The Flavoured BFSS Model at High Temperature
Yuhma Asano, Veselin G. Filev, Samuel Kováčik, Denjoe O'Connor
PPrepared for submission to JHEP
The Flavoured BFSS Model at High Temperature
Yuhma Asano, a Veselin G. Filev, a Samuel Kov´aˇcik, a,b
Denjoe O’Connor, a a School of Theoretical Physics,Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, Ireland. b Faculty of Mathematics, Physics and Informatics,Comenius University Bratislava,Mlynsk´a dolina, Bratislava,842 48, Slovakia.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We study the high-temperature series expansion of the Berkooz-Douglas matrixmodel, which describes the D0/D4–brane system. At high temperature the model is weaklycoupled and we develop the series to second order. We check our results against the high-temperature regime of the bosonic model (without fermions) and find excellent agreement.We track the temperature dependence of the bosonic model and find backreaction of thefundamental fields lifts the zero-temperature adjoint mass degeneracy. In the low-temperaturephase the system is well described by a gaussian model with three masses m tA = 1 . ± . m lA = 2 . ± .
003 and m f = 1 . ± . a r X i v : . [ h e p - t h ] M a y ontents ω ’s. 22B The high-temperature behaviour of energy E , Polyakov loop (cid:104) P (cid:105) , (cid:104) R (cid:105) andmass susceptibility (cid:104)C m (cid:105) for the supersymmetric model. 29 The Berkooz-Douglas model (BD model) [1] was introduced as a non-perturbative formulationof M-theory in the presence of a background of longitudinal M5-branes with the M2-branequantised in light-cone gauge. Its action is written as that of the BFSS model [3] with addi-tional fundamental hypermultiplets to describe the M5-branes. The BFSS model can also beviewed as a many-body system of D0-branes of the IIA superstring. In this framework theBD model is a D0/D4 system with the massless case being the D0/D4 intersection. Whenthe number of D0-branes far exceeds that of the D4-branes the dynamics of the D0-branesis only weakly affected by that of the D4-branes and is captured by the IIA supergravitybackground holographically dual to the BFSS model. In this context the D4-branes, repre-senting the fundamental fields of the BD model, are treated as Born-Infeld probe 4-branes.This holographic set up is a tractable realisation of gauge/gravity duality with flavour.Both the BFSS model and the BD model are supersymmetric quantum mechanical modelswith an SU ( N ) gauge symmetry. When they are put in a thermal bath they become stronglycoupled at low temperature. At finite temperature their gravity duals involve a black holewhose Hawking-temperature is that of the thermal bath. These duals can be used to providenon-perturbative predictions at low temperature. The BFSS and BD models can also be– 1 –tudied by the standard non-perturbative field theory method of Monte Carlo simulation.These models therefore provide excellent candidates for testing gauge/gravity duality non-perturbatively and in a broken supersymmetric setting.There are now several non-perturbative studies of the BFSS model [7–11] and severalrecent reviews[12–14]. Also, the BD model was recently studied non-perturbatively in [15]. Inall cases the predictions from the gauge/gravity duals were found to be in excellent agreementwith that of Monte Carlo simulations of the finite-temperature models.The situation is conceptually simpler at high temperature as the dimensionless inversetemperature, scaled in terms of the BD-coupling, provides a natural small parameter for themodel. In this paper, we obtain the first two terms in the high-temperature expansion of theBD model.In the high-temperature limit only the bosonic Matsubara zero modes survive and theresulting model is a pure potential. This potential, which provides the non-perturbativeaspect of our high-temperature study, also plays a role in the ADHM construction [16]. Westudy the model for adjoint matrix size N between 4 and 32 for N f = 1 (with N f the numberof D4-branes) and for N f between 2 and 16 for N from 9 to 20. For N f ≥ N we find that thesystem has difficulties with ergodicity. In particular, for N f = 2 N and N f = 2 N +1 the systemfailed to thermalise satisfactorily. In contrast the system has no difficulties for N f = 2 N − SU ( N f ) instantons of Chern number N exist only for N ≥ N f [17, 18]. Themoduli space of such instantons is equivalent to the zero locus of the potential with X a = 0and D A = 0 (see equation (2.4)). This moduli space is in general singular and non-singularonly when this bound is satisfied.There is also a natural 1 + 1 dimensional analogue of the BD model, which has N = 4supersymmetry, associated with the D1/D5 system of [5], whose BFSS relative was discussedin [19–21]. When the Euclidean finite-temperature version of this 1 + 1 dimensional quantumfield theory is considered on a torus with the spatial circle of period β and euclidean time ofperiod 1 /T , then at high temperature the fermions decouple and one is left with the purelybosonic version of the BD model. We refer to this model as the bosonic BD model andstudy the small period behaviour (equivalent for us to our high-temperature regime) of themassless version of this model as a check on our high-temperature series. We find the high-temperature series results are in excellent agreement with Monte Carlo simulations of thebosonic BD model. By fitting the dependence, of the expectation values of our observables,on the number of flavour multiplets, N f , we find that extrapolation, to N f = 0, agrees wellwith the corresponding observables of the BFSS model.As β , the inverse temperature, grows the bosonic BD model undergoes a set of phasetransitions. These are the phase transitions of the bosonic BFSS model. We find the high-temperature series expansion is valid up to β ∼ /
2, which is just below the phase transitionregion. Above the transition the bosonic BD model is well described by free massive fields, In this paragraph we avoid using β for 1 /T for simplicity of the comparison. – 2 –here the backreaction of the fundamental fields has lifted the degeneracy of the longitudinaland transverse masses.The principal results of this paper are: • We obtain expansions for observables of the BD model to second order in a high-temperature series. • We tabulate the coefficients of this expansion as functions of N and N f in the range4 ≤ N ≤
32 and 1 ≤ N f ≤ • We measure the expectation values of the composite operator (cid:104) r (cid:105) bos , (see equation(2.8)), and the mass susceptibility (cid:104)C m (cid:105) bos , (see equation (5.7)), of the bosonic BDmodel as a function of temperature down to zero and use it to check our coefficients forthe high-temperature series of the full BD model. • We find that the fundamental fields of the bosonic BD model have mass m f = 1 . ± . • We measure the backreacted mass of the longitudinal adjoint scalars to be m lA = 2 . ± .
