Symmetries in Quantum Field Theory and Quantum Gravity

Abstract

In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any “long-range” bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the “convex hull condition” of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and ’t Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman–Mandula theorem to discrete symmetries, and an improved explanation of the decay $\pi ^0\rightarrow \gamma \gamma$ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in $d+1$ dimensions with horizon topology ${\mathbb {T}}^p\times {\mathbb {S}}^{d-p-1}$.

Appendices

Group Theory

In this appendix we briefly review some standard aspects of Lie group theory which are necessary for our work, but which may not be common knowledge for all physicists. For many more details see eg [119, 189], our discussion of representation theory largely follows [119].

1.1 General structure of Lie groups

A Lie group is a group G which is also a smooth manifold, and for which multiplication and inversion are smooth maps in that smooth structure. A vector field X on G is called left-invariant if for any h in G it is preserved by the pushforward of the map $L_h:g\mapsto hg$. The set of left-invariant vector fields forms a real vector space ${\mathfrak {g}}$, called the Lie algebra of G, whose dimensionality equals that of the manifold, and which is closed under taking vector field commutators (abstractly a Lie algebra is a vector space with a bracket operation which is antisymmetric and obeys the Jacobi identity). If G has dimension zero as a manifold, then ${\mathfrak {g}}$ consists of only the zero vector. There are then two classic results:

Theorem A.1

(Closed subgroup theorem). Let G be a Lie group, and $H\subset G$ a subgroup of G which is topologically closed. Then H is an embedded submanifold, and thus is itself a Lie group.

Theorem A.2

(Lie group–Lie algebra correspondence). Let ${\mathfrak {g}}$ be an abstract real finite-dimensional Lie algebra. Then there exists a unique (up to isomorphism) connected simply-connected Lie group $\widetilde{G}$ whose Lie algebra is isomorphic to ${\mathfrak {g}}$. Moreover any other connected Lie group G whose Lie algebra is isomorphic to ${\mathfrak {g}}$ is itself isomorphic to a quotient of $\widetilde{G}$ by a discrete central subgroup $\Gamma \subset \widetilde{G}$. More generally, any Lie group G with a given Lie algebra is an extension of one of the connected ones by a discrete “component” group C, meaning that there is a surjective homomorphism from G to C which sends each connected component of G to a distinct element of C,⁹¹ and that therefore $C\cong G/G_0$, where $G_0$ is the identity component of G.

The proofs of these theorems use standard geometric techniques (vector flows, Frobenius’s theorem, etc), they are nicely explained in [189] (Ado’s theorem is also needed, which is proven in [119]). We will give the proof of one further result which we will need below:

Theorem A.3

Let G be a connected Lie group, and $H\subset G$ be a subgroup which contains an open neighborhood U of the identity in G. Then $H=G$.

Proof

We will show that H is both open and closed: since G is connected, this implies $H=G$. H is open because for any $h\in H$, the set hU is open in G, it contains h, and it is contained in H. Therefore $H=\bigcup _{h\in H} (hU)$. H is closed because if $g\notin H$, then we also have $gU\cap H=\emptyset$. Indeed if we had $gu=h$ for some $u\in U$ and $h\in H$, then we would have $g=hu^{-1}$, and thus $g\in H$. Therefore we have $G-H=\bigcup _{g\notin H}gU$. $\square$

1.2 Representation theory of compact Lie groups

A representation of a Lie group G on a complex vector space V is a homomorphism $\rho$ from G into the set of linear operators on V, for which the map $\Phi _\rho :G\times V\rightarrow V$ defined by $\Phi _\rho (g,v)=\rho (g)v$ is jointly continuous.⁹² If $\rho$ is injective then it is said to be faithful. $\rho$ is said to be unitary if V admits an inner product with respect to which $\rho (g)$ is unitary for any g, and is said to be finite-dimensional if V is finite-dimensional. The kernel of $\rho$, denoted $\mathrm {Ker}(\rho )$, is the set of elements of G which are mapped to the identity operator on V. $\mathrm {Ker}(\rho )$ is always a closed normal subgoup of G, and $\rho$ is faithful if and only if $\mathrm {Ker}(\rho )=\{e\}$. A subspace $S\subset V$ is called invariant if $\rho (G)S=S$, and $\rho$ is said to be irreducible if the only closed invariant subspaces are V itself and 0. By the closed subgroup theorem any finite-dimensional representation of a Lie group G is automatically smooth, which is why we only required $\rho$ to be continuous, and actually by Theorem A.9 below the same is true for infinite-dimensional unitary representations if G is compact.

The representations of a general Lie group G can be quite sophisticated, but if G is compact and $\rho$ is either unitary or finite-dimensional then there is a simple theory of all representations which can be derived from the existence of the invariant Haar measure dg on G. Indeed there is a simple theorem relating these two conditions:

Theorem A.4

Let G be a compact Lie group, and $\rho$ be a finite-dimensional representation of G. Then $\rho$ is unitary.

Proof

Let $(v,v')_0$ be any inner product on V.⁹³ We may then define a new inner product

$$\begin{aligned} (v,v')\equiv \int \mathrm{d}g(\rho (g)v,\rho (g)v')_0, \end{aligned}$$

(A.1)

which is easily shown to be an inner product with respect to which $\rho (g)$ is unitary for any g. $\square$

For a unitary representation, the orthogonal complement of an invariant subspace is also invariant. Therefore Theorem A.4 shows that any finite-dimensional representation of a compact Lie group can be decomposed into a direct sum of irreducible representations. We next establish a famous technical lemma, which we then use to establish perhaps the most remarkable feature of the representation theory of compact groups: the Schur orthogonality relations.

Theorem A.5

(Schur’s lemma). Let $\alpha$ and $\alpha '$ be finite-dimensional irreducible representations of a group G on V and $V'$ respectively (here G can be an arbitrary group and we assume no continuity properties of $\alpha$ and $\alpha '$). If $L:V\rightarrow V'$ is a linear map obeying $\alpha '(g)L=L\alpha (g)$ for all $g\in G$, then either L is a bijection or $L=0$. Moreover if $\alpha =\alpha '$ and $V=V'$, then L is a multiple of the identity.

Proof

It is easy to see that the kernel and image of L are invariant subspaces of V and $V'$ respectively. Irreducibility of $\alpha$ implies that the kernel of L is either 0 or V: if it is V then $L=0$, while if it is 0 then L is injective. If L is injective, then irreducibility of $\alpha '$ implies that its image must $V'$, so L is surjective. In the case $\alpha =\alpha '$ and $V=V'$, since L is finite-dimensional if it is not equal to zero then it has a nonzero eigenvalue $\lambda$. We may then consider the operator ${\hat{L}}\equiv L-\lambda I$, which again is a linear map which commutes with $\alpha$. But it is not injective so it must be zero. $\square$

Theorem A.6

(Schur orthogonality relations). Let $\alpha$ and $\alpha '$ be irreducible finite-dimensional representations of a compact Lie group G on the vector spaces V and $V'$, which are inequivalent in the sense that there is no invertible linear map $L:V\rightarrow V'$ such that $L\alpha (g)=\alpha '(g)L$ for any $g\in G$. Then in the inner products for which $\alpha$ and $\alpha '$ are unitary we have

$$\begin{aligned} \int \mathrm{d}g (u',\alpha '(g)v')^*(u,\alpha (g)v)&=0 \qquad \forall u,v\in V, u',v'\in V' \end{aligned}$$

(A.2)

$$\begin{aligned} \int \mathrm{d}g (u',\alpha (g)v')^*(u,\alpha (g)v)&=\frac{(u,u')(v,v')^*}{\mathrm {dim}(V)} \qquad \forall u,v,u',v'\in V. \end{aligned}$$

(A.3)

Choosing orthonormal bases for V and $V'$ and reverting to physics notation, we have

$$\begin{aligned} \int \mathrm{d}g D^*_{\alpha ' i' j'}(g)D_{\alpha ,ij}(g)&=0 \end{aligned}$$

(A.4)

$$\begin{aligned} \int \mathrm{d}g D^*_{\alpha ,i'j'}(g)D_{\alpha ,ij}(g)&=d_\alpha ^{-1}\delta _{ii'}\delta _{jj'}. \end{aligned}$$

(A.5)

Proof

Given any $u\in V$, $u'\in V'$, we can define a map $L_{u,u'}:V\rightarrow V'$ via

$$\begin{aligned} (v',L_{u,u'}v)\equiv \int \mathrm{d}g (u',\alpha '(g)v')^*(u,\alpha (g)v). \end{aligned}$$

(A.6)

It is straightforward to verify that $\alpha 'L_{u,u'}=L_{u,u'}\alpha$ using the invariance of the Haar measure, so by Theorem A.5$L_{u,u'}$ must either be a bijection or be zero. Moreover it cannot be a bijection since $\alpha$ and $\alpha '$ are inequivalent, so it must be zero, establishing Eq. (A.2). To establish Eq. (A.3), we can similarly define maps $L_{u,u'}:V\rightarrow V$ and $L_{v,v'}:V\rightarrow V'$ via

$$\begin{aligned} (v',L_{u,u'}v)\equiv (u,L_{v,v'}u')\equiv \int \mathrm{d}g (u',\alpha (g)v')^*(u,\alpha (g)v). \end{aligned}$$

(A.7)

By the invariance of the Haar measure these maps both commute with $\alpha (g)$ for any g, so by Theorem A.5 they both must be multiples of the identity on V. This establishes Eq. (A.3) up to an overall constant, which we may then fix by taking $u=u'$ and summing u over an orthonormal basis for V using the unitarity of $\alpha$. $\square$

The Schur orthogonality relations immediately imply the orthogonality of the characters $\chi _\alpha (g)\equiv \mathrm {Tr}\alpha (g)$ of inequivalent finite-dimensional irreducible representations, as well as the fact that $\int dg \chi _\alpha ^* (g)\chi _\rho (g)$ counts the number of times a finite-dimensional irreducible representation $\alpha$ appears in the direct-sum decomposition of an arbitrary finite-dimensional representation $\rho$. They can be interpreted as saying that the rescaled set of matrix coefficients $\sqrt{d_\alpha }D_{\alpha ,ij}(g)$ give a set of orthonormal states in the Hilbert space $L^2(G)$ of square-normalizable complex-valued functions on G. In fact they are an orthonormal basis:

Theorem A.7

(Peter–Weyl theorem). Let G be a compact Lie group. Then the rescaled matrix coefficients $\sqrt{d_\alpha }D_{\alpha ,ij}(g)$ for all finite-dimensional irreducible representations give an orthonormal basis for $L^2(G)$.

The proof of this theorem is an exercise in functional analysis and can be found in [119], presenting it here would be too much of a digression.

So far all results in this subsection have been essentially topological, and have not actually used the smoothness in the definition of the Lie group G. Indeed Theorems A.4–A.7 are also true if multiplication and inversion are only taken to be continuous and the topology of G is only required to be compact and Hausdorff, since these are sufficient for the existence of the Haar measure. When we do assume that G is a Lie group however we then have the following remarkable result:

Theorem A.8

Any compact Lie group G has a faithful finite-dimensional unitary representation, and thus is isomorphic to a closed subgroup of U(n) for some n.

