Notes on melonic O(N)q−1 tensor models

Journal of High Energy Physics, Jun 2018

Abstract It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N)q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N)q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has \( \left(q-1\right)\frac{N^2}{2} \) new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N2. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ln N. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.

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Notes on melonic O(N)q−1 tensor models

HJE q 1 tensor models Sayantan Choudhury 0 1 2 3 Anshuman Dey 0 1 2 3 Indranil Halder 0 1 2 3 Lavneet Janagal 0 1 2 3 Shiraz Minwalla 0 1 2 3 Rohan R. Poojary 0 1 2 3 0 Am Muhlenberg 1 , 14476 Potsdam-Golm , Germany 1 Max Planck Institute for Gravitational Physics, Albert Einstein Institute 2 Homi Bhabha Rd , Mumbai 400005 , India 3 Department of Theoretical Physics, Tata Institute of Fundamental Research It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N )q 1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N )q 1 theories that di er from their SYK counterparts. We argue that the spectrum of uctuations about the nite temperature saddle point in these theories has (q addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories di er signi cantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the e ective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2. The canonical partition function of the model displays a decon nement or Hawking Page type phase transition at temperatures of order 1= ln N . We derive these results in the large mass limit but argue that they are 1) N22 new light modes in qualitatively robust to small corrections in J=m. 1/N Expansion; Black Holes in String Theory; Gauge-gravity correspondence - 1 Introduction 2 3 1.1 1.2 els 2.1 2.2 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5 4.6 Free Greens function Level zero: free theory Level one: single melon graphs Level 2: 2 melon graphs The in nite sum H0 Thermodynamics Holonomy dynamics and density of states at large mass Classical e ective action E ective action Scaling limit Determination of saddle points Thermodynamics in the canonical ensemble Thermodynamics in the microcanonical ensemble 3.4.1 3.4.2 3.4.3 3.4.4 The saddle at u = 0 The wavy phase The gapped phase Entropy as a function of energy for E N2 2 4 The holonomy e ective action with weak interactions New light modes Holonomy dynamics and the spectrum at large mass Light thermal modes of the Gurau-Witten-Klebanov-Tarnopolsky mod5 Discussion A Conformal kernel B Details of the perturbative computations B.1 Leading power of B.1.1 Two melon graphs B.1.2 n melon graphs B.2 All powers of in a circle diagram B.2.1 Evaluating the integral B.3 Evaluating the subleading correction C The holonomy e ective action from the sigma model { 1 { Introduction It has recently been demonstrated that the dynamically rich Sachdev-Ye-Kitaev model | a quantum mechanical model of fermions interacting with random potentials | is solvable at large N [1{3]. This model is interesting partly because its thermal properties have several features in common with those of black holes. The SYK model self equilibrates over a time scale of order the inverse temperature and has a Lyapunov index that saturates the chaos bound [2, 3]. Moreover the long time behaviour of this model at nite temperature is governed by an e ective action that has been reinterpreted as a particular theory of gravity expanded about AdS2 background solution [1, 4{10]. These facts have motivated the suggestion that the SYK model is the boundary dual of a highly curved bulk gravitational theory whose nite temperature behaviour is dominated by a black hole saddle point. If this suggestion turns out to be correct, the solvability of the SYK model at large N | and its relative simplicity even at nite N | could allow one to probe old mysteries of black hole physics in a manner that is nonperturbative in N1 , the e ective dual gravitational coupling (see e.g. [11{13] for recent progress). There is, however, a potential y in the ointment. While the SYK model | de ned as a theory with random couplings | is an average over quantum systems, it is not a quantum system by itself. One cannot, for instance, associate the SYK model with a Hilbert space in any completely precise manner, or nd a unitary operator that generates time evolution in this model. As several of the deepest puzzles of black hole physics concern con icts with unitarity, this feature of the SYK model is a concern. Of course any particular realisation of the couplings drawn from the SYK ensemble is a genuine quantum theory. It is plausible that several observables | like the partition function | have the same large N limit when computed for any given typical member of the ensemble as they do for the SYK model de ned by averaging over couplings [12, 14, 15]. It might thus seem that every typical realization of random couplings is an inequivalent consistent quantization of classical large N SYK system. As the number of such quantizations is very large, this would be an embarrassment of riches. The potential issue here is that if we work with any given realization of the SYK model, it appears inconsistent to restrict attention to averaged observables for any nite N no matter how large. On the other hand correlators of individual i operators (as opposed to their averaged counterparts) presumably do not have a universal large N limit (and so are not exactly solvable even at large N ).1 In order to address these concerns some authors have recently [16{18] (based on earlier work [19{24]) studied a related class of models. These models are ordinary quantum mechanical systems; in fact they describe the global or gauged quantum mechanics of a collection of fermions in 0+1 dimensions. In this paper we will focus our attention on the model (1.1) 1We thank S. Sachdev for discussion on this point. S = Z dt X[ aD0 a NF a=1 aq + h:c:)]; N { 2 { that was rst proposed | at least in the current context | in [17]. In (1.1) a are a collection of complex gauged fermionic elds in 0 + 1 dimensions that transform in the fundamental of each of the q 1 copies of O(N ). The index a is a avour index that runs from 1 : : : NF . 2 J is a coupling constant with dimensions of mass and q is a schematic for a q vertex generalisation of a `tetrahedronal' interaction term between q copies of the fermionic elds, whose gauge index contraction structure is explained in detail in [16, 17] and will be elaborated on below. The tetrahedral structure of the interaction [16, 17] is such that for any even number of fermions q each fermion has q 1 indices each in a di erent O(N )(or U(N )). The indices among the q fermions are contracted such that every fermion is index contracted with an HJEP06(218)94 index of the same gauge group on one of the remaining fermions. Moreover, given any | and every | 2 fermions have a single index (of some gauge group) contracted between them. For q = 4 it is easy to check that these words de ne a unique contraction structure which may be viewed as a tetrahedral contraction among the 4 fermions each with q 1 = 3 indices(legs) with every fermion(point or vertex of the tetrahedron) connected to 3 di erent coloured legs. For q 6 it is not clear that the words above de ne a unique contraction structure. In case the contraction structure is not unique, we pick one choice | for example the Round-Robin scheduling process to de ne our interaction [25, 26].3 The connection between the quantum mechanical theories (1.1) and the SYK model itself is the following; it has been demonstrated (subject to certain caveats) that sum over Feynman graphs of the theory (1.1) coincides with the sum over Feynman graphs of the SYK model at at leading order at large N (see [16] for the argument in a very similar model), even though these two sums di er at nite values of N (see e.g. the recent paper [27] and references therein). It follows that the quantum mechanical models (1.1) are exactly as solvable as the SYK model at large N ; moreover they also inherit much of the dynamical richness of the SYK model. In other words the models (1.1) are solvable at large N , are unitary and are potentially boundary duals of (highly curved) black hole physics. Motivated by these considerations, in this note we study the e ective theory that governs the long time dynamics of the model (1.1) at nite temperature. We focus attention on dynamical aspects of (1.1) that have no counterpart in the already well studied dynamics In the rest of this introduction we will explain and describe our principal observations of the original SYK model.4 and results. 1.1 New light modes The thermal behaviour of both the theory (1.1) and the original SYK model is determined by the path integral of these theories on a circle of circumference . 2For this simplest case NF = 1 this model was presented in eq. 3.23 of [17]. 3We would like to thank J. Yoon for explaining the Round Robin scheduling process to us and clearing up our misconceptions about uniqueness of the contraction structure for q > 4. 