003 and find that the transverse mass is largely unaffected by backreaction being m tA = 1 . ± . m A = 1 . ± . • We use the measured masses to predict the zero-temperature values of our fundamentalfield observables (cid:104) r (cid:105) bos and mass susceptibility (cid:104)C m (cid:105) bos and find excellent agreementwith direct measurements.The paper is organised as follows: In section 2 we present the finite-temperature BDmodel and describe our notation and observables. In section 3 we set up and implementthe high-temperature series expansion working to second order in the inverse temperature β .Section 4 describes the dependence of our observables on the coefficients in the expansion,which must be determined by numerical simulation of the zero-mode model. In Section 5 weperform lattice simulations of the bosonic BD model and find excellent agreement with thehigh-temperature expansion. We also find the low-temperature phase of the model is welldescribed by a system of gaussian quantum fields. Section 6 gives our concluding remarks.There are two appendices; appendix A gives tables, for different N and N f , of the coefficientsdetermined non-perturbatively while appendix B presents graphs of predictions for the high-temperature behaviour of our observables for the supersymmetric model.– 3 – The Berkooz-Douglas Model
We begin by describing the field content of the model following the notation used in [5]. Theaction of the BFSS model is given by S BFSS = 1 g (cid:90) dt (cid:88) i =1 Tr (cid:26)
12 ( D X i ) + 14 [ X i , X j ] − i T C Γ D Ψ + 12 Ψ T C Γ i [ X i , Ψ] (cid:27) , (2.1)where D · = ∂ t · − i [ A, · ], Ψ is a thirty-two component Majorana–Weyl spinor, Γ µ are tendimensional gamma matrices and C is the charge conjugation matrix satisfying C Γ µ C − = − Γ µT . The fields X i and Ψ are in the adjoint representation of the gauge symmetry group SU ( N ) and A is the gauge field.To describe the addition of the fundamental fields we break the SO (9) vector X i into an SO (5) vector X a and an SO (4) vector which we re-express as X ρ ˙ ρ via X ρ ˙ ρ = i √ (cid:88) m =1 σ mρ ˙ ρ X − m , (2.2)where σ = − i and σ A ’s ( A = 1 , ,
3) are the Pauli matrices. The X ρ ˙ ρ ( ρ, ˙ ρ = 1 , SO (4) which satisfies thereality condition X ρ ˙ ρ = ε ρσ ε ˙ ρ ˙ σ ¯ X σ ˙ σ . The indices ρ and ˙ ρ are those of SU (2) R and SU (2) L ,respectively, where SO (4) = SU (2) L × SU (2) R .The nine BFSS scalar fields, X i , become X a ( a = 1 , · · · ,
5) and X ρ ˙ ρ . The sixteenadjoint fermions of the BFSS model become λ ρ and θ ˙ ρ with λ ρ being SO (5 ,
1) symplecticMajorana-Weyl spinors of positive chirality and satisfying λ ρ = ε ρσ ( λ c ) σ while θ ˙ ρ are sym-plectic Majorana-Weyl spinors of negative chirality satisfying θ ˙ ρ = − ε ˙ ρ ˙ σ ( θ c ) ˙ σ . They combinetogether to form an SO (9 ,
1) Majorana-Weyl spinor in the adjoint of SU ( N ). This SO (9)symmetry is recovered only if the fundamental fields are turned off.To describe the longitudinal M5-branes (or D4-branes), we have Φ ρ and χ , which trans-form in the fundamental representations of both SU ( N ) and the global SU ( N f ) flavoursymmetry. Φ ρ are complex scalar fields with hermitian conjugates ¯Φ ρ , and χ is an SO (5 , T = β − becomes: S = N (cid:90) β dτ (cid:34) Tr (cid:18) D τ X a D τ X a + 12 D τ ¯ X ρ ˙ ρ D τ X ρ ˙ ρ + 12 λ † ρ D τ λ ρ + 12 θ † ˙ ρ D τ θ ˙ ρ (cid:19) + tr (cid:16) D τ ¯Φ ρ D τ Φ ρ + χ † D τ χ (cid:17) − Tr (cid:18)
14 [ X a , X b ] + 12 [ X a , ¯ X ρ ˙ ρ ][ X a , X ρ ˙ ρ ] (cid:19) Here X of [5] is replaced by − X . – 4 – 12 Tr (cid:88) A =1 D A D A + tr (cid:0) ¯Φ ρ ( X a − m a ) Φ ρ (cid:1) − Tr (cid:18) − λ † ρ γ a [ X a , λ ρ ] + 12 θ † ˙ ρ γ a [ X a , θ ˙ ρ ] − √ iε ρσ θ † ˙ ρ [ X σ ˙ ρ , λ ρ ] (cid:19) − tr (cid:16) χ † γ a ( X a − m a ) χ + √ iε ρσ χ † λ ρ Φ σ + √ iε ρσ ¯Φ ρ λ † σ χ (cid:17) (cid:35) , (2.3)where D A = σ A σρ (cid:18)
12 [ ¯ X ρ ˙ ρ , X σ ˙ ρ ] − Φ σ ¯Φ ρ (cid:19) , (2.4)with D τ the covariant derivative which, for the fields of the fundamental multiplet, Φ ρ and χ , acts as D τ · = ( ∂ τ − iA ) · . The trace of SU ( N ) is written as Tr while that of SU ( N f )is denoted by tr. The diagonal matrices, m a , correspond to the transverse positions of theD4-branes.We fix the static gauge: ∂ τ A = 0, so the path integral requires the corresponding ghostfields c and ¯ c with the ghost term N (cid:82) β dτ Tr ∂ τ ¯ cD τ c added to the action (2.3).We will restrict our attention to m a = 0 so that the D4-branes are attached to theD0-branes, and the strings between D0 and D4 are massless, i.e. the fundamental fields aremassless. The factor of N in front of the integral in (2.3) is the remnant of the ’t Hooftcoupling λ = g N which is kept fixed and absorbed into τ and the fields with β = λ / /T .Note that without loss of generality we can set λ = 1.As discussed in the introduction, the BFSS model is the matrix regularization of a su-permembrane theory [2], so the BFSS part of this model can be also interpreted as M2-branedynamics. In this context the D4-branes lift to M5-branes and the model can describe M2-branes ending on longitudinal M5-branes.The BD model is a version of supersymmetric quantum mechanics and could in principlebe treated by Hamiltonian methods. The partition function is then Z = T r (e − βH ) = (cid:90) [ dX ][ dλ ][ dθ ][ d Φ][ d ¯Φ][ dχ ][ dχ † ][ dA ]e − S (2.5)with T r the trace over the Hilbert space of the Hamiltonian restricted to its gauge invariantsubspace and the action S , in the path integral, is given by equation (2.3).The measure in the path integral for the partition function (2.5) has a hidden depen-dence on temperature due to the presence of the Van Vleck-Morette determinant [22] in thedefinition of the path integral measure. This determinant arises from the kinetic contributionto the action (2.3) which, as written, is temperature dependent. To remove the temperaturedependence from the measure, we rescale the variables in the original action (2.3) so thatthe kinetic terms, including the gauge potential, are independent of β . For this τ → βτ , X i → β X i , Φ ρ → β Φ ρ , A → β − A , c → β c and ¯ c → β ¯ c . The fermions do not needrescaling. The path integral measure is now temperature independent and, when the mass– 5 –s zero, the only temperature dependence is β for the bosonic potential and β / for thefermionic potential. If the mass term is included it enters as βm a β in the potential with theoverall scales of β and β / in the bosonic and fermionic contributions respectively. Thetemperature dependence of the model is now explicit.The principal observable of the model is the energy , E = (cid:104) H (cid:105) /N . Once the temperaturedependence of the model has been made explicit, as described above, one can then simplynote that N E is minus the derivative of logarithm of the partition function with respect to β , returning to the original variables one readily sees that in the path integral formulation: E = (cid:104) ε b (cid:105) + (cid:104) ε f (cid:105) , where ε b = 3 N β (cid:90) β dτ (cid:20) Tr (cid:18) −