Proof

The proof begins with the observation that by the Peter–Weyl Theorem A.7, for any $g\in G$ we can find a finite-dimensional irreducible representation $\alpha _g$ for which $\alpha _g(g)$ is not the identity (otherwise we could never approximate a function on G which takes different values at e and g). We may first consider the case where G is discrete, so its identity component $G_0$ consists of only the identity. G is therefore finite, and we can construct a faithful representation via $\oplus _{g\in G}\,\alpha _g$. Alternatively say that there exists a $g_1\ne e$ in $G_0$: then $G_1\equiv \mathrm {ker}(\alpha _{g_1})$ is a closed subgroup of G, so by the closed subgroup Theorem A.1 it is a Lie subgroup whose dimensionality is at most that of G. In fact its dimensionality must be strictly less than that of G, since if they were equal then by Theorem A.3 we would have $(G_1)_0=G_0$, which contradicts the fact that $\alpha _{g_1}(g_1)$ is not the identity. Now say that $G_1$ is zero-dimensional: then as before we see that $\alpha _{g_1}\oplus _{g\in G_1}\alpha _g$ is a faithful finite-dimensional representation of G. Alternatively if $G_1$ has positive dimension then we have $g_2\in (G_1)_0$ such that $g_2\ne e$, so we can take $G_2\equiv \mathrm {ker}(\alpha _{g_1}\oplus \alpha _{g_2})$, which again will be a closed subgroup of $G_1$ of dimension strictly less than that of $G_1$. Continuing in this way we eventually reach a $G_n$ which is zero-dimensional, and we may then take $\alpha \equiv \alpha _{g_1}\oplus \cdots \oplus \alpha _{g_n}\oplus _{g\in G_n}\alpha _g$, which will be faithful. It is unitary by Theorem A.4. $\square$

Thus we see that the structure theory of compact Lie groups and their finite-dimensional representations is quite well understood. In fact their unitary infinite-dimensional representations are also understandable along similar lines, we now note two results in this direction.

Theorem A.9

Let $\rho$ be a unitary representation of a compact Lie group G on a Hilbert space V. Then $\rho$ is the direct sum of a set of finite-dimensional irreducible representations.

The proof of this theorem uses the Peter–Weyl theorem to show that there cannot be any elements of V which are orthogonal to the direct sum of all finite-dimensional invariant subspaces, see [119] for the proof. We then also have

Theorem A.10

Let $\rho$ be a faithful unitary representation of a compact Lie group G on a Hilbert space V. Then there is a finite-dimensional invariant subspace of V on which $\rho$ also acts faithfully, so $\rho$ has a finite-dimensional subrepresentation which is also faithful.

Proof

The faithfulness of $\rho$ ensures that for any element $g\in G$ there is a finite-dimensional irreducible representation $\alpha _g$ appearing in the direct sum decomposition promised by Theorem A.9 for which $\alpha _g(g)$ is not the identity. The remainder of the proof is identical to that of Theorem A.8. $\square$

The last result we will need relates arbitrary irreducible representations of a compact group to any particular faithful finite-dimensional one [190]:

Theorem A.11

Let G be a compact Lie group, $\rho$ be a faithful finite-dimensional representation of G, and $\rho ^*$ be its conjugate representation. Then for any finite-dimensional irreducible representation $\alpha$ of G there exist nonnegative integers n and m such that $\alpha$ appears in the direct sum decomposition of the tensor-product $\rho ^{\otimes n} \otimes \rho ^{*\otimes m}$.

Proof

Consider the representation

$$\begin{aligned} \rho _n\equiv \left( 1\oplus \rho \oplus \rho ^*\oplus \rho \otimes \rho ^*\right) ^{\otimes n}. \end{aligned}$$

(A.8)

It has character

$$\begin{aligned} \chi _n(g)\equiv \mathrm {Tr}\rho _n(g)=|1+\chi _\rho (g)|^{2n}, \end{aligned}$$

(A.9)

where $\chi _\rho (g)\equiv \mathrm {Tr}\rho (g)$ is the character of $\rho$. By Schur orthogonality we can count the number of times any irreducible representation $\alpha$ appears in the direct sum decomposition of $\rho _n$ by

$$\begin{aligned} \int _G \mathrm{d}g \chi _\alpha (g)|1+\chi _\rho (g)|^{2n}. \end{aligned}$$

(A.10)

The quantity $|1+\chi _\rho (g)|$ obeys

$$\begin{aligned} 0\le |1+\chi _\rho (g)|\le 1+d_\rho , \end{aligned}$$

(A.11)

with the maximum attained only when $g=e$ since $\rho$ is faithful. But then we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\int _G \mathrm{d}g\chi _\alpha |1+\chi _\rho (g)|^{2n}}{\int _G \mathrm{d}g |1+\chi _\rho (g)|^{2n}}=d_\alpha , \end{aligned}$$

(A.12)

so at some sufficiently large n we must have

$$\begin{aligned} \int _G \mathrm{d}g \chi _\alpha (g)|1+\chi _\rho (g)|^{2n}>0. \end{aligned}$$

(A.13)

$\square$

If G is connected, much more is known about its representation theory, and indeed both the connected compact Lie groups and their finite-dimensional irreducible representations have been classified long ago using semisimple theory. In this paper however we have striven to treat discrete and continuous groups on equal footing, so we will stop our review here.

Projective Representations

In this appendix we discuss the possibility of extending our definition of global symmetry to include projective representations of the symmetry on Hilbert space, where the multiplication rule (2.1) would be generalized to include a phase

$$\begin{aligned} U(g,\Sigma )U(g',\Sigma )=e^{i\alpha (g,g')} U(gg',\Sigma ). \end{aligned}$$

(B.1)

We now argue that in quantum field theory on ${\mathbb {R}}^d$, any such phase can be removed by a redefinition of the $U(g,\Sigma )$. We first consider the situation where the symmetry is unbroken: then there is an invariant vacuum state, on which the symmetry can at most act with a phase

$$\begin{aligned} U(g,\Sigma )|0\rangle =e^{if(g)}|0\rangle . \end{aligned}$$

(B.2)

But if we act on this state with $U(g,\Sigma )U(g',\Sigma )$, we immediately discover that we must have

$$\begin{aligned} \alpha (g,g')=f(g)+f(g')-f(gg') \qquad \mathrm {mod} \, 2\pi . \end{aligned}$$

(B.3)

We may then define “improved” symmetry operators

$$\begin{aligned} \widetilde{U}(g,\Sigma )\equiv e^{-if(g)}U(g,\Sigma ), \end{aligned}$$

(B.4)

which act in the same way on the local operators but now obey (B.1) with $\alpha =0$. Thus in any quantum mechanical system, nontrivial projective representations are only possible if there is no invariant state: in other words the symmetry must be spontaneously broken. There are indeed quantum mechanical systems where a spontaneously broken global symmetry is represented projectively in a nontrivial way, see “Appendix D” of [144] for an example, but we now argue that in quantum field theory this is impossible.

The reason is that in quantum field theory on ${\mathbb {R}}^d$, spontaneously broken global symmetries (as we have defined them) always lead to the superselection structure described around Eq. (2.10). Since the operators always transform in non-projective representations of the symmetry (the phase $\alpha$ cancels when act on operators by conjugation), and since we can get to all states by acting with operators that do not change the superselection sector on the degenerate vacua, any projectiveness on the states can arise only from phases in the action of the symmetry on the degenerate vacuum states:

$$\begin{aligned} U(g)|b\rangle =e^{if(g,b)}|gb\rangle . \end{aligned}$$

(B.5)

Strictly speaking to have a genuine projective representation we should not allow f to depend on b, but we have allowed this since in any case it will not help: such phases can again be removed by the redefinition

$$\begin{aligned} \widetilde{U}(g)\equiv U(g) e^{-if(g,B_i)}, \end{aligned}$$

(B.6)

where $B_i$ are the operators which diagnose which superselection sector a state is in. Since the $B_i$ commute with all local operators, this modification has no effect on the action of the symmetry on local operators. Thus the $\widetilde{U}(g)$ give a non-projective representation of the symmetry on the Hilbert space.⁹⁴

In Eq. (B.5) we considered a kind of generalized projective representation, where instead of respecting the group multiplication law up to a c-number phase we respect it up to a nontrivial unitary operator which commutes with all of the local operators. One might ask if there are other examples of this kind of thing, where the unitary operator depends on something other than degenerate vacuum data. In a quantum field theory where all states can be obtained by acting on a single ground state with local operators, there can be no nontrivial unitary operator which commutes with all of the local operators. There are two ways we could try to relax this assumption in the hopes of getting something interesting. The first is to have multiple ground states, each of which has on top of it a superselection sector built by acting with local operators. This is the case we just considered, and we saw that allowing the unitary operators to depend on the superselection sector data did not lead to anything worthwhile. The second possibility is to consider theories where not all states can be obtained by acting on the ground state(s) with local operators. The only possibility we are aware of is to have a theory with a “long range gauge symmetry with dynamical charges”, a notion we define in Sect. 3. It basically means that there is a weakly-interacting gauge field and operators charged under the associated gauge symmetry, which must be attached to infinity by Wilson lines to be gauge-invariant. The gauge symmetry is then represented nontrivially on the Hilbert space, in what is sometimes called an asymptotic symmetry group, and since this can be understood as being realized by a surface operator at infinity it will give a set of nontrivial unitary operators that commute with all local operators but act nontrivially on the endpoints of Wilson lines. We could therefore imagine trying to use these long-range gauge symmetries as generalizations of the phases $e^{i\alpha (g,g')}$ in a projective representation of the global symmetry. Indeed in Sect. 3.5 we give a concrete example of a theory that realizes this phenomenon, and in a way in which the unitary cannot be removed by redefining the symmetry operators. One might then wish to say that this is a genuine projective representation of the global symmetry, but as we explain in Sect. 3.5 we find it more natural to instead say that it is a mixing of the global symmetry with a long-range gauge symmetry. Therefore we are not aware of any situation in quantum field theory where the most natural description of the symmetry structure is to say that a global symmetry is represented projectively on the Hilbert space.

Continuity of Symmetry Operators

In this appendix we discuss the continuity of the action of global symmetries in quantum field theory, both on the Hilbert space and on the algebra ${\mathcal {A}}[R]$ of bounded operators in a bounded spatial region R.

First some definitions. Let V be a Hilbert space, which we will always endow with the standard topology induced by the Hilbert space norm

$$\begin{aligned} ||v||\equiv \sqrt{(v,v)}. \end{aligned}$$

(C.1)

One says that a linear operator ${\mathcal {O}}$ on V is bounded if there exists a real constant C such that $||{\mathcal {O}}v||<C||v||$ for all $v\in V$, and we will denote by ${\mathcal {B}}(V)$ the set of bounded operators on V. We will say that a subset $M\subset {\mathcal {B}}(V)$ is uniformly bounded if there exists a single real constant C such that $||{\mathcal {O}}v||<C||v||$ for all $v\in V$ and ${\mathcal {O}}\in M$. The operator norm $||{\mathcal {O}}||$ of any bounded operator ${\mathcal {O}}$ is the smallest real constant C such that $||{\mathcal {O}}v||\le C||v||$ for all $v\in V$.

