Opening Lecture that he taught us is that life is too short to lose one's time on marginal problems One should develop a good scientific taste for what is relevant and important, even if this means not following fashion Another lesson is that doing research is-and should be - enjoyable. It is es- pecially enjoyable when one tries to foresee and anticipate the implications of a plausible physics result before even attempting to prove it, trying to understand whether these implications make sense. Again, life is too short and educated guess is the fastest-and funniest- way to go ahead Another unique aspect of Claudio's style of work is his ability to take advantage of any situation for doing physics. I am sure that all of us have on many occasions discussed physics with him in unthinkable circumstances, be it during a motorbike trip to a lost place in Texas to fetch an angora rabbit, or on a jeep ride on a bumpy lust road to Zapallar, or in a dentist waiting room, or at an airport counter waiting for the airline to accept sending with minimum extra charge oversized and overweight luggage, or even on a risky boat trip in the middle of the night in which we almost sank. Working with Claudio is indeed fun, but requires some capacity of adaptability from his collaborators, which is not part of the standard academic training I will not elaborate more now on Claudio's style and on the characteristics of his work we are all here because we know them! Before letting the fest begin, I would just like to add a few words in Spanish this is a premiere Claudio, vengo a Chile desde hace mas de veinte anos y nunca he hablado espanol en publico. iTengo que empezar a hacerlo! No hay mejor oportunidad que hoy, en tu fiesta cientifica Quiero anadir a lo que dije en ingles que hay otra cosa im- portante que aprendi de ti: es que debemos ser capaces de tomar riesgos, que pueder parecer a veces locos, no solo riesgos en nuestras investigaciones sino tambien ries- gos en la orientacion de nuestra carrera, de nuestra profesion, quizas de nuestra ida. No es una cosa que se aprende en circulos academics. Se puede apren der de exploradores, de poetas. Entonces voy a concluir con dos citas, la primera lel explorador frances Paul-Emile Victor que organiz expediciones al Artico y a la antartica, y la segunda del poeta chileno Vincente Huidobro. Comienzo con Paul-Emile Victor: La unica cosa que estamos seguros de no lograr es la que no intentamos. Y Huidobro: Si yo no hiciera al menos una locura por ao, me volveria loco "Eres un ferviente adept de estos principios y has convertido con tu ejemplo a muchos de tus amigos And now. let the"fest"begin
Opening Lecture 13 that he taught us is that life is too short to lose one’s time on marginal problems. One should develop a good scientific taste for what is relevant and important, even if this means not following fashion. Another lesson is that doing research is – and should be – enjoyable. It is especially enjoyable when one tries to foresee and anticipate the implications of a plausible physics result before even attempting to prove it, trying to understand whether these implications make sense. Again, life is too short and educated guessing is the fastest – and funniest – way to go ahead. Another unique aspect of Claudio’s style of work is his ability to take advantage of any situation for doing physics. I am sure that all of us have on many occasions discussed physics with him in unthinkable circumstances, be it during a motorbike trip to a lost place in Texas to fetch an angora rabbit, or on a jeep ride on a bumpy dust road to Zapallar, or in a dentist waiting room, or at an airport counter waiting for the airline to accept sending with minimum extra charge oversized and overweight luggage, or even on a risky boat trip in the middle of the night in which we almost sank. Working with Claudio is indeed fun, but requires some capacity of adaptability from his collaborators, which is not part of the standard academic training. I will not elaborate more now on Claudio’s style and on the characteristics of his work – we are all here because we know them! Before letting the fest begin, I would just like to add a few words in Spanish – this is a premi`ere. Claudio, vengo a Chile desde hace m´as de veinte a˜nos y nunca he hablado espa˜nol en p´ublico. ¡Tengo que empezar a hacerlo! No hay mejor oportunidad que hoy, en tu fiesta cient´ıfica. Quiero a˜nadir a lo que dije en ingl´es que hay otra cosa importante que aprend´ı de ti : es que debemos ser capaces de tomar riesgos, que pueden parecer a veces locos, no s´olo riesgos en nuestras investigaciones sino tambi´en riesgos en la orientaci´on de nuestra carrera, de nuestra profesi´on, quiz´as de nuestra vida. No es una cosa que se aprende en c´ırculos acad´emicos. Se puede aprender de exploradores, de poetas. Entonces voy a concluir con dos citas, la primera del explorador franc´es Paul-Emile Victor que organiz´o expediciones al Artico y ´ a la Ant´artica, y la segunda del poeta chileno Vincente Huidobro. Comienzo con Paul-Emile Victor : “La ´unica cosa que estamos seguros de no lograr es la que no intentamos.” Y Huidobro : “Si yo no hiciera al menos una locura por ao, me volver´ıa loco.” Eres un ferviente adepto de estos principios y has convertido con tu ejemplo a muchos de tus amigos. And now, let the “fest” begin !
