On the Relation between Dualities and Gauge Symmetries On the Relation between Dualities and Gauge Symmetries Sebastian de Haro, Nicholas Teh, and Jeremy Butterfield*y We make two points about dualities in string theory. The first point is that the conception of duality, which we will discuss, meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. The second point is a re- sult about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than one might expect: each of a certain class of gauge symmetries in the gravity theory, that is, diffeomorphisms, is related to a position-dependent symmetry of the gauge theory. 1. Introduction. In this article, we make two main points about how duality and gauge symmetry are connected. Both points are about dualities in string theory, and both have the ‘flavor’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a du- ality as an ‘isomorphism’ between theories. In section 3, we take a theory to be given by a triple comprising a set of states, a set of quantities, and a dy- namics, so that a duality is an appropriate ‘structure-preserving’ map between such triples. This discussion will be enough to establish our first point, namely, dual theories can indeed ‘say the samething in different words’—which is rem- iniscent of gauge symmetries. Our second point (secs. 4 and 5) is much more specific. We give a result about a specific (complex and fascinating) duality in string theory, gauge/ gravity duality, which we introduce in section 4, using section 3’s concep- tion of duality. We state this result in section 5. (More details are in De Haro, Teh, and Butterfield [2016], and the proof is in De Haro [2016b].) It says, roughly speaking, that each of an important class of gauge symmetries in one of the dual theories (a gravity theory defined on a bulk volume) is mapped *To contact the authors, please write to: Jeremy Butterfield, Trinity College, Cambridge, CB2 1TQ, United Kingdom; e-mail: jb56@cam.ac.uk. yWe thank various audiences, especially the audience at PSA 2014, and two referees. Nicholas Teh thanks the John Templeton Foundation for supporting his work on this ar- ticle. Philosophy of Science, 83 (December 2016) pp. 1059–1069. 0031-8248/2016/8305-0035$10.00 Copyright 2016 by the Philosophy of Science Association. All rights reserved. 1059 This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 1060 SEBASTIAN DE HARO ET AL. All u by the duality to a gauge symmetry of the other theory (a conformal field the- ory defined on the boundary of the bulk volume). This is worth stressing since some discussions suggest that all the gauge symmetries in the bulk the- ory will not map across to the boundary theory but instead be ‘invisible’ to it. To set the stage for these points, section 2 describes the basic similarity be- tween the ideas of duality and gauge symmetry: that they both concern ‘say- ing the same thing in different words’. 2. Saying the Same Thing in Different Words. We use the general term ‘duality’ to denote the formal equivalence between two theories, that is, a bi- jection between the sets of mathematical ‘states’ and ‘quantities’ of a formal theory, without further specifying the physical interpretation of those states and quantities, such as unitary equivalence for quantum theories. At the other extreme, one might be using the duality to describe the “universe” (as in quantum gravity): on an ‘internal interpretation’ (cf. Dieks, van Dongen, and de Haro 2015) the equivalence is then not merely formal or schematic but also empirical and physical. As to gauge symmetry, it has (i) a general philosophical meaning and (ii) a specific physical meaning, as do cognate terms like ‘gauge-dependent’, and so on, as follows: i) (Redundant) If a physical theory’s formulation is redundant (i.e., roughly, it uses more variables than the number of degrees of freedom of the system being described), one can often think of this in terms of an equivalence relation, ‘physical equivalence’, on its states, so that gauge symmetries are maps leaving each class (called a ‘gauge orbit’) invariant. Leibniz’s criticism of Newtonian mechanics provides a pu- tative example: he believed that shifting the entire material contents of the universe by one meter must be regarded as changing only its de- scription and not its physical state. ii) (Local) Ifa physical theoryhasa symmetry(i.e.,roughly,a transforma- tion of its variables that preserves its Lagrangian) that transforms some variables in a way dependent on space-time