4See [25, 28{37] for other recent work on the model (1.1) and its close relatives. { 3 { It was demonstrated in [2, 3] that, in the case of the original SYK model, this path integral is dominated by a saddle point of an e ective action whose elds are the two point function and self energy of the fermions. An extremization of this e ective action determines both the fermionic two point function at nite temperature as well as the free energy of the system at leading order at large N . In a similar manner, the thermal behaviour of the quantum mechanical systems (1.1) is dominated by a saddle point at large N . Under appropriate assumptions it may be shown that resultant e ective action has the same minimum as that of the original SYK theory [16].5 Specialising to the case NF = 1, the leading order fermionic two point function of the quantum mechanical system is given by h a(t) b(t0)i = baGSY K (t t0); (1.2) where a and b denote the (collection of) vector indices for the fermions and GSY K (t) is the thermal propagator of the original SYK model.6 While the thermal behaviour of the model (1.1) is thus indistinguishable from that of the SYK model at leading order in the large N limit, the dynamics of the quantum mechanical model (1.1) di ers from that of the SYK model at subleading orders in 1=N . The rst correction to leading large N thermal behaviour may be obtained by performing a one loop path integral over quadratic uctuations around the saddle point. In the long time limit, correlators are dominated by the lightest uctuations around the saddle point. Recall that in the UV (i.e. as J ! 0) the fermions of (1.1) have dimension zero. The term proportional to q in (1.1) represents a dimension zero relevant deformation of this UV xed point. The resultant RG ow ends in the deep IR in a conformal eld theory in when which the fermions have dimension 1q [2, 3]. In this IR limit (relevant to thermodynamics J ! 1) q is marginal while the kinetic term in (1.1) is irrelevant [2, 3]. The fact that the kinetic term is irrelevant in the IR | and so can e ectively be ignored in analysing the symmetries of (1.1) at large J | has important implications for the structure of light uctuations about the thermal saddle point. The rst implication of the irrelevance of the kinetic term occurs already in the SYK model and was explored in detail in [2, 3, 6]. The main point is that the action (1.1), with the kinetic term omitted, enjoys invariance under conformal di eomorphisms (i.e. di eomorphisms together with a Weyl transformation). However the saddle point solution 5A potential subtlety is that path integral of the quantum mechanical system (1.1) has a degree of freedom that is absent in the original SYK model, namely the holonomy of the gauge group O(N )q 1 . As for the SYK model, integrating out the fermions leads to an e ective action | proportional to N q 1 | whose elds are a two point function of the fermions, a self energy and the holonomy of the gauge group. As in the case of the original SYK model, at leading order in the large N limit the free energy of the system 1 it is highly plausible that this e ective action is minimised when the holonomy is the identity matrix (see 3 below). Under this assumption the saddle point of the quantum mechanical system coincides with that of the SYK model. 6Eq. (1.2) applies both the the case that the group O(N )q 1 is global and local. In the latter case this equation applies in the gauge @0A0 = 0. Assuming that the holonomy degree of freedom is frozen to identity at large N , the gauged and global model coincide. { 4 { for the Greens function GSYK(t) is not invariant under conformal di eomorphisms. It follows immediately that the action of in nitesimal conformal di eomorphisms on this solution generates zero modes in the extreme low energy limit. At any nite temperature, no matter how small, the kinetic term in (1.1) cannot completely be ignored and conformal invariance is broken; the action of conformal di eomorphisms on the SYK saddle point consequently produces anomalously light (rather than exactly zero) modes. The action for these modes was computed in [2, 3, 6] and takes the form of the Schwarzian for the conformal di eomorphisms. A very similar line of reasoning leads to the conclusion that the model (1.1) has (q 1) N22 additional light modes in the large J limit, as we now explain. Let us continue to work in the gauge A0 = 0. In this gauge the action (1.1) is obviously invariant under the global rotations ! V , ! V y where V is an arbitrary time independent O(N )q 1 rotation. In the global model (1.1) the rotation by V is the action of a global symmetry. In gauged model on the other hand, these rotations are part of the gauge group and do not generate global symmetries of our model; the Gauss law in the theory ensures that all physical states are uncharged under this symmetry. Let us now consider the transformation ! V (t) together with ! V (t)y where V (t) is an arbitrary time dependent O(N )q 1 rotation. In the case of the gauged models, this transformation is not accompanied by a change in A0 (A0 = const throughout) so the rotation is not a gauge transformation. At nite J the rotation by a time dependent V (t) is not a symmetry of the action (1.1) in either the global or the gauged theory as the kinetic term in (1.1) is not left invariant by this transformation. As we have explained above, however, the kinetic term is irrelevant in the low temperature limit J ! 1. It follows that the time dependent transformation is an e ective symmetry of dynamics this strict low temperature limit. However the saddle point solution (1.2) is clearly not invariant under the time dependent rotations by V (t). It follows that, as in the discussion for conformal di eomorphisms above, the action of V (t) on (1.2) generates exact zero modes in the strict limit J ! 1 and anomalously light modes at any nite J . We emphasise that this discussion applies both to the global model where O(N )q 1 is a global symmetry, and the gauged model where it is not. In section 2.2 below we argue that the dynamics of our new light modes is governed by the e ective sigma model on the group manifold S = A 2# ; (1.3) where Vl(t) is an arbitrary element of the group O(N ) and A is a number of order unity that we have not been able to determine. The formula (1.3) has appeared before in a closely related context. The authors of [38] (see also [14]) studied the a complex version of the SYK model. Their model had an exact U( 1 ) symmetry at all energies, which | using the arguments presented in the previous paragraphs | was approximately enhanced to a local U( 1 ) symmetry at low energies. The { 5 { authors of [38] argued the long distance dynamics of the new light modes is governed by a sigma model on the group manifold U( 1 ).7 Given these results, the appearance of a low energy sigma model in the large J nite temperature dynamics of the theory (1.1) seems natural. We would, however, like to emphasise two qualitative di erences between the sigma model (1.3) and the model that appeared in [38]. First (1.3) is a sigma model for a group O(N )q 1 whose dimensionality goes to in nity in the large N limit, N ! 1. Second that we nd the new light modes of the action even of the gauged model (1.1) even though O(N )q 1 is not a global symmetry of this theory. The new modes governed by (1.3) are approximately as light | and so potentially as important to long time dynamics | as the conformal di eomorphisms described above. Note, however, that there are (q as we can tell) only one conformal di eomorphism. 1) N22 light time dependent O(N )q 1 modes but (as far We have already remarked above that the light di eomorphism degree of freedom described above has been given an interpretation as a particular gravitational action in an AdS2 background. It seems likely to us that the e ective action (1.3) will, in a similar way, admit a bulk interpretation as a gauge eld propagating in AdS2. The Yang Mills coupling of this gauge eld | like Newton's constant for the gravitational mode | will be of order 1 Nq 1 (this is simply a re ection of the fact that our model has N q 1 degrees of freedom). This means that the t' Hooft coupling of all the gauge elds in the bulk will be of order the bulk gauge elds are classical even though there are so many of them.8 2 gY M N Nq1 2 . The fact that this coupling goes to zero in the large N limit implies that It has been established that the light di eomorphism degree of freedom has a qualitatively important e ect on out of time ordered thermal correlators; it leads to exponential growth in such correlators at a rate that saturates the chaos bound G e2 T t. When we include the contribution of the new light modes described in this subsection, we expect this growth formula to be modi ed to9 G(t) e2 T t + N 2f (t) : (1.4) The factor of N 2 is a re ection of the fact that our new modes are N 2 in number, whereas | as far as we can tell | there is only a single light mode corresponding to conformal di eomorphisms. Given that the solutions of the equations of motion to the Sigma model (1.3) grow no faster than linearly in time, we expect f (t) to grow at most polynomial in time. This suggests it that the light modes (1.3) will dominate correlators up to a time of order 1T ln N . At later times the exponentially growing di eomorphism mode will dominate, leading to exponential growth and a Lyapunov index that saturates the chaos bound. To end this subsection let us return to a slightly subtle point in our discussion. In order to derive the e ective action for V (t) we worked in the gauge A0 = 0. As our theory is on a thermal circle, in the case of the gauged model (1.1) we have missed a degree of freedom | 7They also argued for some mixing between the di eomorphism and U( 1 ) long distance modes. 