Bulk geometry in gauge/gravity duality and color degrees of freedom
DDMUS-MP-21/02
Bulk geometry in gauge/gravity duality and color degrees of freedom
Masanori HanadaDepartment of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK
Abstract U( N ) supersymmetric Yang-Mills theory naturally appears as the low-energy effectivetheory of a system of N D-branes and open strings between them. Transverse spatialdirections emerge from scalar fields, which are N × N matrices with color indices; roughlyspeaking, the eigenvalues are the locations of D-branes. In the past, it was argued thatthis simple ‘emergent space’ picture cannot be used in the context of gauge/gravity duality,because the ground-state wave function delocalizes at large N , leading to a conflict with thelocality in the bulk geometry. In this paper we show that this conventional wisdom is notcorrect: the ground-state wave function does not delocalize, and there is no conflict withthe locality of the bulk geometry. This conclusion is obtained by clarifying the meaning ofthe ‘diagonalization of a matrix’ in Yang-Mills theory, which is not as obvious as one mightthink. This observation opens up the prospect of characterizing the bulk geometry via thecolor degrees of freedom in Yang-Mills theory, all the way down to the center of the bulk.1 a r X i v : . [ h e p - t h ] F e b ontents (3 + 1) -d Yang-Mills 164 Future Directions 17A Relation between BEC and color confinement, and ‘genuine’ gauge in-variance 19 A.1 H ext and H inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.2 BEC, confinement and ‘genuine’ gauge invariance . . . . . . . . . . . . . . . 20A.3 Speculations regarding the Maldacena-Milekhin conjecture . . . . . . . . . . 21 The low-energy effective dynamics of N D p -branes parallel to each other and open stringsconnecting them can be described by ( p + 1)-dimensional maximally supersymmetric Yang-Mills (SYM) theory with the U( N ) gauge group [1]. SYM theory has 9 − p scalar fields X I ( I = 1 , , · · · , − p ) which are N × N Hermitian matrices. When all scalar fields can be(almost) simultaneously diagonalized, diagonal elements of the matrices are interpreted asthe positions of D p -branes in the transverse directions (( X ii , · · · , X ii − p ) ∈ R − p is regardedas the coordinate of the i -th D-brane) and the off-diagonal elements X ijI are interpretedas the amount of the open-string excitations connecting i -th and j -th D p -branes. If thematrices are not (almost) simultaneously diagonalizable but can be taken block diagonal,each block is a bound state of D-branes and strings.Matrix Theory conjecture [2] claims the (0 + 1)-dimensional SYM — D0-brane quantummechanics — is not just a low-energy effective theory, but it actually describes M-theoryin certain parameter region. Bound states, or equivalently non-commutative blocks, areinterpreted as various objects such as graviton, higher-dimensional D-brane and black hole.If we separate the ( N, N ) component from others and see how they interact, we can studythe geometry formed by ( N −
1) D-branes and strings between them, by using the N -thD-brane as a probe. 2 puzzle Can the same geometric picture be valid in the Maldacena-type gauge/gravity duality [3,4]? Naively, we would expect that this simple mechanism of the emergent space works asfollows. In (3 + 1)-dimensional super Yang-Mills theory, there are six scalars with whichthe coordinate in R can be specified. This, and R , along which D-branes are extended,give ten-dimensional spacetime. D-branes and open strings can interact with each otherand nontrivial metric can be induced effectively. R , and the radial coordinate of R formAdS , and the angular part of R is S . We can imagine similar stories for SYM in differentdimensions. It would be nice if such a simple mechanism can actually work. However it iswidely believed that this picture does not work, or at least a more sophisticated approachis required; see e.g., [5, 6, 7, 8, 9]. Such skepticism is based on the observation that thematrices are highly non-commutative in the region where weakly-coupled string theory isvalid, and the notion of ‘location’ is not apparent there. This can also be phrased thatthe bound state of D-branes and matrices are very big compared to the counterpart in thegravity side. Later in this paper, we will show that this argument is not correct and the‘location’ can actually make sense. But for now let us follow the reasoning in the previousreferences. Let us consider the D0-brane matrix model with the following normalization: L = Tr (cid:18)
12 ( D t X I ) + g X I , X J ] + fermion part (cid:19) . (1)Here D t X I = ∂ t X I − i [ A t , X I ] is the gauge-covariant derivative. In the ’t Hooft large- N limit (’t Hooft coupling λ = g N ∼ N , energy E ∼ N ) and at sufficiently strong coupling( λ (cid:29) ( E/N ) ), type IIA supergravity is a good dual description [4]. In the ’t Hooft large- N limit, the expectation value (cid:104) Tr X I (cid:105) is of order N , at any temperature including zero andany coupling. At zero temperature, all contributions are from zero-point fluctuations. If wediagonalize each X I , the eigenvalues are of order √ N . So the bound state is parametricallylarge at large N . If we take the ’t Hooft coupling λ = g N to be large, then the eigenvaluesscale as λ − / √ N at sufficiently low temperature ( T (cid:28) λ / ). This is larger than the size ofthe black hole (black zero-brane) sitting at the center of the bulk geometry, and completelycovers the region where weakly-coupled string theory is valid. When X I =1 is diagonalized, X I =2 , , ··· , are not diagonal at all, and the off-diagonal elements dominate (cid:104) Tr X I =2 , ··· , (cid:105) .Hence the ‘location of D-brane’ is not a crisp notion when weakly-coupled string theory isvalid. The same argument holds in any gauge theory in the ’t Hooft limit, including (3 + 1)-dimensional SYM; when O ( N ) is expected from the gravity picture, O ( √ N ) is obtainedin the gauge theory side. This has been regarded as an obstruction for the sub-AdS-scalebulk reconstruction in AdS/CFT correspondence. We impose the traceless condition for each matrix so that the bound state is centered around the originof R . resolution In this paper, we show that the size of the bound state in the gauge theory side isactually much smaller, and the ‘location’ can make sense. Whether the metric expectedin the holographic duality actually emerges is a separate issue, which will not be discussedin this paper. (We will suggest a few future directions aiming for the verification of theemergence of the local bulk geometry.)The starting point of our discussion is this question:
What do we mean by the ‘diagonalization of a matrix’ ?
Of course, when an N × N Hermitian matrix is given, there is no ambiguity; it is a well-defined linear-algebra problem. However, because we are talking about a physics problem,we have to make sure what is the ‘matrix’ suitable for the problem under consideration.Namely, we have to answer the following question:
What is the ‘matrix’ that characterizes the bulk geometry?
