Noncritical M-theory and the Gross-Neveu model in 2+1 dimensions

We point out a non-trivial connection between the model proposed by Horava and Keeler as a candidate for noncritical M-theory and the Gross-Neveu model with fermionic fields obeying periodic boundary conditions in 2+1 dimensions. Specifically, the vacuum energy of the former is identified with the large-N free-energy of the latter up to an overall constant. This identification involves an appropriate analytic continuation of the subtraction point in noncritical M-theory, which is related to the volume of the Liouville dimension. We show how the world-sheet cosmological constant may be obtained from the Gross-Neveu model. At its critical point, which is given in terms of the golden mean, the values of the vacuum energy and of the cosmological constant are 4/5 and 2/5 of the corresponding values at infinite string coupling constant.


Introduction
Non-relativistic fermi liquid models have been extensively used in the study of non-critical string theories in two dimensions [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In an interesting recent work [15] it has been suggested that fermi liquid model might also be used in the formulation of noncritical Mtheory in three dimensions. The authors of [15] propose to identify the vacuum energy of two-dimensional non-relativistic fermions in an inverted harmonic oscillator potential with the energy of noncritical M -theory in 2 + 1 dimensions.
In this note, we show that the exact vacuum energy of the fermi liquid system calculated in [15] and the large-N free energy of a three-dimensional Gross-Neveu model are given by remarkably similar expressions after an appropriate analytic continuation of the subtraction point needed to define the density of states in the fermi liquid system. In order to make the identification, the fermionic fields of the model must be given periodic boundary conditions along the compactified dimension. This non-trivial relation points to an interesting correspondence between the two models.

Noncritical M -theory
We start with a discussion of the salient features of the 2 + 1-dimensional model proposed by Hořava and Keeler [15] aiming at casting the main results in a form appropriate for comparison with the three-dimensional Gross-Neveu model.
In [15] noncritical M-Theory in 2 + 1 dimensions can be described by the double-scaling limit of non-relativistic fermionic fields ψ(x 0 ,x 1 ,x 2 ) with action [15] where the potential is The semiclassical density of states is obtained from the single particle Hamiltonian The system is equivalent to a two-dimensional inverted harmonic oscillator in the scaling limit of interest. The density of states can be defined in terms of the resolvent It may also be obtained by summing the type-0A contributions of all sectors with integer values of the RR flux [15]. The partition function Z ≡ Tre −iτ H appearing in (4) can be derived from the partition function Z 2 of a normal two-dimensional harmonic oscillator by continuing to imaginary frequency [11]. 1 Namely we consider and we obtain To apply this result to the inverted two-dimensional harmonic oscillator, we rotate ω → −iω 0 and τ → iτ and we deduce Then the density of states (4) reads The integrand has a singularity as τ → 0 which can be regulated in a Pauli-Villars manner, by introducing a subtraction point M and defining The UV cutoff M is related to the volume of the Liouville field in string theory. To calculate the integral, we bring it into the form Making use of [17] ∞ 0 dx sin 2 βx sinh 2 πx = β π(e 2β − 1) we obtain As M → ∞, this is well approximated by in agreement with the result of [15]. Instead of (13), we shall use the more general form (12), which is valid for arbitrary values of M , in order to compare with the Gross-Neveu model. The vacuum energy is given by Integrating, we obtain where we introduced the length scale in order to facilitate comparison with the Gross-Neveu model. The latter will be identified with the length of the finite dimension of the three-dimensional Euclidean space on which the Gross-Neveu model is defined.
To make contact with string theory one may introduce the conjugate variable ∆ (worldsheet cosmological constant) defined as Then the vacuum energy should be viewed as a function of ∆ (F = F [µ(∆)]). One easily obtains In the large-µ limit, which may be thought of as corresponding to weak coupling, 2 we have so that F ∼ ∆ 3/2 , to be compared with the string theory result F ∼ ∆ 2 / ln ∆. Not only is the logarithm absent, we also have a string susceptibility exponent [3][4][5][6][7] γ str > 0, where F ∼ ∆ 2−γstr , indicating that we have crossed the string barrier of central charge c ≤ 1. In the µ → 0 limit, which analogously may be thought of as corresponding to strong coupling, we obtain a behavior reminiscent of an asymptotically free theory [11]. For comparison, in string theory the vacuum energy approaches a finite constant whereas ∆ → 0. Notice also that the UV regulator M contributes in neither limit (µ → 0, µ → ∞).