8We would like to thank J. Maldacena for a discussion of this point. 9See [26] for related work. { 6 { the gauge holonomy | by working in the gauge A0 = 0. This, however, is easily corrected for. Even in the presence of a holonomy, we can set the gauge eld A0 to zero by a gauge transformation provided we allow ourselves to work with gauge transformations that are not single valued on the circle. The net e ect of working with such a gauge transformation is that the matter elds are no longer periodic around the thermal circle but obey the boundary conditions where U is the holonomy around the thermal circle. For the elds of the low energy e ective action (1.3) this implies the boundary conditions HJEP06(218)94 ( ) = U (0); V ( ) = U V (0)U 1 : Recall we are instructed to integrate over all values of the holonomy U . Consequently we must integrate over the boundary conditions (1.6) with the Haar measure. See section C for some discussion of this point. In summary, the discussion of this subsection suggests that the bulk low energy e ective action `dual' to the gauged/global quantum mechanics of (1.1) di ers from the low energy e ective action `dual' to the SYK model in an important way; in addition to the gravitational eld it contains gauge elds of a gauge group whose rank is a positive fractional power of the inverse Newton (and Yang Mills) coupling constant of the theory. In the classical limit in which Newton's constant is taken to zero, the rank of the low energy gauge elds also diverges. Nonetheless the limits are taken in such a way that the e ective bulk theory remains classical. 1.2 Holonomy dynamics and the spectrum at large mass Our discussion up to this point has applied equally to the `global' and `gauged' quantum mechanical models (1.1). In the rest of this introduction we focus attention on the gauged models, i.e. the models in which the O(N )q 1 symmetry algebra is gauged. In this case the thermal path integral of our system includes an integral over gauge holonomies over the thermal circle. We wish to study the e ect of this holonomy integral on the dynamics of our system. O(N )q 1 model In order to do this in the simplest and clearest possible way we deform the model (1.1) in a way that trivializes the dynamics of all non holonomy modes in the theory. This is accomplished by adding a mass to the fermions. For concreteness we work with the S = Z NF dt X a=1 where m, the mass of the fermion is taken to be positive.10 We work the large mass limit, i.e. the limit mJ 1. The e ective interaction between fermions in (1.7), mJ , is small in this 10In the case that the mass is negative, most of our formulae below go through once under the replacement m ! jmj. { 7 { (1.5) (1.6) (1.7) holonomy U .11 limit and can be handled perturbatively. In the strict m ! 1 limit the only interaction that survives in the system is that between the (otherwise free) matter elds and the Let us rst work in the strict limit mJ ! 1. In this limit the dynamics of the holonomy eld U in this theory is governed by an e ective action obtained by integrating out the matter elds at one loop.12 The resultant e ective action is easily obtained and is given Z q 1 i=1 Z = Tr x mH = Y dUi exp( Se (Ui)); Se (Ui) = NF x = e jmj; X n=1 1 ( x)n Qiq=11 Tr Uin n ; (1.8) where H is the Hamiltonian of our theory.13 Each Ui is an O(N ) matrix that represents the holonomy in the ith factor in the gauge group O(N )q 1. dU is the Haar measure over the group O(N )q 1 normalized so that the total group volume is unity. Notice that when x is of order unity, Se N q 1 in (1.8). On the other hand the contribution of the group measure to the `e ective' action is of order N 2. The integral in (1.8) is interesting when these two contributions are comparable. This is the case if we scale temperatures so that with held xed as N is taken to in nity. In this limit the terms in the second of (1.8) with n > 1 are subleading and can be ignored. E ectively x = e jmj = NF N q 3 ; Z(x) = Se = Z q 1 i=1 N q 3 Y dUi exp( Se (Ui)) 11We emphasize that, in the limit under consideration, modes corresponding to di eomorphisms or V (t) are no longer light | and so are irrelevant. However the holonomy continues to be potentially important. 12For orientation, we remind the reader that the integral over the holonomy is the device the path integral uses to ensure that the partition function only counts those states that obey the A0 equation of motion, i.e. the Gauss law constraint. Restated, the integral over holonomies ensures that the partition function only counts those states in the matter Hilbert space that are singlets under the gauge group. 13The generalization of these results to a model with NB bosons and NF fermions yields the holonomy e ective action n=1 Se (Ui) = X(NB + ( 1 )n+1NF ) xn Qiq=11 Tr Uin : 1 n As we will see below, in the scaling limit of interest to this paper, only the term with n = 1 is important. In the strictly free limit it follows that most the results presented above apply also to a theory with NF fermions and NB bosons once we make the replacement NF ! NF + NB. { 8 { In the large N limit the matrix integral (1.11) is equivalent | as we show below | to the well known Gross Witten Wadia model and is easily solved. The solution | presented in detail below | has the following features 1. In the canonical ensemble, the partition function undergoes a decon nement type phase transition at = 1pt where the value of 1pt is given in (3.19). At smaller the system is dominated by the `con ning' saddle point in which U is the clock matrix. At larger values of 1pt the system is dominated by a more complicated `decon ned' or black hole saddle point. The phase transition is reminiscent of the transitions described in [40, 41].14 2. In the microcanonical ensemble, the scaling limit described above captures the density of states of the system at energies less than or of order N 2. Over the range of energies 1 E < N42 , the entropy S is given by the simple formula S(E) = (q 3) + E log NF + (q 3) ln(2): (1.12) E 2 ln E 2 E 2 E 2 The saddle point that governs the density of states of the theory changes in a non analytic manner at E = N42 . For E > N2 the formula for the entropy is more 4 complicated. For energies E (q 2) N42 , however, the entropy simpli es to the formula for nBN q 1 complex bosonic and nF N q 1 free complex fermionic harmonic The complicated formula that interpolates between these special results is presented The formula (1.12) suggests that if a dual bulk interpretation of the theory (1.8) exists, it is given in terms of a collection of bulk elds whose number grows faster than exponentially with energy. It would be fascinating to nd a bulk theory with this unusual behaviour. Moreover the existence of a Hawking Page type phase transition in this model | and in particular the existence of a subdominant saddle point even at temperatures at which the dominant phase is a black hole phase | opens the possibility of the subdominant phase playing a role in e ectively unitarizing correlators about the black hole saddle point by putting a oor on the decay of the amplitude of correlators as in [42]. The results presented above apply only in the limit mJ ! 1. We have also investigated how these results are modi ed at very weak (rather than zero) coupling. We continue to work at low temperatures, in a manner we now describe in more detail. It turns that Se (U ) takes the schematic form 14We note that the rst order phase transitions described in [41] were strongly rst order (i.e. not on the edge between rst and second order) only after turning on gauge interactions. In the current context, in contrast, the phase transition in our system is strongly rst order even in the absence of interactions. { 9 { (1.14) Working to any given order in perturbation theory, the functions fa( ) are all polynomials of bounded degree in . We work at temperatures low enough so that we can truncate (1.14) to its rst term. In other words the terms we keep are all proportional to x multiplied by a polynomial dressing in . takes the form We demonstrate below that within this approximation the partition function (1.14) (1.15) (1.16) HJEP06(218)94 Note that (1.15) asserts that the interacting e ective action has the same dependence on x and U as its free counterpart did. The only di erence between the interacting and free e ective action is a prefactor which is a function of the two e ective couplings mJ and Jm2 . Below we have summed an in nite class of graphs and determined the function H~0. Working at NF = 1 we nd " 1 2 H~0 =2 + 2 (q) ( ) m jJ j2e (q) ( m ) jJj2 2 1 e2 (q) ( m ) jJj2 where (q) is de ned in (4.33). Eqs. (1.15) and (1.16) determine the e ective action of our system whenever the terms proportional to H~m (m = 1; 2 : : :) in the second line of (4.35) can be ignored compared to the term proportional to H~0. This is always the case at weak enough coupling; the precise condition on the coupling when this is the case depends on the nature of the as yet unknown large argument behaviour of the functions H~m. The partition function that follows from the action (1.15) is identical to the free partition function described above under the replacement ! H~0. It follows that the interacting partition function is essentially identical to the free one in the canonical ensemble. The dependence of the e ective value of leads to some di erences in the micorcanonical ensemble that turn out not to impact the main qualitative conclusions of the analysis of the free theory. For instance the super hagedorn growth of the entropy persists upon including the e ects of interaction. Note added. We have recently become aware of the preprint [43] which overlaps with this paper in multiple ways. We hope it will prove possible to combine the results of this paper with the methods of [43] to better understand the new light modes discussed earlier in this introduction. 