To discuss the continuity of maps to and from ${\mathcal {B}}(V)$, we need to give it a topology. There are several possibilities. One obvious one is the norm topology, which has as a basis the set of balls

$$\begin{aligned} B_\epsilon ({\mathcal {O}}_0)\equiv \{{\mathcal {O}}\in {\mathcal {B}}(V)\,\Big |\,||{\mathcal {O}}-{\mathcal {O}}_0||<\epsilon \}, \end{aligned}$$

(C.2)

with ${\mathcal {O}}_0\in {\mathcal {B}}(V)$ and $\epsilon >0$. This topology however is much too strong for our purposes. For example in the norm topology the U(1) global symmetry $\phi '=e^{i\theta }\phi$ of a free complex scalar field has symmetry operators $U(g,\Sigma )$ which are not continuous, since there are states of arbitrarily large charge in the Hilbert space. A topology which is better suited is the strong operator topology, which has as a basis the set of finite intersections of balls of the form

$$\begin{aligned} B_\epsilon ({\mathcal {O}}_0,v_0)\equiv \{{\mathcal {O}}\in {\mathcal {B}}(V)\,\Big |\,||({\mathcal {O}}-{\mathcal {O}}_0)v_0||<\epsilon \}, \end{aligned}$$

(C.3)

with ${\mathcal {O}}_0\in {\mathcal {B}}(V)$, $v_0\in V$, and $\epsilon >0$. This topology is sometimes also called the topology of pointwise convergence, since a sequence ${\mathcal {O}}_n$ of operators converges to an operator ${\mathcal {O}}$ in the strong operator topology if and only if ${\mathcal {O}}_nv\rightarrow {\mathcal {O}}v$ for any $v\in V$. Similarly, if X is a topological space then a map $f:X\rightarrow {\mathcal {B}}(V)$ is continuous in the strong operator topology if and only if the map $f_v:X\rightarrow V$ defined by $f_v(x)=f(x)v$ is continuous for any fixed $v\in V$.

In discussing the continuity of symmetries, there are two maps whose continuity properties we are interested in. The symmetry operators $U(g,\Sigma )$ directly define a map

$$\begin{aligned} U:G\rightarrow {\mathcal {B}}(V), \end{aligned}$$

(C.4)

and also induce an associated map

$$\begin{aligned} f_U:G\times {\mathcal {A}}[R]\rightarrow {\mathcal {A}}[R]\end{aligned}$$

(C.5)

for R any spatial region, defined by $f_U(g,{\mathcal {O}})=U^\dagger (g){\mathcal {O}}U(g)$. As a warmup, we first establish the following theorem

Theorem C.1

Let V be a Hilbert space, G a Lie group, and U a map from G to ${\mathcal {B}}(V)$ for which U(g) is unitary for all $g\in G$. Then the map $\Phi _U:G\times V \rightarrow V$ defined by $\Phi _U(g,v)=U(g)v$ is jointly continuous if and only if U is continuous in the strong operator topology. In particular, if U is a homomorphism which is strongly continuous then it is a representation of G on V in the sense of Sect. A.2.

Proof

If $\Phi _U$ is jointly continuous, then strong continuity of U follows immediately from fixing the second argument. To establish the converse, we need to show that for any ball

$$\begin{aligned} B_\epsilon (v_0)\equiv \{v\in V \,\Big |\, ||v-v_0||<\epsilon \} \end{aligned}$$

(C.6)

in V, $\Phi _U^{-1}(B_\epsilon (v_0))$ is open in $G\times V$. We can do this by showing that any point (g, v) in $\Phi _U^{-1}(B_\epsilon (v_0))$ is contained in an open set $S\times B_\delta (v)$, with S open in G, which is itself contained in $\Phi _U^{-1}(B_\epsilon (v_0))$. We therefore want to show that

$$\begin{aligned} ||U(g')v'-v_0||<\epsilon \qquad \qquad \forall g'\in S, v'\in B_\delta (v). \end{aligned}$$

(C.7)

This follows because by the triangle inequality and the unitary invariance of the Hilbert space norm we have

$$\begin{aligned} \nonumber ||U(g')v'-v_0||&\le ||U(g')(v'-v)||+||(U(g')-U(g))v||+||U(g)v-v_0||\\&=||(v'-v)||+||(U(g')-U(g))v||+||U(g)v-v_0||. \end{aligned}$$

(C.8)

The third term on the second line is less than $\epsilon$ since (g, v) is in $\Phi _U^{-1}(B_\epsilon (v_0))$, and using our freedom to choose $\delta$ and S and the strong continuity of U we can make the first and second terms as small we like. Therefore we can arrange for the sum of all three to be less than $\epsilon$. $\square$

This theorem tells us that in quantum field theory $U(g,\Sigma )$ will be strongly continuous if and only if its action on the Hilbert space gives a continuous representation of G. We saw in the beginning of Sect. 2 that if G is continuous as a Lie group, meaning its dimension as a manifold is greater than zero, then if it is spontaneously broken the $U(g,\Sigma )$ defined by Eq. (2.11) may not be strongly continuous, since elements of g which are arbitrarily close to the identity still send one ground state to another which is orthogonal. If the symmetry is unbroken however, then we take it as a natural postulate that U will indeed be strongly continuous. For example in the free complex scalar example, any particular normalizable state will be acted on continuously even though there are states with arbitrary large charge. More generally the idea is that if the vacuum is invariant, then any particular excited state should only differ from the vacuum in a finite region and by a finite amount of excitation so it should only transform in a representation of limited complexity. We now use the idea that U should be strongly continuous for unbroken symmetries to motivate the continuity clause in condition (b) of our Definition 2.1 of global symmetry.

Theorem C.2

Let V be a Hilbert space, G a Lie group, and U a strongly continuous map from G to the unitary operators on V. Then the restriction to any uniformly bounded subset M of ${\mathcal {B}}(V)$ of the map $f_U:G\times {\mathcal {B}}(V)\rightarrow {\mathcal {B}}(V)$ defined by $f_U(g,{\mathcal {O}})=U^\dagger (g){\mathcal {O}}U(g)$ is strongly continuous.

Proof

We will show that for any ball $B_\epsilon ({\mathcal {O}}_0,v_0)$ in ${\mathcal {B}}(V)$, $f_U^{-1}(B_\epsilon ({\mathcal {O}}_0,v_0))\cap (G\times M)$ is open in $G\times M$. We can do this by showing that for any $(g,{\mathcal {O}})\in f_U^{-1}(B_\epsilon ({\mathcal {O}}_0,v_0))\cap (G\times M)$, there is an open set $S\subset G$ containing g and a ball $B_\delta ({\mathcal {O}},{\hat{v}})$ such that $S\times (B_\delta ({\mathcal {O}},{\hat{v}})\cap M)\subset f_U^{-1}(B_\epsilon ({\mathcal {O}}_0,v_0))\cap (G\times M)$. In other words for any $\epsilon$, ${\mathcal {O}}_0$, and $v_0$, we want to pick S, $\delta$, and ${\hat{v}}$ such that

$$\begin{aligned} ||\left( U^\dagger (g'){\mathcal {O}}' U(g')-{\mathcal {O}}_0\right) v_0||<\epsilon \qquad \forall g'\in S, {\mathcal {O}}'\in B_\delta ({\mathcal {O}},{\hat{v}})\cap M. \end{aligned}$$

(C.9)

By the triangle inequality and the unitary invariance of the Hilbert space norm we have

$$\begin{aligned}&||\left( U^\dagger (g'){\mathcal {O}}' U(g')-{\mathcal {O}}_0\right) v_0||\le ||U^\dagger (g'){\mathcal {O}}'(U(g')-U(g))v_0||+||U^\dagger (g')({\mathcal {O}}'-{\mathcal {O}})U(g)v_0||\nonumber \\&\quad +||(U^\dagger (g')-U^\dagger (g)){\mathcal {O}}U(g)v_0||+||(U^\dagger (g){\mathcal {O}}U(g)-{\mathcal {O}}_0)v_0||\nonumber \\&=||{\mathcal {O}}'(U(g')-U(g))v_0||+||({\mathcal {O}}'-{\mathcal {O}})U(g)v_0||\nonumber \\&\quad +||(U^\dagger (g')-U^\dagger (g)){\mathcal {O}}U(g)v_0||+||(U^\dagger (g){\mathcal {O}}U(g)-{\mathcal {O}}_0)v_0||. \end{aligned}$$

(C.10)

The fourth term on the right hand side is less than $\epsilon$ since $(g,{\mathcal {O}})$ is in $f_U^{-1}(B_\epsilon ({\mathcal {O}}_0,v_0))$, the third term can be made as small as we like using the strong continuity of U and the boundedness of ${\mathcal {O}}$, the second term can be made as small as we like by choosing ${\hat{v}}=U(g)v_0$ and taking $\delta$ to be small, and the first term can be taken to be arbitrarily small by using the strong continuity of U together with the uniform boundedness of M. Therefore for small enough S and $\delta$ we can arrange for the whole right hand side to be less than $\epsilon$. $\square$

Thus we see that strong continuity on any uniformly bounded subset of ${\mathcal {A}}[R]$ is the right continuity requirement on $f_U$ for an unbroken global symmetry. In fact we claim that if the region R is bounded in size, then this should also be the right requirement even if the symmetry is spontaneously broken, since this should not affect the transformation of operators in a finite region, hence our inclusion of it in condition (b) of Definition 2.1. It is worth emphasizing that without the restriction to uniformly bounded subsets the theorem would not apply, since the first term in the right hand side of Eq. (C.10) would not be bounded since there are elements ${\mathcal {O}}'$ of any open ball $B_\delta ({\mathcal {O}},{\hat{v}})$ with arbitrarily large norm.

We can also consider what strong continuity of $f_U$ on uniformly bounded subsets implies in the converse direction about the continuity of U. In general it does not imply anything, which is good since for spontaneously broken symmetries we sometimes do not want U to be continuous. But if we assume that the symmetry is unbroken, by which we mean that there is an invariant ground state $\Omega \in V$, then we have the following theorem:

Theorem C.3

Let V be a Hilbert space, G a Lie group, ${\mathcal {A}}[R]$ a subalgebra of ${\mathcal {B}}(V)$, and U a map from G to the unitary operators on V such that the restriction to any uniformly bounded subset M of ${\mathcal {A}}[R]$ of the map $f_U:G\times {\mathcal {B}}(V)\rightarrow {\mathcal {B}}(V)$ defined by $f_U(g,{\mathcal {O}})=U^\dagger (g){\mathcal {O}}U(g)$ is strongly continuous. Moreover let there exist a state $\Omega \in V$ which is cyclic with respect to ${\mathcal {A}}[R]$,⁹⁵ and which is also invariant in the sense that $U(g)\Omega =\Omega$ for all $g\in G$. Then U is strongly continuous.