The argument in the past implicitly used one of the following two pictures: (1) interpret atypical configuration ( ∼ master field) in the path integral as a ‘bound state’, or (2) interpreta typical result of the measurement of X I,ij (which is a coordinate eigenstate described bythe coordinates X I,ij , i.e., a state | X (cid:105) that satisfies ˆ X I,ij | X (cid:105) = X I,ij | X (cid:105) for all I, i, j ) as a‘bound state’. Either way, there are c-number matrices X I,ij , so we can ‘diagonalize’ oneof them and define ‘eigenvalues of a matrix X I,ij ’. In the picture (2), X I,ij is the ‘eigenvalueof operator ˆ X I,ij ’. Both (1) and (2) fail in more or less the same manner, so let us focuson (2) below for concreteness. Furthermore, we consider the D0-brane theory that has ninescalars X I =1 , , ··· , . (The generalizations to other theories are straightforward.)By assumption, we are interested in low-energy states including the ground state. Acoordinate eigenstate cannot be a low-energy state due to the uncertainty principle; instead,we must consider linear combinations of coordinate eigenstates, such as a wave packet. Ingeneral, a low-energy state | Φ (cid:105) is written in terms of the coordinate eigenstates | X (cid:105) as | Φ (cid:105) = (cid:90) R N dX | X (cid:105)(cid:104) X | Φ (cid:105) ≡ (cid:90) R N dX Φ( X ) | X (cid:105) . (2)The wave function Φ( X ) has to be extended smoothly in R N to some extent. Hence ‘theeigenvalue of operator ˆ X I,ij ’ is not well-defined, and a naive ‘diagonalization’ based on theintuition from coordinate eigenstates is not well-defined either. That (cid:104) Tr X I (cid:105) is of order N does not necessarily mean ‘the eigenvalues of X I ’ are of order √ N ; actually the very notion Strictly speaking, we have to take into account the fermions as well. We emphasize that the wave packet under consideration is in R N and not in R . The bound state ofD-branes and open strings can be extended in R , but it is a completely different story. In the past, thesetwo completely different notions —- a wave packet extended in R N , and a bound state extended in R — were not properly distinguished. f the ‘eigenvalues’ has to be defined more carefully . In order to define the ‘coordinate ofD-branes’ in R , we have to define the ‘coordinate of matrices’ in R N .In fact there is a very standard way to introduce the ‘coordinate of matrices’: if Φ( X )is a wave packet around Y I,ij , the center of the wave packet Y I,ij is a natural ‘coordinate ofmatrices’. This point is explained in Sec. 2. We will see that this Y I,ij is naturally relatedto the locations of D-branes and open-string excitations. It turns out that the ground stateis a wave packet localized around the origin of R N , i.e., Y I,ij = 0. Along each directionof R N , the width of the wave packet is of order N . This is the reason that (cid:104) Tr X I (cid:105) isof order N . The ground state is gauge-invariant, i.e., it is impossible to change the shapeof the wave packet via gauge transformation. It is perfectly consistent with a simple andnatural interpretation: in the ground state, all D0-branes are sitting at the origin of thebulk, and no open string is excited. The organization of this paper
This paper is organized as follows. In Sec. 2, we consider matrix models. To make thelogic transparent, we use the canonical quantization and quantum states, rather than thepath-integral formalism. All the essence which can readily be generalized to Yang-Millstheory in any dimension can be understood just by considering the Gaussian matrix model,which is the subject of Sec. 2.1. We show that the wave function does not delocalize, andprobes can be introduced in a very standard manner. In Sec. 2.2, we will see how simpleresults obtained for the Gaussian matrix model are generalized to interacting theories. InSec. 2.3, we consider the D0-brane matrix model and dual gravity description. Sec. 2.3 israther speculative, because we have not yet fully understood the dynamics of the model. InSec. 3, we consider (3 + 1)-dimensional super Yang-Mills and AdS /CFT correspondence.Potentially interesting future directions are discussed in Sec. 4. In this section, we consider the matrix model. Before studying the D0-brane matrixmodel, let us consider a simpler example, a nine-matrix model with the following Lagrangian(with the Minkowski signature): L = Tr (cid:18)
12 ( D t X I ) − X I + g X I , X J ] (cid:19) . (3)The zero-coupling limit (the Gaussian matrix model) is analytically solvable, and we canunderstand everything explicitly. At strong coupling, the quadratic term − X I is negligibleand this model reduces to the bosonic part of the D0-brane matrix model. This model wasstudied in detail via lattice Monte Carlo simulation [10, 11]. While this theory does not This way of introducing a ‘coordinate of matrices’ does not work for more generic states, such as asuperposition of multiple wave packets. This is not a bug, this is a generic feature of quantum mechanics. H = Tr (cid:18)
12 ˆ P I + 12 ˆ X I − g X I , ˆ X J ] (cid:19) . (4)Because we are studying the gauged matrix model, the physical states are gauge-invariant.Let us denote the Hilbert space of gauge-singlet states as H inv . When the matrices arerelated to D-branes and strings, our brains tend to think in the ‘Higgsed’ picture, namelywe often consider the situation that diagonal elements are large and well-separated. Thisintuition uses non-singlet states. Hence let us also consider the extended Hilbert space H ext that contains non-singlet states as well. The partition function associated with the canonicalensemble at temperature T can be written as Z ( T ) = Tr H inv e − ˆ H/T , where Tr H inv is the traceover gauge singlets. We can also write it by using the trace in the extended Hilbert spaceas Z ( T ) = N ) (cid:82) dU Tr H ext ( ˆ U e − ˆ H/T ). Here U is an element of U( N ), and the integral istaken by using the Haar measure. The operator ˆ U enforces the gauge transformation, and (cid:82) dU ˆ U serves as the projector to H inv . In terms of H ext , ‘gauge fixing’ can naturally beunderstood as in the path integral formalism. See Appendix A.1 for more details.Each state | Φ (cid:105) can be expressed by using the wave function in the coordinate basis, (cid:104) X | Φ (cid:105) = Φ( X ) , (5)where Φ( X ) is a function of 9 N variables X ijI . If Φ( X ) is well-localized wave packet inthe 9 N -dimensional space centered around X ijI = x I,i δ ij , then (cid:126)x i = ( x ,i , · · · , x ,i ) ∈ R isnaturally interpreted as ‘the location of the i -th D-brane’.Let us use the generators of U( N ), which are denoted by τ α and normalized as Tr( τ α τ β ) = δ αβ . We can write ˆ X I and ˆ P I as ˆ X ijI = (cid:80) α ˆ X αI τ ijα and ˆ P ijI = (cid:80) α ˆ P αI τ ijα . The commutationrelation is [ ˆ X αI , ˆ P βJ ] = iδ IJ δ αβ . (6)By using the annihilation operators ˆ a I,α = ˆ X I,α − i ˆ P I,α √ and creation operators ˆ a † I,α , we canconstruct the Fock basis for H ext . For each ( I, α ), the number operator is defined byˆ n I,α = ˆ a † I,α ˆ a I,α , and the Fock state is defined as the eigenstate of the number operator,ˆ n I,α | n (cid:105) I,α = n I,α | n (cid:105) I,α . Specifically, the Fock vacuum | (cid:105) I,α is specified by ˆ a I,α | (cid:105) I,α = 0,and the excited states are constructed as | n (cid:105) I,α = (ˆ a † I,α ) n √ n ! | (cid:105) I,α . Then we can simply takethe tensor product, |{ n I,α }(cid:105) = ⊗ I,α | n I,α (cid:105)
