Grassmann Matrix Quantum Mechanics

We explore quantum mechanical theories whose fundamental degrees of freedom are rectangular matrices with Grassmann valued matrix elements. We study particular models where the low energy sector can be described in terms of a bosonic Hermitian matrix quantum mechanics. We describe the classical curved phase space that emerges in the low energy sector. The phase space lives on a compact Kahler manifold parameterized by a complex matrix, of the type discovered some time ago by Berezin. The emergence of a semiclassical bosonic matrix quantum mechanics at low energies requires that the original Grassmann matrices be in the long rectangular limit. We discuss possible holographic interpretations of such matrix models which, by construction, are endowed with a finite dimensional Hilbert space.


Introduction
Models with matrix like degrees of freedom make numerous appearances throughout physics. Applications range from the study of the spectra of heavy atoms to models of emergent geometry [1,2,3,4,5,6]. In this paper we will concern ourselves with a particular class of quantum mechanical models whose degrees of freedom are purely fermionic rectangular matrices ψ Ai , with A = 1, ..., M and i = 1, ..., N . The matrices transform in the (M, N ) bifundamental representation of a U (M )×SU (N ) symmetry group. In a Lagrangian description of the system, transition amplitudes can be expressed as path integrals over Grassmann valued paths ψ Ai . Grassmann matrices naturally appear as the supersymmetric partners of bosonic Hermitian matrices in supersymmetric matrix quantum mechanical theories such as the low energy worldline dynamics of a stack of N D0-branes in type IIA string theory [3,7] or the Marinari-Parisi matrix model [8]. Our interest is in quantum mechanical models consisting of only the Grassmann matrices.
There, it was shown how the problem of Grassmann matrix integrals at large N , M can be expressed as an eigenvalue problem for the composite N × N matrix Φ ij = Aψ iA ψ Aj , which is effectively bosonic. Unlike bosonic matrices, a Grassmann valued matrix cannot be diagonalized and characterized in terms of eigenvalues. Instead, the authors were able to analyze the model by diagonalizing Φ ij .
Certain features of the Φ ij integral, such as a contribution to the potential of the form tr log Φ, were shown to be universal and specifically related to the Grassmann nature of the original problem. Along a similar vein, emergent bosonic matrices from spin systems were considered in [12,13]. The models of interest in our work can be viewed as multi-particle quantum mechanical models of fermions which can occupy a finite set of single particle states |A, i, α , labeled by the matrix indices. In particular the Hilbert space is finite dimensional. Fermionic multi-particle models often arise as lattice models in condensed matter physics, where there is typically an assumption about some sort of nearest-neighbour interaction between the fermions reflecting spatial locality. In contrast, the class of models of interest in our paper have no such notion of spatial locality. They are described by actions of the form: The potential V (x) is an N ×N matrix valued function. The index α is an spinor index associated to the d-dimensional rotation group, but we will focus on the particular case of d = 3 and take the σ αβ to be the ordinary Pauli matrices. We will also demand that the potential V (x) be SO(3) invariant. 1 An example of such a model was studied in [14]. The objects we wish to understand are path integrals over {ψ α iA (t), ψ α Ai (t)} rather than simple integrals. In particular, we study to what extent the Grassmann matrix models at large N and M can be described in terms of a composite bosonic matrix degree of freedom. We then describe several features of the emergent bosonic matrix quantum mechanical systems. We focus on the case where V (x) is quartic in the Grassmann matrices, but the techniques we develop can be used more generally.
As mentioned, our models have a finite dimensional Hilbert space. In this sense they differ from many of the quantum mechanical models studied in the context of holography, such as the D0-brane quantum mechanics or N = 4 super Yang-Mills, where the systems have an infinite space of states, even at finite N . On the other hand, several proposals have been made throughout the literature suggesting that the holographic dual of a de Sitter universe (or at least its static patch) is indeed a system with a finite dimensional Hilbert space [15,16,17,18,19,20]. Our considerations are particularly similar, in spirit, to those of [15,16] where the basic building blocks are also taken to be a large collection of fermionic operators. Part of our motivation is to understand to what extent systems with a finite Hilbert space can give rise to a holographic description with a dual gravitational theory in an appropriate large N type limit. In order for this to be the case, bosonic variables (such as the Hermitean matrices) should emerge from the discrete variables, at least at low energies and in an appropriate large N limit. The models studied in this work serve as toy models where this can be seen explicitly, and we can examine to what extent the bosonic effective degrees of freedom adequately capture the physics and when this description breaks down.
The first part of the paper provides a detailed study for the N = 1 case, in which the degrees of freedom are organized as vectors. We derive several results regarding the physics of the effective composite degree of freedomψ α A σ αβ ψ β A . We show to what extent the theory is described by three bosonic degrees of freedom x = (x, y, z) transforming as an SO(3) vector. The Euclidean path integral is expressed as a path integral over x and a low velocity expansion is developed at large M . We study the theories at finite temperature and note a breakdown of the bosonic description at high temperatures. We describe the structure of the emergent classical phase space for the effective bosonic theory, which is the compact Kähler manifold CP 1 . Some of the results in this section have appeared in several contexts (see for example [21,22,25]).
However, certain aspects of our treatment are novel and furthermore our treatment naturally generalizes to the matrix case. This is studied in the second part of the paper, where now the effective theory becomes that of three bosonic Hermitian N ×N matrices Σ a ij , with a ∈ {x, y, z}. The matrix Σ a ij transforms in the adjoint of SU (N ) and is an SO(3) vector. The matrix analogue of the emergent classical phase space is identified as a compact Kähler manifold, first introduced by Berezin [26]. The Kähler metric is parameterized by a complex N × N matrix Z ij . We discuss how the Z ij and Z † ij relate to the description of the system in terms of the Σ a ij as well as the original Grassmann matrices. The volume of the Kähler metric computes the dimension of the Hilbert space captured by the (quantized) classical phase space. It is shown to precisely match the dimension of the U (M ) invariant Hilbert space of the original Grassmann theory. We end with an outlook discussing speculative connections of our models to holography.