2 Light thermal modes of the Gurau-Witten-Klebanov-Tarnopolsky models In this section we consider the Gurau-Witten-Klebanov-Tarnopolsky model at nite temperature. The Lagrangians for the speci c theories we study was listed in (1.1). As we have explained in the introduction, this model has a new set of light modes parameterized by V (t), an arbitrary group element as a function of time, where V belongs to O(N )q 1. In this section we will present an argument that suggests that the dynamics of these light modes is governed by a (quantum mechanical) sigma model on the group manifold. We will also present an estimate for the coupling constant of this sigma model. That the dynamics of V (t) should be governed by a sigma model is very plausible on general grounds. Recall that in the formal IR limit, V (t) is an exact zero mode of dynamics. It follows that V (t) picks up dynamics only because of corrections to extreme low energy dynamics. From the point of view of the low energy theory these corrections are UV e ects, and so should lead to a local action for V (t). The resultant action must be invariant under global shifts V (t) ! V0V (t). We are interested in the term in the action that will dominate long time physics, i.e. the action with this property that has the smallest number of time derivatives. Baring a dynamical coincidence (that sets the coe cient of an apparently allowed term to zero) the action will be that of the sigma model. In the rest of this section we will put some equations to these words. We would like to emphasise that the `derivation' of the sigma model action presented in this section is intuitive rather than rigorous | and should be taken to be an argument that makes our result highly plausible rather than certain. 2.1 Classical e ective action In [3] the e ective large N dynamics of the SYK model was recast as the classical dynamics of two e ective elds; the Greens function G(t) and the self energy (t). The action for and G derived in [3] was given by ~ ) + Z S = N q 1 dt1 dt2 ~ (t1; t2)G~(t2; t1) G~q(t1; t2) : (2.1) q The utility of the action (2.1) was twofold. First, the solutions to the equations of motion that follow from varying (2.1) are the saddle point that govern thermal physics of the SYK model. Second, an integral over the uctuations in (2.1) also correctly captures the leading order (in N1 ) correction to this saddle point result. In order to obtain these corrections, one simply integrates over the quadratic uctuations about this saddle point. In particular the action (2.1) was used to determine the action for the lightest uctuations about the saddle point (2.1), namely conformal di eomorphism [3]. In this section we wish to imitate the analysis of [3] to determine the action for uctuations of the new zero modes | associated with time dependent O(N )q 1 rotations | described in the introduction. The action (2.1) is not su cient for this purpose. As explained in the introduction, the low energy uctuations we wish to study are obtained by acting on the saddle point Greens function with time dependent O(N )q 1 rotations; however the elds G and that appear in (2.1) have no indices and so cannot be rotated. As the rst step in our analysis we proceed to generalise the e ective action (2.1) to an action whose variables are the matrices Gba and b a fundamental indices of the group O(N )q 1. Our generalised action is given by . The indices a and and b are both S = log P f (D0 ~ ) + Z dt1 dt2 ~ ab(t1; t2)G~ba(t2; t1) q jgj2 G~q(t1; t2) : (2.2) HJEP06(218)94 In this action, the expression G~q is a product of q copies of G~ba where all gauge indices are contracted in a manner we now describe. Recall that a and b are fundamental indices for the group O(N )q 1. Each of these indices may be thought of as a collection of q 1 fundamental indices a = (a1a2 : : : aq 1); b = (b1b2 : : : bq 1); where ai and bi are fundamental indices in the (ith factor of) O(N ). In the contraction G~q, a type indices are contracted with each other while b type indices are also contracted with each other | there is no cross contraction between a and b type indices. The structure of contractions is as follows; the a indices of precisely one of the O(N ) factors of the gauge group are contracted between any two (and every two) Gs and, simultaneously, the b indices of the same two O(N ) factors are also contracted between the same two G~s.15 As a quick check note that the total number of contraction of a (or b) indices, according to our rule, is the number of ways of choosing two objects from a group of q, or, q(q 1) . As each pair hit two indices, we see that the pairing rule described in this paragraph saturates the indices present q copies of G~ (there are a total of q(q 1) a type indices). 2 The contraction structure described for a type indices in the previous paragraph is precisely the contraction structure for the interaction term q in the action (1.1). We regard (2.2) as a phenomenological action with the following desirable properties. First it is manifestly invariant under global O(N )q 1 transformations. Second if we make the substitutions G~ba ! G~ ba, ~ ba ! ~ ba into (2.2) we recover the action (2.1). It follows in particular that, if G and denote the saddle point values of (2.1) then are saddle points of (2.2). This point can also be veri ed directly from the equations of motion that follow from varying (2.2), i.e. Gba = baG; ba = ba ; Gab(t1; t2) =((D0 ) 1)ab(t1; t2); ab(t1; t2) =jgj2 (Gq 1)ab(t1; t2): While (2.2) correctly reproduces nite temperature saddle point of the the model (1.1), it does not give us a weakly coupled description of arbitrary uctuations about this saddle point. The fact that (2.2) has N 2(q 1) elds makes the action very strongly coupled. The key assumption in this section | for which we will o er no detailed justi cation beyond its general plausibility | is that the action (2.2) can, however, be reliably used to obtain the e ective action for the very special manifold of con gurations described in the introduction, namely G~ba(t1; t2) =Vb b0 (t1)G(t1; t2)Vb0a(t2); ~ ba(t1; t2) =Vb b0 (t1) (t1; t2)Vb0a(t2); 15These rules have their origin in the generalized `tentrahedronal' contraction structure described in the introduction. For values of q at which the basic interaction structure has an ambiguity, we make one choice; for instance we adopt the `Round Robin' scheme to x the ambiguities. As far as we can tell, none of our results depend on the details of the choice we make. (2.3) (2.4) (2.5) where the index free functions G(t;t2) and V (t1; t2) are the solutions to the SYK gap equations and V (t) is an arbitrary O(N )q 1 group element. The r.h.s. in (2.5) is the result of performing a time dependent O(N )q 1 rotation on the saddle point solution (2.3). The fact that we have only (q elds (V (t)) on this manifold of solutions | at least formally makes the action restricted to this special manifold weakly coupled, as we will see below. In the rest of this section we will use the action (2.2) to determine the e ective action that controls the dynamics of the matrices V (t) at leading order in the long wavelength limit. In order to study quadratic uctuations about (2.3), we follow [3] to insert the expansion16 G~ab(t1; t2) = Gab(t1; t2) + jG(t1; t2)j q 2 2 gab(t1; t2); ~ ab(t1; t2) = ab(t1; t2) + jG(t1; t2)j 2 2 q ab(t1; t2); into (2.2) and work to quadratic order in gab(t1; t2) and ab(t1; t2). Integrating out ab(t1; t2) using the linear equations of motion, we nd an e ective action of the general structure S(G~; ~ ) = S(G; ) + dt1::dt4 gab(t1; t2)K~ 1(t1; t2; t3; t4)gba(t3; t4) jgj2 q q 2 N 21 (q 1)(q 4)+1 Z dt1 dt2 g(t1; t2)g(t1; t2): The expression in the rst line of (2.7) results from varying the rst two terms in (2.2), while the second line is the variation of the G~q term in (2.2). This term denotes the a sum of di erent contraction of indices between the two gs g(t1; t2)g(t1; t2) = X gcc11cc22::::::cckk 11bakkcckk++11::::::::ccqq 11 gdd11dd22::::::ddkk 11bakkddkk++11::::::::ddqq 11 : In the special case that the uctuation elds g are taken to be of the form g matrix contractions in (2.7) give appropriate powers of N , and (2.7) reduces to the e ective ba = bag, the action for g presented in [3]. It was demonstrated in [3] that In the long distance limit the Greens function can be expanded as G = Gc + G + : : : ; G(t1; t2) Gc(t1; t2) f0(t1; t2); 16Note that we have scaled G uctuations and uctuations with factors that are inverses of each other ensures that our change of variables does not change the path integral measure. The scalings of uctuations in (2.6) are chosen to ensure that the second line of (2.7) takes the schematic form gg rather than gK0G where K0 is an appropriate Kernel. We emphasise that the scaling factor jG(t1; t2)j q 2 2 in (2.6) represents the power of a function; no matrices are involved. (2.6) (2.7) (2.8) (2.9) (2.10) HJEP06(218)94 q 2 2 K~ (t1; t2; t3; t4) = K~c(t1; t2; t3; t4) (f0(t1; t2) + f0(t3; t4)) + f0(t1; t3) + f0(t2; t4) : The rst two contributions have their origin in the factors of G 2 in (2.9) and were called rung contributions