I,α , (7)to obtain the orthonormal basis of H ext . If we take a specific set of { n I,α } in which thediagonal elements are highly excited while the off-diagonal elements are not, then such state6s analogous to the ‘Higgsed’ states. Indeed, by taking a linear combination of such states,we can build a wave packet localized about X I,ii (cid:54) = 0 ( i = 1 , , · · · , N ) and X I,ij = 0 ( i (cid:54) = j ).The U( N ) transformation is defined byˆ X I,ij −→ ( U ˆ X I U − ) ij = N (cid:88) k,l =1 U ik ˆ X I,kl U − lj (8)and ˆ P I,ij −→ ( U ˆ P I U − ) ij = N (cid:88) k,l =1 U ik ˆ P I,kl U − lj . (9)Creation and annihilation operators are transformed in the same manner. With the adjointindex α , the transformation rule isˆ X α = Tr( ˆ Xτ α ) −→ ˆ X ( U ) α = Tr(( U ˆ XU − ) τ α ) , ˆ P α = Tr( ˆ P τ α ) −→ ˆ P ( U ) α = Tr(( U ˆ P U − ) τ α ) , ˆ a α = Tr(ˆ aτ α ) −→ ˆ a ( U ) α = Tr(( U ˆ aU − ) τ α ) . (10)The Fock vacuum |{ }(cid:105) = ⊗ I,α | (cid:105) I,α is U( N )-invariant, and the excited states transform as |{ n I,α }(cid:105) = (cid:32)(cid:89) I,α (ˆ a † I,α ) n I,α (cid:112) n I,α ! (cid:33) |{ }(cid:105) −→ (cid:32)(cid:89) I,α (ˆ a ( U ) † I,α ) n I,α (cid:112) n I,α ! (cid:33) |{ }(cid:105) . (11)From each non-singlet state, a U( N )-invariant state is obtained by averaging over all groupelements of U( N ) and then properly normalizing the norm. Let us consider the case of g = 0, i.e., the Gaussian matrix model. This example issolvable, and contains all the essence. In the free limit ( g = 0), the Hamiltonian isˆ H Gaussian = N (cid:88) α =1 (cid:18)
12 ˆ P I,α + 12 ˆ X I,α (cid:19) . (12)The ground state is the Fock vacuum: | ground state (cid:105) = |{ }(cid:105) = ⊗ I,α | (cid:105) I,α . (13)7he vacuum expectation value of (cid:80) I Tr ˆ X I is N due to the zero-point fluctuation. Hencebased on the conventional wisdom one would conclude that the size of the ground-state wavefunction is of order √ N . However this is actually not the case. Because the Fock vacuum ofeach harmonic oscillator is represented by the Gaussian wave function (cid:104) X I,α | (cid:105) I,α = e − X I,α/ (2 π ) / ,the wave function describing all matrix entries is (cid:104) X | ground state (cid:105) = 1(2 π ) N / exp (cid:32) − (cid:88) I,α X I,α (cid:33) = 1(2 π ) N / exp (cid:32) − (cid:88) I Tr X I (cid:33) . (14)This is manifestly U( N )-invariant. The size of the wave function is the same for all matrixentries. We cannot arrange the ground-state wave function such that we can observe alarge value of a diagonal element (more specifically, of order √ N ) with large probability,in any ‘gauge’. Typically Tr X I is of order N , but this is because all the entries can takeorder N values, and the probability of at least one eigenvalue becomes of order √ N scalesroughly as e − N , which is negligible at large N . This state is a well-localized wave packet inthe 9 N -dimensional space centered around X ijI = 0. Namely, all D-branes are sitting atthe origin, and no open string is excited. Note that the full U( N )-invariance is a naturalproperty of N -coincident D-branes without open string excitations [1].Of course, each | X (cid:105) is not U( N )-invariant, and we can ‘choose a gauge’ e.g., in which X is diagonal, if we like. However, the linear combination (cid:82) dX | X (cid:105)(cid:104) X | ground state (cid:105) is U( N )-invariant and there is no way to choose any ‘gauge’. As far as we consider low-energy states,it is meaningless to talk about the eigenvalue distribution of the coordinate eigenstate | X (cid:105) .It may be instructive to emphasize the difference between two kinds of the ‘size of boundstate’ that were not properly distinguished in the past. The first one is the distributionof D-branes (diagonal elements) that can be read off from the center of the wave packet.This is defined in R . The second one is the width of the wave packet in R N . These twonotions correspond to the ‘slow modes’ and ‘fast modes’ in the references, respectively. Wehave seen that, for the ground state, the latter is of order N . All D-branes are sitting atthe origin, so the former is zero. Perhaps it is not easy to grasp the essence of the statement just by looking at the groundstate. Let us examine the coherent states, which nicely illuminate the important points. It may be instructive to rephrase it as follows. Imagine a uniform probability distribution on a spherewith radius R in D dimensions parametrized by x , · · · , x D . By integrating out x , · · · , x D , we obtain thedistribution of x scaling as ρ ( x ) ∼ (cid:16) − x R (cid:17) ( D − / . In the matrix model, we roughly have a situationthat D ∼ N → ∞ , R ∼ N → ∞ , which leads to ρ ( x ) ∼ e − Dx R ∼ e − x . Therefore, if the radius increaseswith dimensions as D ∼ R , large radius does not mean the delocalization.
8e can put the wave packet at any point in { Y I,α } ∈ R N , just by acting the translationoperator: | wave packet at { Y I,α }(cid:105) = e − i (cid:80) I =1 (cid:80) N α =1 Y I,α ˆ P I,α | ground state (cid:105) = e − i (cid:80) I =1 Tr( Y I ˆ P I ) | ground state (cid:105) . (15)A more generic wave packet with nonzero momentum is e − i (cid:80) I =1 Tr( Y I ˆ P I − Q I ˆ X I ) | ground state (cid:105) . (16)Below we mainly focus on the case of Q I = 0 for simplicity.The center of the wave packet { Y I,α } ∈ R N describes the D-brane configuration, whichcorresponds to the ‘slow mode’ in the references. It can change via the U( N ) transformationas | wave packet at { Y I,α }(cid:105) −→ | wave packet at { Y ( U ) I,α }(cid:105) , (17)where Y ( U ) I,ij = ( U − Y I U ) ij . (18)See Fig. 1 for a visual sketch. Therefore it makes sense to talk about the diagonalization ofthe slow mode Y . However the width of the wave packet, that comes from | ground state (cid:105) ,does not change via the U( N ) transformation; see Fig. 1 again. This is because (cid:104) wave packet at { Y I,α }| ( ˆ X I,α − Y I,α ) k | wave packet at { Y I,α }(cid:105) = (cid:104) ground state | ˆ X kI,α | ground state (cid:105) (19)holds for each ( I, α ) and any k , and the right hand side is gauge-invariant. This part picksup the quantum fluctuation, which corresponds to the ‘fast mode’ in the references; henceit does not make sense to talk about the diagonalization of the fast mode.If we measure the coordinate in R N , we get a localized distribution around { Y I,α } .The width of the fluctuation along each of 9 N directions is of order N . Therefore, thelocation of the center of the wave packet can be distinguished from the origin if (cid:112)(cid:80) I Tr Y I is sufficiently larger than 1. If there are two wave packets around { Y I,α } and { Y (cid:48) I,α } , theycan be distinguished if (cid:112)(cid:80) I Tr( Y I − Y (cid:48) I ) is sufficiently larger than 1.The problem with the past treatment [5, 6] was that they took a typical configurationin the path integral, or a typical result in the measurement of ˆ X I,ij , and diagonalized itwithout separating the slow modes, that can actually be diagonalized, and the fast modes,that cannot really be diagonalized. A better procedure is to diagonalize the center of the The eigenvalues of Y I,ij are gauge-invariant, and the distance from the origin in R N , that can beexpressed as (cid:112)(cid:80) I Tr Y I , is also gauge-invariant. To their credit, they clearly pointed out the necessity of the separation of slow and fast modes, but didnot identify a concrete procedure for the separation. R N defined by (15). Each gray disk and black pointrepresent a wave packet and its center, respectively. Under the gauge transformation, thelocation of the center moves, but the shape of the wave packet in R N does not change.wave packet, or equivalently, to diagonalize the expectation value of ˆ X . This procedure hasa well-defined meaning at the level of the quantum states in the Hilbert space.Another way to phrase it is that the past treatment was the gauge fixing of | X (cid:105) ratherthan that of | ground state (cid:105) or | wave packet at { Y I,α }(cid:105) . The position-eigenstate | X (cid:105) is notthe low-energy state relevant for physics under consideration; the uncertainty principleforces us to consider a wave packet.Let us see a few special cases whose meanings are obvious. • Let us separate one of the D-branes from others sitting at the origin, without excitingany open string. Specifically, we can construct a wave packet centered around a point (cid:126)Y ij = δ iN δ jN (cid:126)y ∈ R N , as e − i(cid:126)y · ˆ (cid:126)P NN | ground state (cid:105) . (20)As long as | (cid:126)y | (cid:38)