Vector model
In this section we discuss a quantum mechanical model in which the degrees of freedom are a vector ψ α A of complex Grassmann numbers, with A = 1, . . . , M and α = 1, 2 a spinor index of SU (2), the double cover of the rotational group SO(3). Our system has a 2 2M complex-dimensional Hilbert space of states. The purpose of the section is to analyze a simplified version of the matrix model studied in the next section, which however still retains some of the salient features.
We focus on an action with quartic interactions of the specific form: where it is understood that the A and α indices are summed over and the σ a αβ = {σ x αβ , σ y αβ , σ z αβ } are the three Pauli matrices. The model has an SU (2) × U (M ) global symmetry group. The (ψ α A ) ψ α A transform in the (anti-)fundamental representation of U (M ) and SU (2).
Upon canonical quantization, the non-vanishing anti-commutation relations between the fermionic operators are given by {ψ α A , ψ β B } = δ αβ δ AB . The SU (2) genera-tors working on these operators are given byĴ a =ψ α A σ a αβ ψ β A /2. The U (M ) generators are given by:Ĵ The T n AB with n > 0 are the traceless generators of SU (M ) subgroup of U (M ), and T 0 AB = δ AB generates the U (1) subgroup of U (M ). c is a normal ordering constant that appears as a possible central extension of the U (1). As expected, [Ĵ n ,Ĵ a ] = 0.
We take g > 0 in what follows and measure quantities in units of g so that g = 1.

Spectrum
The Hamiltonian of the system is proportional to the normal ordered square of the angular momentum operator: wheren ≡ψ α A ψ α A , commutes with theĴ a . If we view the index A as a lattice site, the system above is describing two-body SU (2) spin-spin interactions of spin-1/2 fermions between all M possible lattice sites, each with equal strength. From (2.3), it follows that the the eigenstates |J, m; n can be labeled by their total angular momentum J, their angular momentum m in the z-direction and their eigenvalue n with respect to then operator. The energy of |J, m; n is simply E = −4J(J + 1) + 3n. For M > 1, the ground states |g are the (M + 1) states in the maximally spinning spin-M/2 multiplet, whereas the J = 0 state with n = 2M has maximal energy. We can construct the full Hilbert space by acting with theψ α A operators on the particular J = 0 state |0 , defined to be the state annihilated by all the ψ α A . For instance the ground state with maximal spin-z angular momentum is |M/2, M/2; M = Aψ For each A we have two states with vanishing angular momentum in the zdirection, and a spin-1/2 doublet. The full Hilbert space can thus be written succinctly as H = (0 ⊕ 1/2 ⊕ 0) ⊗M . The degeneracies for a given angular momentum in the z-direction can be obtained from the partition function: From the above partition function, we can also obtain the degeneracies of the multi- plets with total spin J: .