position (and is thus ‘local’), then this symmetry is called ‘gauge’. In the context of Yang-Mills the- ory, these variables are ‘internal’, whereas in the context of general rel- ativity, they are space-time variables—both types of examples will occur in sections 4 and 5. Although Local is often a special case of Redun- dant, it will be important to us that this is not always so. For we will be concerned with Local gauge symmetries (specifically diffeomorphisms) that do not tend to the identity at space-like infinity and that can thus change the state of a system relative to its environment.1 1. Compare the discussion in Greaves and Wallace (2014) and Teh (2016). This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DUALITIES AND GAUGE SYMMETRIES 1061 These sketches are enough to suggest that duality and gauge symmetry are likely to be related. The obvious suggestion to make is that the differences between two dual theories will be like the differences between two formula- tions of a gauge theory: they ‘say the same thing, despite their differences’. Indeed, for several notable dualities—including some string theory dualities (and our case of gauge/gravity duality) in which the dual theories are strik- ingly disparate—this is the consensus among physicists. Besides, several phil- osophical commentators endorse this consensus (e.g., Rickles 2011, secs. 2.3, 5.3; 2015; Dieks et al. 2015, sec. 3.3.2; De Haro 2016a, sec. 2.4; Huggett 2016, secs. 2.1, 2.2). But beware of a false impression: roughly, that dualities between gauge theories must ignore Local gauge structure. To see how this impression arises, assume for the moment (falsely) that all Local gauge sym- metries exemplify Redundant. Then given a duality between two theories’ gauge-invariant structures, it would be surprising if the duality also mapped their gauge-dependent structures (gauge symmetries and gauge-dependent quantities) into each other. For, think of the everyday analogy in which (i) a duality is like a translation scheme between languages, and so (ii) gauge struc- ture, a theory allowing several gauges, is like a language having several syn- onyms for one concept. (Indeed, this analogy is entrenched in physics: phys- icists call the definition of the duality transformation the ‘dictionary’, etc.) One would not expect a translation scheme to match the languages’ synonym structures, that is, to translate each of a set of synonyms in language L1 by just one of the corresponding (synonymous) set of synonyms in L2 and vice versa. Analogously, it seems that for a duality between gauge theories, the gauge- dependent structures on the two sides will not be related by the duality. Each such structure, on one side, will be ‘invisible’ to the other side—at least, ‘in- visible’ if you are using just the duality map ‘to look through’. Thus, the impression is tempting. Indeed we think the impression is wide- spread because of this line of thought: as we will see, no lesser authors than Horowitz and Polchinski seem to endorse it. It is this impression that we will rebut. We of course admit that in general, Local gauge-dependent structures may be ‘invisible from the other side’. But surprisingly, for the case of gauge/gravity duality, some Local gauge struc- ture is visible. This is our result in section 5: roughly, that each of a certain class of Local gauge symmetries of the gravity/bulk theory is mapped by the duality to Redundant gauge symmetries of its dual, that is, the position- dependent conformal symmetries of the boundary conformal field theory. 3. Duality as a Symmetry between Theories. In this section, we propose schematic definitions of a physical theory and of a duality. The definitions will be general enough to apply to both classical and quantum physics, al- though we of course have quantum physics, especially quantum field theory and string theory, mostly in mind. This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 1062 SEBASTIAN DE HARO ET AL. All u The idea of the definitions is that duality is a symmetry between theories. But while symmetry is a matter of sameness between distinct situations, ac- cording to a theory, duality is a matter of sameness between theories. So a du- ality is, roughly, an ‘isomorphism’ between two theories or from a theory to itself. To make these ideas more precise, we define, first, theories (sec. 3.1) and, then, duality maps (sec. 3.2). 