1, the position of the probe is a legitimate notion. • By using the U(1)-part we can easily make a U( N )-invariant state describing N -coincident D0-branes at point (cid:126)y , as e − i(cid:126)y · ( (cid:80) Nk =1 ˆ (cid:126)P kk ) | ground state (cid:105) . (21)Note that this full U( N )-invariance is exactly what we expect when N D-branes aresitting on top of each other [1]. • We can construct a state describing ‘diagonal matrices’ (cid:126)Y ij = (cid:126)y i δ ij as e − i (cid:80) Nk =1 (cid:126)y k · ˆ (cid:126)P kk | ground state (cid:105) . (22)10f some (cid:126)y i ’s take the same value, say N of them are (cid:126)x , N of them are (cid:126)x (cid:48) and so on,then such a state is invariant under U( N ) × U( N ) × · · · . This symmetry enhancementis consistent with the interpretation that (cid:126)x i is the location of i -th D-brane [1].We can add further justification for the interpretation that the center of the wave packetshould be identified with the ‘location of D-branes’, and more generally, ‘matrices’. TheHamiltonian ˆ H is a polynomial of ˆ P I and ˆ X I , so let us write it as ˆ H = H ( ˆ P , ˆ X ). Then wecan show that e i (cid:80) I Tr( Y I ˆ P I ) H ( ˆ P , ˆ X ) e − i (cid:80) I Tr( Y I ˆ P I ) = H ( ˆ P , ˆ X + Y ) . (23)Therefore, instead of acting H ( ˆ P , ˆ X ) on | wave packet at { Y I,α }(cid:105) , we could act H ( ˆ P , ˆ X + Y )on | ground state (cid:105) , if we like. In the latter treatment, when the coupling constant g isnonzero (which will be studied in Sec. 2.2), if we take Y to be diagonal, the mass terms forthe off-diagonal elements are generated from the commutator-squared term in the action.They are identified with the mass terms for open strings [1].We emphasize that the coherent state discussed here is just one of many possible real-izations of the wave packets. When the interaction is introduced, it may or may not be astable wave packet. If we consider strongly-coupled theories with gravity duals, D-braneprobes in the gravity side may not be described by the coherent state precisely, and a largemodification may be needed. We will discuss this point further in later sections.Note also that, generically, these states are not U( N )-invariant, that is, they belong to H ext but not to H inv . If we want to discuss everything in terms of H inv , we have to projectthem to the singlet sector. Equivalently, we can take a superposition of all wave packetsalong the gauge orbit; , see Fig. 2. On the other hand, the ground state is automaticallygauge-invariant without performing such a projection. In this sense, the ground state is‘genuinely’ U( N )-invariant [12]. The N -coincident-D-brane state (21) is also genuinelyU( N )-invariant. In Appendix A, we explain how such ‘genuine’ gauge singlets can bedistinguished from the other kind of singlets.The same situation appears in a system of N identical bosons, which can be regarded asa gauged quantum mechanics of N -component vectors [12]. That the bosons are ‘identical’means the physical states have to be invariant under the S N permutation, hence this systemis a gauge theory with S N gauge group. This system can be analyzed by using the extendedHilbert space, and Bose-Einstein condensation [13] is characterized by the same ‘genuine’gauge invariance [14, 15, 16, 17, 18]. The author would like to thank Alexey Milekhin for useful discussion regarding this point. In (16), we should choose Q I appropriately such that the configuration does not move along the gaugeorbit. An implicit but important assumption here is that | Φ (cid:105) and its U( N )-symmetrized version have similarproperties, in that the expectation values of gauge-invariant operators are identical up to small corrections. auge orbit of Figure 2: Gauge-symmetrized version of the wave packet shown in Fig. 1 (gray ring) . Thewave function is localized near the gauge orbit of { Y I,α } (dotted circle) . The ground state(gray disk) is localized around the origin. Note that the shape and volume of the gaugeorbit depends on { Y I,α } . Let us take a generic wave packet, by taking Y I and Q I in (16) to be generic matriceswhose eigenvalues are of order N . (More generally, we can take a superposition of suchstates.) Generically, such a state is not invariant under any U( N ) transformation (11),except that any state is trivially invariant under the adjoint action of the U(1) part. There-fore we can choose a gauge such that the diagonal elements are more highly excited thanthe off-diagonal elements. In this case the size of the bound state is actually of order √ N .The same holds for other kinds of excited states such as the Fock state |{ n I,α }(cid:105) ∈ H ext at sufficiently high energy.
More interesting physics can be observed at the intermediate energy scale. As shownin Ref. [19], there are two phase transitions at EN = 0 (Hagedorn transition) and EN = (Gross-Witten-Wadia transition). In between these two phase transitions, at E = M , theU( M ) subgroup of U( N ) is deconfined. This is a particular example of partial deconfine-ment [20, 21, 22, 19, 11] that is conjectured to be a generic feature among various large- N gauge theories. We can fix a gauge such that deconfinement is taking place in the M × M upper-left block. Equivalently, we restrict Y I and Q I to be M × M . This fixes U( N ) down We subtracted the zero-point energy N .
12o U( M ) × U( N − M ). We can further fix U( M ) such that the diagonal entries of the decon-fined block becomes as large as O ( √ M ). The ‘genuine’ symmetry U( N − M ) is left unfixed.Hence we obtain a bound state whose radius is ∼ √ M . This bound state is conjectured tobe the gauge-theory realization of the small black hole [20].As a probe, we can excite the ( N, N ) component. The notion of a ‘location’ can makesense if the distance from the origin is sufficiently larger than the ‘BH radius’ ∼ √ M . Even at finite coupling ( g > N limit, the distributionof the phases of the Polyakov loop can be used to see if a given state in H inv is ‘genuinely’gauge-invariant [12]. As a starting point, let us write the canonical partition function attemperature T as Z ( T ) = 1volU( N ) (cid:90) dU Tr H ext ( ˆ U e − ˆ H/T ) , (24)where Tr H exit is the trace in the extended Hilbert space. Here U is an element of U( N ),and the integral is taken by using the Haar measure. The operator ˆ U enforces the gaugetransformation. This U corresponds to the Polyakov line in the path integral formulation.The contribution of the ground state is1volU( N ) (cid:90) dU e − E /T (cid:104) ground state | ˆ U | ground state (cid:105) , (25)where E is the energy of the ground state. If the ground state is not genuinely U( N )-invariant, there are degenerate vacua in H ext related by gauge transformation, and we needto sum them up. Either way, only such U ∈ U( N ) that leaves | ground state (cid:105) invariant cancontribute to the partition function. This U is the Polyakov loop. That the phases of thePolyakov loop is uniform at zero temperature is consistent with the genuine U( N )-invarianceof the ground state, i.e., it is invariant under any U( N ) transformation. For details, seeRef. [12]. Note that this argument is essentially identical to the characterization of Bose-Einstein condensation of N indistinguishable bosons via the S N -invariance [14, 15, 16].In the ’t Hooft large- N limit ( λ = g N ∼ N , T ∼ N , E ∼ N ), the expectation value (cid:104) Tr X I (cid:105) is proportional to N . At low temperature T (cid:28) λ / and strong coupling λ (cid:29)
1, itscales as (cid:104) Tr X I (cid:105) ∼ λ − / N . From this, in the past it has been interpreted that the size ofthe ground state wave function is λ − / N / . However, with a natural assumption that theground-state wave function is gauge-invariant, this scaling simply means that the width ofthe ground-state wave function is proportional to λ − / with respect to each direction of R N . Just as in the free theory, we can introduce a probe by exciting the ( N, N ) componentby using (20), by taking | ground state (cid:105) to be the vacuum of the interacting theory. Such13robe is well outside the bound state of N − λ − / . We can use (15) to construct various other wave packets. Correction to the coherent state