Effective theory
We would now like to recast the Euclidean path integral of the theory as a Euclidean path integral of a bosonic (mesonic) variable and understand several features of the model in terms of the bosonic degree of freedom. The Euclidean path integral computes features in the low energy sector the system. For instance, the generating function of vacuum correlation functions is given by: where the Euclidean action S E is obtained from −iS by a Wick rotation t = −iτ .
Upon introducing an auxiliary three-vector x and integrating out the Grassmann variables, this can be recast as: where r = |x|. From the partition function we can read off the effective action for the x degree of freedom: As it stands, the above action is highly non-local in τ . We would like to understand under what conditions this action can approximated by a small velocity expansion.
Generally speaking there is no a priori reason for this to be the case in a quantum system, given that the spectrum is discrete and one cannot continuously change the kinetic energy. However, one may hope that it would be a valid approximation at large M . We will see that this is the case.

Small velocity expansion
It is useful to diagonalize the 2 × 2 Hermitian matrix x · σ for each τ . Since the σ are traceless, we take some U ∈ SU (2) such that U † σ · x U = r σ z for each τ . The U matrix is parameterized by a unit vector n = (sin θ cos φ, sin θ sin φ, cos θ). Explicitly: (2.11) It then follows that: Notice that we can transform the above functional determinant under the time reparameterization symmetry 14) The first factor on the right-hand side of (2.14) is independent of U and r and can be absorbed into the overall normalization of the path integral. The above symmetry can therefore be used to set r to a constant in performing a small velocity expansion of the functional determinant. 3 It follows from this that no time derivatives will be generated for r.
We expand (2.12) in powers of υ a σ a = i U †U by expanding the logarithm. The zeroth order term is the effective potential governing r. Going to Fourier space, the computation becomes: where we have regulated the ω-integral by differentiating once with respect to r and re-integrating it back while setting the constant of integration to zero. Note that the effective potential is minimized at r = 2M for which V The first order term in the velocity expansion is given by: whereυ a (l) is the Fourier transform of υ a at frequency l. The linear velocity piece kin is the phase picked up by a unit charge moving on the surface of a two-sphere, in the presence of a magnetic monopole of strength M/2 at the origin.
Similarly, the quadratic kinetic term is found to be: where in the right-hand side we have expressed the answer in terms of x, but now written in spherical coordinates. The higher order terms can be similarly computed and they contain even powers of time derivatives of the angular variables divided by one less power of r. 4 Denoting the characteristic frequency for some particular motion of θ and φ by ω c , the condition that there is a small derivative expansion is: For r near the minimum of the effective potential, we have ω c M . Hence, for large M there is a parametrically large range of frequencies allowing for a small velocity expansion. 4 In appendix B we consider a modified vector model where the leading kinetic piece is (2.17).

Finite temperature
As was previously noted, the whereas at small β we have simply the dimension of the Hilbert space: The transition between these two behaviors occurs at β ∼ 1/M .
We now consider the finite temperature partition function as a Euclidean path integral over x. We must integrate out the Grassmann numbers with anti-periodic boundary conditions along the thermal circle. In analogy to previous calculations, we can compute the thermal effective potential. What changes is that the ω-integrals are replaced by sums over the thermal frequencies ω n = 2π(n + 1/2)/β with n ∈ Z.
The thermal effective potential thus becomes: As before, the sum has been regulated by differentiating with respect to r.
For large β, the minimum of V ef f is at r = 2M as for the zero temperature analysis. We can find the critical point for r in a large β expansion. To first order: From this we see the tendency of r to decrease upon increasing the temperature. At small β, we can Taylor expand: We see that for β 1/M the thermal potential is minimized at r = 0. In figure 2 we show a plot for the values of r minimizing V ef f (β) as we vary β.
When r is near zero, we can no longer assume that the kinetic contributions are small and thus our analysis breaks down. This as an indication that the high temperature phase does not have a reliable small velocity description in terms of x.
Instead, the correct description requires taking into account the full set of Grassmann degrees of freedom.