3.1. The Definition of ‘Theory’. We take a theory as a triple T 5 hS, Q, Di of state-space S, set of quantities Q, and dynamics D. States and quantities are assignments of values to each other. So there is a natural pairing, and we write hQ; si for the value of Q in s. In classical physics, we think of this as the system’s intrinsic possessed value for Q, when in s; in quantum physics, we think of it as the (orthodox, Born-rule) expectation value of Q, for the system in s. As to the dynamics, D, this can here be taken as the deterministic time evolution of states. Our construal of theory allows us to recognize that usually there are many token systems of the type treated by a theory in our sense. This prompts an important point: we of course recognize (as everyone must) that there can be cases of two disjoint parts of reality—in Hume’s phrase, two ‘distinct exis- tences’—whose formal or schematic structure matches exactly, ‘are isomor- phic’, in the taxonomy used by some theory but are otherwise different, that is, distinct and known to be distinct. This point has an obvious corollary for what in section 1 we announced as our first main point: that dual theories ‘can say the same thing’. For it is natural to take ‘saying the same thing’ to mean saying (i) the same assertions about (ii) the same objects. Then we must beware that a definition of dual theories as ‘isomorphic’ (as in sec. 3.2 below) will cue in only to i, and the possibility of ‘distinct but iso- morphic existences’ implies that this does not secure ii. This leads in to a fi- nal point. There is one scientific context in which the idea of distinct but isomorphic existences falls by the wayside, namely, when we aim to write down a cos- mology, that is, a theory of the whole universe. For such a theory, there will be, ex hypothesi, only one token of its type of system, that is, the universe. Agreed, this scientific context is very special and very ambitious: we rarely aim to write down a cosmology. But of course, it is the context of much work in string theory and quantum gravity more generally. So it will apply when we turn, in section 4, to string theory and gauge/gravity duality.2 2. In this context, one might go further than setting aside the possibility of distinct but isomorphic existences. One might also hold that the interpretation of our words, i.e., of the symbols in the cosmological theory, must be fixed from within the theory: this view is endorsed, under the label ‘internal point of view’, in Dieks et al. (2015, sec. 3.3.2) and De Haro (2016a, sec. 2.4). We should note, however, that a lot of work on gauge/gravity duality concerns systems that are much smaller than the universe. For gauge/gravity ideas This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DUALITIES AND GAUGE SYMMETRIES 1063 3.2. The Definition of ‘Duality’. We now define a duality as a ‘meshing’ map between two theories: T1 5 hS1, Q1, D1i is dual to T2 5 hS2, Q2, D2i if and only if there are bijections ds : S1 → S2, dq : Q1 → Q2 (d for ‘duality’) that give matching values of quantities on states in the following sense:3 hQ1; s1i1 5 hdq(Q1); ds(s1)i2, 8 Q1 ∈ Q1, 8 s1 ∈ S1: (1) This definition of ‘duality’ is obvious and simple, given our conception of ‘theory’. One could strengthen the definition in various ways, for example, to require that ds be a symplectomorphism for Hamiltonian theories, unitary for quantum theories. And there is a whole tradition of results relating the requirement of matching values (eq. [1]) to such strengthenings. But we do not need to pursue such strengthenings. It is not just that a simple defini- tion is clearer and can be weakened and qualified, as needed. Also, with this definition, we immediately establish our first main point: that two dual the- ories can be gauge related, in section 2’s general philosophical sense of be- ing physically equivalent. More precisely, the point follows immediately, when we bear in mind our preceding discussion and the “can be” in ‘can be gauge related’ (i.e., sec. 3.1’s allowance of distinct but isomorphic ‘exis- tences’) and how this allowance falls by the wayside for a cosmology, that is, theory of the whole universe. And more important, we will see in section 4 that this definition of duality is indeed instantiated, albeit formally, by gauge/ gravity duality and using maps ds, dQ that are (formally) unitary. 