14 [ X i , X j ] (cid:19) + tr (cid:18) ¯Φ ρ X a Φ ρ − ¯Φ ρ [ ¯ X σ ˙ ρ , X ρ ˙ ρ ]Φ σ −
12 ¯Φ ρ Φ σ ¯Φ σ Φ ρ + ¯Φ ρ Φ ρ ¯Φ σ Φ σ (cid:19) (cid:21) ,ε f = 32 N β (cid:90) β dτ (cid:20) Tr (cid:18) λ † ρ γ a [ X a , λ ρ ] − θ † ˙ ρ γ a [ X a , θ ˙ ρ ] + √ iε ρσ θ † ˙ ρ [ X σ ˙ ρ , λ ρ ] (cid:19) + tr (cid:16) − χ † γ a X a χ − √ iε ρσ χ † λ ρ Φ σ − √ iε ρσ ¯Φ ρ λ † σ χ (cid:17) (cid:21) . (2.6)We see only the potential contributes and the coefficients 3 and 3 / R = 1 N β (cid:90) β dτ Tr X i , P = 1 N Tr (exp [ iβA ]) . (2.7)Here R is a hermitian operator whose expectation value is a measure of the extent of theeigenvalue distribution of the scalars X i and P is the Polyakov loop. Note: Path-ordering isnot needed here for the Polyakov loop as we consider A in the static gauge.Since the model has new degrees of freedom it is important to consider other observablesthat capture properties of these new fields. The natural candidates are r = 1 βN f (cid:90) β dτ tr ¯Φ ρ Φ ρ , (2.8)which is the analogue of R for the fundamental degrees of freedom, and the condensatedefined as c a ( m ) = ∂∂m a (cid:18) − N β log Z (cid:19) = (cid:28) β (cid:90) β dτ tr (cid:110) ρ ( m a − X a )Φ ρ + χ † γ a χ (cid:111)(cid:29) . (2.9)However, for us, with m a = 0, c a will be zero. So our focus will be on the mass susceptibility (cid:104)C m (cid:105) := ∂c a ∂m a (0) , (2.10) We divide by N so that E remains finite in the large- N limit. – 6 –.e. the derivative with respect to m a with a fixed (not summed over) and evaluated at m a = 0where C m = 2 β (cid:90) β dτ tr ¯Φ ρ Φ ρ − N β (cid:18)(cid:90) β dτ tr (cid:110) − ρ X a Φ ρ + χ † γ a χ (cid:111)(cid:19) . (2.11)Here a in (2.11) is summed over a = 1 , · · · , In this section, we develop the high-temperature expansion of the BD model. For very hightemperatures only the Matsubara zero modes, i.e. the zero modes in a Fourier expansion,survive and the model reduces to a bosonic matrix model for these modes. A high-temperatureseries expansion is therefore obtained by developing a perturbative expansion of the modelin the non-zero modes. The zero-modes must then be treated non-perturbatively and this isdone by Monte Carlo simulation.Our strategy is therefore to expand the model in Matsubara modes, show that the tem-perature can be seen as a coupling constant for these modes and then integrate out thenon-zero modes order by order in perturbation theory to obtain an effective action for thezero modes, which can then be treated non-perturbatively.To obtain the series to second order we will only need one loop computations. The non-zero mode integration can be done analytically and yields an effective action and observablesin terms of the zero temperature variables. As a final step the integration over these zeromodes must then be performed non-perturbatively via Monte Carlo simulation.The Fourier expansion of the fields is given by X i ( τ ) = (cid:88) n ∈ Z X in e πinτ/β , λ ρ ( τ ) = (cid:88) r ∈ Z + λ rρ e πirτ/β , θ ˙ ρ ( τ ) = (cid:88) r ∈ Z + θ r ˙ ρ e πirτ/β ,c ( τ ) = (cid:88) n ∈ Z ,n (cid:54) =0 c n e πinτ/β , ¯ c ( τ ) = (cid:88) n ∈ Z ,n (cid:54) =0 ¯ c n e − πinτ/β , Φ ρ ( τ ) = (cid:88) n ∈ Z Φ nρ e πinτ/β , χ ( τ ) = (cid:88) r ∈ Z + χ r e πirτ/β , (3.1)where thermal boundary conditions require that the bosons and ghosts are periodic in τ whilethe fermions are anti-periodic.The action (2.3) now takes the form of the sum of a zero mode action, the kinetic termfor the non-zero modes and an interaction term. As discussed above, in the high-temperaturelimit, only the zero-modes play a role. As the temperature is lowered one can integrate out thenon-zero modes perturbatively with β playing the role of a perturbation parameter. Usingthis procedure, the first two terms in the high-temperature expansion of E , (cid:104) R (cid:105) and (cid:104) P (cid:105) for the BFSS model were obtained in [6]. We follow the same method here and obtain thecorresponding expansion of these observables for the BD model and for them the novel feature– 7 –ill be the additional dependence on N f , the number of flavour multiplets. In addition wehave the new observable (cid:104) r (cid:105) and (cid:104)C m (cid:105) .In order to develop the high-temperature series it is convenient to rescale the scalar fieldsin (2.3) as follows X i → β − X i , A → β − A , Φ → β − Φ ,X in (cid:54) =0 → β X in (cid:54) =0 , Φ n (cid:54) =0 → β Φ n (cid:54) =0 , c n → β c n , ¯ c n → β ¯ c n , (3.2)while the fermions remain unchanged. This rescaling makes the coefficients of the zero-modeterms and the kinetic terms independent of β so that one can concentrate on the β -dependence,which now appears only in the interaction terms.The rescaling (3.2) can be understood as the rescaling of section 2, which was necessaryto remove the temperature dependence of the measure, followed by the further zero-moderescaling X i → β − X i , Φ ρ → β − Φ ρ and A → β A . This zero-mode rescaling means thepartition function becomes Z = β − (8( N − NN f ) ¯ Z , (3.3)where ¯ Z is the partition function in terms of the rescaled fields of (3.2) and the only remainingtemperature dependence is in S int . We can now develop the high-temperature series bydiagrammatic techniques with β playing the role of a coupling.The action is then written in terms of the variables of (3.2), which we will use for theremainder of the paper, as S = S + S kin + S int , (3.4)where S is a zero-mode action S = − N (cid:16) [ X i , X j ] + 2[ A, X i ] (cid:17) + N tr (cid:18) ¯Φ ρ A Φ ρ + ¯Φ ρ ( X a ) Φ ρ − ¯Φ ρ [ ¯ X σ ˙ ρ , X ρ ˙ ρ ]Φ σ −