On the symmetries of Classical String Theory Constantin P Bacha Abstract I discuss some aspects of conformal defects and conformal interfaces in two spacetime dimensions. Special emphasis is placed on their role as spectrum generating symmetries of classical string theory 1 Loop Operators in 2d CFT Wilson loops [47] are important tools for the study of gauge theory. They describe worldlines of external probes, such as the heavy quarks of QCD, which transform in some representation of the gauge group and couple to the gauge fields minimally More general couplings, possibly involving other fields(e.g, scalars and fermions), are in principle also allowed. They are, however, severely limited by the requirement f infrared relevance or, equivalently, of renormalizability. In four dimensions this only allows couplings to operators of dimension at most one, i.e., linear in the gauge and the scalar fields. An example in which the scalar coupling plays a role is the supersymmetric Wilson loop of N= 4 super-Yang Mills theory [38, 43] The story is much richer in two space-time dimensions. Power-counting rene malizable defects in a two-dimensional non-linear sigma model, for example, are described by the following loop operators try(C)=try Pe fc Hdef where V is the n-dimensional space of quantum states of the external probe, whose Hamiltonian is of the general form B=/(B0y+6(00)2+T0小3 CP Baches Laboratoire de Physique Theorique, Ecole Normale Su 24 rue Lhomond. 75231 Paris France e-mail: bacha @lpt.ens. fr M. Henneaux, J Zanelli(eds ) Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity, DOI 10.1007/978-0-387-87499.9-3 C Springer Science+ Business Media LLC 2009
On the Symmetries of Classical String Theory Constantin P. Bachas Abstract I discuss some aspects of conformal defects and conformal interfaces in two spacetime dimensions. Special emphasis is placed on their role as spectrumgenerating symmetries of classical string theory. 1 Loop Operators in 2d CFT Wilson loops [47] are important tools for the study of gauge theory. They describe worldlines of external probes, such as the heavy quarks of QCD, which transform in some representation of the gauge group and couple to the gauge fields minimally. More general couplings, possibly involving other fields (e.g., scalars and fermions), are in principle also allowed. They are, however, severely limited by the requirement of infrared relevance or, equivalently, of renormalizability. In four dimensions this only allows couplings to operators of dimension at most one, i.e., linear in the gauge and the scalar fields. An example in which the scalar coupling plays a role is the supersymmetric Wilson loop of N = 4 super-Yang Mills theory [38, 43]. The story is much richer in two space–time dimensions. Power-counting renormalizable defects in a two-dimensional non-linear sigma model, for example, are described by the following loop operators trVW(C) = trV Pei C Hdef , (1) where V is the n-dimensional space of quantum states of the external probe, whose Hamiltonian is of the general form C Hdef = dsBM(Φ)∂αΦM +εαβBM(Φ)∂β ΦM d ˆ ζα ds +T(Φ) . (2) C.P. Bachas Laboratoire de Physique Th´eorique, Ecole Normale Sup´ ´ erieure, 24 rue Lhomond, 75231 Paris, France e-mail: bachas@lpt.ens.fr M. Henneaux, J. Zanelli (eds.), Quantum Mechanics of Fundamental Systems: The Quest 17 for Beauty and Simplicity, DOI 10.1007/978-0-387-87499-9 3, c Springer Science+Business Media LLC 2009
Here s is the length along the defect worldline C, and the hamiltonian is a hermitean n x n matrix which depends on the sigma-model fields p(s )and on their first ves evaluated at the of the defect 5(s) specified by two matrix-valued one-forms, Bydd and BMdd, and by a matrix- valued function, T, all defined on the sigma-model target space l. Because Hdef is a matrix, the path-ordering in(1)is non-trivial even if the bulk fields are treated classical, and hence commuting c-numbers The non-linear sigma model is classically scale-invariant. The function T, on he other hand, has naive scaling dimension of mass, so(classical) scale-invariance requires that we set it to zero. The reader can easily check that, in this case, the operator (1)is invariant under all conformal transformations that preserve C. This symmetry is further enhanced if, as a result of the field equations, the induced one-form B=(B()+pBy(0)0)d 3) is a flat U(n)connection, i.e., if in laconic notation dB+[B, B=0. The loop op- rator is in this case invariant under arbitrary continuous deformations of C, as follows from the non-abelian Stokes theorem Such defects can therefore be called topological. The eigenvalues of topological loops W(C), with C winding around compact space, are charges conserved by the time evolution. The existence of a one- (spectral-) parameter family of flat connections is, for this reason, often tantamount to classical integrability, see [6] Quantization breaks, in general, the scale invariance of the defect loop even when the bulk theory is conformal. This is because the