4. Gauge/Gravity Duality. We first (sec. 4.1) give a brief introduction to the original and most studied case of gauge/gravity duality: AdS/CFT, the duality between a gravity theory on anti–de Sitter space-time (AdS) and a conformal field theory (CFT) on its boundary. Then in section 4.2, we argue that section 3.2’s simple definition of duality is indeed instantiated, albeit formally, by AdS/CFT. For a philosophically informed introduction to gauge/ gravity dualities, see De Haro, Mayerson, and Butterfield (2016). 4.1. Introducing AdS/CFT. The general idea of gauge/gravity duality is that some gauge quantum field theories (QFTs) in d space-time dimensions are dual to some quantum theories of gravity in a (d 1 1)-dimensional space- 3. We also require the dynamics to mesh in the obvious sense: i.e., ds commutes with (is equivariant for) the two theories’ dynamics. In sec. 4.2 we add a stronger condition, i.e., that the values of the quantities on any pair of states match: h Q1; s1, s2 i1 5 h dq(Q1); ds(s1), ds(s2) i2, 8 s1, s2 ∈ S1. For quantum theories, this strengthening of this section’s simple approach is natural because it amounts to unitary equivalence. turn out to be very useful for understanding strongly coupled microscopic systems, like a quark-gluon plasma as explored by instruments like the Relativistic Heavy Ion Collider; cf. McGreevy (2010). This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 1064 SEBASTIAN DE HARO ET AL. All u time that has the d-dimensional manifold of the QFTas its conformal bound- ary. Pictorially, AdSd11 is a family of copies of a d-dimensional Minkowski space-time of varying sizes. The family is parameterized by the coordinate r. The boundary is the locus r → 0. The original case (Maldacena 1999) takes the QFT to be a strong cou- pling regime of a supersymmetric gauge theory (SYM, for ‘super Yang- Mills’) in four space-time dimensions (d 5 4), with gauge group SU(N). To display the AdS/CFT correspondence in a simple case, take the path in- tegral for the scalar field, in Euclidean signature, as a function of the bound- ary conditions:4 Z½f(0)� ≔ ð f(r,x)jbdy5f(0)(x) Df exp 2Sbulk½f�ð Þ: (2) The AdS/CFT correspondence now states that this is the generating func- tional in the CFT: Z½f(0)� 5 hexp 2 ð d4xf(0)(x)O(x) � � i, (3) where f(0)(x) is a ‘source’ that couples to a certain gauge-invariant operator O(x), whose scaling dimension D is determined by the mass of the bulk sca- lar field (in a CFT, the scaling dimension uniquely determines a gauge- invariant, scalar operator). The exponential is evaluated in the vacuum state of the theory. In the CFT, the vacuum correlation functions hO(x1) ⋯ O(xn)i are cal- culated from the generating functional Z½f(0)�: one takes functional deriva- tives of the right-hand side of (3) with respect to the source and sets the source to zero. But according to the correspondence (eqq. [2] and [3]), the correlation functions can also be calculated using the bulk theory. For instance, to cal- culate the two-point function, one can use the leading classical approxima- tion to (2), which is just the classical action evaluated on solutions that sat- isfy the prescribed boundary conditions. Up to normalization, the result is hO(x)O(y)i 5 1jx 2 yj2D , (4) where D is the scaling dimension of the operator, and Fx 2 yF the distance between the two boundary points. This result matches the CFT result pre- cisely. 4. For simplicity, we are now taking the metric to be fixed, and we are suppressing it in the notation. This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DUALITIES AND GAUGE SYMMETRIES 1065 4.2. AdS/CFT Exemplifies the Definition of Duality. We now turn to how (3) instantiates our definition of duality (i.e., eq. [1]). We begin with the boundary, CFT, side. In a CFT, the quantities one is interested in are the correlation functions: hs O(x1) ⋯ O(xn)j js0i, (5) of products of gauge-invariant operators, in some states s, s0. In section 4.1 we showed how to calculate the two-point function in the vacuum state, leading to (4). Thus, the gauge theory side of the duality is, by construction, framed in the language of states and quantities that was used in section 3.1 to define a theory. We admit that in the absence of proper nonperturbative methods for rigorously defining higher-dimensional theories such as SYM, one should be wary of expressions such as (5). To assume that SYM makes sense nonperturbatively is to assume that expressions such as (5) make sense. Similar remarks hold for the path-integral expressions (2) used in the bulk. In the present state of knowledge, one simply assumes that these formal structures will some day be defined with rigorous mathematics: de- fining these