12 ¯Φ ρ Φ σ ¯Φ σ Φ ρ + ¯Φ ρ Φ ρ ¯Φ σ Φ σ (cid:19) , (3.5) S kin is the kinetic part of the action for non-zero modes S kin = (cid:88) n (cid:54) =0 (2 πn ) N (cid:20) Tr (cid:16) X a − n X an + ¯ X ρ ˙ ρ − n X nρ ˙ ρ + 2¯ c − n c n (cid:17) + tr (cid:0) ρ − n Φ nρ (cid:1) (cid:21) + (cid:88) r πirN (cid:104) Tr (cid:16) λ † ρ − r λ rρ + θ † ˙ ρ − r θ r ˙ ρ (cid:17) + tr (cid:16) χ †− r χ r (cid:17)(cid:105) , (3.6)and S int is the interaction part of the action. The terms quadratic in non-zero modes presentin S int but not present in the BFSS model are ∆ S int = − N β ( V ( A )1 + V ( B )1 ) − N β ( V ( A )2 + V ( B )2 + V ) + O ( β ) , (3.7)where V ( A )1 = 4 π (cid:88) n (cid:54) =0 n tr( ¯Φ ρ − n A Φ nρ ) + (cid:88) r tr( iχ †− r Aχ r ) , – 8 – ( B )1 = (cid:88) r tr( χ †− r γ a X a χ r + √ iε ρσ χ †− r λ rρ Φ σ + √ iε ρσ ¯Φ ρ λ † σ − r χ r ) ,V ( A )2 = − (cid:88) n (cid:54) =0 tr( ¯Φ ρ − n A Φ nρ ) ,V ( B )2 = − (cid:88) n (cid:54) =0 tr (cid:0) ¯Φ ρ − n X a Φ nρ + ¯Φ ρ X a − n X an Φ ρ − ¯Φ ρ − n [ ¯ X σ ˙ ρ , X ρ ˙ ρ ]Φ nσ − ¯Φ ρ [ ¯ X σ ˙ ρ − n , X nρ ˙ ρ ]Φ σ − ¯Φ ρ − n Φ σ ¯Φ σ Φ nρ − ¯Φ ρ − n Φ nσ ¯Φ σ Φ ρ + 2 ¯Φ ρ − n Φ ρ ¯Φ σ Φ nσ + 2 ¯Φ ρ − n Φ nρ ¯Φ σ Φ σ (cid:1) ,V = − (cid:88) n (cid:54) =0 tr (cid:104) ( ¯Φ ρ − n X an X a Φ ρ + ¯Φ ρ − n X a X an Φ ρ + ¯Φ ρ X a − n X a Φ nρ + ¯Φ ρ X a X a − n Φ nρ ) − ( ¯Φ ρ − n [ ¯ X σ ˙ ρn , X ρ ˙ ρ ]Φ σ + ¯Φ ρ − n [ ¯ X σ ˙ ρ , X nρ ˙ ρ ]Φ σ + ¯Φ ρ [ ¯ X σ ˙ ρ − n , X ρ ˙ ρ ]Φ nσ + ¯Φ ρ [ ¯ X σ ˙ ρ , X − nρ ˙ ρ ]Φ nσ ) −
12 ( ¯Φ ρ − n Φ σ ¯Φ σn Φ ρ + ¯Φ ρ Φ − nσ ¯Φ σ Φ nρ ) + ( ¯Φ ρ − n Φ ρ ¯Φ σn Φ σ + ¯Φ ρ Φ − nρ ¯Φ σ Φ nσ ) − (cid:88) r ( χ †− r γ a X a − n χ r + n + √ iε ρσ χ †− r λ r + n ρ Φ − nσ + √ iε ρσ ¯Φ ρ − n λ † σ − r χ r + n ) (cid:105) . (3.8) V does not contribute to the expectation values of operators at next-leading order. Two suchvertices would be required and the resultant contribution would therefore be of higher orderin β . Similarly, fermionic terms that involve only non-zero modes also scale as β , and againcontribute at a two and higher loop order to the expectation values of observables.The zero-mode action (3.5) corresponds to the bosonic part of the original model (2.3)dimensionally reduced to a point and plays an important role in the ADHM construction asthe solutions to S = 0 with D A = 0, where D A is given in (2.4), provide the ADHM data[16].This zero-mode model is the flavoured bosonic version of the IKKT model[4]. We use thenotation (cid:104)· · · (cid:105) DR for the expectation value calculated with this dimensionally reduced model.Thus for a generic observable, O which is a function of X i , A and Φ ρ we denote (cid:104) O (cid:105) DR = 1¯ Z (cid:90) [ dX ][ d ¯Φ ][ d Φ ] O e − S . (3.9)Furthermore we denote (cid:104) AB (cid:105) DR,c = (cid:104) AB (cid:105) DR − (cid:104) A (cid:105) DR (cid:104) B (cid:105) DR , (3.10)and the subscript ‘c’ denotes connected part.Identities such as (cid:82) ddX ( X Oe − S ) = 0 for X and similar identities for A and Φ ρ yieldthe Ward-type identities:9( N − (cid:104) O (cid:105) DR − (cid:28) λ dS ( λX ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 O (cid:29) DR + (cid:28) λ dO ( λX ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 (cid:29) DR = 0 , ( N − (cid:104) O (cid:105) DR − (cid:28) λ dS ( λA ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 O (cid:29) DR + (cid:28) λ dO ( λA ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 (cid:29) DR = 0 , N f N (cid:104) O (cid:105) DR − (cid:28) λ dS ( λ Φ ρ ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 O (cid:29) DR + (cid:28) λ dO ( λ Φ ρ ) dλ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 (cid:29) DR = 0 . (3.11)– 9 –hese identities can be used to simplify various expressions and in particular one can see thatone never needs to consider the insertion of S or s = N Tr (cid:16) −
14 [ X i , X j ] (cid:17) + N tr (cid:16) ¯Φ ρ X a Φ ρ − ¯Φ ρ [ ¯ X σ ˙ ρ , X ρ ˙ ρ ]Φ σ −
12 ¯Φ ρ Φ σ ¯Φ σ Φ ρ + ¯Φ ρ Φ ρ ¯Φ σ Φ σ (cid:17) (3.12)with other correlators as they can be eliminated by use of these identities. The simplestidentities resulting from (3.11) are that4 (cid:104) S (cid:105) DR = 10( N −
1) + 4 N f N and 4 (cid:104) s (cid:105) DR = 8( N −
1) + 4 N f N , (3.13)the latter of which establishes the equivalent leading order expression for the energy using(2.6) as discussed in the next section.We will next present the leading high-temperature expansion of our observables.
The expectation values of our observables to leading order are determined solely by the zeromodes. To this order the partition function is given by (3.3) with ¯ Z a constant. We thereforehave E = 34 β − (cid:26) (cid:18) − N (cid:19) + 4 N f N (cid:27) + O ( β ) . (3.14)The direct expression using (2.6) gives E = N β − (cid:104) s (cid:105) DR + O ( β / ) and using the secondidentity of (3.13) we see that this agrees with (3.14).Also, the leading terms in the β -expansion of (cid:104) R (cid:105) and the expectation value of thePolyakov loop are (cid:104) R (cid:105) = β − (cid:28) N Tr X i (cid:29) DR + O ( β ) , (cid:104) P (cid:105) = 1 − β (cid:28) N Tr A (cid:29) DR + O ( β ) . (3.15)For our new observables (cid:104) r (cid:105) and (cid:104)C m (cid:105) we have the leading contributions (cid:104) r (cid:105) = β − (cid:28) N f tr ¯Φ ρ Φ ρ (cid:29) DR + O ( β ) (3.16)and (cid:104)C m (cid:105) = 2 β − (cid:18)(cid:10) tr ¯Φ ρ Φ ρ (cid:11) DR − N (cid:10) (tr ¯Φ ρ X a Φ ρ ) (cid:11) DR,c (cid:19) + O ( β ) . (3.17)Note: All the leading order contributions are purely bosonic, since fermions decouple at hightemperature. The necessary expectation values are computed numerically via Monte Carlosimulation with the action S of equation (3.5) and given in the tables in appendix A fordifferent values of N and N f . – 10 – .2 Next-leading order The higher order contributions in the high-temperature expansion come from integrating outthe non-zero modes in (3.4). The first subleading order is obtained by performing the gaussianintegrals over the non-zero modes, where the potential is truncated as in (3.7), and expandingthe resulting exponential and ratio of determinants in terms of β .Examining the action (2.3) we see the fermionic terms can be written in the form (cid:90) dτ (cid:20) Tr (cid:18) λ † ρ ( D τ + γ a ad X a ) λ ρ + 12 θ † ˙ ρ ( D τ − γ a ad X a ) θ ˙ ρ (cid:19) + tr (cid:16) χ † ( D τ − γ a X a ) χ (cid:17) + Tr (cid:16) θ † ˙ ρ J ˙ ρ (cid:17) + tr (cid:16) χ † J + J † χ (cid:17) (cid:21) , (3.18)and the commutator action of X a is denoted by ‘ad X a ’. Since J ˙ ρ and J are fermioniccurrents that depend linearly on λ σ , integrating out θ ˙ ρ and χ gives the additional contributions − (cid:82) J † ˙ ρ G θ J ˙ ρ and − (cid:82) J † G χ J to the quadratic form for λ ρ . Here G λ , G θ and G χ are Green’sfunctions for λ ρ , θ ˙ ρ and χ , respectively. These current-current terms will be of order β / andcontribute at the sub-leading order under consideration.The non-zero modes can now be integrated out and to one loop we obtain S eff = S + (cid:88) n (cid:54) =0