The Gross-Neveu model
Next, we consider the Gross-Neveu model in three Euclidean dimensions, one of which has finite length β. The action is [18][19][20][21] whereψ a , ψ a (a = 1, 2, . . . , N ) are here taken to be two-component Dirac fermions satisfying periodic boundary conditions along the finite dimension. 3 The γ-matrices are defined in terms of the Pauli matrices as γ i = σ i (i = 1, 2) and γ 0 = σ 3 . The path integral of the model gives the subtracted partition function as where F denotes the free energy, V 2 denotes the spatial volume and G = gN is the rescaled coupling which is held fixed as N → ∞ and g → 0. In the second line of (22) we have introduced the auxiliary scalar field σ (with periodic boundary conditions) and in the third line we have integrated out the fermions. We can evaluate the last path integral in (22) by a saddle point expansion setting 4 σ = µ + λ/ √ N . Standard manipulations then give the leading-N result where, due to the periodic boundary conditions we have From (23) we read the large-N free energy F defined in (22), and the gap equation ( ∂F ∂µ = 0), To arrive at the second line of (26) we used a large momentum cutoff Λ and denoted The expression (25), up to an overall negative sign, coincides with the large-N free energy density of the three-dimensional O(N ) vector model [20] and the same applies also to the gap equation (26) up to the overall µ factor. In the O(N ) vector model, −F (β) in (25) is interpreted as a physical free energy. In the case at hand, (25) can be a positive quantity and may be interpreted as energy.
To proceed, let us define a critical parameter M, with dimensions of mass as This will be kept finite (zero, negative or positive) in the scaling limit Λ → ∞ and G → G * . M essentially quantifies the distance of 1/G from a critical value 1/G * and determines the parameter µ by 2 sinh 4 We use on purpose the same notation µ here and in Section 2 to facilitate the comparison of results.
The parameter µ is the mass of the effective λλ propagator, or alternatively it may be viewed as an inverse correlation length ξ ∼ 1/µ. In the O(N ) vector model at zero temperature, it is an order parameter for the O(N ) → O(N − 1) symmetry breaking [19]. If we substitute 1/G from the gap equation (26) into (25) and take the Λ → ∞ limit, we find after some straightforward manipulations, Crucial in obtaining (30) was the fact that the linear divergence in 1/G * canceled the linear divergence of the integral containing the logarithms in (25). Comparing (30) with the result (15) for the vacuum energy in the Hořava-Keeler model, we observe a remarkable non-trivial correspondence provided we identify the length scale β with the corresponding one in the Hořava-Keeler model (eq. (16)) and the inverse correlation length µ with the opposite of the fermi energy in the fermi liquid of the Hořava-Keeler model. The identification of length scales amounts to identifying the harmonic oscillator frequency ω 0 in the Hořava-Keeler model with the fundamental frequency ω 1 of the Gross-Neveu model (24) as The two expressions (30) of the Gross-Neveu model and (15) of the Hořava-Keeler model differ in two respects.
(a) In an overall constant dimensionless factor coming from the ratio V 2 /β 2 , and (b) to obtain agreement, the subtraction point needed to define the vacuum energy (15) ought to be analytically continued to the imaginary value The free energy (30) should be viewed as a function of the critical parameter M (29). The latter is conjugate to the parameter µ 2 ; at fixed temperature we have to be contrasted with the Legendre equation (17) defining the world-sheet cosmological constant ∆ in the Hořava-Keeler model. We may similarly obtain a two-dimensional "cosmological constant" in the Gross-Neveu model starting from the expression (30) for the free energy. Instead of introducing a Legendre transform for µ 2 , let us define D by conjugating with respect to µ, i.e., A short calculation yields which agrees with the the Hořava-Keeler model parameter ∆ (18) for the special choice (32) of the subtraction point M .
In the limit µ → 0, we obtain Notice that F (β) reduces to the free energy of a free fermion and D to the strong-coupling expression for the cosmological constant (20).

Discussion
We have shown that the fermi liquid model of noncritical M -theory introduced in [15] yields an expression for the vacuum energy which agrees (up to an overall multiplicative constant) with the expression for the free energy one obtains in the Gross-Neveu model, after analytical continuation of the subtraction point M (which is related to the Liouville field volume) to one of the values (32). To achieve agreement, we had to make two identifications: We could then obtain string dynamics in the Gross-Neveu model as in the Hořava-Keeler model by introducing a Legendre transform with respect to µ. However, one normally thinks of the critical parameter M in the Gross-Neveu model in terms of a Legendre transform with respect to µ 2 , which yields the gap eq. (26). For M > 0 and fixed finite β, we cannot go to the µ → 0 limit. Instead, we obtain a non-trivial critical point in the scaling limit Λ → ∞ and M → 0, which corresponds to a non-trivial three-dimensional CFT [20]. Solving the gap equation (26) for M = 0, we obtain the critical value given in terms of the golden mean. The free energy at the critical point is i.e., 4/5 of its value at µ = 0 (free fermion). This is the result of non-trivial polylogarithm identities (see Appendix A). The same rational relationship holds for the vacumm energy in the Hořava-Keeler model, if we allow M to take on a value given by eq. (32). We deduce from (15), Moreover, the world-sheet cosmological constant at the critical point µ = µ * is related to its value at µ = 0 (strong coupling limit) by which is obtained from (18) for M given by (32) and using dilogarithm identities (see Appendix A). It would be interesting to understand what two-dimensional critical model this critical point might correspond to.
Finally, it would be of great interest to investigate the possibility of extending the observed correspondence between noncritical M -theory and the Gross-Neveu model in three dimensions to correlation functions. We hope to report on progress in this direction in the near future. Eq. (39) follows from eq. (15) (with M given by eq. (32)) and eq. (A.12).