3.1. The Definition of ‘Theory’

We take a theory as a triple $T=〈S,Q,D〉$ 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 〈Q; s〉 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 existences’—whose formal or schematic structure matches exactly, ‘are isomorphic’, 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 isomorphic existences’ implies that this does not secure ii. This leads in to a final 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 cosmology, 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.Footnote ²

3.2. The Definition of ‘Duality’

We now define a duality as a ‘meshing’ map between two theories: $T_1=〈S_1,Q_1,D_1〉$ is dual to $T_2=〈S_2,Q_2,D_2〉$ if and only if there are bijections $d_s:S_1\to S_2,d_q:Q_1\to Q_2$ (d for ‘duality’) that give matching values of quantities on states in the following sense:Footnote ³

(1)$\begin{array}{}{〈Q_1;s_1〉}_1={〈d_q\left(Q_1\right);d_s\left(s_1\right)〉}_2,\forall Q_1\in Q_1,\forall s_1\in S_1.\end{array}$

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 d_s 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 definition is clearer and can be weakened and qualified, as needed. Also, with this definition, we immediately establish our first main point: that two dual theories can be gauge related, in section 2’s general philosophical sense of being 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 ‘existences’) 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 d_s, d_Q that are (formally) unitary.

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)-dimensional space-time that has the d-dimensional manifold of the QFT as its conformal boundary. Pictorially, AdS_d+1 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 Reference Maldacena1999) takes the QFT to be a strong coupling regime of a supersymmetric gauge theory (SYM, for ‘super Yang-Mills’) in four space-time dimensions (d = 4), with gauge group SU(N). To display the AdS/CFT correspondence in a simple case, take the path integral for the scalar field, in Euclidean signature, as a function of the boundary conditions:Footnote ⁴

(2)$\begin{array}{}Z\left[{ɸ}_{\left(0\right)}\right]≔{\int }_{ɸ(r,x{\left)\right|}_{\mathrm{bdy}}={ɸ}_{\left(0\right)}(x)}Dɸ\exp \left(-S_{\mathrm{bulk}}\left[ɸ\right]\right)\mathrm{.}\end{array}$

The AdS/CFT correspondence now states that this is the generating functional in the CFT:

(3)$\begin{array}{}Z\left[{ɸ}_{\left(0\right)}\right]=〈\exp \left(-\int d^{4}x{ɸ}_{\left(0\right)}\left(x\right)O\left(x\right)\right)〉,\end{array}$

where ɸ₍₀₎(x) is a ‘source’ that couples to a certain gauge-invariant operator $O\left(x\right)$, whose scaling dimension Δ is determined by the mass of the bulk scalar 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 $〈O\left(x_1\right)\cdots O\left(x_n\right)〉$ are calculated from the generating functional $Z\left[{ɸ}_{\left(0\right)}\right]$: one takes functional derivatives 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 calculate the two-point function, one can use the leading classical approximation to (2), which is just the classical action evaluated on solutions that satisfy the prescribed boundary conditions. Up to normalization, the result is

(4)$\begin{array}{}〈O\left(x\right)O\left(y\right)〉=\frac{1}{|x-y{|}^{2\Delta }},\end{array}$

where Δ is the scaling dimension of the operator, and |x − y| the distance between the two boundary points. This result matches the CFT result precisely.

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:

(5)$\begin{array}{}〈s\left|O\right(x_1)\cdots O(x_n\left)\right|s^{\prime }〉,\end{array}$

of products of gauge-invariant operators, in some states s, s′. 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: defining these expressions nonperturbatively would amount to proving AdS/CFT. This is why the AdS/CFT correspondence still has the status of a conjecture.

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 correspondence between path integral quantization and the Hilbert space description of states and operators, provided one can adapt that correspondence 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 (Reference De Haro, Mayerson and Butterfield2016, secs. 5, 6.1) and De Haro, Teh, and Butterfield (Reference De Haro, Teh and Butterfield2016, sec. 4.2).

So to sum up, the bulk side of the duality gives rise to the theory $T_{\mathrm{bulk}}=〈{\mathscr{H}}_1,Q_1〉$, whereas the boundary dual gives rise to $T_{\mathrm{bdy}}=〈{\mathscr{H}}_2,Q_2〉$, where ${\mathscr{H}}_i$ is a Hilbert space, $Q_i$ is some algebra of operators, and unitary dynamics is implicit. The above remarks show that (3) yields the duality maps d_s and d_q, which are isomorphisms of Hilbert spaces and operators, respectively, satisfying (1).