As explained in the paragraph that contains (23), the open-string mass term is naturallyinduced by considering a wave packet (15). For example, if we put the (
N, N ) componentat (cid:126)y = ( L, , , · · · , g L (cid:80) I =2 (cid:80) N − i =1 | ˆ X I,iN | . A caveathere is that we did not touch the off-diagonal elements. If the off-diagonal elements acquiremass due to the Higgsing, the energy of the wave packet becomes large unless the width ofthe wave packet along these directions in R N (in the example above, ˆ X I,iN , I = 2 , · · · , i = 1 , , · · · , N −
1) shrink. We did not take into account such effects. In order to obtaina stable, low-energy wave packet, probably we should fix the location of the center of thewave packet and then minimize the energy: A natural construction of wave packet | Φ (cid:105) at finite coupling (cid:19) (cid:16) Minimize (cid:104) Φ | ˆ H | Φ (cid:105) satisfying the constraints (cid:104) Φ | ˆ X I | Φ (cid:105) = Y I and (cid:104) Φ | ˆ P I | Φ (cid:105) = Q I . (cid:18) (cid:17) The symmetry enhancement we observed for the coherent states in Sec. 2.1.2 persists here:if ( Y I , Q I ) is invariant under the action of a subgroup of U( N ), the corresponding quantumstate | Φ (cid:105) is also invariant. Partially-deconfined states
The strong coupling limit ( λ → ∞ ) has been studied numerically via lattice MonteCarlo simulation and partial deconfinement has been demonstrated [11, 10]. Therefore, theargument provided in Sec. 2.1.4 can be repeated. The deconfined sector in the partially-deconfined state is interpreted as a thermally-excited bound state that is analogous to thesmall black hole in string theory. The argument given above applies to the D0-brane matrix model (1) as well. Modulo anatural assumption that the ground state is genuinely gauge-invariant, the scaling (cid:104) Tr X I (cid:105) ∼ λ − / N at low temperature simply means that the width of the ground-state wave functionwith respect to each direction of R N is λ − / .We repeat that the coherent state (15), and more generally (16), is merely one of manypossible realizations of the wave packet. An apparent issue when we try to relate thecoherent state to the probe D-brane in gravity side is supersymmetry: the wave packetought to be supersymmetric when Y I ’s are simultaneously diagonalizable and Q I ’s vanish. It would be better to use the U( N )-symmetrized version of | Φ (cid:105) (Fig. 2) to evaluate the energy. We may have to remove the flat direction in order to pick up the gauge-invariant vacuum. It can beachieved e.g., by adding a small mass to scalars, take the large- N limit and then removing the mass. | Φ (cid:105) is obtained by minimizing (cid:104) Φ | ˆ H | Φ (cid:105) with theconstraints (cid:104) Φ | ˆ X I | Φ (cid:105) = Y I and (cid:104) Φ | ˆ P I | Φ (cid:105) = Q I ; see Sec. 2.2. After the gauge transformation,the wave packet is localized about Y ( U ) I and Q ( U ) I .We expect that the D0-brane matrix model has significantly richer dynamics than thebosonic theories. Dual gravity analysis of this system [4], combined with the analogy tothe partial-deconfinement proposal for 4d SYM [20], leads to the following speculations regarding finite-temperature physics: • At λ / N − / (cid:46) T (cid:28) λ / , the system is dual to black zero-brane in type IIA su-perstring theory [4]. The black zero-brane is analogous to the large black hole inAdS which has positive specific heat. According to the proposal in Ref. [20, 22], suchstates should be completely-deconfined. (Still, at very low energy, the off-diagonalelements should be highly suppressed; otherwise the energy cannot be parametricallysmall. In this sense, this state may be almost block-diagonal, and the size of the blockshrinks at low temperature. See Appendix A.3 for a related material.) If we simplyidentify the size of the bound state R and the radius of black zero-brane, we obtain R ∼ ( λT ) / N / . • As the energy goes down, the finite extent of the M-theory circle becomes non-negligible. Around T ∼ λ / N − / , the transition to eleven-dimensional Schwarzschildblack hole takes place [4]. Below this energy scale, the specific heat is negative, i.e.,temperature goes up as the energy goes down and black hole shrinks. Such phase isnaturally described by partially-deconfined states [20, 22]. , It would be natural toidentify the size of the M × M deconfined block with the radius of the Schwarzschildblack hole.In principle, these speculations can be tested via lattice Monte Carlo simulation, orperhaps also via the machine-learning approach along the line of Ref. [24].By generalizing the probe picture, and by following the philosophy of the Matrix Theoryconjecture [2], it would be natural to interpret the small bound states as physical objectssuch as a graviton or tiny black hole. For example, a simple operatorTr( ˆ X I ˆ X J ˆ X K ) = N (cid:88) p,q,r =1 ˆ X pqI ˆ X qrJ ˆ X rpK (26) There is a subtle difference from the original proposal [20]: in the original proposal N − M D-branes notcontributing to black hole were supposed to be hovering somewhere outside black hole, but in the currentproposal they are sitting at the center of the bulk. The same holds for a proposal on the small black holein AdS × S discussed in Sec. 3. The idea that a nontrivial M -dependence may explain the negative specific heat of the eleven-dimensional Schwarzschild black hole was proposed in Ref. [23], though that reference contains a fewapparent mistakes and confusions. See Ref. [11] for the analysis of the partially-deconfined phase in the bosonic matrix model.
15s the U( N )-symmetrized version ofˆ X N,N − I ˆ X N − ,N − J ˆ X N − ,NK , (27)and hence it can be regarded as a small bound state occupying a 3 × (3 + 1) -d Yang-Mills The same puzzle regarding the size of the bound state existed for quantum field theoriesincluding (3 + 1)-d maximal SYM compactified on S (see e.g., Ref. [7]). The resolutionprovided for the matrix models can work for quantum field theories as well, because thekey ingredient — ‘genuine’ gauge-invariance of the ground state — is not specific to thematrix models. The only difference is that D3-brane can take a nontrivial shape, namely X I,ij can be a nontrivial function on S .The weak-coupling limit of (3 + 1)-d maximal Yang-Mills on S can be studied analyt-ically via technologies introduced in Refs. [26, 27], regardless of the details of the theorysuch as supersymmetry or matter content. We can explicitly confirm the ‘genuine’ gauge-invariance of the ground state [12] and partial deconfinement in the intermediate-energyregime [19] with the size of the U( M )-deconfined states scaling as √ M .Strong coupling region is challenging, nonetheless let us make a crude, heuristic estimate.(The following is essentially the argument in Ref. [20], with slight improvement.) Forconcreteness, we take the radius of S to be R S = 1. We use the normalization L = g Tr (cid:0) F µν + · · · (cid:1) , in which the ’t Hooft counting is straightforward.Our hypothesis is that the thermal bound state (deconfined block) is a black hole, andwe identify the radius of the thermal bound state with the radius of black hole up to amultiplicative factor. Hence let us first estimate the radius of the thermal bound state. Wefocus on the U( M )-partially-deconfined state, and assume that the radius and the energyof the thermal bound state can roughly be estimated by truncating N × N matrices to M × M , with the effective ’t Hooft coupling λ M ≡ g M . This truncated system is stronglycoupled when λ M (cid:29)