12 ln
Det [(1 + β / A πn ) + β / M X (2 πn ) ] Det [(1 + β / A πn ) + β / M φ (2 πn ) ] Det [1 + ad A πn ] − ∞ (cid:88) n = −∞ ln (cid:20) Pf (cid:40) (cid:15) (cid:32) β / − i ad A + γ a ad X a πi ( n + ) (cid:33)(cid:41) × Pf (cid:40) (cid:15) (cid:32) β / − i ad A − γ a ad X a πi ( n + ) (cid:33)(cid:41) Det (cid:32) − β / − iA + γ a X a πi ( n + ) (cid:33) (cid:21) + ∞ (cid:88) n = −∞ β / (cid:0) Tr [ G nλ ad( X ) G nθ ad( ¯ X )] + 2 Tr [ G nλ Φ G nχ ¯Φ ] (cid:1) , (3.19)with G nθ = G nλ = G nχ = 12 πi ( n + ) . Equations (3.7) with details in (3.8) specify the quadratic forms whose determinants andPfaffians enter in (3.19), and here Tr is the operator trace.In detail one has:( M X X ) an = −
12 [ X b , [ X b , X an ]] + 12 [ X b , [ X a , X bn ]] + 12 [ ¯ X ρ ˙ ρ , [ X ρ ˙ ρ , X an ]] − Φ ρ ¯Φ ρ X an , ( M X X ) nρ ˙ ρ = −
12 [ X σ ˙ ρ , [ ¯ X σ ˙ σ , X nρ ˙ σ ]] + 12 [ X ρ ˙ σ , [ ¯ X σ ˙ σ , X nσ ˙ ρ ]] −
12 [ X a , [ X a , X nρ ˙ ρ ]] − [Φ ρ ¯Φ σ , X nσ ˙ ρ ] , ( M Φ Φ) nρ = − X a Φ nρ + [ ¯ X σ ˙ ρ , X ρ ˙ ρ ]Φ nσ – 11 – Φ σ ¯Φ σ Φ nρ + Φ nσ ¯Φ σ Φ ρ − ρ ¯Φ σ Φ nσ − mρ ¯Φ σ Φ σ . (3.20)Expanding (3.19) in β we will obtain S eff with S eff = S + β / S . (3.21)Let us look at the individual contributions in more detail.After straightforward computations we find the contribution from the first determinantin (3.19) due to the integration over non-zero X a modes gives S X = N (cid:0)
16 Tr X i −
18 Tr A (cid:1) + 512 ( N − N tr ¯Φ ρ Φ ρ . (3.22)The ghost contribution similarly expanded gives S g = 2 N
24 Tr A , (3.23)and the contribution from the fundamental scalars is S Φ1 = 112 (cid:0) N f Tr( X a − A ) + 3 N tr ¯Φ ρ Φ ρ (cid:1) . (3.24)Putting the bosonic contributions together we have S bos = 2 N (cid:26) Tr X i − Tr A + (cid:18) − N (cid:19) tr ¯Φ ρ Φ ρ (cid:27) + N f (cid:0) Tr X a − Tr A (cid:1) . (3.25)The fermionic contributions can similarly be evaluated to give S θ = S λ = − N (Tr X a − Tr A ) (3.26)for the Pfaffian contribution to the integration over θ ˙ ρ and λ ρ . The θ current-current contri-bution gives S J ˙ ρ G θ J ˙ ρ = − N Tr( ¯ X ρ ˙ ρ X ρ ˙ ρ ) . (3.27)These three contributions together recombine to give an SO (9) invariant term, which is thefermionic part of S BF SS .Next considering the χ determinant we find S χ = − N f X a − Tr A ) , (3.28)and its current-current contribution is S J † G χ J = − N − N tr ¯Φ ρ Φ ρ . (3.29)Putting all these fermionic contributions together we find S fer = − N (cid:26) Tr X i − Tr A + (cid:18) − N (cid:19) tr ¯Φ ρ Φ ρ (cid:27) − N f (cid:0) Tr X a − Tr A (cid:1) . (3.30)– 12 –inally, defining e − β / S = 1 + β / O , so for the supersymmetric model adding (3.25) and(3.30) we have O = − N (cid:26) Tr X i − Tr A + (cid:18) − N (cid:19) tr ¯Φ ρ Φ ρ (cid:27) + 13 N f (cid:0) Tr X a − Tr A (cid:1) . (3.31)Similarly using (3.25) we can define O bos = − S bos , which we can write O bos = − O − N tr ¯Φ ρ Φ ρ . (3.32)These expressions will be useful in section 4.The partition function including the next to leading corrections is now given by Z = β − (8( N − NN f ) (1 + β / (cid:104)O(cid:105) DR ) ¯ Z , (3.33)and the temperature dependence is explicit. We immediately have the subleading correctionto the energy E by straightforward differentiation of (3.33) to obtain that E = 34 β − (cid:26) (cid:18) − N (cid:19) + 4 N f N (cid:27) − N β / (cid:104)O(cid:105) DR . (3.34)Turning to the high-temperature behaviour of R and the Polyakov loop, the resultingexpectation values are given by (cid:104) R (cid:105) = β − (cid:28) N Tr X i (cid:29) DR + β (cid:32) (cid:18) − N (cid:19) + (cid:28) N (Tr X i ) O (cid:29) DR,c (cid:33) + O ( β ) , (3.35)and (cid:104) P (cid:105) = 1 − β (cid:34) (cid:28) N Tr A (cid:29) DR − β (cid:40) (cid:28) N Tr A (cid:29) DR − (cid:28) N (Tr A ) O (cid:29) DR,c (cid:41) + O ( β ) (cid:35) . (3.36)The constant (1 − N ) is the contribution due to the expectation value of the non-zeromodes, which are traceless.Our observable (cid:104) r (cid:105) is similarly given by (cid:104) r (cid:105) = β − (cid:28) N f tr ¯Φ ρ Φ ρ (cid:29) DR + β (cid:32)
16 + (cid:28) N f (cid:0) tr ¯Φ ρ Φ ρ (cid:1) O (cid:29) DR,c (cid:33) + O ( β ) , (3.37)and its bosonic version is again obtained by replacing O with O bos .In terms of Fourier modes we have c a ( m ) = (cid:42) tr (cid:18) β − m a ¯Φ ρ Φ ρ + 2 βm a (cid:88) n (cid:54) =0 ¯Φ ρ − n Φ nρ + (cid:88) r χ † r γ a χ r − β − ¯Φ ρ X a Φ ρ – 13 – β (cid:88) n ( ¯Φ ρ − n X a Φ nρ + ¯Φ ρ X a − n Φ nρ + ¯Φ ρ − n X an Φ ρ ) − β (cid:88) n,m ¯Φ ρ − n X an − m Φ mρ (cid:19)(cid:43) . (3.38)However, we will restrict ourselves to the massless case and as discussed SO (5) invarianceguarantees that this observable is zero so we focus on the mass susceptibility, (cid:104)C m (cid:105) .Calculating C m in the high-temperature expansion to the next to leading order yields (cid:104)C m (cid:105) = 2 β − (cid:18) (cid:104) tr ¯Φ ρ Φ ρ (cid:105) DR − N (cid:104) (tr ¯Φ ρ X a Φ ρ ) (cid:105) DR,c (cid:19) + 2 β (cid:18) − N f (cid:10) (tr ¯Φ ρ Φ ρ ) O (cid:11) DR,c − N (cid:10) (tr ¯Φ ρ X a Φ ρ ) O (cid:11) DR,c (cid:19) + O ( β ) . (3.39)The contribution − N f / − N f /
2, while the bosonic con-tribution is N f /