Proof

We want to show that for any $\epsilon >0$, $v_0\in V$, ${\mathcal {O}}_0\in {\mathcal {B}}(V)$, we have that $U^{-1}(B_\epsilon ({\mathcal {O}}_0,v_0))$ is open in G. We do this by showing that for any g such that $U(g)\in B_{\epsilon }({\mathcal {O}}_0,v_0)$, there is a neighborhood S of g in G such that U(S) is also contained in $B_{\epsilon }({\mathcal {O}}_0,v_0)$. In other words we want

$$\begin{aligned} ||(U(g')-{\mathcal {O}}_0)v_0||<\epsilon \quad \forall g'\in S. \end{aligned}$$

(C.11)

We first note that by the cyclicity of $\Omega$, we have

$$\begin{aligned} v_0=\widetilde{O}\Omega +\widetilde{v} \end{aligned}$$

(C.12)

for some $\widetilde{O}\in {\mathcal {A}}[R]$, with the norm of $\widetilde{v}$ being as small as we like. From the triangle inequality and the invariance of $\Omega$ we then have

$$\begin{aligned} ||(U(g')-{\mathcal {O}}_0)v_0||&\le ||\left( U^\dagger (g'^{-1})\widetilde{{\mathcal {O}}}U(g'^{-1})-U^\dagger (g^{-1})\widetilde{{\mathcal {O}}}U(g^{-1})\right) \Omega ||+||U(g')\widetilde{v}|| \nonumber \\&\quad +||U(g)\widetilde{v}||+||(U(g)-{\mathcal {O}}_0)v_0||. \end{aligned}$$

(C.13)

The fourth term will be less than $\epsilon$ since U(g) is in $B_{\epsilon }({\mathcal {O}}_0,v_0)$, by cyclicity we can take $||U(g)\widetilde{v}||=||U(g')\widetilde{v}||=||\widetilde{v}||$ as small we like, and since $\widetilde{O}$ will always be part of some uniformly-bounded subset of ${\mathcal {A}}[R]$ the first term can be made arbitrarily small using the joint strong continuity of $f_U$ on uniformly-bounded subsets. Therefore the sum of all three terms can be taken to be less than $\epsilon$. $\square$

Thus we can be reassured that our continuity requirement in condition (b) of Definition 2.1 is not too weak.

Finally we point out that if we do have an invariant ground state which is both cyclic and separating with respect to ${\mathcal {A}}[R]$, then actually there is a different topology in which the situation is even nicer. This topology is defined by noting that we can actually use the state $\Omega$ to define an inner product on ${\mathcal {A}}[R]$ via

$$\begin{aligned} ({\mathcal {O}}_1,{\mathcal {O}}_2)_\Omega \equiv ({\mathcal {O}}_1\Omega ,{\mathcal {O}}_2\Omega ), \end{aligned}$$

(C.14)

which gives ${\mathcal {A}}[R]$ the structure of a Hilbert space. Here $(\cdot ,\cdot )$ is the usual Hilbert space inner product on V, and $(\cdot ,\cdot )_\Omega$ is a good inner product on ${\mathcal {A}}[R]$ because $({\mathcal {O}},{\mathcal {O}})_\Omega \ge 0$, with equality only when ${\mathcal {O}}=0$ due to the fact that $\Omega$ is separating with respect to ${\mathcal {A}}[R]$. We may then use this inner product to define an alternative topology on ${\mathcal {A}}[R]$, which we call the vacuum topology, using as a basis the balls $B_\epsilon ({\mathcal {O}}_0,\Omega )$. Since these are a subset of the balls used in defining the strong operator topology, this topology is weaker than the strong operator topology. We then have the following theorem:

Theorem C.4

Let V be a Hilbert space, G a Lie group, ${\mathcal {A}}[R]$ a subalgebra of ${\mathcal {B}}(V)$, and U a map from G to the unitary operators on V such that the restriction to any uniformly bounded subset M of ${\mathcal {A}}[R]$ of the map $f_U:G\times {\mathcal {B}}(V)\rightarrow {\mathcal {B}}(V)$ defined by $f_U(g,{\mathcal {O}})=U^\dagger (g){\mathcal {O}}U(g)$ is strongly continuous. Moreover let there exist a state $\Omega \in V$ which is cyclic and separating with respect to ${\mathcal {A}}[R]$, and which is also invariant in the sense that $U(g)\Omega =\Omega$ for all $g\in G$. Then the restriction to ${\mathcal {A}}[R]$ of $f_U$ is jointly continuous in vacuum topology on ${\mathcal {A}}[R]$, without any uniform-boundedness requirement, and in particular if U is a homomorphism then $f_U$ gives a representation of G on the Hilbert space ${\mathcal {A}}[R]$ with inner product $(\cdot ,\cdot )_\Omega$. Moreover this representation is unitary.

Proof

We can first invoke Theorem C.3 to learn that U is strongly continuous. We may then imitate the proof of Theorem C.2, noting however that now we only need the inequality (C.10) to hold when $v_0=\Omega$. But then the first term on the righthand side is automatically zero since $(U(g')-U(g))\Omega =0$, so we have no need of a uniform boundedness requirement. Finally to see that the representation of G on ${\mathcal {A}}[R]$ furnished by $f_U$ is unitary, we simply note that

$$\begin{aligned} (U^\dagger (g){\mathcal {O}}_1U(g),U^\dagger (g){\mathcal {O}}_2U(g))_\Omega&=(U^\dagger (g) {\mathcal {O}}_1\Omega ,U^\dagger (g){\mathcal {O}}_2\Omega )=({\mathcal {O}}_1\Omega ,{\mathcal {O}}_2\Omega ) \nonumber \\&=({\mathcal {O}}_1,{\mathcal {O}}_2)_\Omega . \end{aligned}$$

(C.15)

$\square$

In particular this theorem tells us that if a global symmetry is unbroken, then the map D defined by Eq. (2.5) gives a unitary representation of G. And in particular if G is compact, then by Theorem A.9D should decompose into a direct sum of finite-dimensional unitary representations. Moreover not only did we not need a uniform-boundedness requirement in the proof of Theorem C.4, in fact we did not even need to assume that the elements of ${\mathcal {A}}[R]$ are bounded! As long as we restrict to operators whose domain includes the invariant state $\Omega$, we still may use $\Omega$ to define an inner product on these operators in terms of which the action of $f_U$ is unitary and continuous, and thus gives a unitary representation.

It is interesting to note that if we drop the assumption that the symmetry is unbroken, there are easy examples where the action $f_U$ of G on local operators is not unitary. For example in a free scalar field theory in $d>2$, there is a spontaneously-broken global symmetry which acts on the scalar $\phi$ and the identity 1 as

$$\begin{aligned} \begin{pmatrix} \phi ' \\ 1' \end{pmatrix} =\begin{pmatrix} 1 &{}\quad a \\ 0 &{}\quad 1\end{pmatrix}\begin{pmatrix}\phi \\ 1\end{pmatrix}, \end{aligned}$$

(C.16)

which is a non-unitary representation of the symmetry group ${\mathbb {R}}$. In this kind of situation it is sometimes said that the symmetry “acts non-linearly” on $\phi$, but in fact $f_U$ always gives a linear action of G on the set of local operators, and this is manifest in (C.16).

Building Symmetry Insertions on General Closed Submanifolds

Consider a $(d-1)$-dimensional compact connected oriented manifold $\Sigma$ embedded in ${\mathbb {R}}^d$. Since $H_{d-1}({\mathbb {R}}^d)$ is trivial, there is a d-dimensional compact connected oriented submanifold M in ${\mathbb {R}}^d$ such that $\Sigma = \partial M$. In this appendix we show that the insertion of a symmetry operator on $\Sigma$ into the path integral can always be understood in operator language as conjugating all operators in M by $U(g,{\mathbb {R}}^{d-1})$, as shown in Fig. 2 for the special case of $d=3$ and $\Sigma ={\mathbb {T}}^2$.

Fig. 20

Illustrations of $\Sigma _t = f^{-1}((-\infty , t])= \{ x \in \Sigma : ~ f(x) \le t \}$ at two different values of t when $\Sigma$ is a torus

Indeed by generically choosing a “time” direction in ${\mathbb {R}}^d$, with a linear coordinate t, we can define a Morse function f on $\Sigma$ such that $f(p)=t$ at $p \in \Sigma$ (a Morse function is a smooth map from a manifold $\Sigma$ to ${\mathbb {R}}$ which has no degenerate critical points; such functions are dense in the set of smooth maps from $\Sigma$ to ${\mathbb {R}}$, so a generic orientation of the time direction will give us one). For each t, define,

$$\begin{aligned} \Sigma _t = f^{-1}((-\infty , t]) = \{ p \in \Sigma : ~ f(p) \le t \} . \end{aligned}$$

(D.1)

See Fig. 20 for its illustration. We also define,

(D.2)

where ${\mathbb {R}}^{d-1}_t$ and $M_t$ are sections of ${\mathbb {R}}^d$ and M at t. Let us glue $\Sigma _t$ with at their common boundaries $f^{-1}(t)$, to get a surface we call $C_t$. In the following, we will use Morse theory to study how $U(g, \mathcal{C}_t)$ behaves as we increase t from $-\infty$ to $+ \infty$.

The Morse function f has isolated non-degenerate critical points on $\Sigma$. The fundamental theorems (Theorems 3.1 and 3.2 in [192]) of the Morse theory say:

Theorem D.1

Suppose $t_1 < t_2$ and $f^{-1}([t_1, t_2])$ is compact and contains no critical points of f. Then $\Sigma _{t_1}$ is diffeomorphic to $\Sigma _{t_2}$ and the inclusion map $\Sigma _{t_1} \rightarrow \Sigma _{t_2}$ is a homotopy equivalence.

Fig. 21

When p is a critial point of f with index n, $\Sigma _{f(p)+\epsilon }$ is homotopic to $\Sigma _{f(p) - \epsilon }$ with an n-cell attached, provided we choose $\epsilon >0$ to be sufficiently small

The second fundamental theorem tells us what happens at critical points. Before stating the theorem, let us note that according to Morse’s lemma, each critical point p of f is characterized by its index n, which means that we can choose coordinates $(x_1, \ldots , x_{d-1})$ around p such that p is at $x=0$ and,

$$\begin{aligned} f (x) = f(p) - x_1^2 - \cdots - x^2_n + x_{n+1}^2 + \cdots + x_{d-1}^2, \end{aligned}$$

(D.3)

holds throughout the coordinate patch (these coordinates are obtained by diagonalizing the Hessian matrix at p). We can choose $\epsilon > 0$ sufficiently small so that f has no other critical point in $[t- \epsilon , t+ \epsilon ]$, where $t=f(p)$.

Theorem D.2

If p is a critical point of f with $f(p)=t$ and index n, and if there is no other critical point in $f^{-1}([t-\epsilon , t+\epsilon ])$ for some $\epsilon > 0$, $\Sigma _{t+\epsilon }$ is homotopic to $\Sigma _{t - \epsilon }$ with an n-cell attached.⁹⁶ See Fig. 21 for illustration.