1, and there the interaction term g Tr[ X I , X J ] = Mλ M Tr[ X I , X J ] dominates the dynamics. By noticing that the dependence on λ M disappears when ˜ X I ≡ λ − / M X I is used, we can see that the eigenvalues of ˜ X I are of order M , and those of X I areproportional to λ / M . Hence we estimate that the radius of black hole R BH is proportionalto M / . In our setup R BH is of order 1 when the transition between large and small blackholes takes place, and this transition should be at M ∼ N . Therefore, R BH ∼ (cid:0) MN (cid:1) / , and T BH ∼ (cid:0) MN (cid:1) − / . This is a highly nontrivial assumption, given that we are studying the strongly-coupled region. Here by the ‘eigenvalues’ we mean the slow-mode contribution. S BH . From the ’t Hooft counting, the entropy S BH should be written as S BH ∼ f ( λ M ) · M , with some function f . To determine f , weagain look at the transition between large and small black holes takes place. There theentropy is simply proportional to N as long as λ = g N is large, and hence, we conclude f ( λ M ) is just constant at λ M (cid:29)
1, and the entropy is S BH ∼ M .By combining R BH ∼ (cid:0) MN (cid:1) / , T BH ∼ (cid:0) MN (cid:1) − / and S BH ∼ M , we obtain S BH ∼ N R ∼ R G N ∼ G N T , (28)where G N is the ten-dimensional Newton constant. This is the expected behavior of thesmall black hole.If the effective coupling describing the thermal bound state is small ( g M (cid:46) g M ∼
1, that translates to S BH ∼ g − . This is the same as the expectation from the dual gravity analysis [28, 26].This argument is based on many nontrivial assumptions (including that partial decon-finement takes place at the strongly-coupled region of 4d SYM), and hence we do not claimit is a ‘derivation’. Our purpose here was to show how the bulk geometry, including blackhole, might be described by color degrees of freedom. A better test might be doable byusing the index [29] with complex chemical potential [30]. In this paper we suggested that a classic way of seeing the emergent bulk geometry,analogous to the Matrix Theory proposal by Banks, Fischler, Shenker and Susskind [2] —roughly speaking, ‘eigenvalues are coordinates’ — can make sense in the Maldacena-typegauge/gravity duality [3, 4]. The key was to understand the meaning of ‘matrices’ and‘eigenvalues’ precisely. Because we are interested in low-energy states, we need to considera wave packet whose center is identified with ‘matrices’. The genuine gauge invariance ofthe ground state [12] played the important role for the determination of the size of theground-state wave function.A natural expectation would be that probe D-branes, whose locations are identifiedwith the diagonal elements of the ‘matrices’, are described by the Dirac-Born-Infeld actionin the black-brane spacetime as in Maldacena’s original proposal [3]. (Note however thatthe determination of the appropriate wave packet is a nontrivial problem that requiresfurther study, as we emphasized a few times in this paper.) An ideal way to test it is torealize supersymmetric gauge theories on a quantum computer [25] and then perform the D-brane-scattering experiments. See Refs. [31, 32, 33, 34, 35] for analytic calculations relatedto such scattering processes. Another interesting approach is the machine-learning methodto obtain the wave function [24], which might be useful for determining the potential energy17s a function of the location of the probe D-brane. Such an approach is analogous to theanalysis based on the probe effective action via path integral [36]. Monte Carlo simulationbased on Euclidean path integral can also be a powerful tool. In the past, similar but slightlydifferent setups were studied. In Ref. [37] the (
N, N ) component was Higgsed by adding anextra term to the potential, and the interaction between the probe and thermal bound statewas studied. The parameter region studied in that paper was T (cid:38) λ / , where the subtletyassociated with ‘delocalization’ in the path-integral picture was not the important issue.Ref. [38] used the D0/D4-system described by the Berkooz-Douglas matrix model [39]. Atheoretically cleaner setup would be to use Z ( T ; Y ) ≡ (cid:88) | E (cid:105)∈H inv (cid:104) E | e − H ( ˆ P , ˆ X + Y ) /T | E (cid:105) (29)to estimate the interaction between the probe and black hole, by using a coherent state as aprobe. Although the coherent state may not be an ideal probe, there may be a qualitativechange when it goes into the thermal bound state.Another interesting direction is to understand the relationship to other approaches tothe emergent space. This is very important toward the understanding of the interior of theblack hole, where a simple geometric picture discussed in this paper may not be applicable.Recently there are several attempts to use the entanglement between color degrees of free-dom for this purpose [40, 41, 42, 43, 44, 45, 46]. It would be useful to study the meaningsof these proposals, or to make a better proposal, based on the geometric picture discussedin this paper. For example, for the D0-brane quantum mechanics, we can consider a wavepacket localized about (cid:126)Y ij = (cid:126)y i δ ij , where (cid:126)y , · · · , (cid:126)y M ∈ A ⊂ R and (cid:126)y M +1 , · · · , (cid:126)y N ∈ ¯ A ⊂ R .Then we can integrate out the upper-left M × M block to define the entanglement entropybetween the probes in a region A and those in a region ¯ A .How can we see the ‘shape’ of a bound state? One natural approach is to make it‘maximally diagonal’, for example by fixing U ∈ U( N ) such that (cid:80) I =1 (cid:80) Ni =1 | ( U X I U − ) ii | is maximized [47, 48]. In the past this procedure was applied by using typical configurationsin the path integral as ‘matrices’. Obviously, we should apply this procedure to the slowmodes.The IKKT matrix model [49] is another interesting model that may exhibit the emer-gence of spacetime. It is more ambitious than the class of theories discussed in this paper,in that even time direction should emerge from color degrees of freedom. The argument inthis paper does not apply to the IKKT matrix model because we assumed the existence oftime when we defined the Hamiltonian. It would be interesting to think about a properdefinition of ‘diagonalization’ and ‘eigenvalue distribution’ in this model. Acknowledgement
The author would like to thank D. Anninos, T. Anous, S. Das, S. Funai, X. Han,G. Ishiki, J. Maldacena, A. Milekhin, G. Mandal, E. Rinaldi, H. Shimada, B. Swingle,18. Trivedi, T. Wiseman and T. Yoneya for discussions and comments. He thanks the Inter-national Centre for Theoretical Sciences (ICTS) for hosting the online program “Nonper-turbative and Numerical Approaches to Quantum Gravity, String Theory and Holography”(code: ICTS/numstrings2021/1), which gave him a valuable opportunity of discussing thematerials presented in this paper with several participants. He was supported by the STFCErnest Rutherford Grant ST/R003599/1.
A Relation between BEC and color confinement, and‘genuine’ gauge invariance
A.1 H ext and H inv Let us consider generic gauge group G . The canonical partition function of gauge theoryis defined as Z ( T ) = Tr H inv ( e − ˆ H/T ) . (30)Let us show that this can also be written as Z ( T ) = 1vol( G ) (cid:90) G dg Tr H ext (ˆ ge − ˆ H/T ) , (31)where vol( G ) is the volume of G .Let | Φ (cid:105) ∈ H ext be an energy eigenstate in certain gauge. It can be projected to a singletstate | Φ (cid:105) inv ∈ H inv as | Φ (cid:105) inv = 1 √ C Φ (cid:90) G dg (ˆ g | Φ (cid:105) ) , (32)where the integral is taken over the gauge group G by using the Haar measure, and ˆ g generates the gauge transformation associated with the group element. The normalizationfactor C Φ is C Φ = (cid:90) G dg (cid:90) G dg (cid:48) (cid:0) (cid:104) Φ | ˆ g − (cid:1) (ˆ g (cid:48) | Φ (cid:105) ) = vol( G ) · (cid:90) G dg (cid:104) Φ | ˆ g | Φ (cid:105) = vol( G ) · vol( G Φ ) , (33)where G Φ is a subgroup of G that leaves | Φ (cid:105) invariant.When the trace is taken over the extended Hilbert space, the over-counting factor asso-ciated with an energy eigenstate | Φ (cid:105) is vol( G )vol( G Φ ) . Therefore,Tr H inv ( e − ˆ H/T ) = (cid:88) Φ vol( G Φ ) · e − E Φ /T vol( G ) , (34)19here the sum with respect to energy eigenstates | Φ (cid:105) is taken over H ext . We can also showthat (cid:90) G dg Tr H ext (ˆ ge − ˆ H/T ) = (cid:90) G dg (cid:88) Φ e − E Φ /T (cid:104) Φ | ˆ g | Φ (cid:105) = (cid:88) Φ vol( G Φ ) · e − E Φ /T . (35)Therefore, (30) and (31) are equivalent. A.2 BEC, confinement and ‘genuine’ gauge invariance