expressions nonperturbatively would amount to proving AdS/ CFT. This is why the AdS/CFT correspondence still has the status of a con- jecture. So in order to substantiate our claim—that section 3.2’s simple definition of duality is instantiated by AdS/CFT—we need to argue that the bulk side in (2) can be written in the language of states and operators. Accepting the comments in the previous paragraph, the point follows readily from the cor- respondence between path integral quantization and the Hilbert space de- scription of states and operators, provided one can adapt that correspon- dence to take into account the fact that (2) contains boundary, rather than bulk, sources. This can in fact be done: for details, compare De Haro, Mayerson, and Butterfield (2016, secs. 5, 6.1) and De Haro, Teh, and But- terfield (2016, sec. 4.2). So to sum up, the bulk side of the duality gives rise to the theory Tbulk 5 hH1, Q1i, whereas the boundary dual gives rise to Tbdy 5 hH2, Q2i, where Hi is a Hilbert space, Qi is some algebra of operators, and unitary dynamics is implicit. The above remarks show that (3) yields the duality maps ds and dq, which are isomorphisms of Hilbert spaces and operators, respectively, satisfying (1). 5. A Pandora’s Box for Gauge Invariance 5.1. Gauge Invariance and Duality in AdS/CFT. Let us now turn to the topic of gauge invariance in AdS/CFT. One typically identifies Local gauge symmetry at the level of the classical Lagrangian, which enters into the path integral formulas of (3). In the bulk theory, the main Local gauge This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 1066 SEBASTIAN DE HARO ET AL. All u symmetry is diffeomorphism symmetries. Furthermore, as mentioned at the end of section 4.1, one can add U(1) (Yang-Mills type) gauge symmetry by coupling a U(1) gauge field to the bulk metric. In the boundary CFT, how- ever, one has an SU(N) gauge symmetry acting on the gauge fields. What about gauge invariance at the quantum level? Were we dealing (al- beit perturbatively) with a simple case of gauge theory (e.g., QED), we would obtain the quantum state space by constructing a nilpotent Hermitian (BRST) operator Q and defining the physical (gauge-invariant) states as those annihilated by Q: cohomology then ensures that each state represents only a gauge orbit.5 Thus, for two such quantum theories T1 and T2, there can be no question about whether a duality map relates their classical gauge symmetries: these symmetries are simply not represented to begin with. In contrast, one cannot proceed in this way for the gauge theories in- volved in AdS/CFT (and many other dualities) because each side of the du- ality typically contains a nonperturbative sector, for which we cannot directly construct the quantum state space. It is precisely here that the duality rela- tion (3) is strikingly useful; for example, by exchanging the nonperturbative sector of Tbdy with the perturbative sector of Tbulk (i.e., semiclassical super- gravity), it allows us to indirectly construct the quantum state space of Tbdy by means of Tbulk. But this also opens up a Pandora’s box for gauge invariance. For, prima facie, it allows that the Local gauge symmetries of Tbulk might be related to the symmetries of Tbdy. As we will now explain, Horowitz and Polchinski have argued against this possibility, that is, for what section 2 called ‘invis- ibility’. Recall that (naively) a duality is a bijection between the gauge-invariant content (states and quantities) of two theories T1 and T2. Now suppose one is skeptical that a duality would match elements of a gauge orbit G1(G1 ⊂ S1) in theory T1 with elements of the dual gauge orbit G2 ≔ ds(G1) in T2. That is, in terms of section 2’s everyday analogy with translations between languages, one doubts that a translation will match each synonym in a set of synonyms in T1 with a synonymous member of an equinumerous set of synonyms in T2. Then one would naturally expect that the duality mapping (e.g., as given for AdS by eq. [3]) secures 5. Fa woul se sub (Invisibility) Each gauge symmetry of T1, that is, permutation of S1 that leaves each gauge orbit invariant (in the analogy, permutation of T1’s words that leaves each synonymy equivalence class invariant) carries over to only the identity permutation on S2 and vice versa. In this sense, the duality only relates the gauge orbits of T1 and T2. iling to perform this move, i.e., naively quantizing the gauge fields of the theory, d lead to an indefinite Fock space (i.e., a