definition of w(C)requires the in- troduction of a short-distance cutoff E. As the cutoff is being removed the couplings run to infrared fixed points,B()→B"andB()→B*ase→0. I will explain later that this renormalization-group flow can be described perturbatively [8]by generalized Dirac-Born-Infeld equations. The fixed-point operators commute with diagonal conformal algebra. More specifically, if C is the circle of a cylindrical the left-and right-moving Virasoro generators, then [LN-L-N, trvW'(C)=0 VN Topological quantum defects satisfy stronger conditions: they must commute sepa rately with the LN and with the L These facts can be illustrated with the symmetry-preserving defect loops of the wZW model [8]. Consider the following chiral, symmetry-preserving defect or(c)=x(Pe'fcAJp) where J are the left-moving Kac-Moody currents, r" the generators of the global group G, and X the character of the G-representation, r, carried by the state-space of the defect. In the classical theory O(C)is topological for all values of the pa rameter n. But upon quantization, the spectral parameter runs from the UV fixed ointλ=0 to an IR fixed pointλ*≈1/k, where k is the level of the Kac-Moody algebra(and k > 1 for perturbation theory to be valid). It is interesting here to
18 C.P. Bachas Here s is the length along the defect worldlineC, and the Hamiltonian is a hermitean n × n matrix which depends on the sigma-model fields ΦM(ζα ) and on their first derivatives evaluated at the position of the defect ˆ ζα (s). The loop operator is thus specified by two matrix-valued one-forms, BMdΦM and BMdΦM, and by a matrixvalued function, T, all defined on the sigma-model target space M. Because Hdef is a matrix, the path-ordering in (1) is non-trivial even if the bulk fields are treated as classical, and hence commuting c-numbers. The non-linear sigma model is classically scale-invariant. The function T, on the other hand, has naive scaling dimension of mass, so (classical) scale-invariance requires that we set it to zero. The reader can easily check that, in this case, the operator (1) is invariant under all conformal transformations that preserve C. This symmetry is further enhanced if, as a result of the field equations, the induced one-form B ≡ BM(Φ)∂αΦM +εαβBM(Φ)∂β ΦM dζα (3) is a flat U(n) connection, i.e., if in laconic notation dB + [B ,B ] = 0. The loop operator is in this case invariant under arbitrary continuous deformations of C, as follows from the non-abelian Stoke’s theorem. Such defects can therefore be called topological. The eigenvalues of topological loops W(C), with C winding around compact space, are charges conserved by the time evolution. The existence of a one- (spectral-) parameter family of flat connections is, for this reason, often tantamount to classical integrability, see [6]. Quantization breaks, in general, the scale invariance of the defect loop even when the bulk theory is conformal. This is because the definition of W(C) requires the introduction of a short-distance cutoff ε. As the cutoff is being removed the couplings run to infrared fixed points, B(ε) → B∗ and B(ε) → B∗ as ε → 0. I will explain later that this renormalization-group flow can be described perturbatively [8] by generalized Dirac–Born–Infeld equations. The fixed-point operators commute with a diagonal conformal algebra. More specifically, if C is the circle of a cylindrical spacetime, and LN, LN the left- and right-moving Virasoro generators, then [LN −L−N, trVW∗ (C)] = 0 ∀N. (4) Topological quantum defects satisfy stronger conditions: they must commute separately with the LN and with the LN. These facts can be illustrated with the symmetry-preserving defect loops of the WZW model [8]. Consider the following chiral, symmetry-preserving defect: Or(C) = χr(Pei C λJat a ), (5) where Ja are the left-moving Kac–Moody currents, t a the generators of the global group G, and χr the character of the G-representation, r, carried by the state-space of the defect. In the classical theory Or(C) is topological for all values of the parameter λ. But upon quantization, the spectral parameter runs from the UV fixed point λ∗ = 0 to an IR fixed point λ∗ 1/k, where k is the level of the Kac–Moody algebra (and k 1 for perturbation theory to be valid). It is interesting here to