in [3] (2.9). The remaining two contributions have their origin in the factors of G in (2.9) and were called rail contributions in [3]. We note that for rung contributions f0 appears with either rst two times or last two times of the kernel. On the other hand the two times in rail contributions are one from the rst set and one from the Our discussion so far has applied to general uctuations about the saddle point, and has largely been a review of the general results of [3] with a few extra indices sprinkled in. In the rest of this subsection we now focus attention on the speci c uctuations of interest to us, namely those generated by the linearized form of (2.5) around conformal solution q 2 (gc)ab(t1; t2) = jGc(t1; t2)j 2 Gc(t1; t2) Hab(t1) h Hab(t2) : i (2.13) Notice that the uctuations (2.13) represent the change of the propagator under a time dependent O(N )q 1 rotation. The form of (2.13) is similar in some respects to the variation of the propagator under di eomorphisms, studied in [3], with one important di erence; the factors of Hab(t1) and Hab(t2) appear with a relative negative sign in (2.13), whereas the in nitesimal di eomorphism elds in the light uctuations of [3] appeared with a relative positive sign in [3]. The fact that our uctuations are `antisymmetric' rather than` symmetric' will play an important role below. Specialising to this particular uctuation, It can be shown (see appendix A) that gc is an eigenfunction of K~c 1 with eigenvalue jJ j2 more clearly where Gc is the Greens function in the conformal limit and G is the rst correction to Gc in a derivative expansion. It follows that f0 is an even function of the time di erence, an approximate form of which is given in [3]. Plugging this expansion into (2.9) it follows that K^ can be expanded as K~ = K~c + K~ + : : : ; gc K~c 1 gc = jgj2 gc gc: q (2.11) (2.12) dt3 dt4 K~c 1(t1; t2; t3; t4)(gc)ab(t3; t4) = jJ j2 (gc)ab(t1; t2): It follows immediately from (2.14) that Using this equation it may be veri ed that for the for the particular uctuations under study | the second line of (2.7) simply cancels the part of the term in the rst line obtained by replacing K~ with K~c. It follows that the action (2.7) evaluated on the modes (2.13) is nonzero only because K 1 di ers from Kc 1. Recall K = Kc + K (see (2.11)). Using K 1 = K KK 1 that the action for our special modes evaluates at quadratic order to Using the fact that K~ 1 is hermitian ([3]) and the eigenvalue equation (2.14), the action simpli es to Se = 1 2 J 4 j j Z dt1::dt4 (gc)ab(t1; t2) K~ (t1; t2; t3; t4) (gc)ba(t3; t4): (2.17) Plugging the speci c form of our uctuations (2.13) into this expression we nd17 Se = 1 2 N q 2 X Z l=1 (i;k) pair where i 2 (1; 2), k 2 (3; 4) and Z A(t1; ::t4) = jJ j4 Gc(t1; t2)jGc(t1; t2)j 2 q 2 K~ (t1; t2; t3; t4)jGc(t3; t4)j q 2 2 Gc(t3; t4): (2.19) The expression (2.18) is not yet completely explicit, as Lik in (2.19) is given in terms of K which is given in terms of the rst correction to the conformal propagator Gc which, in turn, is not explicitly known. Luckily G can be eliminated from (2.18) as we now demonstrate.18 While we do not know the explicit form of the correction to the conformal two-point function G(t1; t2), we know that it satis es the equation c G + Gc + s Gc = 0: (2.22) This is simply the gap equation expanded around the conformal point. Here s(t1; t2) = t2) is a local di erential operator. summed up to 17Here factors of N comes from trace over other colour index -functions that multiply Hl of any colour. 18Using the fact that gc is an eigenfunction of K~c with eigenvalue jJj2 rung contributions can easily be 1 Srung = e 1 2 (q 2) jJ12j Z (gc)ab(t1; t2) f0(t1; t2) (gc)ba(t1; t2) dt1dt2: This expression is not by itself useful as the integral that appears in it has a log divergence once numerically determined form of f0( 1; 2) j 1 2j!!0 j 1 1 2j (from [3]) is used; follows from q 2 gc( 1; 2) = jGc( 1; 2)j 2 Gc( 1; 2)[H( 1 ) H( 2)] j 1 2j!!0 sgn( 1 j 1 2j 2) H0( 1 )( 1 2) O(j 1 (2.20) 2j0): (2.21) In order to make the expression (2.18) explicit we rst simplify the formulae (2.19) for Lij . Plugging the expansion G = Gc + G into (2.9), and using properties of conformal solutions, it may be veri ed after some algebra that for odd i The fact that Lik is proportional to a function establishes that the contribution of terms with odd i k to the action is local. Eq. (2.23) may be further simpli ed using the relation Lik(ti; tk) = q (ti tk) c G(ti; tk): Multiplying -function on both sides of (2.22) and using (2.24), we nd Se = N q 2 X Z q 1 l=1 dti dtkGc(ti tk) (ti tk) Tr Hl(ti)Hl(tk) (2.31) 20As before q 2 2 2 comes from rung part. 19Here overall factor of 2 comes from symmetry of the integrations and q 2 2 comes from rung part. (2.24) (2.25) (2.26) (2.28) (2.29) (2.30) and to give to give (ti tk)Gc q 1 (ti; tk) = (ti tk) c G(ti; tk); k is even, using properties of conformal solutions20 Eq. (2.27) can be further simpli ed by substituting and then using the linearized form of the gap equation c G c = + s; Gc = (q 1) G ; (ti Adding together the contributions of i k even and i k odd we have a manifestly local e ective action, whose structure accounts for the fact that we have worked beyond the purely conformal limit (recall that in the purely conformal limit our uctuation action simply vanished) even though the nal expression makes no reference to the explicit form of the correction G to the conformal propagator Gc. Expanding Hl(tk) in a Taylor series expansion about ti Hl(tk) = 1 X n=0 ti)n n! allows us to recast (2.31) into the form where follows that function and assumed that G valid for t1 t2 1 J Se = Z N q 2 dt Cn = dt Gc(t) (t)tn: The term in the sum (2.32) with n = 0 is a total derivative and so can be ignored. It Se = Z dt Our nal result (2.34) for the e ective action, has now been arranged as an expansion over terms with increasing numbers of derivatives. Recall that all the results of this section have been obtained after expanding the Greens G(t1; t2) = Gc(t1; t2) + G(t1; t2); Gc. This assumption is only valid when t1 t2 J1 , but are not . All potential non localities in the e ective action for H presumably have their origin in regions where our approximations are valid. It thus seems plausible that the central result of this section | namely the absence of nonlocalities in the e ective action on length scales large compared to J1 | which therefore takes the form (2.34) | is a reliable result. On the other hand the precise expressions for the coe cient functions Cn involve integrals over a function | namely the delta function | which varies over arbitrarily small distances | and so is not reliable (it uses our approximations in a regime where they are not valid). We would expect the correct versions of (2.33) to be given by smeared out versions of the integrals in (2.33). On general dimensional grounds it follows that We will make the replacement (2.36) in what follows. The numbers An could presumably be computed by studying four point correlators of appropriate operators at nite temperature. We will not attempt this exercise in this paper. For the purposes of long time physics we are interested only in the term with the leading number of derivatives, i.e. with the term with n = 1 in (2.34). The coe cient of Cn ! An j j J n : (2.32) (2.33) (2.34) (2.35) (2.36) HJEP06(218)94 our action in this case is proportional to A1 at leading order in the derivative expansion takes the form A.21 and the e ective action of our theory S = A j j In the analysis presented so far we have determined the form of the e ective action for in nitesimal group rotations H. The group invariant extension of our result to nite group rotations is the sigma model action S = A 1 Z J j j dt q 1 X Tr l=1 " 2# ; HJEP06(218)94 V (t) is, indeed, local. under which where Vl 2 SU(N ) whose in nitesimal form is Vl = 1 + Hl + O(Hl2). Eq. (2.29) is simply the action for a free particle moving on the group manifold O(N )q 1.22 As explained in the introduction, the structure of this action could have been anticipated on general grounds. The fact that the action is proportional to J1 follows largely on grounds of dimensional analysis. As we have already seen in the introduction, once we have established that the action for V (t) is local the form of the low energy e ective action (1.3) for our system is almost inevitable using the general principles of e ective eld theory. The main accomplishment of the algebra presented in this section is the demonstration that the e ective action for Note that the Sigma model action (2.29) has an O(N )q 1 O(N )q 1 global symmetry Vl ! AVlB; where A and B both belong to O(N )q 1. The rotations by A are simply the global symmetry that the microscopic SYK model possesses. Rotations by B are an emergent symmetry of the low energy e ective action. The corresponding conserved quantities are Ll = V_lVl 1, and Rl = Vl 1V_l.23 Choosing a basis (Ta),24 of Lie algebra O(N ) it can be 21Note that Plugging the formula into (2.37) we nd, formally, that Z C1 = dt (t)Gc(t) t: Gc = b sgn(t2) ; jJtj q C1 / Z dtjtj1 q2 (t) = 0; (2.40) (2.41) (2.42) (2.37) (2.38) (2.39) (where we have used the fact that q > 2). As explained above, we expect that the vanishing of C1 is not a physical result but rather is a consequence of inappropriate use of approximations. We assume that C1 ! jAJj in what follows where A is an unknown dimensionless number. 