Exact partition function for N = ∞

We now show that the exact partition function of our free gauge theory in the strict N = ∞ limit (where no trace relations exist) may be obtained by simple counting argu-ments, following [4,5].

We consider a model with m ≥ 2 matrix-valued bosonic harmonic oscillators; the number m can be finite or infinite, as in the example of a compactified d-dimensional gauge theory. Let the i^th oscillator have energy Ei. We encode the spectrum of oscillators in a

“single-particle partition function” z(x)≡P

ix^Ei where the sum goes over all oscillators²³. Examples of such single-particle partition functions for compactified 3 + 1 dimensional theories will be provided in §3.5 below.

Following the discussion above, the partition function of single-trace states with k oscillators is given by

Zk = z(x)^k

k + (positive) (3.4)

where the factor of 1/k compensates for cyclicity in the trace, and the +(positive) acknowl-edges that this is an over-compensation in cases where the trace breaks up into repeated sequences of oscillators. Since in the large N limit a single-trace state can have any number of oscillators, we need to sum this result over k to find a partition function for single-trace states,

Z_ST = X∞ k=1

Z_k =− ln(1 − z(x)) + (positive). (3.5)

22 For example, for a single N × N matrix, tr(A^{N +1}) may be expressed as a combination of products of traces of lower powers of A.

23 To be precise, this is the single-particle partition function for the U (1) theory, since in general the single oscillator states are non-physical.

Actually, it is not too difficult to promote (3.5) to an exact expression which correctly accounts for repetitions inside the trace. This careful counting utilizes Polya theory and was already performed in [4,5]. For the convenience of the reader we reproduce it in appendix B. The exact result in the large N limit is

Z_ST =− X∞ q=1

ϕ(q)

q ln(1− z(x^q)), (3.6)

where ϕ(q) represents the number of positive integers which are not larger than q and are relatively prime to q. Note that the first term in (3.6) is precisely the term we explicitly wrote down in (3.5).

Using (3.6), we may now write the full partition function of our model, summing over states with arbitrarily many traces. The space of multi-trace states is simply the Fock space of single-trace states, so the full multi-trace partition function Z is easily obtained from Z_ST (taking into account the fact that the single-trace states behave as identical bosonic particles),

where we have used the formula P

q|kϕ(q) = k. It is easy to generalize this result to include also fermionic oscillators; denoting by z_B the single-particle partition function of the bosonic oscillators, and by z_F the single-particle partition function of the fermionic oscillators, we obtain

ln(Z) = − X∞ k=1

ln(1− zB(x^k) + (−1)^kz_F(x^k)). (3.8)

The inclusion of fermions does not change the qualitative behavior, so it is sufficient to assume that all oscillators are bosonic for the purposes of qualitative discussions.

Equation (3.8) gives the exact partition function in the strict N =∞ theory. It is not correct for the finite N theory because we have assumed all traces to be independent.²⁴ However, as long as the thermodynamics is dominated by states for which the number of oscillators in a trace is small compared to N² (which will certainly be true for low enough temperatures), trace relations will be unimportant and (3.8) should give a very good approximation to the correct behavior.

In order to understand the dynamics of (3.7) we note some obvious properties of the

“single-particle partition function” z(x), namely that z(0) = 0, z(x) is a monotonically increasing function of x, and that (since m ≥ 2) z(1) > 1. It follows that the equation z(x) = 1 has a unique solution (which we will denote by x = x_H) for 0 < x < 1. If we denote the corresponding inverse temperature by β_H (e^−βH = x_H), then ln(Z(β)) is well defined for β > β_H, but diverges like − ln(β − βH) as β tends to β_H from above. For β < β_H (high temperatures) Z(β) in (3.7) is ill defined. All these features are reproduced by the formula ²⁵

Z(β) = Z

dEρ(E)e^−βE, (3.11)

where ρ(E)≈ e^βHE at high energies. Consequently, our system of oscillators has Hagedorn-like thermodynamics with the inverse Hagedorn temperature β_H. Note that z(x) = 2x for the two oscillator model of the previous subsection, yielding x_H = 1/2 and β_H = ln(2) in agreement with the discussion in that subsection.

The fact that Z(β) diverges for T > T_H = β_H⁻¹ indicates that T_H is a limiting temperature if N is set strictly equal to infinity. Of course, this cannot be correct at any finite N , no matter how large. Indeed, a system of mN² harmonic oscillators (or even a compactified field theory for which m is infinite but there is a finite number of oscillators

24 When the number of fields in a trace is of order N² trace identities relate various states that are assumed to be different in (3.7). For example, any 2× 2 Hermitian matrix M obeys tr(M³) = ¹₂(3tr(M )tr(M²)− tr(M)³).

25 Let

ln(Z) = Z

dEρ^′(E)e^−βE (3.9)

and

ZST = Z

dEρST(E)e^−βE. (3.10)

Then, ρ^′(E) ∼ ρ^ST(E) ∼ e^βHE/E at high energies. Note that (3.9) and (3.10) match (2.2) and (2.1), respectively, upon setting d = def f = 1 in those formulas.

below any fixed energy) can certainly be heated to any temperature. It must instead be the case that at high temperatures (T > T_H), the condition for the validity of (3.8) (namely, that the number of fields contained in the typical state is much smaller than N²) fails;

we will see in §5 that this is indeed the case. Thus, in the high temperature phase, trace relations invalidate the analysis above, and traces no longer provide a useful basis for U (N ) singlets in the high temperature phase. This is true at any finite N and so it is also true in the large N limit of principal interest to us. In the next subsection we therefore turn to a more generally applicable method of dealing with the U (N ) singlet condition.