space containing states with negative norm). This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM ject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DUALITIES AND GAUGE SYMMETRIES 1067 This is indeed the definition of ‘invisibility’ that Horowitz and Polchinski (2006, sec. 1.3.2) use. Polchinski (2016, secs. 2.3 and 2.4) demonstrates this property for the duality between p-form gauge theories, and Horowitz and Polchinski (2006, sec. 1.3.2) claim that in AdS/CFT, even the Local gauge symmetry of bulk, that is, diffeomorphism symmetry, displays such ‘invis- ibility’, thus showing that diffeomorphism symmetry (and relatedly, space- time) is an ‘emergent’ property: “the gauge variables of AdS/CFT are triv- ially invariant under the bulk diffeomorphisms, which are entirely invisible in the gauge theory” (12). Is this latter claim correct? To be sure, there exist simple examples of ‘in- visibility’ in AdS/CFT, from both the bulk and the boundary perspective. For example, from the bulk: Since all correlation functions defined by (3) are invariant under boundary Local gauge symmetry, this symmetry is not seen in the bulk. Thus, one might expect that the fundamental Local gauge symmetry of the bulk theory, that is, diffeomorphism symmetry, is also invisible in the boundary theory. In the next section, we argue that not all the diffeomor- phisms of the bulk theory are invisible, and we sketch criteria to distinguish visible from invisible diffeomorphisms. Furthermore, there is a sense in which the duality map (3) maps Local gauge symmetries in the bulk to position- dependent Redundant symmetries in the boundary. 5.2. AdS/CFT’s Visible and Invisible Diffeomorphisms. As mentioned in section 5.1, we will scrutinize the statement that all the bulk diffeomorphisms are invisible to the boundary theory. We construe the notion of ‘visible’ dif- feomorphism, along the lines of Horowitz and Polchinski’s own examples, as one that does not restrict to the identity map on the boundary, whereas ‘invisible’ means that it does restrict to the identity. First, consider the class of diffeomorphisms that satisfy the following three conditions: i) (Fixed) Leave the form of the bulk metric fixed; ii) (Invisibility) Are equal to the identity, at the boundary; iii) (Existence) Are nontrivial (i.e., not the identity map) in the bulk. One can show that if the boundary dimension d is odd, then there are no such diffeomorphisms.6 More precisely, there are no diffeomorphisms that are equal to the identity at infinity and extend nontrivially to the bulk. The desiderata i–iii cannot all be met. 6. In the case of even d, diffeomorphism invariance is broken by the presence of the conformal anomaly. For a discussion, see De Haro, Mayerson, and Butterfield (2016, sec. 6.2) and De Haro, Teh, and Butterfield (2016, secs. 4.2.1 and 5.2.2). This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 1068 SEBASTIAN DE HARO ET AL. All u We sketch the argument for this in a moment. But to do so, we need to consider relaxing the above conditions because this will allow us to identify the relevant visible diffeomorphisms. And this relaxation will also lead in to our final point about Local diffeomorphisms that are Redundant. We thus replace Invisibility by the weaker ii0) (Invariance) Leave all the boundary quantities (in particular, the metric) invariant; that is, the bulk diffeomorphisms can be nonvan- ishing (hence, visible) on the boundary but must leave all the CFT quantities invariant. A sketch of the argument that shows the incompatibility of i–iii is as fol- lows. Let the bulk metric be that of an Einstein space (i.e., a solution of gen- eral relativity’s field equations with a negative cosmological constant), so that an arbitrary metric is induced at the boundary. Requiring that the infin- itesimal diffeomorphisms leave the asymptotic form of the metric fixed, one finds the following condition for them: ∇iyj(x) 1 ∇jyi(x) 2 2 d gij(x) ∇ k yk(x) 5 0: (6) Here, yi is a reparameterization of the boundary coordinates x (a d-dimensional vector). There are two important points about this equation, which is the math- ematical representation of Invariance: a) This equation is the condition for an infinitesimal boundary coordi- nate transformation yi to give a local scale transformation: thus, the yi generate the boundary conformal group. b) As a consequence of a, when one requires in addition to Invariance also Invisibility, that is, that the boundary coordinate transformations