.0188 [hep-th]]. 16. I. Brunner, H. Jockers and D. Roggenkamp, arXiv:0806.4734 [hep-th] (unpublished). 17. C. G. Callan, C. Lovelace, C. R. Nappi and S. A. Yost, Nucl. Phys. B 293 (1987) 83. 18. O. DeWolfe, D. Z. Freedman and H. Ooguri, Phys. Rev. D 66 (2002) 025009 [arXiv:hep-th/ 0111135]. 19. J. Ehlers, Konstruktionen und Charakterisierung von Lsungen der Einsteinschen Gravitationsfeldgleichungen, Dissertation, Hamburg University, Hamburg, Germany (1957). 20. J. Erdmenger, Z. Guralnik and I. Kirsch, Phys. Rev. D 66 (2002) 025020 [arXiv:hep-th/ 0203020]. 21. S. Ferrara, R. Kallosh and A. Strominger, Phys. Rev. D 52 (1995) 5412 [arXiv:hep-th/ 9508072]. 22. S. Fredenhagen, Nucl. Phys. B 660 (2003) 436 [arXiv:hep-th/0301229]. 23. J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Phys. Rev. Lett. 93 (2004) 070601 [arXiv:cond-mat/0404051]. 24. J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Nucl. Phys. B 763 (2007) 354 [arXiv: hep-th/0607247]. 25. J. Fuchs, M. R. Gaberdiel, I. Runkel and C. Schweigert, J. Phys. A 40 (2007) 11403 [arXiv:0705.3129 [hep-th]]. 26. J. Fuchs, I. Runkel and C. Schweigert, arXiv:math/0701223 (unpublished). 27. R. Geroch, J. Math. Phys. 12 (1971) 918. 28. K. Graham and G. M. T. Watts, JHEP 0404 (2004) 019 [arXiv:hep-th/0306167]. 29. S. Gukov and E. Witten, arXiv:hep-th/0612073 (unpublished). 30. J. A. Harvey and G. W. Moore, Commun. Math. Phys. 197 (1998) 489 [arXiv:hep-th/ 9609017]