6. Therefore, the condensate susceptibility for the bosonic model is obtainedfrom (3.39) by replacing the numerical constant − N f by N f and O by O bos . The resultingexpression is (cid:104)C m (cid:105) bos = 2 β − (cid:18) (cid:104) tr ¯Φ ρ Φ ρ (cid:105) DR − N (cid:104) (tr ¯Φ ρ X a Φ ρ ) (cid:105) DR,c (cid:19) + 2 β (cid:18) N f (cid:10) (tr ¯Φ ρ Φ ρ ) O bos (cid:11) DR,c − N (cid:10) (tr ¯Φ ρ X a Φ ρ ) O bos (cid:11) DR,c (cid:19) + O ( β ) . (3.40)An alternative to the above treatment is to work directly with perturbation theory in β , but we believe the structure of the computations is simpler in the above treatment. Thecontributions to O , E , R and P from the pure BFSS model were derived in [6] and when N f and the fundamental fields are set to zero our results reproduce the results reported there. In this section we express the coefficients, Ξ i ,used in the high-temperature expansion ofobservables (see equations (4.3) and (4.4)), in terms of the primitive observables ω i , ω i,j and ω i,j,k ( i, j, k = 1 , ...,
6) defined in equations (4.2) below. We have the following expressions:Ξ = − ω + ω ) − N f N (cid:26) ω + (cid:18) − N (cid:19) ω (cid:27) , Ξ = 5 ω + 4 ω , Ξ = 34 (cid:18) − N (cid:19) + 4 + N f N ω , + 15 ω , + 16 ω , ) + 163 (5 ω , + 4 ω , )+ 4 N f N (cid:18) − N (cid:19) (5 ω , + 4 ω , ) , – 14 – = 12 ω , Ξ = 14! ω − (cid:40) N f N ω , − ω , ) + 163 ω , + 4 N f N (cid:18) − N (cid:19) ω , (cid:41) , Ξ = 2 N f ( ω − ω , ) , Ξ = − N f N f (cid:40) N f N ω , + 43 ω , + 13 (cid:18) − N (cid:19) ω , (cid:41) , Ξ = − N f (cid:40) N f N ω , , + 3 ω , , ) + 163 ω , , + 43 (cid:18) − N (cid:19) ω , , (cid:41) , Ξ = ω , Ξ = 14! ω , Ξ bos1 = −
12 Ξ − N f N ω , Ξ bos3 = 34 (cid:18) − N (cid:19) − N f N ω , + 15 ω , + 16 ω , ) −
83 (5 ω , + 4 ω , ) − N f N (cid:18) − N (cid:19) (5 ω , + 4 ω , ) , Ξ bos5 = 14! ω + 14 (cid:40) N f N ω , − ω , ) + 163 ω , + 4 N f N (cid:18) − N (cid:19) ω , (cid:41) , Ξ bos7 = N f − N f (cid:40) N f N ω , + 43 ω , + 13 (cid:18) − N (cid:19) ω , (cid:41) , Ξ bos8 = 2 N f (cid:40) N f N ω , , + 3 ω , , ) + 163 ω , , + 43 (cid:18) − N (cid:19) ω , , (cid:41) , (4.1)where ω = 1 N (cid:10) Tr A (cid:11) DR , ω = 15 N (cid:10) Tr( X a ) (cid:11) DR = ω , ω = 14 N (cid:68) Tr ¯ X ρ ˙ ρ X ρ ˙ ρ (cid:69) DR ,ω = 1 N f (cid:10) tr ¯Φ ρ Φ ρ (cid:11) DR , ω = 1 N (cid:10) tr ¯Φ ρ X Φ ρ (cid:11) DR = 0 , ω = 1 N (cid:10) Tr A (cid:11) DR ,ω , = (cid:68)(cid:0) Tr A (cid:1) (cid:69) DR,c , ω , = 15 (cid:10) Tr A Tr( X a ) (cid:11) DR,c ,ω , = 14 (cid:68) Tr A Tr ¯ X ρ ˙ ρ X ρ ˙ ρ (cid:69) DR,c , ω , = NN f (cid:10) Tr A tr ¯Φ ρ Φ ρ (cid:11) DR,c ,ω , = 116 (cid:28)(cid:16) Tr ¯ X ρ ˙ ρ X ρ ˙ ρ (cid:17) (cid:29) DR,c , ω , = N N f (cid:68) Tr ¯ X ρ ˙ ρ X ρ ˙ ρ tr ¯Φ σ Φ σ (cid:69) DR,c ,ω , = NN f (cid:68)(cid:0) tr ¯Φ ρ Φ ρ (cid:1) (cid:69) DR,c , ω , = NN f (cid:68)(cid:0) tr ¯Φ ρ X Φ ρ (cid:1) (cid:69) DR,c , – 15 – , , = N N f (cid:68)(cid:0) Tr A (tr ¯Φ ρ X Φ ρ (cid:1) (cid:69) DR,c ,ω , , = N N f (cid:68)(cid:0) Tr( X ) (tr ¯Φ ρ X Φ ρ (cid:1) (cid:69) DR,c ,ω , , = N N f (cid:68) Tr ¯ X ρ ˙ ρ X ρ ˙ ρ (cid:0) tr ¯Φ σ X Φ σ (cid:1) (cid:69) DR,c ,ω , , = N N f (cid:68)(cid:0) tr ¯Φ ρ Φ ρ (tr ¯Φ σ X Φ σ (cid:1) (cid:69) DR,c . (4.2)In terms of the Ξ i the observables of the full BD model (2.3) become: E = 34 β − (cid:26) (cid:18) − N (cid:19) + 4 N f N (cid:27) + β Ξ + O ( β ) , (cid:104) R (cid:105) = β − Ξ + β Ξ + O ( β ) , (cid:104) P (cid:105) = 1 − β (cid:34) Ξ − β Ξ + O ( β ) (cid:35) , (cid:104) r (cid:105) = β − Ξ + β (cid:18)
12 + Ξ N f (cid:19) + O ( β ) , (cid:104)C m (cid:105) = β − Ξ + β (Ξ + Ξ ) + O ( β ) . (4.3)For the bosonic BD model we have E bos = 34 β − (cid:26) (cid:18) − N (cid:19) + 4 N f N (cid:27) + β Ξ bos1 + O ( β ) , (cid:104) R (cid:105) bos = β − Ξ + β Ξ bos3 + O ( β ) , (cid:104) P (cid:105) bos = 1 − β (cid:34) Ξ − β Ξ bos5 + O ( β ) (cid:35) , (cid:104) r (cid:105) bos = β − Ξ + β N f Ξ bos7 + O ( β ) , (cid:104)C m (cid:105) bos = β − Ξ + β (Ξ bos7 + Ξ bos8 ) + O ( β ) . (4.4)The observables of interest for the high-temperature expansion are all expressed in terms ofΞ i and Ξ bos i listed above. As discussed they are temperature independent and depend onlyon N , the matrix dimension of the BFSS fields and N f , the number of flavour multiplets.We computed their values for a range of N and N f by hybrid Monte Carlo simulation withthe action S given in (3.5). We tabulate our results for different N and N f . We choose N = 4 , , , , , , , , , ,
32 for N f = 1 and tabulate ω ’s, the building blocks of Ξ i ,in Table 1.From the results of Table 1 we extrapolate the N -dependence of the ω ’s by fitting themwith a function , a + b/N + c/N (see Figure 3 and 4). The limiting extrapolated values are Note that as expected we find it necessary to include a linear fall of in 1 /N for large N . This is in contrastto the BFSS model, where the fall off is 1 /N – 16 –ncluded as the row N = ∞ in Table 1.Ξ i =1 , ··· , , and10 naturally reduce to counterparts in the BFSS model when the fundamentalfields are removed. We extrapolate Ξ i =1 , ··· , , and10 for N f = 1 , , , , , , , ,
16 for fixed N = 12 , , ,
18 and 20 to N f = 0 and find good agreement, to within the quoted errors,with the measured values for their BFSS counterparts as quoted in [6] .Figure 5 shows plots of the ω ’s against N f for each N and we fit the dependence on N f basically with a quartic polynomial. However, we find that higher order terms contribute forsome ω ’s and by using the fitting function a + bN f + ce dN f we capture the dependence on N f over the range considered. We are in the process of making a direct comparison of both the high-temperature regimeof the BD model as determined by the above predictions and the low-temperature regimeas predicted by gauge/gravity with results from a rational hybrid Monte Carlo simulationusing the code used in [15]. We will present those results in a separate paper as, apart fromtheir value as a check on the code and the computations presented here, they have additionalphysics that merits a separate discussion.For this paper we restrict our considerations to a comparison of the results obtainedhere with those obtained from the bosonic Berkooz-Douglas model, whose Euclidean actionis given by S bos = N (cid:90) β dτ (cid:34) Tr (cid:18) D τ X a D τ X a + 12 D τ ¯ X ρ ˙ ρ D τ X ρ ˙ ρ −
14 [ X a , X b ] + 12 [ X a , ¯ X ρ ˙ ρ ][ X a , X ρ ˙ ρ ] (cid:19) + tr (cid:0) D τ ¯Φ ρ D τ Φ ρ + ¯Φ ρ ( X a − m a ) Φ ρ (cid:1) + 12 Tr (cid:88) A =1 D A D A (cid:35) . (5.1)Our comparison is presented in Figure 1, where we restrict our considerations to a highprecision test with N = 10 and N f = 1. As one can see from the figure the agreementis excellent. Furthermore, the high T expansion remains valid at temperatures as low as T ∼ .