22Non-trivial holonomy can be turned on for these new light modes, details of contribution of these light modes to e ective action for holonomy is presented in appendix C. 23A dot over a quantity indicates derivative with respect to time. 24It is assumed in what follows that this basis puts the Killing form in a form proportional to identity. shown that Hamiltonian vector elds corresponding to group functions Ll;a = T r (TaLl), Rl;a = T r (TaRl) give two copies of O(N ) (at both classical and quantum level), both of which commutes with the Hamiltonian which is the quadratic Casimir of the algebra. 3 Holonomy dynamics and density of states at large mass We now switch gears; in this section and next we discuss a the mass deformed SYK theory (1.7) in the large mass limit. We work with the theory based on the O(N )q 1 symmetry where this symmetry is gauged. The large mass limit is of interest because it allows us to focus on the dynamics of the holonomy at nite temperature, and also allows us to compute the growth of states in the theory as a function of energy in a very simple setting. As explained in the introduction, in this section we will compute the nite temperature Z = Tr x m ; H for the mass deformed gauged O(N )q 1 melonic theory (1.7). In the large mass limit all elds in (1.7) except the holonomies of the gauge group can be integrated out at quadratic order. The result of this integration is easily obtained using the formulae of [39], and is given by (1.8). Notice that the e ective action Se (Ui) presented in (1.8) is invariant under the global `gauge transformations' Ui ! ViUiVi 1 for arbitrary orthogonal matrices Vi. This invariance may be used to diagonalize each Ui. The integral in (1.8) may then be recast as an integral over the eigenvalues of each of the holonomy matrices Ui with the appropriate measure. As Um are each unitary, their eigenvalues take the form ei mn where n runs from 1 to N . We de ne the eigen value density functions m( ) = 1 N X N n=1 ( mn): even function. Note that As we are dealing with orthogonal matrices, the eigenvalues of our matrix occurs in equal and opposite pairs ( a; a) and so the eigenvalue density function de ned in (3.1) is an As usual the rather singular looking sum over delta functions in (3.1) morphs into an e ectively smooth function at large N as the individual eigenvalues merge into a continuum. Tr Umn = N PjN=1 ein mj N = Z i( )ein n ; m where the last equality de nes the symbol in . Note that the subscript m on runs from 1 : : : q 1 and labels the O(N ) factor under study, while the superscript n runs from 1 : : : 1 and labels the Fourier mode of the eigenvalue distribution. Using the fact that i( ) = i ) it follows that in = Z d ( ) cos n : It follows that in are all real numbers and that in = i n. (3.1) (3.2) (3.3) (4.29) (4.30) current subsection, therefore (where we wish to ignore terms at order x2 or higher and only keep highest power of ) this denominator can be dropped, and we can work with the simpli ed propagator41 h a(t) b(0)i =e (m+i a)t h (t) ( t) x e i a i : In this subsection we assume m > 0; the case m < 0 can be argued in a completely analogous manner with the role of and reversed in the analysis below. In the computation of Feynman diagrams on the circle we will need to choose a `fundamental domain' on the circle; our (arbitrary but convenient) choice of fundamental domain is to px) at t = magnitude from p minimum (i.e. px) at t = 2 2 . Finally some terminology. We will call the part of the propagator (4.29) that is proportional to (t) the `forward' (`normal') part of the propagator, and the part of the propagator proportional to ( t) the `reverse' part of the propagator. Note that the normal part of the propagator ranges is modulus from 1 to px; it is maximum (i.e. unity) at t = 0 and . The modulus of the reverse part of the propagator varies in x to x. It is minimum (i.e. equal to x) at t = 0 and maximum (i.e. equal pxq 1 < x (recall q With all this preparation we are now ready to isolate the parts of the level n diagrams whose contribution is proportional to x n . To start with let us consider the simple nth level ring diagram depicted in gure 4. In this diagram we have n a type vertices and n b type vertices. In this graphs we have q 1 propagators connecting adjacent a and b type vertices, but only a single propagator connecting b to a type vertices. Consider any propagator between a and b type vortices | which has a type vertex A at time t1 and its adjacent b type vertex B at time t2. Depending on whether t1 > t2 or t1 < t2, all the q 1 propagators from A to B are either simultaneously all reverse or simultaneously all normal. If all propagators are reverse, the modulus of these propagators is less than 4). It follows that con gurations in which the propagators from A to B do not contribute in the scaling limit, and so all propagators from A to B must be normal. Given that these propagators are all normal their modulus is proportional to e m(q 1)jt1 t2j. It is intuitively clear that separating t1 from t2 over a nite fraction of the circle forces us to pay a high cost in factors of x; it can be shown (this will be clearer in 41The role that of the overall holonomy dependent phase factors above is quite subtle. Naively these overall factors can be dropped in their contribution to free energy diagrams. The naive argument for this is that the net contribution to of these phase factors at any interaction vertex is proportional to Q a ei( a)t1 where the sum runs over the phases a of all the q propagators that end at that interaction vertex. As the interaction vertex is a gauge singlet, P a vanishes, so it might at rst seem that the contribution of all these phase factors drops out. This is in general incorrect. The subtlety is that t1 is not single valued on the circle. In diagrams in which propagators `wind' as they go around the circle, one of the factors in the product may e ectively be evaluated at, e.g. t1 + and so the net contribution of this phase factor could turn out to be ei a . While this contribution is constant (independent of t1), it is nontrivial in nonzero winding sectors. Such a contribution will play an important role in our computation below. to . The diagram is drawn for q = 4. a bit) that such con gurations do not contribute to the result in the scaling limit. In the scaling limit we only receive contributions from con gurations in which jt1 t2j is of order 1 . It follows that for parametric purposes, we can simply regard t1 and t2 as the same m point, replacing the integral over t1 t2 by m1 . For parametric purposes, in other words, each of the melons in gure 4 can be thought of as a single interaction vertex, inserted at a single `self energy vertex', inserted at a single time, with e ective an e ective insertion factor of order Jm2 . Now let us turn to the propagators between b and a type vertices. These are now n di erent propagators connecting the e ective self energy blobs described in the previous paragraph. Let the e ective times of insertions of these self energy blobs be T1, T2 : : : Tn. Our graph is proportional to the product of n propagators, the rst from T1 to T2, the second from T2 to T3 : : : and the last from Tn to T1 + w reverse propagator contributes a factor of at least p of these propagators can be reverse. where w is an integer. As each x to the integrand, no more than two Let us rst consider diagrams in which all propagators are forward. As all propagators move forward in time, the nal propagator in the sequence must end not at time T1 but at time T1 + w where w is a positive integer. The modulus of the product of these propagators is then easily seen to be proportional to e wm = xw. In the scaling limit of interest to us, the only option is w = 1. Once we set w = 1, the integrand of the diagram is now independent of the e ective insertion times Ti. The integral over these n insertion times thus gives a factor n, and the contribution of the graph in question is proportional to x n as desired. Now let us consider diagrams in which one of the propagators between the e ective self energy vertices is reverse, and the rest are forward. It is easy to verify that the modulus of the product of propagators in such a graphs is proportional to xe wm where w = 0; 1; : : :. In the scaling limit under consideration we are interested only in w = 0. Once again the modulus of these graphs is independent of the insertion positions of the e ective self energy vertices, and integration over their locations produces a result proportional to x n of Diagrams in which two of the propagators are reverse are kinematically very constrained. Similar argument as above shows these graphs are proportional to x only if w = 1 ) and so all the forward propagators have length zero, again up to corrections of order 1, i.e, if the two reverse propagators each have length 2 (up to corrections of order 1 . These constraints ensure that such graphs are proportional to but no higher power (certainly not n) and so are not of interest to the current section. In summary, graphs of the form depicted in gure 4 only contribute at order x n if all propagators from a to adjacent b type vertices are normal, if the separation between a and adjacent b type vertices is of order m1 , and if the propagators between adjacent melons are either all normal with net winding number one or one reverse and the rest normal with net winding number zero. Once we have identi ed the parts of these graphs that contribute at order x n, the computation of these contributions is very simple (see below). Let