Bloch coherent state path integral
So far we have introduced the variable x as a convenient integration variable to capture correlations in the vacuum state and thermal properties. Here we would like to point out that in a fixed large angular momentum sector, there is some more significance to x.
Following Bloch, we define a collection of coherent states built from the state |v , which has the lowest angular momentum in the z-direction and hence is also a minimal energy state. In other words |v = Aψ 2 A |0 . We can act on |v with the spin raising operatorĴ + =Ĵ x + iĴ y to generate states in the maximally spinning multiplet, These states are not orthogonal, but they constitute an over-complete basis of the Hilbert space of the maximally spinning multiplet, The purpose of these states is to describe, with minimal uncertainty, points on the S 2 of spin directions. Indeed, the angular momentum expectation value defines a point on S 2 -through the stereographic projection -with decreasing uncertainty in the One may ask about transition amplitude between two such states: z N |e −iTĤ |z 0 for some given HamiltonianĤ built out of theĴ a . The result is [23,24]: This is the Fubini-Study metric on CP 1 ∼ = S 2 , and we occasionally refer to it as the Bloch sphere. The symplectic form is given by the Kähler form and the large M limit plays the role of the small Planck constant limit. Time evolution of a function A(z,z) in the emergent classical phase space is governed by the Poisson bracket, i.e. (2) symmetry of the original Grassmann model acts on z as: Since the classical phase space has finite volume, we recover the fact that the underlying system has a finite number of ground states. The complex coordinate (z,z) can

Matrix model
The goal of this section is to analyze a matrix version of the vector model studied above. Given that the model is more complicated, we will not be able to attain as explicit a description, however we will uncover and generalize several of the features found in the vector model.

Action and Hamiltonian
Our degrees of freedom are now 2M N complex rectangular Grassmann matrices,ψ α iA and ψ α Ai , with A = 1, . . . , M and i = 1, . . . , N . As before, α is an SU (2) spinor index. The dimension of the Hilbert space now becomes 2 2N M . The Grassmann elements obey the anti-commutation relations {ψ α Ai ,ψ β jB } = δ αβ δ ij δ AB . We will focus on the following action: 5 S = dt iψ iA ∂ t ψ Ai + g (ψ iA σ a ψ Aj )(ψ jB σ a ψ Bi ) . We will analyze g > 0 and from now on choose units setting g = 1. Unlike the vector case previously studied, the combinatorial problem of finding the exact spectrum ofĤ seems to be rather difficult and we have not solved it. Instead, we will try to extract information about the low energy sector of the theory by going to an effective description in terms of bosonic matrices. Before doing so, we will establish some further properties about the operator algebra.

U (2N ) operator algebra
The analogues of the spin operatorsĴ a = Aψ A σ a ψ A /2 studied in the previous section are the U (M ) invariant N ×N spin matrix operators:Ŝ a ij = A (ψ iA σ a ψ Aj )/2. These operators transform as vectors in the three-dimensional real representation of SU (2), as well as in the adjoint of the SU (N ). Introducing an additional operator S 0 ij = A (ψ iA σ 0 ψ Aj )/2, with σ 0 the 2 × 2 identity matrix, we have the following closed operator algebra: The N diagonal components of theŜ a ij generate N copies of the usual su(2) algebra. The above operators can be arranged in a 2N × 2N Hermitian matrix σ µ αβ ⊗Ŝ µ ij (with µ = {0, x, y, z} summed over) and hence they generate a u(2N ) algebra. They