0. Below this temperature the figure shows evidence of a phase transition. This is thephase transition of the bosonic BFSS model.From studies of the bosonic BFSS model [10, 20, 21] we know that it undergoes twophase transitions at T c = 0 . ± .
002 and T c = 0 . ± . A , which at high temperature behaves as one of the X i , while at low temperatureit effectively disappears from the system and can be gauged away at zero temperature. Asthe temperature is increased through T c there is a deconfining phase transition with thesymmetry A ( t ) → A ( t ) + α broken and the distribution of eigenvalues of the holonomy becomes non-uniform. When the temperature reaches T c the spectrum of the holonomybecomes gapped and above this temperature the eigenvalues no longer cover the entire [0 , π ] The Polyakov loop, P = N Tr( U ), where U is the holonomy. – 17 – ! r " bo s ! ! c " ! m Figure 1 : Comparison of the high-temperature predictions for the fundamental observable (cid:104) r (cid:105) bos and the derivative of the condensate at zero mass, ( ∂c/∂m ) = N (cid:104)C m (cid:105) bos , with aMonte Carlo simulation of the bosonic BD model. The simulation is for N f = 1 and N = 10.range. In the low-temperature phase the bosonic BFSS model becomes a set of massivegaussian matrix models with Euclidean action S eff = N (cid:90) β dτ Tr (cid:18)
12 ( D τ X i ) + m A ( X i ) (cid:19) , (5.2)with the mass m A = 1 . ± . X i becomemassive they induce a mass for the fundamental scalars and the induced bare mass for theseis estimated by integrating out the adjoint fields and expanding it to quadratic order in Φ ρ .This gives a mass m f = (cid:113) m A ∼ . m f , in which case (cid:104) r (cid:105) bos = (cid:28) βN f (cid:90) dτ tr ¯Φ ρ Φ ρ (cid:29) bos (cid:39) m f . (5.3)Note that the right-hand side of equation (5.3) is independent of β and from Figure 1 wesee that (cid:104) r (cid:105) bos is more or less constant below the transition. A direct measurement of theexpectation value (5.3) at T = 0 . . (cid:39) m f , which gives the estimate m f (cid:39) . β isnow just the length of the time circle and not an inverse temperature. Because the SO (9)symmetry of the bosonic BFSS model is broken down to SO (5) × SO (4) there are now two– 18 – ! t r " ! Α " ! Α " t $ ! T r " X " X " t $ Figure 2 : Plots of the Green’s functions equations (5.4) and (5.5) for β = 10, Λ = 144, N = 10 and N f = 1. The fits correspond to m f = 1 .
461 and m lA = 2 . m lA , for the four X ρ ˙ ρ and a transverse mass, m tA , forthe five matrices X a . In figure 2 we present results for the Green’s functions: (cid:104) tr ¯Φ ρ (0)Φ ρ ( τ ) (cid:105) = N f m f e − m f τ + e − m f ( β − τ ) − e − β m f , (5.4) (cid:104) Tr X (0) X ( τ ) (cid:105) = N m lA e − m lA τ + e − m lA ( β − τ ) − e − β m lA , (5.5)where we have chosen the last of the four SO (4) adjoint scalars. We have also measuredthe longitudinal mass m lA by measuring the correlator for X defined similarly to (5.5). Theresults for β = 10, Λ = 144, N = 10 and N f = 1 are m lA = 2 . ± . m tA = 1 . ± . m f = 1 . ± . (cid:104) R (cid:105) bos (cid:39) m tA + 42 m lA = 2 . ± . , (5.6)which agrees well with the direct measurement where we find (cid:104) R (cid:105) bos = 2 . ± . m f using (5.3) predicts that (cid:104) r (cid:105) bos = 0 . ± . m f is very close to the one obtained from equation(5.3). Also the slightly different values of the adjoint masses m tA and m lA from the purelyBFSS case considered in equation (5.2) reflect the presence of backreaction at N f /N = 0 . (cid:104)C m (cid:105) bos at zero temperature.Assuming that both X a and Φ ρ are well approximated by massive gaussians and using Wick’s– 19 –heorem on C m bos = 2 β (cid:90) β dτ tr ¯Φ ρ Φ ρ − N β (cid:18)(cid:90) β dτ tr (cid:8) ¯Φ ρ X a Φ ρ (cid:9)(cid:19) (5.7)to perform the contractions, we obtain (cid:104)C m (cid:105) bos (cid:12)(cid:12)(cid:12) T =0 = 2 N f m f − N f m f m tA (2 m f + m tA ) = 1 . ± . . (5.8)Finally, a direct measurement of the measured condensate shown in Figure 1 for T =0 . , . , . T = 0 . T = 0 gives (cid:104)C m (cid:105) bos (cid:12)(cid:12) T =0 = 1 . ± . We have obtained the first two terms in the high-temperature series expansion for the Berkooz-Douglas model (BD model) for general adjoint matrix size, N and fundamental multipletdimension, N f . These results should prove useful for future studies of this model. The modelis an ideal testing ground for many ideas of gauge/gravity duality. The system is stronglycoupled at low temperature while at high temperature it is weakly coupled, aside from theMatsubara zero-modes, which remain strongly coupled even at high temperature. It is thesemodes that provide the residual non-perturbative aspect of the current study. Their effectcan be captured in numerical coefficients that depend only on N and N f .Once the coefficients are determined and tabulated (see appendix A) they can be usedas input for the high-temperature expansion of the observables of the BD model. We havechecked these coefficients by comparing with a high precision simulation of the bosonic versionof the BD model which we simulated using the Hybrid Monte Carlo approach. The coefficientsin the high-temperature expansion of the bosonic model’s observables are similarly determinedby the tables presented in appendix A. In fact the observable (cid:104) r (cid:105) bos (see equation (2.8)) andmass susceptibility (5.7) of the model, shown in Figure 1, show that the agreement is excellenteven down to temperature one. Below this temperature the system undergoes a set of phasetransitions. These are essentially the two phase transitions of the bosonic BFSS model.We found that for N f /N = 0 . m tA = 1 . ± .