us now turn to more general graphs than those drawn in gure 4. All graphs that contribute to the free energy at leading order in the large N limit are of the general structure depicted in 4, but with the melons in gure 4 replaced by e ective melons or `cactus graphs'. The net e ect of this is to replace the bare propagators between a and b type vertices in gure 4 by exact propagators. Recall that we are only interested in the propagator corrections at times t = jt1 t2j forward propagator at short times takes the schematic form 1 m . The kth order correction to the G(t) jJ j2ktk k mk X Cn n=0 m n t (4.31) J 2n m2n F2n = 2x N q 1 where As all values of t that contribute to our integrand in the low energy scaling limit of interest to this paper are of order m1 , it follows that all terms on the r.h.s. of (4.31) are of order mJ22kk . As compared to the contribution of the graphs of gure 4, in other words, these graphs have extra powers of J 2 but no compensating factors of . It follows that The contribution of such graphs at level n is always of the form x h with h strictly less than n. Consequently all such graphs can be ignored. In summary, the only graphs that contribute at terms proportional to x n at level n are the very simple `necklace' graphs depicted in gure 4. We have already explained above that the contribution of each of these graphs is easily evaluated in the scaling limit. It follows that the computation of the sum of these graphs is a relatively simple job. Relegating all further details to the appendix B.1.2 we simply list our results. The contribution of order x n to Se (U ) from graphs of level n is given, for n 2 by q 1 Y m=1 1 m ! 1 (n 1)! (q) ( ) m j j 2 2 n + O( n 1); (4.32) (4.33) Summing these contributions over all n = 2 to in nity and adding the separate contribution of n = 1 we nd H0. H0 = 2x N q 1 Y m=1 1 !" 1 X Y m=1 1 2 given by H0 in (4.34). 1 (n 1)! (q) ( ) m j j 2 2 q m J 2 # m ; (4.34) so that the free holonomy e ective action takes the form (4.3) with F0 in that equation follows that the exponential terms in (4.34) decay at large J2 q but blow up for the second set of values of q. It would be interesting to better understand the meaning and consequences of this observation. m for the rst set of values of At su ciently weak coupling we have demonstrated in the previous subsection that the free result for Se (U ) in the scaling limit, (3.6), is replaced by the formula (4.35) (4.36) q 1 Y m=1 1 m ! xH~0; where H0 was computed in the previous subsection. Note that (4.35) has the same structure of U dependence as (3.6); it follows that the partition function obtained by integrating e Seff (U) over U is simply Z(x~) (where the function Z(x) was de ned in (3.6)). At small enough coupling x~ is close to x, and the structure of the canonical partition function generated by (4.35) is very similar to the results described in detail for the free theory in the previous section. What does consequence does the replacement of x by x~ have for the micro canonical partition function? Let us rst recall a simple formal result. Let e m ! e m (1 + h0( )): By linearizing the usual thermodynamical formulae it is easy to show that this replacement results in the replacement S(E) = S0(E) + E m h0 + O( 2); (this result holds provided we expand about an analytic point in the phase diagram, i.e. away from phase transitions). Clearly in our context this result applies if mJ22 ln N and is taken to be small. However our results for the partition function are valid over a larger parametric regime; they are de nitely valid whenever mJ22 In order to understand the e ect H~0 has on the entropy as a function of energy at such ln N even at nite values of . we take a slightly di erent route. De ne a `particle mass probability function' p(m) by the following requirement Z dm0e m0 p(m0) = xH0: (4.37) Intuitively p(m) denotes a spread in the mass density function (which was a function for the free theory) that mimics the e ects of interactions in thermodynamics. A little thought demonstrates that the following ansatz for p(m0) reproduces the structure of our perturbative expansion for xH0 p(m0) = X1 1 k=0 m gk m0 jJ j2=m J j j 2k 2 ; (4.38) where the functions gk(y) do not depend on J . Working with the probability distribution (4.38) is equivalent to replacing x by x ! Z 1 0 m0 p(m0)dm0 = x 1 X n;k=0 1 J j j 2k Z 1 m2=J2 ungk(u)du: (4.39) The lower limit of the integration in (4.39) can safely be approximated by the r.h.s. of (4.39) to equal x~ we must choose 1. If we want Z 1 1 Z 1 1 g0(u)du = 1 ; u g0(u)du = (q); un g0(u)du = 4n 1 (q)n ; n 2 2 n 2 n p(m0) =2 (m0 m0 m 4 (q) j j 0 m0 m 2 q 2 (q) jJmj2 (q) jJmj2 These relationships determine the moments the as yet unknown g0. Inverting these relations we nd (4.40) (4.41) Z 1 1 Recall that the function p(m0) in the free theory was just a function localised at m0 = m. The interaction e ects considered in this section split this function into a set of 4 localised (or 0) spikes, distributed in a width of order Jm2 around m0 = m. As an aside we note the striking fact that interaction e ects | at least at the order we have computed them | do not smoothen the free spectral function out. It is not di cult to convince oneself that the function S(E) that follows from (4.41) is qualitatively similar to the entropy as a function of energy derived in detail for the free theory in the previous section, and in particular displays faster than Hagedorn growth. 5 In these notes we have argued that the quantum mechanical model (1.1) | which is known to agree with the SYK model in the strict large N limit | displays qualitatively new dynamics at subleading orders in N1 . We argued that the uctuation spectrum about the nite temperature saddle point in this theory has new light modes | that originate in time dependent O(N )q 1 transformations | in addition to the modes that arise from conformal di eomorphisms and that were present also in the original SYK theory. The total number of new light modes is (q 1) N22 and so is very large in the large N limit. We have also proposed that the dynamics of these new modes is governed by the sigma action (1.3), with a normalisation constant A whose value we have not been able to calculate. Assuming that our proposal for the new light modes is correct, it raises several interesting questions. It should be possible to check our proposal for the structure for the e ective action (1.3) by performing an independent computation of the four point function of four operators in the theory (1.1) (by summing ladder diagrams) and comparing the long time behaviour of this computation with what one obtains directly from (1.3). Such a procedure should also permit the direct computation of the as yet unknown constant A. It is also natural to attempt to nd a bulk interpretation of our new modes. One natural suggestion is that these modes are dual to gauge elds in AdS242 If this is the case it is interesting that the rank of the bulk gauge elds diverges in the e ectively classical N ! 1 limit. In other words the bulk classical dual of this theory is given in terms of a weakly coupled theory of an in nite number of classical elds. The situation is somewhat reminiscent of the proliferation of `light states' in the duality of [48], and also the situation with ABJ `triality' in the ABJM limit [49] (although in this context the number of bulk Vasiliev elds is never both parametrically large and parametrically weakly coupled). It would be very interesting to investigate this further. We have also shown that the density of states in an extreme mass deformation of the model (1.1) displays a faster than Hagedorn growth at energies of order N 2. In our opinion this is also a very striking result; the phase that displays this rapid growth is the `thermal graviton' or `string gas' phase. The rapid growth in the density of states of this phase presumably means it cannot thermally equilibriate with another system. It would be interesting to understand what consequences this rapid growth has for potential bulk duals of mass deformed versions of the theory (1.1). Finally we have performed detailed calculations for the holonomy e ective action of the mass deformed theory (1.1) away from the strict large mass limit. In a particular scaling limit that zooms in on the dynamics of the theory at energies of order N 2 we demonstrated that the holonomy e ective action of our theory, Se (U ) takes a simple universal form. We were able to capture the leading interaction e ects by summing the appropriate in nite class of graphs and obtain a very simple e ective action that captures the leading deviation away from free behaviour. It should certainly be possible to generalise our perturbative computation of H~0 to a computation of H~1. More ambitiously, it may eventually prove possible to completely sum this perturbative expansion. We leave investigation of this possibility to the future. Acknowledgments We would like to thank S. Jain, I. Klebanov, C. Krishnan, J. Maldacena, G. Mandal, P. Nayak, S. Sachdev, S. Shenker, D. Stanford, J. Yoon and E. Witten for useful discussions. 