On the Symmetries of Classical String note [8]that one can regularize(5)while preserving the following symmetries:(a) chirality,i.,o(C),JN=0 for all right-moving Kac-Moody(and Virasoro)gen- erators,(b)translations on the cylinder, i.e., O(C), Lo+Lo=0, and(c) global Gleft-invariance. These imply, among other things, that the RG flow can be restricted to the single parameter n, and that the IR fixed-point loop operator is topological This fixed-point operator is the quantum-monodromy matrix of the WZW model [4] It can be constructed explicitly, to all orders in the 1/k expansion, as a central ele- ment of the enveloping algebra of the Kac-Moody algebra 3, 32] The above renormalization-group flow describes, for G= SU(2), the screening of a magnetic impurity interacting with the left-moving spin current in a quantum wire. This is the celebrated Kondo problem[48] which can be solved exactly by the Bethe ansatz [5, 46]. It was first rephrased in the language of conformal field theory by Affleck [1]. Close to the spirit of our discussion here is also the work of Bazhanov et al. [11-13], who proposed to study quantum loop operators in minimal models using conformal(as opposed to integrable lattice-model) techniques. Topological loop operators were first introduced and analyzed in CFT by Petkova and Zuber [40] Working directly in the CFT makes it possible to use the powerful(geometric and algebraic) tools that were developed for the study of D-branes 2 Interfaces as Spectrum-Generating Symmetries Conformal defects in a sigma model with target .l can be mapped to conformal boundaries in a model with target . l by the folding trick [10, 39],i.e, by folding space so that all bulk fields live on the same side of the defect. Confor- mal boundaries can, in turn, be described either as geometric D-branes [41 ,or algebraically as conformal boundary states on the cylinder [17, 42]. In the latter de- cription space is taken to be a compact circle, and the boundary state is a(generally entangled) state of the two decoupled copies of the conformal theory l∞》=∑ana1,a1)|,2) (6 Here a(a) labels the state of the left-(right-)movers in the jth copy Unfolding reverses the sign of time for one copy, and thus transforms the corresponding states by hermitean conjugation. This converts 98)to a formal operator, 6, which acts on the Hilbert space. of the conformal field theory. The fixed-point operators of the previous section are all, in principle, unfolded boundary states This discussion can be extended readily to the case where the theories on the left and on the right of the defect are different, CFTI# CF more properly, called interfaces or domain walls. They can be described similarly by a boundary state of CFTl@ CFT2, or by the corresponding unfolded operator Strictly-speaking, in the Kondo setup the magnetic impurity interacts with the s-wave conduction electrons of a 3D metal. This is mathematically identical to the problem discussed here
On the Symmetries of Classical String Theory 19 note [8] that one can regularize (5) while preserving the following symmetries: (a) chirality, i.e., [Oε r (C),J a N] = 0 for all right-moving Kac–Moody (and Virasoro) generators, (b) translations on the cylinder, i.e., [Oε r (C),L0 ± L0] = 0, and (c) global Gleft-invariance. These imply, among other things, that the RG flow can be restricted to the single parameter λ, and that the IR fixed-point loop operator is topological. This fixed-point operator is the quantum-monodromy matrix of the WZW model [4]. It can be constructed explicitly, to all orders in the 1/k expansion, as a central element of the enveloping algebra of the Kac–Moody algebra [3, 32]. The above renormalization-group flow describes, for G = SU(2), the screening of a magnetic impurity interacting with the left-moving spin current in a quantum wire. This is the celebrated Kondo problem1 [48] which can be solved exactly by the Bethe ansatz [5,46]. It was first rephrased in the language of conformal field theory by Affleck [1]. Close to the spirit of our discussion here is also the work of Bazhanov et al. [11–13], who proposed to study quantum loop operators in minimal models using conformal (as opposed to integrable lattice-model) techniques. Topological loop operators were first introduced and analyzed in CFT by Petkova and Zuber [40]. Working directly in the CFT makes it possible to use the powerful (geometric and algebraic) tools that were developed for the study of D-branes. 2 Interfaces as Spectrum-Generating Symmetries Conformal defects in a sigma model with target M can be mapped to conformal boundaries in a model with target M ⊗M by the folding trick [10, 39], i.e., by folding space so that all bulk fields live on the same side of the defect. Conformal boundaries can, in turn, be described either as geometric D-branes [41], or algebraically as conformal boundary states on the cylinder [17, 42]. In the latter description space is taken to be a compact circle, and the boundary state is a (generally entangled) state of the two decoupled copies of the conformal theory: ||B = ∑Bα1α˜ 1a2α˜ 2 |α1,α˜ 1 ⊗|α2,α˜ 2 . (6) Here αj (α˜ j) labels the state of the left- (right-) movers in the jth copy. Unfolding reverses the sign of time for one copy, and thus transforms the corresponding states by hermitean conjugation. This converts ||B to a formal operator, O, which acts on the Hilbert space H of the conformal field theory. The fixed-point operators of the previous section are all, in principle, unfolded boundary states. This discussion can be extended readily to the case where the theories on the left and on the right of the defect are different, CFT1 = CFT2. Such defects should be, more properly, called interfaces or domain walls. They can be described similarly by a boundary state of CFT1⊗ CFT2, or by the corresponding unfolded operator 1 Strictly-speaking, in the Kondo setup the magnetic impurity interacts with the s-wave conduction electrons of a 3D metal. This is mathematically identical to the problem discussed here