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 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, however, one has an SU(N) gauge symmetry acting on the gauge fields.

What about gauge invariance at the quantum level? Were we dealing (albeit 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.Footnote ⁵ Thus, for two such quantum theories T ₁ and T ₂, 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 involved in AdS/CFT (and many other dualities) because each side of the duality typically contains a nonperturbative sector, for which we cannot directly construct the quantum state space. It is precisely here that the duality relation (3) is strikingly useful; for example, by exchanging the nonperturbative sector of T _bdy with the perturbative sector of T _bulk (i.e., semiclassical supergravity), it allows us to indirectly construct the quantum state space of T _bdy by means of T _bulk.

But this also opens up a Pandora’s box for gauge invariance. For, prima facie, it allows that the Local gauge symmetries of T _bulk might be related to the symmetries of T _bdy. As we will now explain, Horowitz and Polchinski have argued against this possibility, that is, for what section 2 called ‘invisibility’.

Recall that (naively) a duality is a bijection between the gauge-invariant content (states and quantities) of two theories T ₁ and T ₂. Now suppose one is skeptical that a duality would match elements of a gauge orbit $G_1\left(G_1\subset S_1\right)$ in theory T ₁ with elements of the dual gauge orbit $G_2≔d_s\left(G_1\right)$ in T ₂. 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 T ₁ with a synonymous member of an equinumerous set of synonyms in T ₂. Then one would naturally expect that the duality mapping (e.g., as given for AdS by eq. [3]) secures

This is indeed the definition of ‘invisibility’ that Horowitz and Polchinski (Reference Horowitz, Polchinski and Oriti2006, sec. 1.3.2) use. Polchinski (Reference Polchinski2016, secs. 2.3 and 2.4) demonstrates this property for the duality between p-form gauge theories, and Horowitz and Polchinski (Reference Horowitz, Polchinski and Oriti2006, sec. 1.3.2) claim that in AdS/CFT, even the Local gauge symmetry of bulk, that is, diffeomorphism symmetry, displays such ‘invisibility’, thus showing that diffeomorphism symmetry (and relatedly, space-time) is an ‘emergent’ property: “the gauge variables of AdS/CFT are trivially 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 ‘invisibility’ 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 diffeomorphisms 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’ diffeomorphism, 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.Footnote 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.

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

• ii′) (Invariance) Leave all the boundary quantities (in particular, the metric) invariant; that is, the bulk diffeomorphisms can be nonvanishing (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 follows. Let the bulk metric be that of an Einstein space (i.e., a solution of general relativity’s field equations with a negative cosmological constant), so that an arbitrary metric is induced at the boundary. Requiring that the infinitesimal diffeomorphisms leave the asymptotic form of the metric fixed, one finds the following condition for them:

(6)$\begin{array}{}{\nabla }_i{\xi }_j\left(x\right)+{\nabla }_j{\xi }_i\left(x\right)-\frac{2}{d}{g}_{ij}\left(x\right){\nabla }^k{\xi }_k\left(x\right)=0.\end{array}$

Here, ξ_i is a reparameterization of the boundary coordinates x (a d-dimensional vector). There are two important points about this equation, which is the mathematical representation of Invariance:

• a) This equation is the condition for an infinitesimal boundary coordinate transformation ξ_i to give a local scale transformation: thus, the ξ_i 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., ξ_i = 0), then one can subsequently show that the diffeomorphism 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 ξ_i to be nonzero. In this case, the bulk diffeomorphism has two parts: (1) a nontrivial boundary 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 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 ξ_i considered here are Local: they are position-dependent coordinate transformations, but they represent overall transformations of the correlation functions (5) derived from either (2) or (3). The correlation functions are indeed covariant under such coordinate transformations.Footnote 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 symmetries. But, the contrary ‘external point of view’ interprets (3) as coupled to an already interpreted physical background, represented by ${ɸ}_{\left(0\right)}\left(x\right)$, and in that case, the diffeomorphisms would not be Redundant but would become physical. When we describe Galileo’s ship relative to the quay, its state of motion is indeed physically meaningful.