We emphasized the importance of the ‘genuine’ gauge invariance throughout this paper.A crisp characterization of this notion can be illuminated via the close connection betweenBose-Einstein condensation and color confinement at large N [12].Let us consider a system of N free bosons in the harmonic trap. The Hamiltonian isˆ H = N (cid:88) i =1 (cid:32) ˆ (cid:126)p i m + mω (cid:126)x i (cid:33) , (36)where ˆ (cid:126)x i = (ˆ x i , ˆ y i , ˆ z i ) and ˆ (cid:126)p i = (ˆ p x,i , ˆ p y,i , ˆ p z,i ) are the coordinate and momentum of i -thparticle.Because N bosons are indistinguishable, this is a gauged quantum mechanics withthe gauge group S N . As the basis of the extended Hilbert space H ext , we can use theFock states | (cid:126)n , · · · , (cid:126)n N (cid:105) , which are energy eigenstates with the energy E = (cid:80) Ni =1 E (cid:126)n i = (cid:80) Ni =1 (cid:0) ( n x,i + n y,i + n z,i ) ω + (cid:1) . The partition function is given by Z ( T ) = 1 N ! (cid:88) σ ∈ S N (cid:88) (cid:126)n , ··· ,(cid:126)n N (cid:104) (cid:126)n , · · · , (cid:126)n N | ˆ σe − ˆ H/T | (cid:126)n , · · · , (cid:126)n N (cid:105) = 1 N ! (cid:88) (cid:126)n , ··· ,(cid:126)n N e − ( E (cid:126)n + ··· + E (cid:126)nN ) /T (cid:32) (cid:88) σ ∈ S N (cid:104) (cid:126)n , · · · , (cid:126)n N | (cid:126)n σ (1) , · · · , (cid:126)n σ ( N ) (cid:105) (cid:33) . (37)The factor (cid:80) σ ∈ S N (cid:104) (cid:126)n , · · · , (cid:126)n N | (cid:126)n σ (1) , · · · , (cid:126)n σ ( N ) (cid:105) counts the number of σ ∈ S N that leaves | (cid:126)n , · · · , (cid:126)n N (cid:105) invariant. (This is corresponds to vol( G Φ ) in (35).) If all (cid:126)n i ’s are the same(e.g., the ground state, (cid:126)n = · · · (cid:126)n N = (cid:126) N ! appears. Letus call such states ‘genuinely S N -invariant states’. For generic states most permutations σ ∈ S N change the state and hence such enhancement factor does not appear.Bose-Einstein condensation [13] is a phenomenon that many particles fall into the groundstate. It is triggered by the enhancement mechanism mentioned above: if M particles areexcited while N − M particles are in the ground state, then the enhancement factor ( N − M )!appears from the latter. The same mechanism triggers color confinement: vol( G Φ ) in (35)serves as the enhancement factor, and genuinely gauge-invariant state that satisfies G = G Φ becomes dominant at low temperature. Partially-BEC phase corresponds to the partially-confined phase (= partially-deconfined phase).20s we mentioned in Sec. 2.2, that the distribution of the Polyakov loop in U( N ) gaugetheory becomes uniform at low temperature is the consequence of the genuine gauge invari-ance of the ground state. Exactly the same holds for the Bose-Einstein condensation; seeRef. [12] for details.For BEC, the off-diagonal long-range order (ODLRO) [17, 18] is often used to detect thegenuine S N invariance [14, 15, 16]. Let ˆ ρ = | Φ (cid:105)(cid:104) Φ | be the N -particle density matrix madeof the state | Φ (cid:105) ∈ H inv . From ˆ ρ , the one-particle density matrix ˆ ρ is defined by tracingout N − ρ = Tr , ··· ,N ˆ ρ . We perform the spectral decomposition of ˆ ρ asˆ ρ = n max | Ψ (cid:105)(cid:104) Ψ | + (cid:88) i n i | Ψ i (cid:105)(cid:104) Ψ i | , (38)where n max is the largest eigenvalue of ˆ ρ . If n max is of order one, (cid:104) x | ˆ ρ | y (cid:105) does not vanish atlong distance, and the system has the ODLRO. This n max counts the number of degrees offreedom in BEC. For example, for the ground state | Φ (cid:105) = | (cid:126) , · · · , (cid:126) (cid:105) , we obtain ˆ ρ = | (cid:126) (cid:105)(cid:104) (cid:126) | ,hence n max = 1, becauseˆ ρ = Tr , ··· ,N ( | (cid:126) , · · · , (cid:126) (cid:105)(cid:104) (cid:126) , · · · , (cid:126) | ) = | (cid:126) (cid:105)(cid:104) (cid:126) | . (39)On the other hand, if all (cid:126)n i ’s are different, n max = N , becauseˆ ρ = Tr , ··· ,N (cid:32) N ! (cid:88) σ,σ (cid:48) | (cid:126)n σ (1) , · · · , (cid:126)n σ ( N ) (cid:105)(cid:104) (cid:126)n σ (cid:48) (1) , · · · , (cid:126)n σ (cid:48) ( N ) | (cid:33) = 1 N N (cid:88) i =1 | (cid:126)n i (cid:105)(cid:104) (cid:126)n i | . (40)This n max is related to the Polyakov loop as follows [12]. Firstly note that the groupelement σ in (37) is the Polyakov loop operator. In the thermodynamic limit, such anelement σ ∈ S N that leaves a typical state dominating the partition function invariant givesthe expectation value. If N = N − M particles are in the Bose-Einstein condensate, atypical state has a S N -permutation symmetry. Any element of S N is a product of cyclicpermutations. The eigenvalues of a cyclic permutation of length k is e πil/k , l = 1 , , · · · , k .As N → ∞ , dominant cyclic permutations becomes infinitely long [14, 15, 16], and theconstant offset of the distribution of Polyakov loop phases proportional to N appears.When N = N → ∞ , the phase distribution becomes completely uniform. See Ref. [12] formore details. A.3 Speculations regarding the Maldacena-Milekhin conjecture
In Ref. [8], Maldacena and Milekhin conjectured that the gauge-singlet constraint is notimportant at the low-energy regime of the D0-brane matrix model. That is, the ‘gauged’partition function we have been discussing, Z gauged ( T ) = Tr H inv ( e − ˆ H/T ) = 1volU( N ) (cid:90) dU Tr H ext ( ˆ U e − ˆ H/T ) , (41)21hould be exponentially close to the ‘ungauged’ partition function Z ungauged ( T ) = Tr H ext ( e − ˆ H/T ) = 1volU( N ) (cid:90) dU Tr H ext ( e − ˆ H/T ) (42)at large- N . Specifically, the difference of the free energy should decay as ∼ exp( − Cλ / /T ),where C is of order 1. Therefore, the gauged and ungauged theory should be almost in-distinguishable at T (cid:28) λ / , where weakly-coupled string or M-theory is a legitimate dualdescription. (Actually this conjecture was developed based on the intuition in the gravityside.) Monte Carlo simulation provided a result consistent with this conjecture [50].A natural mechanism in the matrix model side that can lead to this relation is that eachlow-energy state in H ext is invariant under a large subgroup of U( N ). In usual confininggauge theory that has a mass gap of order N , this happens trivially at the energy scalewell below the gap, because the ground state dominates the partition function. A highlynontrivial point in the Maldacena-Milekhin conjecture is that they claim it happens eventhough the D0-brane matrix model does not have such gap; namely the non-singlet sectorshould be gapped while the singlet sector is not gapped. But perhaps we should not find ittoo surprising, because the same light mode, represented by a small block, can be excitedmultiple times. If the multiplicities of light modes are n , n , · · · , then such a state in H ext isinvariant under U( n ) × U( n ) × · · · . If the multiplicities grow sufficiently fast as the energygoes down, it would be hard to distinguish the gauged and ungauged theories. References [1] E. Witten, “Bound states of strings and p-branes,”
Nucl. Phys. B (1996)335–350, arXiv:hep-th/9510135 .[2] T. Banks, W. Fischler, S. Shenker, and L. Susskind, “M theory as a matrix model: AConjecture,”
Phys. Rev. D (1997) 5112–5128, arXiv:hep-th/9610043 .[3] J. M. Maldacena, “The Large N limit of superconformal field theories andsupergravity,” Int. J. Theor. Phys. (1999) 1113–1133, arXiv:hep-th/9711200 .[4] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, “Supergravityand the large N limit of theories with sixteen supercharges,” Phys. Rev. D (1998)046004, arXiv:hep-th/9802042 . A similar characterization is valid for the microcanonical partition function, which can be applied tothe M-theory parameter region where the gravity dual is the eleven-dimensional Schwarzschild black hole. More naive answer would be that only the U(1) subgroup, which acts on all the states trivially, leaveslow-energy states invariant. However this possibility is excluded due to the mismatch of the distribution ofthe phases of the Polyakov loop: it gives the delta-function-like distribution, which is rather different fromactual distribution.