Since $\Sigma$ is compact, there is $t_0$ such that $\Sigma _t$ is empty for $t < t_0$. For such t, $\mathcal{C}_t = {\mathbb {R}}^{d-1}_t$ and $U(g, \mathcal{C}_t)$ is the symmetry generator. Let us choose $t_0$ to be the largest such $t_0$. Increasing t continuously, we reach $t=t_0$ where ${\mathbb {R}}_{t_0}^{d-1}$ touches $\Sigma$. Clearly, $\Sigma _{t_0+\epsilon }$ is homotopic to $\Sigma _{t_0 - \epsilon }$ (which is empty) with a 0-cell (the point of the first contact) attached, as expected from Theorem D.2. We can then continously deform $\mathcal{C}_{t_0 - \epsilon } ={\mathbb {R}}^{d-1}_{t-\epsilon }$ to $\mathcal{C}_{t_0 + \epsilon }$ and $U(g, \mathcal{C}_{t_0 + \epsilon })$ is still a symmetry generator.

Fig. 22

Since symmetry insertions on the same cell with opposite orientations (in red and blue in the figure) cancel, $U(g, \mathcal{C}_{t-\epsilon })$ can be continuously deformed to $U(g, \mathcal{C}_{t+\epsilon })$

As we increase t further, we will inevitably encounter a critical point with non-zero index n at some t. According to Theorem D.2, we can homotopically deform $\Sigma _{t - \epsilon }$ to $\Sigma _{t + \epsilon }$ by attaching an n-cell. We can also deform to by attaching the same n-cell with opposite orientation. Since symmetry insertions on the pair of n-cells with opposite orientations has no effect, $U(g, \mathcal{C}_{t-\epsilon })$ can be continuously deformed to $U(g, \mathcal{C}_{t+\epsilon })$. See Fig. 22 for illustration.

Since $\Sigma$ is compact, there is $t_1$ suth that $\Sigma _{t} = \Sigma$ for $t > t_1$. Choosing $t_1$ to be the smallest such $t_1$, $\mathcal{C}_{t_1} = \Sigma \cup {\mathbb {R}}^{d-1}_{t_1}$.

We conclude that the symmetry generator $U(g, \mathcal{C}_t) = U(g, {\mathbb {R}}^{d-1}_t)$ for $t < t_0$ can be deformed to $U(g, \mathcal{C}_{t_1}) =U(g, \Sigma \cup {\mathbb {R}}^{d-1}_{t_1})$ at $t = t_1$. Since $U(g, \Sigma ) = U(g, \Sigma \cup {\mathbb {R}}^{d-1}_{t_1}) U(g, {\mathbb {R}}^{d-1}_{t_1})^\dagger$, this is what we wanted to show.

Lattice Splittability Theorem

In this appendix we give a proof of Theorem 2.1, which says that a unitary which acts locally on each tensor factor of a tensor product Hilbert space must itself be a tensor product of local unitaries.⁹⁷

Proof

We first note that it is enough to establish the theorem for the case of two tensor factors, ${\mathcal {H}}={\mathcal {H}}_A\otimes {\mathcal {H}}_B$, with a unitary $U_{AB}$ which send operators on A to operators on A, and operators on B to operators on B, since we can then iterate the argument to obtain the desired result for any finite number of tensor factors. We thus just need to show that $U_{AB}=(U_A\otimes I_B)(I_A\otimes U_B)$.

The basic idea is to double the size of the system, introducing copies ${\mathcal {H}}_{{\hat{A}}}$ and ${\mathcal {H}}_{{\hat{B}}}$ of ${\mathcal {H}}_A$ and ${\mathcal {H}}_B$, and then consider the state

$$\begin{aligned} |\phi \rangle \equiv \frac{1}{\sqrt{|A||B|}}\sum _{ab}|a\rangle _{{\hat{A}}}|b\rangle _{{\hat{B}}}U_{AB}|ab\rangle _{AB}. \end{aligned}$$

(E.1)

Here $|a\rangle _A$, $|a\rangle _{{\hat{A}}}$ are orthonormal bases for ${\mathcal {H}}_A$ and ${\mathcal {H}}_{{\hat{A}}}$, and similarly for $|b\rangle _B$, $|b\rangle _{{\hat{B}}}$. Noting that $U_{AB}^\dagger (I_A\otimes {\mathcal {O}}_B)U_{AB}=(I_A\otimes {\mathcal {O}}'_B)$ for any ${\mathcal {O}}_B$, and that any operator ${\mathcal {O}}_{B{\hat{B}}}$ can be expanded as a sum of tensor products of operators on ${\mathcal {H}}_B$ and ${\mathcal {H}}_{{\hat{B}}}$, a simple calculation shows that for any operators ${\mathcal {O}}_{{\hat{A}}}$ and ${\mathcal {O}}_{B{\hat{B}}}$ on ${\mathcal {H}}_{{\hat{A}}}$ and ${\mathcal {H}}_B\otimes {\mathcal {H}}_{{\hat{B}}}$ respectively, we must have

$$\begin{aligned} \langle \phi |{\mathcal {O}}_{{\hat{A}}}{\mathcal {O}}_{B{\hat{B}}}|\phi \rangle&=\langle \phi |U_{AB}{\mathcal {O}}'_A{\mathcal {O}}'_{B{\hat{B}}}U_{AB}^\dagger |\phi \rangle \nonumber \\&=\langle \phi |U_{AB}{\mathcal {O}}_{{\hat{A}}}U_{AB}^\dagger |\phi \rangle \langle \phi |U_{AB}{\mathcal {O}}'_{B{\hat{B}}}U_{AB}^\dagger |\phi \rangle \nonumber \\&=\langle \phi |{\mathcal {O}}_{{\hat{A}}}|\phi \rangle \langle \phi |{\mathcal {O}}_{B{\hat{B}}}|\phi \rangle . \end{aligned}$$

(E.2)

In other words there is no correlation between ${\hat{A}}$ and $B{\hat{B}}$, so the partial trace of $|\phi \rangle \langle \phi |$ over A factorizes:

$$\begin{aligned} \rho _{{\hat{A}}B{\hat{B}}}(\phi )\equiv \mathrm {Tr}_{A}|\phi \rangle \langle \phi |=\rho _{{\hat{A}}}(\phi )\otimes \rho _{B{\hat{B}}}(\phi ). \end{aligned}$$

(E.3)

Moreover from (E.1) we have

$$\begin{aligned} \rho _{{\hat{A}}}(\phi )=\frac{I_{{\hat{A}}}}{|A|}, \end{aligned}$$

(E.4)

where |A| denotes the dimensionality of ${\mathcal {H}}_A$.

Now the key point is that the state $\rho _{{\hat{A}}B{\hat{B}}}(\phi )$ must be purified into $|\phi \rangle$ by adding back the A system, which means that its rank can be at most |A|. But since the rank of $\rho _{{\hat{A}}}(\phi )$ is already |A|, this means that $\rho _{B{\hat{B}}}(\phi )$ must have unit rank, or in other words must be a pure state $|\chi \rangle \langle \chi |_{B{\hat{B}}}$. We may then observe that since any two purifications of a mixed state onto a given system differ at most by a unitary transformation on that system, and since the state

$$\begin{aligned} |\psi \rangle =\frac{1}{\sqrt{|A|}}\sum _{a}|a\rangle _{{\hat{A}}}|a\rangle _A|\chi \rangle _{B{\hat{B}}} \end{aligned}$$

(E.5)

is a purification of ${\hat{A}}B{\hat{B}}$ onto A, it must be that $|\phi \rangle$, which is another such purification, is given by

$$\begin{aligned} |\phi \rangle =U_A|\psi \rangle =\frac{1}{\sqrt{|A|}}\sum _{a}|a\rangle _{{\hat{A}}}U_A|a\rangle _A |\chi \rangle _{B{\hat{B}}} \end{aligned}$$

(E.6)

for some $U_A$. Moreover since again from (E.1) we have $\rho _{{\hat{B}}}(\phi )=\frac{I_{{\hat{B}}}}{|B|}$, by the same argument we must have

$$\begin{aligned} |\chi \rangle _{B{\hat{B}}}=\frac{1}{\sqrt{|B|}}\sum _b |b\rangle _{{\hat{B}}}U_B|b\rangle _B \end{aligned}$$

(E.7)

for some $U_B$. We then finally have that

$$\begin{aligned} |\phi \rangle =\frac{1}{\sqrt{|A||B|}}\sum _{ab}|a\rangle _{{\hat{A}}}|b\rangle _{{\hat{B}}}U_A|a\rangle _A U_B|b\rangle _B, \end{aligned}$$

(E.8)

which is compatible with (E.1) if only if $U_{AB}=U_A\otimes U_B$. $\square$

Hamiltonian for Lattice Gauge Theory with Discrete Gauge Group

In this appendix we sketch how to derive the lattice gauge theory Hamiltonians (3.25), (3.31) from the continuous-time limit of the Wilson action. The Euclidean Wilson action on a spacetime cubic lattice with lattice spacing a is [115]

$$\begin{aligned} S_E=-\frac{a^{d-4}}{g^2}\sum _{\gamma \in {\hat{\Gamma }}} W_\alpha (\gamma ), \end{aligned}$$

(F.1)

where ${\hat{\Gamma }}$ is the set of (oriented) plaquettes in Euclidean spacetime and $\alpha$ is a faithful representation of G. This action makes sense for any gauge group G, discrete or continuous. To extract a Hamiltonian, we need to take the lattice spacing in the time direction, which we’ll denote as $a_0$, to be much smaller than the lattice spacing in the space directions, which we’ll continue to call a. In this case the Wilson action becomes

$$\begin{aligned} S_E&=-\frac{a^{d-4}}{g^2}\left( \frac{a}{a_0}\sum _{\gamma \in {\hat{\Gamma }}_0}W_\alpha (\gamma )+\frac{a_0}{a}\sum _{\gamma \in {\hat{\Gamma }}_s}W_{\alpha }(\gamma )\right) \nonumber \\&\equiv -A \sum _{\gamma \in {\hat{\Gamma }}_0}W_\alpha (\gamma )-B\sum _{\gamma \in {\hat{\Gamma }}_s}W_{\alpha }(\gamma ), \end{aligned}$$

(F.2)

where ${\hat{\Gamma }}_0$ denotes the set of plaquettes which have a time component and ${\hat{\Gamma }}_s$ denotes the set of plaquettes with no time component.