Effective theory
We introduce three N × N Hermitian bosonic matrices Σ a ij = (Σ x ij , Σ y ij , Σ z ij ). In analogy with the vector case, we introduce them as auxiliary variables which are given on-shell by Σ a ij = 2Ŝ a ij . Upon integrating out the ψ α Ai , the generating function of vacuum correlations of ψ andψ can be expressed as a Euclidean path integral over the Σ ij : where J a ij are sources for theŜ a ij . It is worth noting that, unlike the N = 1 case, thê S a ij no longer commute with the Hamiltonian and thus non-trivial time correlations amongst them may exist.
We now proceed to study the validity and properties of the 'small velocity' expansion of det (−∂ τ + R) = exp [Tr log (−∂ τ + R)]. Since R is a 2N × 2N Hermitian matrix, we can diagonalize it as U † RU = λ with λ = diag [λ 1 , . . . , λ 2N ] , U ∈ U (2N ) and λ n ∈ R. Note that due to the tracelessness of R, not all λ n can have the same sign. Similar to the N = 1 case, in the diagonal R frame, we can write the functional determinant as: Tr log (−∂ τ + R) = Tr log −∂ τ − U †U + λ . (3.8) With the above expression we can again use the time reparameterization symmetry to see that the effective action will be independent ofλ n , analogous to how the vector model is independent ofṙ. Using the propagator: we can expand the logarithm in powers of the Hermitian matrix υ = iU †U . Each term in the expansion will be endowed with a U (2N ) symmetry taking U †U → The linear velocity contribution to the effective action is: Theυ(l) is the Fourier transform of υ at frequency l. To define the above ω-integral we have put a cutoff at large ω, performed the exact integration and then taken the large cutoff limit. The kinetic piece containing two time derivatives in U (τ ) is given by: with Λ mn = 1/|λ m − λ n | and the sum running only over the pairs (n, m) for which λ n and λ m have opposite signs. The reason why only pairs of λ m with opposite sign appear in the sum is that the integral appearing in (3.12): vanishes whenever λ n and λ m have the same sign. It is interesting to note that the effective kinetic piece of the theory, and hence what we mean by the dynamical content, depends on the particular distribution of eigenvalues λ n .
Having obtained expressions for the first few velocity dependent terms in the effective action, we can estimate when the low velocity expansion is valid. Denoting the characteristic frequency for some motion as ω c , then in order for S (1) kin to be large compared to S (2) kin one requires: ω c λ n N . (3.14) The factor of N stems from the fact that S (2) kin has an additional matrix index to be summed over that was not present in the vector model previously studied. In what follows we will see that the effective potential is minimized for λ m ∼ M . Thus, in the limit M N , we can have a large range of allowed ω c (in units where g = 1).
If instead M does not scale with N and we take the large N limit, the window of allowed ω c shrinks to zero.
Since the global symmetry group of the theory, for our choice of Hamiltonian, is

Effective potential
We would now like to focus on the effective potential V ef f for Σ. In order to compute this we can take Σ to be time independent. V ef f must respect the SU (N ) × SU (2) symmetries. For instance it can contain a piece which is the trace of a function of the SU (2) invariant matrix Σ · Σ. Moreover, when the Σ are diagonal (or when they all commute with each other), it must reproduce N copies of the potential (2.15) we found in the vector model. Finally, the piece of V ef f originating from the functional determinant must scale linearly in Σ. We can write a general expression by noting that: is the characteristic polynomial for matrix R with eigenvalues λ n . We must also take the product over all ω, a procedure which must be regulated. For each λ n , we can express the product over the ω as the exponential of an integral over the logarithm: To define the above integral, 6 we have subtracted the integral of log(ω 2 ). Putting things together: As expected, V ef f is invariant under both the SU (N ) and SU (2) global symmetries.
It is instructive to write the 2N × 2N matrix R 2 explicitly: From the above expression, it immediately follows that trR 2 = 2 tr Σ · Σ. However, this does not imply that tr The indices (a, b) run over all distinct pairs of (x, y, z), thus rendering the expression SO(3) invariant. Since the Hermitian matrix Σ · Σ has positive eigenvalues, and the commutator i[Σ a , Σ b ] is Hermitean, we see that non-zero commutations cost potential energy. Thus, at least locally the potential (3.17) is minimized when the Σ mutually commute (which means, in turn, that we can mutually diagonalize the Σ). In this approximation, we can estimate the minimum value of V ef f as the first term in the expansion (3.19). The problem we want to solve becomes a saddle point approximation of the following matrix integral for M N : In order to obtain the saddle point equation for the eigenvalues, we first introduce a delta function δ(ρ − Σ · Σ) and integrate out the Σ, such that we remain with an integral over the N × N Hermitian ρ matrix. Upon diagonalizing ρ, and including the 6 One may be concerned about the discontinuity of the first derivative at λn = 0. However, the expression agrees with what we expect of the determinant ω (1 + λ 2 n /ω 2 ). Namely, it should equal one when λn = 0, it should be symmetric under λn → −λn and have an exponent linear in λn. Moreover, one can check that at any non-zero temperature T for which ω → 2πT (n + 1/2) with n ∈ Z, the kink at λn = 0 smoothens out.
Vandermonde contribution, we can obtain the potential for its eigenvalues ρ i ≥ 0. It is convenient at this point to rescale ρ i = M 2ρ i . We find: To leading order in a large M expansion (taking M to be much larger than N ) we can considerρ i to be peaked aroundρ i ∼ 4. Expanding aboutρ i = 4 + δ i for small δ i , and keeping the leading term only, we have: There is a slightly more efficient way to see the above. Using the property tr R 2 = 2 tr Σ · Σ we can write the effective potential (3.17) completely in terms of the eigenvalues of R as: Again, at least in the limit M N where we can ignore the effects of the matrix measure, we find V (min) ef f ≈ −M 2 N as before.
We now proceed to study the kinetic contribution linear in velocity. 7 We are considering here the situation where both M and N are large but M N .