AIP Conf. Proc. no. 1, (1999)98–112, arXiv:hep-th/9901079 .[6] J. Polchinski, “M theory and the light cone,”
Prog. Theor. Phys. Suppl. (1999)158–170, arXiv:hep-th/9903165 .[7] I. Heemskerk, D. Marolf, J. Polchinski, and J. Sully, “Bulk and TranshorizonMeasurements in AdS/CFT,”
JHEP (2012) 165, arXiv:1201.3664 [hep-th] .[8] J. Maldacena and A. Milekhin, “To gauge or not to gauge?,” JHEP (2018) 084, arXiv:1802.00428 [hep-th] .[9] N. Iizuka, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Probing black holes innonperturbative gauge theory,” Phys. Rev. D (2002) 024012, arXiv:hep-th/0108006 .[10] G. Bergner, N. Bodendorfer, M. Hanada, E. Rinaldi, A. Sch¨afer, and P. Vranas,“Thermal phase transition in Yang-Mills matrix model,” JHEP (2020) 053, arXiv:1909.04592 [hep-th] .[11] H. Watanabe, G. Bergner, N. Bodendorfer, S. Shiba Funai, M. Hanada, E. Rinaldi,A. Sch¨afer, and P. Vranas, “Partial Deconfinement at Strong Coupling on theLattice,” JHEP, to appear , arXiv:2005.04103 [hep-th] .[12] M. Hanada, H. Shimada, and N. Wintergerst, “Color Confinement and Bose-EinsteinCondensation,” arXiv:2001.10459 [hep-th] .[13] A. Einstein, “Quantentheorie des einatomigen idealen gases,” S-B Preuss. Akad.Berlin (1924) .[14] R. P. Feynman, “Atomic theory of the λ transition in helium,” Phys. Rev. (1953)1291.[15] R. P. Feynman, “Atomic theory of liquid helium near absolute zero,” Phys. Rev. (1953) 1301–1308.[16] R. P. Feynman, “Atomic theory of the two-fluid model of liquid helium,” Phys. Rev. (1954) 262–277.[17] O. Penrose and L. Onsager, “Bose-Einstein Condensation and Liquid Helium,” Phys.Rev. (1956) 576–584.[18] C. N. Yang, “Concept of off-diagonal long-range order and the quantum phases ofliquid he and of superconductors,”
Rev. Mod. Phys. (1962) 694–704.[19] M. Hanada, A. Jevicki, C. Peng, and N. Wintergerst, “Anatomy of Deconfinement,” JHEP (2019) 167, arXiv:1909.09118 [hep-th] .2320] M. Hanada and J. Maltz, “A proposal of the gauge theory description of the smallSchwarzschild black hole in AdS × S ,” JHEP (2017) 012, arXiv:1608.03276[hep-th] .[21] D. Berenstein, “Submatrix deconfinement and small black holes in AdS,” JHEP (2018) 054, arXiv:1806.05729 [hep-th] .[22] M. Hanada, G. Ishiki, and H. Watanabe, “Partial Deconfinement,” JHEP (2019)145, arXiv:1812.05494 [hep-th] . [Erratum: JHEP 10, 029 (2019)].[23] E. Berkowitz, M. Hanada, and J. Maltz, “Chaos in Matrix Models and Black HoleEvaporation,” Phys. Rev. D no. 12, (2016) 126009, arXiv:1602.01473 [hep-th] .[24] X. Han and S. A. Hartnoll, “Deep Quantum Geometry of Matrices,” Phys. Rev. X no. 1, (2020) 011069, arXiv:1906.08781 [hep-th] .[25] H. Gharibyan, M. Hanada, M. Honda, and J. Liu, “Toward simulatingSuperstring/M-theory on a quantum computer,” arXiv:2011.06573 [hep-th] .[26] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk,“The Hagedorn - deconfinement phase transition in weakly coupled large N gaugetheories,” Adv. Theor. Math. Phys. (2004) 603–696, arXiv:hep-th/0310285 .[27] B. Sundborg, “The Hagedorn transition, deconfinement and N=4 SYM theory,” Nucl. Phys. B (2000) 349–363, arXiv:hep-th/9908001 .[28] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N fieldtheories, string theory and gravity,”
Phys. Rept. (2000) 183–386, arXiv:hep-th/9905111 .[29] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An Index for 4 dimensionalsuper conformal theories,”
Commun. Math. Phys. (2007) 209–254, arXiv:hep-th/0510251 .[30] S. Choi, J. Kim, S. Kim, and J. Nahmgoong, “Comments on deconfinement inAdS/CFT,” arXiv:1811.08646 [hep-th] .[31] U. H. Danielsson, G. Ferretti, and B. Sundborg, “D particle dynamics and boundstates,”
Int. J. Mod. Phys. A (1996) 5463–5478, arXiv:hep-th/9603081 .[32] D. N. Kabat and P. Pouliot, “A Comment on zero-brane quantum mechanics,” Phys.Rev. Lett. (1996) 1004–1007, arXiv:hep-th/9603127 .[33] M. R. Douglas, D. N. Kabat, P. Pouliot, and S. H. Shenker, “D-branes and shortdistances in string theory,” Nucl. Phys. B (1997) 85–127, arXiv:hep-th/9608024 . 2434] K. Becker, M. Becker, J. Polchinski, and A. A. Tseytlin, “Higher order gravitonscattering in M(atrix) theory,”
Phys. Rev. D (1997) 3174–3178, arXiv:hep-th/9706072 .[35] Y. Okawa and T. Yoneya, “Multibody interactions of D particles in supergravity andmatrix theory,” Nucl. Phys. B (1999) 67–99, arXiv:hep-th/9806108 .[36] A. Jevicki, Y. Kazama, and T. Yoneya, “Generalized conformal symmetry in D-branematrix models,”
Phys. Rev. D (1999) 066001, arXiv:hep-th/9810146 .[37] E. Rinaldi, E. Berkowitz, M. Hanada, J. Maltz, and P. Vranas, “Toward HolographicReconstruction of Bulk Geometry from Lattice Simulations,” JHEP (2018) 042, arXiv:1709.01932 [hep-th] .[38] V. G. Filev and D. O’Connor, “A Computer Test of Holographic Flavour Dynamics,” JHEP (2016) 122, arXiv:1512.02536 [hep-th] .[39] M. Berkooz and M. R. Douglas, “Five-branes in M(atrix) theory,” Phys. Lett. B (1997) 196–202, arXiv:hep-th/9610236 .[40] E. A. Mazenc and D. Ranard, “Target Space Entanglement Entropy,” arXiv:1910.07449 [hep-th] .[41] S. R. Das, A. Kaushal, S. Liu, G. Mandal, and S. P. Trivedi, “Gauge Invariant TargetSpace Entanglement in D-Brane Holography,” arXiv:2011.13857 [hep-th] .[42] S. R. Das, A. Kaushal, G. Mandal, and S. P. Trivedi, “Bulk Entanglement Entropyand Matrices,”
J. Phys. A no. 44, (2020) 444002, arXiv:2004.00613 [hep-th] .[43] H. R. Hampapura, J. Harper, and A. Lawrence, “Target space entanglement inMatrix Models,” arXiv:2012.15683 [hep-th] .[44] T. Anous, J. L. Karczmarek, E. Mintun, M. Van Raamsdonk, and B. Way, “Areasand entropies in BFSS/gravity duality,” SciPost Phys. no. 4, (2020) 057, arXiv:1911.11145 [hep-th] .[45] F. Alet, M. Hanada, A. Jevicki, and C. Peng, “Entanglement and Confinement inCoupled Quantum Systems,” arXiv:2001.03158 [hep-th] .[46] A. Mollabashi, N. Shiba, and T. Takayanagi, “Entanglement between TwoInteracting CFTs and Generalized Holographic Entanglement Entropy,” JHEP (2014) 185, arXiv:1403.1393 [hep-th] .[47] T. Azeyanagi, M. Hanada, T. Hirata, and H. Shimada, “On the shape of a D-branebound state and its topology change,” JHEP (2009) 121, arXiv:0901.4073[hep-th] . 2548] T. Hotta, J. Nishimura, and A. Tsuchiya, “Dynamical aspects of large N reducedmodels,” Nucl. Phys. B (1999) 543–575, arXiv:hep-th/9811220 .[49] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, “A Large N reduced model assuperstring,”
Nucl. Phys. B (1997) 467–491, arXiv:hep-th/9612115 .[50] E. Berkowitz, M. Hanada, E. Rinaldi, and P. Vranas, “Gauged And Ungauged: ANonperturbative Test,”
JHEP (2018) 124, arXiv:1802.02985 [hep-th]arXiv:1802.02985 [hep-th]