We now study the thermal partition function

$$\begin{aligned} Z(\beta )\equiv \int {\mathcal {D}}g e^{-S_E}, \end{aligned}$$

(F.3)

where we are integrating over an element of g assigned to each edge of a cubic Euclidean spacetime lattice with periodic time. We can use gauge transformations to set the temporal edges all to the identity except for at one time, and the integral over the temporal edges at that time simply imposes a projection onto gauge-invariant states. The thermal partition function then has the form [117]

$$\begin{aligned} Z(\beta )=\mathrm {Tr}(T^N), \end{aligned}$$

(F.4)

where the trace is over only gauge-invariant states and T is called the transfer matrix; it is given by

$$\begin{aligned} \langle g'|T|g\rangle =\exp \left( A \sum _{e\in E}\mathrm {Tr}\Big (D_{\alpha }(g_eg_e^{\prime -1})+D_\alpha (g_e'g_e^{-1})\Big )+B\sum _{\gamma \in \Gamma }W_\alpha (\gamma )\right) . \end{aligned}$$

(F.5)

Here $|g\rangle$ and $|g'\rangle$ are elements of gauge-field part of the Hilbert space (3.12). As in the main text, E denotes the set of edges in a time slice and $\Gamma$ denotes the set of plaquettes in a timeslice. Note that $\Gamma$ is not equal to ${\hat{\Gamma }}_s$, which is the set of spatial plaquettes at all times. We may re-express T using our lattice gauge theory operators:

$$\begin{aligned} T=\prod _{e\in E}\left( \int dh e^{A\mathrm {Tr}\left( D_\alpha (h)+D_\alpha (h^{-1})\right) }L_h(e)\right) e^{B\sum _{\gamma \in \Gamma }W_\alpha (\gamma )}, \end{aligned}$$

(F.6)

where we have written $L_h(e)$ instead of $L_h(\ell )$ since this expression does not care which way we orient the link $\ell$ on edge e. Finally to extract the Hamiltonian we take the limit $a_0\rightarrow 0$, identifying the Hamiltonian via

$$\begin{aligned} T=e^{-a_0 H}. \end{aligned}$$

(F.7)

To proceed, we now need to decide if G is continuous or discrete. If it is continuous, in the limit $a_0\rightarrow 0$ the integral over h will be dominated by the region near the identity. We may then use a Gaussian approximation to evaluate it, which directly leads to the Kogut–Susskind Hamiltonian (3.25) up to an additive c-number renormalization [117]. When G is discrete things are a little more subtle, to obtain an interesting theory we need to forget the expressions for A and B in terms of a, $a_0$, and g, which after all came from trying to reproduce the Yang–Mills action in the continuum, and instead simply view A and B as parameters to vary as we like. For G continuous we took A to infinity and B to zero such that their product was finite, but for G discrete the right limit is instead to take A to infinity and B to zero such that $Be^A$ is finite: it is only in this limit that (after another c-number renormalization) we have that $T\approx 1-\epsilon H$ with $\epsilon$ small and H a Hamiltonian with both “electric” and “magnetic” terms [118]. In this limit the identity contribution to the sum over h is set to one by the c-number renormalization, which replaces $\mathrm {Tr}\left( D_\alpha (h)+D_\alpha (h^{-1})\right)$ by $\mathrm {Tr}\left( D_\alpha (h)+D_\alpha (h^{-1})\right) -2d_\alpha$ for each edge, and the other terms in the sum over h which survive in the continuous-time limit are those which maximize $\mathrm {Tr}\left( D_\alpha (h)+D_\alpha (h^{-1})\right)$. This finally leads to the Hamiltonian (3.31), with the normalization of the new gauge coupling g being chosen in a somewhat arbitrary manner.

Stabilizer Formalism for the ${\mathbb {Z}}_2$ Gauge Theory

The stabilizer formalism is a useful technique for defining nontrivial subspaces of the Hilbert space of n qubits [124]. In this appendix we explain how it may be used to compute the ground state degeneracy of the ${\mathbb {Z}}_2$ lattice gauge theory with charged matter in the limit of small g and large $\lambda$, with Hamiltonian (3.38). In fact in these ground states the charges are never excited, so our result also gives the ground state degeneracy of the pure ${\mathbb {Z}}_2$ gauge theory, which is one of the simplest topological quantum field theories. In the main text we are primarily interested in cubic lattices which discretize the $d-1$-dimensional ball $B^{d-1}$, but, mostly for fun, we will use a few tools from algebraic topology to compute the ground state degeneracy for any spatial lattice with the structure of a $d-1$-dimensional CW complex.⁹⁸ In the continuum limit, this will give a formula for the Hilbert space dimension of the ${\mathbb {Z}}_2$ gauge theory on any spatial $d-1$-manifold, with or without boundary. In particular we will show that the Hamiltonian (3.38) has a unique ground state on any lattice whose CW complex is homeomorphic to $B^{d-1}$, on which the operators $Z(\gamma )$ and $\prod _{{\delta }}X({x},{\delta })$ act as the identity for any plaquette $\gamma$ and site ${x}$, while more generally the ground state degeneracies for any connected CW complex (or connected manifold) are given by (G.5) if there is no boundary and (G.11) if there is a boundary.

The basic idea of the stabilizer formalism is to consider the $+1$ eigenspace of an abelian subgroup S of the n-qubit Pauli group $P_n$. $P_n$ is the multiplicative group of operators on the Hilbert space of n qubits which is generated by all single-qubit Pauli operators together with iI, where I is the identity operator and $i=\sqrt{-1}$. The stabilizer formalism then rests on the following theorem:

Theorem G.1

Let S be an abelian subgroup of $P_n$, not containing $-I$, which is generated by m independent generators $\{g_1, \ldots , g_m\}$. Then the subspace of states on which all elements of S act as the identity has dimension $2^{n-m}$.

We refer the reader to [194] for a proof, but the basic idea is that the projection onto the $+1$ subspace of each generator decreases the dimensionality of the subspace by a factor of two.

We can apply this theorem to the lattice ${\mathbb {Z}}_2$ gauge theory with charged matter by noting that in unitarity gauge the Hilbert space is just the tensor product of a qubit on each edge of the lattice. The set of plaquettes $Z(\gamma )$ and “stars” $\prod _{{\delta }}X({x},\delta )$ generate an abelian subgroup S of the Pauli group on this Hilbert space, and it is easy to see that no product of plaquettes and stars can give $-I$. In fact, below we will classify all the relations among plaquettes and stars. Hermitian elements of the Pauli group can only have eigenvalues $\pm 1$, so states where all plaquettes and stars act as the identity will necessarily be ground states of the Hamiltonian (3.38). We may thus apply Theorem G.1 to identify the dimensionality of the ground state subspace. To show that the ground state is unique, we need to show that the number of independent generators of S is equal to the number of edges in the lattice.

Fig. 23

Stabilizer generators for a cubic lattice with two spatial dimensions. The nine red circles indicate plaquettes, and the four star constraints live at the black dots. Since the product of all the plaquettes is the identity, the number of independent generators is $8 + 4 = 12$, which agrees with the number of edges. By Theorem G.1, the ground state is unique

Counting the number of independent generators of S is nontrivial because there are relations among stars and plaquettes. For example consider the situation in Fig. 23. Since stars and plaquettes commute with each other, and since the only relations among Pauli generators that reduce their numbers are $X^2 = Z^2 = 1$, any relation among stars and plaquettes can be expressed as the product of a relation among stars only and a relation among plaquettes only. Thus, it is sufficient to treat stars and plaquettes separately when counting their relations. There are no relations among the four stars, since it is not possible to cancel the X(e) on boundary-piercing edges, but the product of the nine red plaquettes is equal to the identity. Therefore, the number of independent generators (plaquettes and stars) is equal to twelve, which indeed equals the number of edges. It is easy to see that this counting works out more generally for a two-dimensional rectangular square lattice with some numbers of rows and columns. We now explain how to generalize this counting to arbitrary dimension and topology.

For simplicity we first discuss the case where the lattice has no boundary, for example it could be a discretization of a Riemann surface. We will refer to the CW complex associated to the lattice as X, and we will denote by $N_n(X)$ the number of n-cells in X. We will take X to be connected, since in the disconnected case the ground state subspace just tensor factorizes component by component. The number of stars is $N_0(X)$, the number of edges is $N_1(X)$, and the number of plaquettes is $N_2(X)$. There is however one relation between the stars: the product of all of them is the identity. There can be no further relations, as can be seen by the following argument. Any relation between the stars can be expressed by saying that the product of some subset of them is equal to the identity. To get a nontrivial relation, at least one star must be included. Consider any loop of edges which includes an edge attached to that star. Each edge of the loop must appear in either zero or two stars in the relation in order for it to be equal to the identity, and moreover they must all appear in zero or all appear in two. Since one of them appears in two, they all must. But since this true for any loop containing that edge, to get a nontrivial relation we need to include all the stars. Thus we have

$$\begin{aligned} \#(\mathrm {independent\,\,stars})=N_0(X)-1. \end{aligned}$$

(G.1)

Counting the relations between the plaquettes is more nontrivial, we claim that

$$\begin{aligned} \#(\mathrm {independent\,\, plaquettes})&=N_2(X)-(N_3(X)+b_2(X))+(N_4(X)+b_3(X))-\cdots \nonumber \\&\quad +(-1)^{d-1}(N_{d-1}(X)+b_{d-2}(X))-(-1)^{d-1} b_{d-1}(X), \end{aligned}$$

(G.2)

where $b_m(X)$ is the dimensionality of the homology group $H_m(X,{\mathbb {Z}}_2)$. The idea of this is as follows: the product of any set of plaquettes living on a two-cycle in ${\mathbb {Z}}_2$ homology is the identity, and so gives a relation between the plaquettes. The set of two-cycles which are boundaries of three-chains is generated by products of three-cells, of which there are $N_3(X)$. We also need to include one representative of each nontrivial homology class of two-cycles, hence our subtraction of $(N_3(X)+b_2(X))$. But there aren’t actually $N_3(X)$ independent homologically-trivial two-cycles, since those collections of three-cells which form three-cycles have trivial boundary and thus do not generate two-cycles. So we need to add back the number of three-cycles, which is given by $(N_4(X)+b_3(X))$, except then some collections of the four cells are five cycles, which we need to resubtract, and so on. In the last step we need to add or subtract the number of $d-1$-cycles, which are clearly never boundaries of d cycles, so we are left with only $b_{d-1}$. In stabilizer parlance, we have

$$\begin{aligned} n=\#(\mathrm {edges})=N_1(X) \end{aligned}$$

(G.3)

qubits and

$$\begin{aligned} m=\#(\mathrm {independent\,\,stars})+\#(\mathrm {independent\,\, plaquettes}) \end{aligned}$$

(G.4)

generators of ${\mathcal {S}}$, so the groundstate degeneracy is

$$\begin{aligned} 2^{n-m}=2^{b_1(X)}, \end{aligned}$$

(G.5)

where we have used the expressions

$$\begin{aligned} \chi (X)\equiv \sum _{n=0}^d (-1)^nN_n(X)=\sum _{n=0}^d (-1)^n b_n(X). \end{aligned}$$

(G.6)

for the Euler characteristic of X, and also that $b_0(X)=1$ since X is connected. The expression (G.5) has a natural interpretation: the ground state subspace is labeled by the eigenvalues of the Wilson lines on the topologically distinct one-cycles of X [123].