vanish (i.e., yi 5 0), then one can subsequently show that the diffeo- morphism in fact is the identity map throughout the bulk. Requiring Invisibility of the diffeomorphism thus deprives it of Existence. But, Invariance is compatible with Existence, that is, allowing yi to be non- zero. In this case, the bulk diffeomorphism has two parts: (1) a nontrivial bound- ary coordinate transformation (of a special kind: a conformal transformation) and (2) a compensating reparameterization of the bulk coordinate r, so that the overall boundary metric is invariant, yet the diffeomorphism nontrivial. The above is exactly what one expects: we have identified the ‘residual’ diffeomorphisms, that is, the ones that preserve the bulk form of the metric (as per Fixed) and preserve the boundary metric (as per Invariance) as the boundary conformal group. Thus, the CFT’s group of invariances arises in this way explicitly from bulk diffeomorphisms. This content downloaded from 131.111.184.102 on February 07, 2017 03:31:16 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). DUALITIES AND GAUGE SYMMETRIES 1069 This technical discussion returns us to our distinction between Local and Redundant and the ‘internal point of view’ mentioned in footnote 2. Thus, the boundary coordinate transformations yi considered here are Local: they are position-dependent coordinate transformations, but they represent over- all transformations of the correlation functions (5) derived from either (2) or (3). The correlation functions are indeed covariant under such coordinate transformations.7 Therefore, on the ‘internal point of view’ mentioned in footnote 2, two states of the universe related by such transformations are physically equivalent: the transformations count as Redundant gauge symme- tries. But, the contrary ‘external point of view’ interprets (3) as coupled to an already interpreted physical background, represented by f(0)(x), and in that case, the diffeomorphisms would not be Redundant but would become phys- ical. When we describe Galileo’s ship relative to the quay, its state of motion is indeed physically meaningful. REFERENCES De Haro, Sebastian. 2016a. “Dualities and Emergent Gravity: Gauge/Gravity Duality.” Studies in History and Philosophy of Modern Physics, forthcoming. doi:10.1016/j.shpsb.2015.08.004. ———. 2016b. “Is Diffeomorphism Invariance Emergent?” Unpublished manuscript, University of Cambridge. De Haro, Sebastian, Daniel Mayerson, and Jeremy Butterfield. 2016. “Conceptual Aspects of Gauge/ Gravity Duality.” Foundations of Physics, forthcoming. doi:10.1007/s10701-016-0037-4. De Haro, Sebastian, Nicholas Teh, and Jeremy Butterfield. 2016. “Comparing Dualities and Gauge Symmetries.” Studies in History and Philosophy of Modern Physics, forthcoming. doi:10.1016 /j.shpsb.2016.03.001. Dieks, Dennis, Jeroen van Dongen, and Sebastian de Haro. 2015. “Emergence in Holographic Sce- narios for Gravity.” Studies in History and Philosophy of Modern Physics B 52:203–16. Greaves, Hillary, and David Wallace. 2014. “Empirical Consequences of Symmetries.” British Jour- nal for the Philosophy of Science 65:59–89. Horowitz, Gary, and Joseph Polchinski. 2006. “Gauge/Gravity Duality.” In Towards Quantum Grav- ity? ed. Daniele Oriti. Cambridge: Cambridge University Press. Huggett, Nick. 2016. “Target Space ≠ Space.” Studies in History and Philosophy of Modern Phys- ics, forthcoming. Maldacena, Juan. 1999. “The Large N Limit of Superconformal Field Theories and Supergravity.” International Journal of Theoretical Physics 38 (4): 1113–33. McGreevy, John. 2010. “Holographic Duality with a View to Many-Body Physics.” Advances in High Energy Physics. http://dx.doi.org/10.1155/2010/723105. Polchinski, Joseph. 2016. “Dualities of Fields and Strings.” Studies in History and Philosophy of Modern Physics, forthcoming. Rickles, Dean. 2011. “A Philosopher Looks at String Dualities.” Studies in History and Philosophy of Modern Physics 42 (1): 54–67. ———. 2015. “Dual Theories: ‘Same but Different’ or ‘Different but Same’?” Studies in History and Philosophy of Modern Physics, forthcoming. http://dx.doi.org/10.1016/j.shpsb.2015.09 .005. Teh, Nicholas. 2016. “Galileo’s Gauge: Understanding the Empirical Significance of Gauge Sym- metry.” Philosophy of Science 83 (1): 93–118. 7. Of course, they are invariant under the full bulk diffeomorphisms considered, i.e., the boundary coordinate transformations and the compensating rescaling of r. 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