Linear velocity term
We consider the linear velocity term for the matrix model. The simplest case occurs when the Σ ij matrix is diagonal, i.e. Σ ij = x i δ ij with i = 1, . . . , N . In this case, we simply find a sum of N terms (one for each x i ) each identical with the vector case.
Each will have their own M + 1 lowest Landau levels. Generally, however, the Σ a will not be mutually diagonalizable. Inspired by the expression (2.28), we claim that the linear velocity term is given by: where Z ij is a complex N × N matrix. The stereographic map (2.26) relating z to a point on the Bloch sphere is generalized to: (3.28) In order to verify that Σ a = (Σ a ) † it is useful to take advantage of identities such

Berezin coherent states
As in the vector case, the matrix action (3.25) can stem from a curved phase space endowed with a Kähler structure. These compact Kähler manifolds were studied extensively by Berezin [26]. The Kähler metric is given by: where c is a normalization constant. The Kähler potential is given by: This potential transforms under the U (2N ) isometry (3.29) as More precisely, what Berezin shows [26] is that there exist a collection of coherent states, analogous to the Bloch coherent states, parameterized by a complex matrix Z ij . Explicitly: where the state |v is the state annihilated by all ψ 1 Ai andψ 2 iA operators. It can be expressed as |v = A,iψ 2 iA |0 , where |0 is the state that is annihilated by all the ψ α Ai operators. Consequently |v is annihilated byŜ − ij . The overlap between two Berezin coherent states is given by: states was computed in [27]. The result reads: We can study the behavior of dim H K in various limits. When N M 1 we find dim H K ∼ 2 2M N to leading order. Thus in this limit, the dimension of the effective Hilbert space closely approximates the full Hilbert space of the original Grassmann where α is fixed in the large N limit, we have: with: Similarly, in the α → ∞ limit, f (α) ∼ log α for which log dim H K ∼ N 2 log M . As

Hamiltonian and path integral
In the vector case, the HamiltonianĤ (2.3) we studied was constant along the Bloch two-sphere given that all the Bloch coherent states had the same total angular momentum. In this regard our matrix model differs from the vector case. Given our Hamiltonian operator (3.2), the Hamiltonian H[Z, Z † ] ≡ Z|Ĥ|Z † governing time evolution on the emergent classical phase space is found to be: to leading order in M . We have defined:   We end with some speculative remarks on this question.

Outlook
We have discussed systems with a finite dimensional Hilbert space, whose constituents Holographically, large N matrix models might be associated with a gravitational theory. For the quantum mechanical model [7] dual to the ten-dimensional geometry near a collection of N D0-branes, one has nine N ×N Hermitian bosonic matrices X I ij and their Fermionic superpartners. The index I is an SO (9) index, corresponding to the rotational symmetry of the eight-sphere in the near horizon of a stack of N D0branes in type IIA string theory. The indices i and j run from 1 to N . The Hilbert space is infinite dimensional and there are states with indefinitely high energy. In these models, the emergent radial direction has been argued to be captured by the energy scale. At high energies, the quantum mechanics is weakly coupled. One manifestation of this, from the bulk viewpoint, is that the size (in the string frame) of the eight-sphere shrinks indefinitely at large radial distances, eventually leading to a stringy geometry.
Consider now a system where the spectrum is capped, as occurs in the deep infrared of a CFT living on a spatial sphere (due to the curvature coupling of the fields). In such a situation we expect the emergent sphere to cap off. This is indeed what happens in global anti-de Sitter space where the sphere at fixed r and t smoothly caps off in the deep interior. 8 Consider now the geometry of the static patch of fourdimensional de Sitter space: Notice that the size of the two-sphere resides on a finite interval. It smoothly caps off at r = 0 and is largest at r = 1 where the cosmological horizon resides. If, somehow, r was an emergent holographic direction related to the energy scale [28], then it would seem we have to cap the spectrum both in the infrared as well as the ultraviolet.
This would indicate a holographic quantum mechanical dual with a finite number of states [15,16,17,18,19,20], so long as the spectrum is discrete. If moreover we require the holographic model to have a matrix-quantum mechanical sector described by ordinary bosonic matrices, perhaps the systems we have considered above are natural candidates. We postpone the examination of this proposal and the relation to other approaches of de Sitter holography (for an overview see [29]) to future work.