We now turn to lattices where $\partial X$ is nontrivial. In order to allow a nontrivial long-range gauge symmetry, we had to choose boundary conditions on our gauge theory with matter fields as in Figs. 9 and 23, where boundary edges are not included since we do not have degrees of freedom there and there are no star constraints on boundary sites. For X to be a CW complex however, we need to include these boundary edges as one-cells and boundary sites as zero-cells, since otherwise the boundaries of plaquettes which are adjacent to the boundary will not be part of the set of zero-cells and one-cells. Similarly X needs to include all higher cells in $\partial X$ as well. The number of edges which carry qubits is thus now given by

$$\begin{aligned} \#(\mathrm {edges})=N_1(X)-N_1(\partial X). \end{aligned}$$

(G.7)

There are no longer any relations between the star constraints, since given any edge in a star involved in such a relation we can construct a path to the boundary on which all edges would need to appear in two stars, but this is impossible for boundary-piercing edges since there are no star constraints on boundary sites. Therefore we have

$$\begin{aligned} \#(\mathrm {independent \,\, stars})=N_0(X)-N_0(\partial X). \end{aligned}$$

(G.8)

Counting the number of independent plaquettes is again more difficult, we claim that

$$\begin{aligned} \#(\mathrm {independent\,\, plaquettes})&=N_2(X)-N_2(\partial X)\nonumber \\&\quad -\left( (N_3(X)-N_3(\partial X)+b_2(X)-b_2^{NT}(\partial X)+b_1^T(\partial X)\right) \nonumber \\&\quad +\left( N_4(X)-N_4(\partial X)+b_3(X)-b_3^{NT}(\partial X)+b_2^T(\partial X)\right) \nonumber \\&\quad -\cdots \nonumber \\&\quad +(-1)^{d-1}\left( N_{d-1}(X)+b_{d-2}(X)-b_{d-2}^{NT}(\partial X)+b_{d-3}^T(\partial X)\right) \nonumber \\&\quad -(-1)^{d-1}\left( b_{d-1}(X)+b^T_{d-2}(\partial X)\right) . \end{aligned}$$

(G.9)

In this formula we use a notation where we have split the n-cycles in $\partial X$ which are not boundaries in $\partial X$ into a set which are boundaries in X, which have $b_n^{T}(\partial X)$ independent representatives, and a set which aren’t boundaries in X, which have $b_n^{NT}(\partial X)$ independent representatives. By definition, we have

$$\begin{aligned} b_n(\partial X)=b_n^T(\partial X)+b_n^{NT}(\partial X). \end{aligned}$$

(G.10)

To understand Eq. (G.9), we begin as before: there are $N_2(X)-N_2(\partial X)$ plaquettes, but the product of plaquettes on any two-cycle in ${\mathbb {Z}}_2$ homology vanishes identically. This again imposes relations on the plaquettes. The set of two-cycles which are boundaries is generated by the three-cells, of which there are $N_3(X)$, but the three cells which lie in the boundary are automatically trivial, so we should subtract $N_3(\partial X)$. In counting two-cycles we should include a representative of each nontrivial class in $H_2(X,{\mathbb {Z}}_2)$, hence adding $b_2(X)$, but now we need to account for the fact that nontrivial two-cycles in X which are homologous to nontrivial two-cycles in the boundary can still be generated by the three-cells, so we should subtract $b_2^{NT}(\partial X)$. Finally in addition to the two-cycles, there are also relations from two-chains whose boundaries lie in $\partial X$, since these again are the identity. When the boundary of such a two-chain is a boundary in $\partial X$, then the relation associated to it is equivalent to one from a two-cycle in X which contains some boundary two-cells, so we only get new relations from those two-chains in X whose boundary is in $\partial X$ but is not a boundary there. These are counted precisely by $b_1^T(\partial X)$, hence we add this to our list of relations, finally subtracting the whole set as the second line of (G.9). We then observe that collections of three-cells which generate three-cycles or three-chains whose boundary is in $\partial X$ do not actually define two-cycles, and so we need to add back the third line of (G.9). And so on. Combining (G.7), (G.8), and (G.9), and again using (G.6) and $b_0(X)=1$, we at last have a ground state degeneracy

$$\begin{aligned} 2^{n-m}=2^{b_0(\partial X)-1+b_1(X)-b_1^{NT}(\partial X)}. \end{aligned}$$

(G.11)

This formula again has an elegant interpretation in terms of Wilson lines:

$$\begin{aligned} b_0(\partial X)-1 \end{aligned}$$

(G.12)

counts the number of independent Wilson lines stretching from one component of $\partial X$ to another, while

$$\begin{aligned} b_1(X)-b_1^{NT}(\partial X) \end{aligned}$$

(G.13)

counts the number of independent homologically-nontrivial Wilson loops which are not homologous to boundary one-cycles, since those which are must be trivial by the boundary conditions. In particular if X is homeomorphic to $B^{d-1}$, then (G.12) and (G.13) both vanish ($\partial B^{d-1}={\mathbb {S}}^{d-2}$ is connected and there are no nontrivial one-cycles in $B^{d-1}$), so the ground state is unique.

Multiboundary Wormholes in Three Spacetime Dimensions

In this appendix we review some of what is known about multiboundary wormholes in $\hbox {AdS}_3/\hbox {CFT}_2$, focusing on the feasibility of constructing geometries which can be used in our second proof of Theorem 4.2. The great advantage of $d=2$ is that there are no gravitational waves, so all solutions of the Einstein equation with negative cosmological constant and no matter are locally isometric to $\hbox {AdS}_3$. More precisely, they are quotients of $\hbox {AdS}_3$ by a discrete subgroup $\Gamma$ of its isometry group $\hbox {SO}(2,2)$. In $\hbox {AdS}_3/\hbox {CFT}_2$ such states can often be prepared by cutting the path integral of the CFT on a Riemann surface [195,196,197,198], we now review this construction.

Fig. 24

A genus two Riemann surface constructed using four Schottky discs. On the left the surface is the union of the green and blue regions, with the indicated identifications and the marked points identified. Performing the CFT path integral over just the blue region below the cut prepares a state in the Hilbert space of the CFT on three circles, labeled 1, 2, 3. On the right we show a heuristic picture of the cut geometry embedded into ${\mathbb {R}}^3$

We begin by recalling the Schottky construction of an arbitrary Riemann surface. Viewing the complex plane as the Riemann sphere, we place an even number of non-intersecting discs and then identify their boundaries in pairs with opposite orientation: the Riemann surface is the region to the exterior of all the discs. Each identified pair can be viewed as adding a handle to the Riemann sphere, so if we place 2g discs we get a genus g Riemann surface. The moduli of the Riemann surface arise from the locations and sizes of the discs, as well as a possible twist in each identification. By an $\hbox {SL}(2,{\mathbb {C}})$ transformation we can always choose one of the discs to be centered at infinity, and if we restrict to geometries which are time-reversal invariant then we can take all discs to be centered on the real axis with no twists. A $g=2$ example is shown in Fig. 24, where we cut to get a state of the CFT on three circles. More generally by cutting a genus g surface we can produce a pure state in the Hilbert space of the CFT on $g+1$ spatial circles.

Fig. 25

A time-symmetric genus two handlebody. In the left diagram, the handlebody lies above the small hemispheres and below the large hemisphere, with the indicated identifications of hemispheres. The dashed boundary-time slice is extended straight up to give a symmetric timeslice of the bulk geometry. In the right diagram this bulk timeslice is shaded yellow as a cut through the heuristic representation of the genus two handlebody embedded in ${\mathbb {R}}^3$. Note that the three asymptotic boundaries are connected through a wormhole, as in Fig. 17

In order to find the bulk geometry of a state constructed in this manner, one needs to minimize the Euclidean Einstein–Hilbert action with negative cosmological constant over all solutions whose asymptotic boundary is the Riemann surface in question. Assuming that this minimum has a time-symmetric slice whose boundary lies in the real axis of the Schottky construction (if not then the bulk interpretation of the state is unclear), one then takes that slice as initial data for the Lorentzian Einstein equation to construct the real-time bulk geometry. The full set of these Euclidean solutions is rather complex, but there is an especially simple subset referred to as the handlebodies, which are obtained by “filling in” the Riemann surface embedded in ${\mathbb {R}}^3$. Given a Schottky presentation of a Riemann surface, there is a natural way to do this by viewing the complex plane in the Schottky construction as the boundary of the three-dimensional upper half plane, with metric

$$\begin{aligned} \mathrm{d}s^2=\frac{\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2}{z^2} \end{aligned}$$

(H.1)

and $z>0$, and then contracting the boundary of each disc using a hemisphere in the bulk. We illustrate this for genus two in Fig. 25. It is important to emphasize however that there can be different Schottky presentations of the same Riemann surface, which differ by acting with an element of the mapping class group of “large” diffeomorphisms that exchange the various cycles, eg $\hbox {PSL}(2,{\mathbb {Z}})$ for genus one, and these different presentations lead to different handlebodies in the bulk since different cycles are contracted. Moreover in general the Schottky presentation in which the time-symmetric slice is the real axis is not the Schottky presentation from which the handlebody is constructed, unlike in Fig. 25 where it is. At genus one there are only two time-symmetric handlebodies, the “Euclidean BTZ” and “thermal AdS” solutions, which differ by which of the two cycles is contracted in the bulk, and it is the Euclidean BTZ solution which is constructed as in Fig. 25.

Fig. 26

The time-symmetric bulk slice of a three-boundary wormhole. On the left we give the upper-half-plane presentation, while on the right we give the Poincare-disc presentation. The “interior” region is shaded grey, while the three “exterior” regions are shaded green, blue, and yellow. The dashed lines are the minimal length curves between the identification semicircles, and in Lorentzian signature they are the bifurcate horizons

In fact at any genus we are especially interested in the particular handlebody where the Schottky presentation with time-symmetry about the real axis does coincide with the Schottky presentation where the disc boundaries are contracted in the bulk, as shown in Fig. 25. The reason is that this is the only handlebody for which the time-symmetric bulk slice is connected, so in Lorentzian signature it is the one that describes a wormhole connecting all of the asymptotic boundaries. For example at genus one the bulk timeslice of the “thermal AdS” handlebody is two disconnected discs. We can understand better the structure of this wormhole by looking in more detail at the geometry of the time-symmetric slice, obtained by cutting through the geometry in the left diagram of Fig. 25 directly above the dashed boundary cut. This slice has the geometry of a quotient of the upper-half plane by a discrete subgroup, and in fact for this particular handlebody it is the Fuchsian presentation of the same cut Riemann surface on which the CFT path integral was evaluated to prepare the state. Moreover the intersection of the bifurcate horizons in the Lorentzian solution with this timeslice are given precisely by the minimal length curves between the identification semicircles [199], which gives an elegant way of splitting the time-symmetric slice into “interior” and “exterior” regions. We illustrate this for genus two in Fig. 26. In general whenever this spatial slice connects n asymptotic boundaries without any additional interior handles we can compute its volume using the Gauss–Bonnet theorem: it is an n-punctured sphere with a metric of constant negative curvature $R=-2$, and whose punctures are bounded by geodesics with $K=0$, so (in units where $\ell _{ads}=1$) we just have [200]

$$\begin{aligned} \mathrm {Interior \,\, spatial \,\, volume}=2\pi (n-2), \end{aligned}$$

(H.2)

which is independent of the moduli. Notice that indeed for $n>2$ (and therefore $g>1$) we have a nontrivial interior which grows in size as we increase n. Moreover it will not be in the entanglement wedge of any one of the boundaries, which is the key property for our wormhole-based proof of Theorem 4.2.