003 while the longitudinal mass was lifted to m lA = 2 . ± . m f = 1 . ± . S . It is the bosonic sector of the dimensional reduction of theBD model to zero dimensions and is equivalent to a flavoured version of the bosonic sectorof the IKKT model. For this reason we refer to the model as the flavoured bosonic IKKTmodel. The potential is always positive semi-definite and the Higgs branch of its zero-locusis isomorphic to the instanton moduli space [16].There was some evidence for peculiar behaviour in the zero mode model for N f ≥ N . Wefound that simulations required significant fine tuning for N f ≥ N , in that when using thesame leapfrog step length which gave 95% acceptance rate for N f = 2 N − N f ≥ N fell to a fraction of this within a couple of thousand sweeps and Wardidentities we use as checks on the simulations were not fulfilled. After tuning the simulationwe found the generated configurations had very long auto-correlation time. Also, in fittingthe dependence of the observables Ξ i on N for a given N f we found evidence for a simple poleat N = 2 N f . Furthermore, one can see from the results tabulated in appendix A that theygrow rapidly when the region N f = 2 N is approached. We expect that these difficulties andthe growth of observables as N f = 2 N is approached are related to the singular structure ofthe instanton moduli space, i.e. the minimum of the potential in (2.3) with X a = 0, D A = 0.We have not pursued this further in the current study as it would take us too far afield,however, we believe it merits further attention.Finally, our preliminary studies of the supersymmetric BD model show [23] that, for someobservables, the high-temperature series expansion remains valid to lower temperatures thanone might expect. This validity of the high T expansion at lower T could provide alternativequasi-analytic estimates for observables in the window where gauge/gravity duality is valid. Acknowledgment
D.O’C. thanks Stefano Kovacs and Charles Nash for helpful discussions on the ADHM con-struction. Support from Action MP1405 QSPACE of the COST foundation is gratefullyacknowledged. – 21 –
Tables for the ω ’s. In this appendix we gather the numerical data from Monte Carlo simulations for differentmatrix sizes, N and different numbers of flavour multiplets N f and present it in tabular form. N f = 1 N ω ω ω ω ω , ω , ω , ω , ∞ N f = 1 N ω , ω , ω , ω , ω , , ω , , ω , , ω , , ∞ Table 1 : Mean values of the ω ’s for N f = 1 were obtained from 3 × Monte Carlosamples for all values of N but 10 (3 × samples) and 20 ,
32 (6 × samples). Errorsare estimated with the Jackknife resampling. N = ∞ values are the one extrapolated byquadratic polynomials: a + b/N + c/N . The quoted errors of this extrapolation are thefitting errors of the parameter a . – 22 –n the remaining tables we tabulate fixed N = 9 , , , ,
18 and 20 while we vary N f .Mean values of observable ω i , ω i,j and ω i,j,k ( i, j, k = 1 , ...,
6) were obtained from 3 × samples generated by hybrid Monte Carlo simulation of flavoured bosonic IKKT model withthe action specified in (3.5). Errors are estimated with the Jackknife resampling. N = 9 N f ω ω ω ω ω , ω , ω , ω , N = 9 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , N = 12 N f ω ω ω ω ω , ω , ω , ω , – 23 – = 12 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , N = 14 N f ω ω ω ω ω , ω , ω , ω , N = 14 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , – 24 – = 16 N f ω ω ω ω ω , ω , ω , ω , N = 16 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , N = 18 N f ω ω ω ω ω , ω , ω , ω , – 25 – = 18 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , N = 20 N f ω ω ω ω ω , ω , ω , ω , N = 20 N f ω , ω , ω , ω , ω , , ω , , ω , , ω , , Table 2 : The tables for N = 9 to N = 20 were obtained from Monte Carlo simulations with3 × samples and errors are estimated using Jackknife resampling.– 26 – .00 0.05 0.10 0.15 0.20 0.250.220.230.240.250.26 1 / N 〈 ω 〉 / N 〈 ω 〉 / N 〈 ω 〉 / N 〈 ω 〉 / N 〈 ω , 〉 - - - / N 〈 ω , 〉 - - - / N 〈 ω , 〉 - - - - - - / N 〈 ω , 〉 / N 〈 ω , 〉 / N 〈 ω , 〉 / N 〈 ω , 〉 / N 〈 ω , 〉 - - - / N 〈 ω , , 〉 / N 〈 ω , , 〉 - - / N 〈 ω , , 〉 / N 〈 ω , , 〉 Figure 3 : Mean values of the ω ’s plotted against N with N f = 1. Dashed lines correspondto fits of the form a + b/N + c/N while vertical lines correspond to N → ∞ values obtainedfrom those fits. Errors are estimated with the Jackknife resampling. Its values are quite smalland it is determined rather precisely in the tables.– 27 – .00 0.05 0.10 0.15 0.20 0.25 - / N 〈 ω 〉 , 〈 ω 〉 , 〈 ω / 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , 〈 ω 〉 , Figure 4 : Mean values of the ω ’s plotted against N with N f = 1. Dashed lines correspondto fits of the form a + b/N . Errors are estimated with the Jackknife resampling.– 28 – 〈 ω 〉 〈 ω 〉 〈 ω 〉 〈 ω 〉 〈 ω , 〉 - - - - - 〈 ω , 〉 - - - - - - - - 〈 ω , 〉 - - - - - - - 〈 ω , 〉 〈 ω , 〉 〈 ω , 〉 〈 ω , 〉 〈 ω , 〉 - - 〈 ω , , 〉 〈 ω , , 〉 - 〈 ω , , 〉 〈 ω , , 〉 N =
12, N =
14, N =
16, N =
18, N = Figure 5 : Mean values of the ω ’s plotted against N f for different values of N . Dashed linescorrespond to either fits of the form a + bN f , a + bN f + cN f + dN f + eN f or a + bN f + ce dN f . B The high-temperature behaviour of energy E , Polyakov loop (cid:104) P (cid:105) , (cid:104) R (cid:105) and mass susceptibility (cid:104)C m (cid:105) for the supersymmetric model. In this appendix we graphically present the high-temperature predictions for the BD-modelobservables the energy E , the Polyakov loop (cid:104) P (cid:105) , the extent of the eigenvalues of the adjointfields X i given by (cid:104) R (cid:105) and the mass susceptibility (cid:104)C m (cid:105) . Figure 6 shows the predictedhigh-temperature behaviour of the BD-model observables.– 29 – E 〈 P 〉 〈 R 〉 〈 r 〉 ∂ c a ∂ m a Figure 6 : Temperature dependence of physical observables for the supersymmetric BD modelas defined in (4.3) and with the values of ω ’s from table 1. The solid line is the leading orderprediction for N = ∞ , N f = 1 while the long dashed line is up to the next to leading orderfor N = ∞ , N f = 1. The third curve with short dashes is N = 10, N f = 1. Note that incontrast to the bosonic model the high-temperature dependence of the Polyakov loop turnsupwards, as T decreases, between T = 1 . .
0. This indicates that the high-temperatureseries for (cid:104) P (cid:105) is not reliable in this region. – 30 – E 〈 P 〉 〈 R 〉 〈 r 〉 ∂ c a ∂ m a N = N f =
1, 2, 4, 6, 8, 10, 12, 14, 16
Figure 7 : Temperature dependence of physical observables as defined in (4.3) with ω ’s fromtable 2 for N = 20 and different values of N f .– 31 – E 〈 P 〉 〈 R 〉 〈 r 〉 ∂ c a ∂ m a N f =
1, N =
4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 32
Figure 8 : Temperature dependence of physical observables of the supersymmetric model asdefined in (4.3) with ω ’s from table 1 for N f = 1 with different values of N .– 32 – E , N = E , N = E , N = E , N = E , N = E , N = N f =
1, 2, 4, 6, 8, 10, 12, 14, 16
Figure 9 : Dependence of the energy on the temperature for the supersymmetric model asdefined in (4.3) for N = 9 , , , , ,
20 with different values of N f . Note that for eachvalue of N the curves (approximately) intersect at a crossing temperature T x . At this pointthe energy is essentially independent of N f . Extrapolating the crossing value to large N wefind T x = 0 . ± .
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