A Counting U (M ) gauge invariant states
In this appendix we present the derivation of the formula for the dimension of the Hilbert space of two complex Grassmann matrices χ i A and θ i A with indices ranging from i = 1, . . . , N and A = 1, . . . , M .
Therefore we consider the action: Integrating out the gauge field gives us M 2 constraints: We define the vacuum state |0 of the theory to be annihilated by all χ and θ operators.
Note that it obeys the gauge constraint and is thus gauge invariant. Moreover, acting with gauge invariant operators always increases the energy, hence |0 is unique.
We wish to find the thermal partition function and extract the entropy S(T ) at infinite temperature. We can then use the fact that lim T →∞ S(T ) = log dim H to find the dimension of the Hilbert space with a U (M ) singlet constraint imposed. In the absence of the gauge field A t , we would have dim H = 2 2N M .

A.1 Euclidean path integral
We can compute the thermal partition function as a Euclidean path integral. Wick rotate time t → −iτ such that The Grassmann variables obey anti-periodic boundary conditions around the thermal circle. The Euclidean path integral of interest is: The gauge transformations acting on A τ are given by A τ → U A τ U † + i∂ τ U · U † . Due to the non-contractible thermal circle, we can only fix the gauge up to the holonomy around the thermal circle [30]. The Fadeev-Popov procedure in doing so gives us the following action for the (time independent upon gauge fixing) eigenvalues of A τ which we denote α A : We have dropped an overall constant which we must later recover by computing the zero temperature entropy, which should vanish because the ground state is unique.
We have yet to calculate the contribution to the action of the fundamental matter fields. We first expand them in a Fourier expansion: χ(τ ) = n∈Z e i2π(n+1/2)τ /β χ n , θ(τ ) = n∈Z e i2π(n+1/2)τ /β θ n . (A.7) Thus we obtain the thermal eigenvalues: λ A n = 2π(n + 1/2)/β + im 1 + α A ,λ A n = 2π(n + 1/2)/β + im 2 − α A . (A.8) The determinant to be evaluated is given by n λ A nλ A n . It is UV divergent. We regulate the logarithm of the determinant by taking two derivatives with respect to m and integrating m twice while setting the integration constants to zero. The result is: n log λ A nλ A n = log cos Our remaining integral becomes (we are rescaling the eigenvalues by a factor of the temperature in obtaining the below formula): Our task has been reduced to solving a multi-variable integral for the N variables

B Modified vector model
In this appendix we briefly mention a slight modification of the vector model considered in the main body of the text. The degrees of freedom are given by two sets of M complex fermion spinors {ψ α A , θ α A }. We consider the following Euclidean action: (B.1) Following the procedure outlined in the main text, we end up with an effective action for a bosonic three-vector x: The reason for the cancellation is that this model has a Hamiltonian given by the difference in angular momentum. The ground state is given by the configuration where the two angular momenta, whose operators are given byĴ 1 =ψ A σψ A /2 and J 2 =θ A σθ A /2, are anti-aligned. In the language of the charged particle on the twosphere, it is as if we have added a positron on top of the electron, thus canceling the effect of the Lorentz force, leaving an ordinary kinetic term for the bound neutral particle. The configuration space is still parameterized by the angles on a two-sphere.
The mass of the neutral particle is twice that of the original one, explaining the 1/4 as opposed to the 1/8 in (B.3). As before, at large M we have a controlled low velocity expansion. At high energies, the two angular momenta can fluctuate independently and this simple picture is lost. A similar modification can be made for the matrix model.