Fig. 27

The time-symmetric bulk slice of a genus five wormhole with six exterior regions. The interior is shaded grey, while the exteriors are shaded in various colors. The horizons are the dashed lines, and on this line in moduli space the horizon lengths are equal for boundaries 2, 3, 4 and 5, each of which is twice the length of the horizons for boundaries 1 and 6. On the left we show a geometry where these length are all finite, while on the right we show the limiting configuration as the lengths go to infinity

In order for that proof to be valid however, we need to check that these connected-wormhole handlebodies do actually dominate the Euclidean path integral, at least somewhere in moduli space. For genus one the handlebodies are all the solutions, and we know that at high temperature the Euclidean BTZ geometry is dominant. For $g\ge 2$ they are not: the others are usually called non-handlebodies, and they are less well-understood. Fortunately there is some evidence that non-handlebodies are always subleading to at least one handlebody in the Euclidean path integral [198, 201], and in what follows we will assume this to be the case. We are then left with the following question: at any particular point in moduli space, which choice of handlebody minimizes the Euclidean action? Unfortunately even this question has not been systematically addressed, since evaluating the Euclidean action of a handlebody amounts to computing the classical action of a solution of the Liouville equation on the boundary Riemann surface [195], which so far is only possible analytically in very restricted cases.⁹⁹ Recently a numerical algorithm has been developed for computing the Liouville action on arbitrary Riemann surfaces [198], specifically with the goal of clarifying which handlebodies dominate the Euclidean gravitational path integral with a boundary Riemann surface in various regions of moduli space, but so far it has only been applied in a few special cases. We also will not solve this problem, but will instead just suggest a limit in moduli space where we find it plausible that the connected wormhole should be the dominant handlebody.

Our proposal is most natural in the Poincare disk representation of the bulk time-slice, shown for genus two as the right diagram in Fig. 26. The idea is to introduce 2g equally-spaced and equally-sized semicircles around the edge of the Poincare disk, oriented such that there is a reflection symmetry across the real axis, and then identify the semicircles which are related by this reflection. We leave the size of the semicircles as a free parameter, which means we are looking at a one-dimensional slice through the moduli space. We illustrate this construction for genus five in Fig. 27, notice in particular the increased size of the interior region compared to Fig. 26, which is consistent with (H.2). Our conjecture is then that as we take the radii of the identification semicircles to zero, shown in the right diagram of Fig. 27, this handlebody will be the dominant solution in the Euclidean gravity path integral. Our conjecture is based on the observation that the Euclidean action is essentially the renormalized volume of spacetime, indeed evaluated on any solution which is a quotient of the hyperbolic three-plane we have we have

$$\begin{aligned} S_E=\frac{1}{4\pi G}\left( \int _M d^3x \sqrt{g}-\frac{1}{2}\int _{\partial M} d^2 x\sqrt{\gamma }(K-1)\right) . \end{aligned}$$

(H.3)

Given a choice of which boundary cycles to contract in the bulk, it is natural to expect that this action will tend to want to contract the smallest cycles, since most likely this can be done at the cost of the least volume in the bulk. For the family of handlebodies we have constructed, in the limit of small identification semicircles, and therefore large horizon length, the cycles in the boundary which correspond to spatial circles in the time-symmetric slice become parametrically larger than their dual cycles, which are the cycles which appear as the boundaries of the Schottky discs. At genus one and genus two we can confirm that this is indeed the case: the transition from thermal AdS to Euclidean BTZ indeed happens right when the thermal circle becomes smaller than the spatial one, and the numerical results of [198] confirm that our limiting family of Riemann surfaces, which corresponds to the line $\ell _3=2\ell _{12}$ in their Fig. 7, dominates over the other possible handlebodies (and also one non-handlebody they were able to check analytically) in the limit of large horizon length. Assuming this conjecture is also correct at higher genus, the connected wormhole will always dominate at sufficiently large horizon length, and any quasilocal bulk operator can fit into the interior region for sufficiently high genus.¹⁰⁰ We are then able to run our second proof of Theorem 4.2.

Sphere/Torus Solutions of Einstein’s Equation

In this appendix we discuss in more detail the solutions of Einstein’s equation with negative cosmological constant used in Sect. 8.3, with metric of the form

$$\begin{aligned} \mathrm{d}s^2=-\alpha (r)\mathrm{d}t^2+\frac{\mathrm{d}r^2}{\alpha (r)\beta (r)}+e^{\gamma (r)}\mathrm{d}x_p^2+r^2 \mathrm{d}\Omega _{d-p-1}^2. \end{aligned}$$

(I.1)

The time, planar, radial, and spherical components of Einstein’s equations with negative cosmological constant for metrics of the form (I.1) are given respectively by¹⁰¹

$$\begin{aligned}&r(\alpha \beta '+\alpha ' \beta )\left( 2(d-p-1)+pr \gamma '\right) +2(d-p-1)\alpha \beta \left( d-p-2+pr \gamma '\right) \nonumber \\&\qquad +pr^2\alpha \beta \left( \frac{p+1}{2}\gamma '^2+2\gamma ''\right) \nonumber \\&\quad =2\big ((d-p-2)(d-p-1)+d(d-1)r^2\big ) \end{aligned}$$

(I.2)

$$\begin{aligned}&r\beta '(r\alpha '+\alpha (2(d-p-1)+(p-1)r\gamma '))+2\beta \Big ((d-p-2)(d-p-1)\alpha +2(d-p-1)r\alpha '\nonumber \\&\qquad +r^2\alpha ''\Big )+2(p-1)\beta \Big (r\gamma '\big (d-p-1+r\alpha '+\frac{p}{4}r \gamma '\big )+r\gamma ''\Big ) \nonumber \\&\quad =2\big ((d-p-2)(d-p-1)+d(d-1)r^2\big ) \end{aligned}$$

(I.3)

$$\begin{aligned}&p(p-1)r^2\alpha \beta \gamma '^2+2pr\beta (2(d-p-1)+r\alpha ')\gamma '+4\beta (d-p-1)((d-p-2)\alpha +r\alpha ') \nonumber \\&\quad =4\big ((d-p-2)(d-p-1)+d(d-1)r^2\big ) \end{aligned}$$

(I.4)

$$\begin{aligned}&r^2\alpha '\beta '+r(2\alpha '\beta +\alpha \beta ')(2(d-p-2)+pr\gamma ')+2r^2\beta \alpha ''+2(d-p-3)(d-p-2)\alpha \beta \nonumber \\&\qquad +2p(d-p-2)r\alpha \beta \gamma '+pr^2 \alpha \beta \left( \frac{p+1}{2}\gamma '^2+2\gamma ''\right) \nonumber \\&\quad =2\big ((d-p-3)(d-p-2)+d(d-1)r^2\big ). \end{aligned}$$

(I.5)

We first consider the vacuum solution, where it is the sphere that contracts in the bulk. We can then assume a further symmetry between the time and planar directions, setting

$$\begin{aligned} \gamma =\log \alpha . \end{aligned}$$

(I.6)

The first two equations of motion become redundant, and the third simplifies so that we can solve for $\beta$:

$$\begin{aligned} \beta =\frac{4\alpha \left( (d-p-2)(d-p-1)+d(d-1)r^2\right) }{4(d-p-2)(d-p-1)\alpha ^2+4(d-p-1)(p+1)r\alpha \alpha '+p(p+1)r^2\alpha '^2}.\nonumber \\ \end{aligned}$$

(I.7)

After this substitution, the first, third, and fourth equations of motion each give the same second order ordinary differential equation for $\alpha$.

To find the right boundary conditions, we can expand $\alpha$ in a power series near $r=0$ and then substitute into this differential equation. The result is that if we want $\alpha (0)>0$ then we must have

$$\begin{aligned} \alpha (r)=\alpha (0)\left( 1+\frac{1}{d-p}r^2+O(r^3)\right) . \end{aligned}$$

(I.8)

This then tells us that we must impose $\alpha '(0)=0$, which from (I.7) then implies that $\beta (0)\alpha (0)=1$, as needed to avoid a singularity at $r=0$. The overall scale of $\alpha$ can be absorbed into a rescaling of the time coordinate, so we thus have a unique vacuum solution, as found by Horowitz and Copsey for $d=4$ and $p=1$.

Fig. 28

Numerical plots of the vacuum solution for $p=2$ and $d=5$

Fig. 29

Numerical plots of the wormhole solution, for $r_s=2$, $d=5$, and $p=2$

The differential equation for $\alpha$ can only be solved numerically, which we’ve written a mathematica file (included in the arxiv submission) to do. We’ve checked for a variety of d and p that, with these boundary conditions, the solutions for $\alpha$ and $\beta$ are positive, and behave as $\alpha \beta =r^2+o(r^2)$ at large r, as required for the geometry to be asymptotically AdS. We plot a typical example in Fig. 28.

We now consider the wormhole solutions, where $\alpha$ vanishes at some $r_s>0$. In this case we cannot assume symmetry between t and x, so we must treat $\alpha$, $\beta$, and $\gamma$ independently. We first observe that the third equation of motion is quadratic in $\gamma '$, and can be solved to give an expression for $\gamma '$ in terms of $\alpha$, $\alpha '$, and $\beta$:

$$\begin{aligned} \gamma '&=\frac{1}{p(p-1)r^2\alpha \beta }\Bigg (-pr\beta (r\alpha '+2(d-p-1)\alpha )\nonumber \\&\quad +\Big [p^2r^2\beta ^2(r\alpha '+2(d-p-1)\alpha )^2 \nonumber \\&\quad +4p(p-1)r^2\alpha \beta \left( (d-p-2)(d-p-1)\right. \nonumber \\&\quad \left. + d(d-1)r^2-(d-p-1)\beta (r\alpha '+(d-p-2)\alpha )\right) \Big ]^{1/2}\Bigg ) \end{aligned}$$

(I.9)

This expression then may be substituted into the other equations, to produce a pair of independent differential equations which are second order in $\alpha$ and first order in $\beta$. One nice simplification occurs if we take the difference of the first and fourth equations, which tells us that

$$\begin{aligned}&-2r^2\beta \alpha ''+2r\alpha \beta -r^2\alpha '\beta '+2\alpha \beta (2(d-p-2)\\ {}&+pr\gamma ')-r\alpha '\beta (2(d-p-3)+pr\gamma ')=4(d-p-2). \end{aligned}$$

We can pair this equation with, say, the first equation, and then solve them numerically. We now need three boundary conditions: one is provided by $\alpha (r_s)=0$, and another can be fixed by rescaling time so that $\alpha '(r_s)$ takes any value we choose. Finally by inspecting the form of the equations at a point where $\alpha =0$, we can see that we must have

$$\begin{aligned} \beta (r_s)=\frac{d-p-2+\mathrm{d}r_s^2}{r_s\alpha '(r_s)}. \end{aligned}$$

(I.10)

The parameter $r_s$ is physical, and sets the temperature parameter in the thermofield double state.

We’ve again written mathematica code (included in the arxiv submission) to solve these equations numerically, and again confirmed for a variety of d, p, and $r_s$ that $\alpha$ and $\beta$ are positive, and they have the right large-r asymptotics. We plot an example in Fig. 29.