021: 41-56. Conformal interfaces correspond to operators that intertwine the action of the diagonal virasoro algebra. 23)-72)O21=O21(L-) while topological interfaces intertwine separately the action of the left-and right movers. In the string-theory literature conformal interfaces were first studied as holographic duals [10, 18, 20, 37] to codimension-one anti-de sitter branes [9, 36 Note that conformal boundaries are special conformal interfaces for which CFT2 is the trivial theory, i. e, a theory with no massless degrees of freedom. If 01e is the corresponding operator(where the empty symbol denotes the trivial theory) ther conformal invariance implies that (LO)-E_ )O Let me now come to the main point of this talk. Consider a closed-string back ground described by the worldsheet theory CFTl, and let 010 correspond to a D-brane in this background. Take the worldsheet to be the unit disk, or equiva lently the semi-infinite cylinder, with the boundary described by the above D-brane. Consider also a conformal interface 021, where CFt2 describes another admissi ble closed-string background Now insert this interface at infinity and push it to the boundary of the cylinder, as in Fig. 1. The operation is, in general, singular except when 021 is a topological interface in which case it can be displaced freely. Let us assume, more generally, that this fusion operation can be somehow defined and yields a boundary state of CFT2 which we denote by 021 0 010. We assume that he Virasoro generators commute past the fusion symbol. It follows then from(7) that the new boundary state is conformal whenever the old one was. Since con- formal invariance is equivalent to the classical string equations, one concludes that 021 acts as a spectrum-generating symmetry of classical string theory. Conformal interfaces could, in other words, play a similar role as the Ehlers-Geroch transfor mations [19, 27]of General Relativity CFT2 Fig. 1 An interface brought from infinity to the boundary of a cylindrical D-branes of one bulk cft to those of the other Conformal interfaces betwee eories with the same central charge act thus as spectrum-generating symmetries of classi theory. In many worked-out examples these include and extend the perturbative dualities, and other classical symmetries, of the open- and closed-string action
20 C.P. Bachas O21 : H1 → H2. Conformal interfaces correspond to operators that intertwine the action of the diagonal Virasoro algebra, (L(2) N −L (2) −N)O21 = O21(L(1) N −L (1) −N), (7) while topological interfaces intertwine separately the action of the left- and rightmovers. In the string-theory literature conformal interfaces were first studied as holographic duals [10, 18, 20, 37] to codimension-one anti-de Sitter branes [9, 36]. Note that conformal boundaries are special conformal interfaces for which CFT2 is the trivial theory, i.e., a theory with no massless degrees of freedom. If O1/0 is the corresponding operator (where the empty symbol denotes the trivial theory) then conformal invariance implies that (L(1) N −L (1) −N)O1/0 = 0. Let me now come to the main point of this talk. Consider a closed-string background described by the worldsheet theory CFT1, and let O1/0 correspond to a D-brane in this background. Take the worldsheet to be the unit disk, or equivalently the semi-infinite cylinder, with the boundary described by the above D-brane. Consider also a conformal interface O21, where CFT2 describes another admissible closed-string background. Now insert this interface at infinity and push it to the boundary of the cylinder, as in Fig. 1. The operation is, in general, singular except when O21 is a topological interface in which case it can be displaced freely. Let us assume, more generally, that this fusion operation can be somehow defined and yields a boundary state of CFT2 which we denote by O21 ◦ O1/0. We assume that the Virasoro generators commute past the fusion symbol. It follows then from (7) that the new boundary state is conformal whenever the old one was. Since conformal invariance is equivalent to the classical string equations, one concludes that O21 acts as a spectrum-generating symmetry of classical string theory. Conformal interfaces could, in other words, play a similar role as the Ehlers–Geroch transformations [19, 27] of General Relativity. CFT2 CFT1 CFT2 D2 D1 Fig. 1 An interface brought from infinity to the boundary of a cylindrical worldsheet maps the D-branes of one bulk CFT to those of the other. Conformal interfaces between two theories with the same central charge act thus as spectrum-generating symmetries of classical string theory. In many worked-out examples these include and extend the perturbative dualities, and other classical symmetries, of the open- and closed-string action