42We thank J. Maldacena for this suggestion. We would like to thank S. Mazumdar, Y. Dandekar and S. Wadia for collaborations at the initial stages of this work. The work of all authors was supported in part by a UGC/ISF Indo Israel grant, and the Infosys Endowment for Research into the Quantum Structure of Spacetime. Finally we would all like to acknowledge our debt to the steady support of the people of India for research in the basic sciences. A Conformal kernel In this appendix following main result is proved Z 1 j j j j 1 Z J 2 j j j j 1 j j q 2 Z Z dt3 Gc(t1; t3)H(t3) dt4 Gc(t2; t4)H(t4) dt4 Gc(t2; t4) jJ j2Gc(t4; t3)q 1 dt3 Gc(t1; t3) jJ j2Gc(t3; t4)q 1 dt3 Gc(t1; t3)H(t3)( (t2 t3)) dt4 Gc(t2; t4)H(t4)( (t1 t4)) j j where relevant quantities are de ned by Z dt3 dt4 K~c(t1; t2; t3; t4)(gc)ab(t3; t4) = 1 j j J 2 (gc)ab(t1; t2); K~c(t1; t2; t3; t4) = jGc(t1; t2)j 2 Gc(t1; t3)Gc(t2; t4)jGc(t3; t4)j 2 ; q 2 q 2 gc(t1; t2) =jGc(t1; t2)j 2 Gc(t1; t2)[H(t1) H(t2)]: Important part of the integration is given by: Q(t1; t2) dt3 dt4 Gc(t1; t3)Gc(t2; t4)Gc(t3; t4)q 1[H(t3) H(t4)] J 2 [Gc(t1; t2)H(t2) + Gc(t2; t1)H(t1)] J 2 Gc(t1; t2) [H(t1) H(t2)] : This proves claimed result when multiplied with B Details of the perturbative computations jGc(t1; t2)j 2 . q 2 B.1 B.1.1 Leading power of Two melon graphs In this subsection we consider the contribution to the free energy given by gure 5. First non-trivial e ect of winding is seen at this level as explained below. The term whose Wick contraction is calculated is 41! (J 4 + h:c:)4 | where each of 4C4=2 terms contribute the same. The symmetry factor is calculated as follows. Any one of q number of 's of rst -vortex contracts with any one of q number of 's of any one of two -vortex to give a factor of 2q2. Any one of q number of 's of second -vortex contracts with any one of q (A.1) (A.2) (A.3) HJEP06(218)94 b1 θ b2 θ b3 θa1 θa2 θa3 θc2 1)]2I(4); ! (B.1) (B.2) number of 's of remaining -vortex to give a factor of q2. In large-N only non-suppressed diagram is obtained by joining to (of same vortex) of same common colour. Choice of external propagator gives q 1 possibilities at each blob. Sign of the symmetry factor comes from noticing as there are two identical 'blobs' sign of contraction of each blob cancel and overall sign is just because of contraction between two 'blobs', it turns out to be -1. Contribution of symmetry factor at this order becomes F4 = 4! 1 4C4=2 ( 1 )2[q2(q 1 2 where I(4) = Z 4 Y dti i=1 q 1 i=1 i=1 Y G0(t12; ai ) Y G0(t34; bi ) G0(t32; c2 )G0(t14; c1 ): Where s are holonomies on di erent propagators. Here time di erences are not necessarily single valued and to satisfy the constraint t12 + t23 + t34 + t41 = w ; where w = 0; 1; 2 (note that each tik is in ( 2 ; 2 ), and this restricts allowed values of n) we introduce dimensionless Lagrange multiplier integration P Z +1 ds eis(t12+t23+t34+t41 w ) = t12 + t23 + t34 + t41 w : (B.3) In the scaling limit (assuming m > 0), the propagator becomes G0(t) = e (m+i a)t (t) xe i a e (m+i a)t (t) x1=2e m =2e i a e (m+i a)t ( t): (B.4) of G0 as x0; x; x1=2 contributions. This way of writing ensures in each of three parts of G0 excluding explicit x dependence integration over 2 to 2 gives only positive powers of x. We will refer to these three parts In the scaling limit of interest I(4) can receive contribution from 5 di erent types of integration I(4) = x0 everywhere + x1=2 on one of the outer ( c1 ; c2 ) lines + x1=2 on both of the outer lines + x on one of the outer lines (B.5) + x on one of the inner lines ( a1 ; a2 : : : aq 1 ; b1 ; b2 : : : bq 1 ): HJEP06(218)94 Note that choosing x1=2 on one of the inner propagators will force choosing all the inner propagators in the same blob to be x1=2 term due to unit step function. Therefore this choice is ignored in scaling limit calculation. Here we'll present the calculation corresponding to the rst one and mention results for others. Consider x1=2 on c1 say, and on all others we choose x independent part of G0. This ensures following time ordering for non-zero integrand t12 > 0; t32 > 0; t34 > 0; t41 > 0, with which only consistent values of n are 0; 1. Contribution to I(4) becomes, omitting ( x1=2)e i c1 e+i c1 w (for a contribution like F0 we must have n = 0 which is shown to e (m(q 1) is)t34 e+(m+is)t41 m =2e (m+is)t32 ( x1=2)e i c1 e+i c1 w Z ds (eis =2 x1=2)(x1=2e is =2 1) 2 (s + i(q 1)m)2(s im)2 e isw + O(x3=2); where we ignored higher order contributions in x. Simplifying the numerator gives 3 terms: x independent piece that comes with a non-zero phase factor eis =2 (which will give a factor of upon integration because only w = 0 will contribute), x1=2 term that comes with no non-trivial phase (cannot give a upon integration), x term drops out in scaling limit. Rest of the integration can be done easily choosing proper contour (semi-circle on upper or lower half plane as required by convergence) to ensure only w = 0 term contributes to give the following result w;0x1=2 2 All other integrations can be performed similarly to give leading order contribution to free energy F4 = 4! 1 4C4=2 2q4 (q 1)2 q2 m2 2 N (q 1)2 x Y 1m + O( ): q 1 m=1 (B.6) (B.7) (B.8) B.1.2 n melon graphs Here a circle diagram with n 2 blobs ( gure 6) is considered and leading term in is calculated using methods demonstrated in previous sub-section. ψ¯(t2) (B.9) (B.10) HJEP06(218)94 Symmetry factor for the diagram in large N limit is43 The leading order contribution in comes from two distinct choices | i) considering x1=2 in any one of the n external propagators (with holonomy a say) with x0 part of the free propagator in all others and ii) x0 part of the free propagator in all propagators. Contribution from the integral due to choice (i) is easily seen to be x1=2jgjn e i a +iw a 2x jgjn e i a 1 (n Z ds 2 e i(w 12 )s w;0; 1 on which x1=2 is considered. with one another. where we have kept only highest power of . Note that extra powers of beta n 1 came from the integration because of evaluation of residue around a pole of order n. This contribution is to be multiplied with a factor of n due to freedom in choosing one external propagator 43Here an extra factor of (n 1)! comes as compared to n = 2 case because of freedom of joining n blobs Now we turn to the choice (ii). In this case contribution to the integral is jgjn eiw a = 2x jgjn e i a Z ds 2 e iws 1 (1 x1=2e is 2 )n 2 n 1 1)! 2mq n = 2. where F2n = 2x N q 1 q 1 Y m=1 1 m 1 (n 1)! m j j J 2 2 n 1 + O( n 1); (B.12) In this subsection we shall compute explicitly the integral involved in computing the contribution to the free energy in the scaling limit linear in x = e m . The free fermionic Green's function at any nite temperature is given by, As before we have kept only highest power of . Note that this contribution vanishes for After summing over the holonomies, and canceling loop N's with that of scaling of g, contribution to free energy becomes (B.11) (B.13) (B.14) (B.16) where, x = e m function at nite temperature as, 1 (scaling limit). Hence, one can also write the `reversed' Green's h (0) (t)i = G0( m) = 2 1 e(m+i j)t sgn(t) 2 tanh (m + i j ) : (B.15) Here j are holonomies, satisfying the following constraint Now in the computation we use discrete representation of the delta function h (t) (0)i G0(t) 1 2 = e (m+i j)t sgn(t) + tanh = e (m+i j)th (t) xe i j i ; 2 (m + i j ) q X j=1 j = 0 : t2n 1 2n + t1 2n = 2 1 1 X != 1 e 2 i ! (t21+t32+t43+t54+t65+ : : : t2n 1 2n+t1 2n ): (B.17) 1 X != 1 Z =2 =2 Let us focus on the diagram which can be computed as using the integral, I(2n) = 2 1 J 2n 4 dt1 e t1(m+i q)e 2 i ! t1 sgn(t1) + tanh m +i q 2 dtq etq((q 1)m i q)e 2 i ! tq (A sgn(tq) #n B)) : Z =2 =2 q 1 Y h Here the rst integral inside the sum is a single propagator while the second one represents 1 propagatrs, where A and B are de ned as tanh m +i j 2 i = ( 1 )q (A sgn(tq) B) : on a circle of length the propagators as shown in (B.18). We integrate over the time intervals of these propagators in (B.18) and since there are n of them we raise it to the power n. However, we would also have to implement the constraint that the times add up to an integral of . This is achieved by repesenting the delta function as an in nite sum. This contributes a factor of e2 i ti in each of ! Now we would like to focus on the integrals within the box brackets in (B.18) Since z = 2 i! we see that these reduce to (B.18) (B.19) (B.20) (B.21) (B.22) (B.23) F (2n) = 1 X != 1 hI(q)in : ! Upon integrating over t1 and tq one nds that I(q) = ! f1 + f2 1)m ; z = Here f1 consists of terms with e kz where k 2 Zodd. Its is evident that upon raising I! where k 2 Zeven while f2 consists of terms e kz =2 (q) to n one would have to evaluate sums in z of the form S1 = S2 = 1 X != 1 1 X != 1 S1 = S2 = != 1 1 X != 1 1)m 1)m e kz e kz =2 i q + z)(m + i q z))n ; k 2 Zeven z))n ; k 2 Zodd: 1)m 1)m 1 ez =2 z))n z))n ; : We will use the technique of Matsubara summation to evaluate the above, where a weighting function is included to replace the sum by a contour integral. So, rst let us evaluate S1. With a weighting function f (z) = 1 1ez , one can replace the above summation with the following contour integral, S1 = I dz (1 ez )(((q 1)m i q + z)(m + i q z))n Notice that the integrand has two poles at z za = (q 1)m + i q and z zb = m + i q and both are of n-th order. Using the residue theorem, one can evaluate the above S1 = zl!imza (n 1 1)! z Now, it is very easy to verify that for any function f (z), ez )(z zb)n + lim z!zb (n 1 1)! z (1 ez )(z za)n (B.25) (z 1 = X( 1 )k (n 1)Ck (n 1)! (z za)n+k 1 In the present case, taking f (z) = (1 e z) , one can evaluate where, A(n) is the Eulerian polynomial in e z, given by, 1 1 e z ne z (e z 1)n+1 A(n) n 1 m+1 m=0 k=0 A(n) = X X ( 1 )k n+1Ck(m + 1 k)n e zm Using equation (B.27) and (B.28), one can easily obtain, 1 1 e z Finally Substituting equation (B.29) into equation (B.26), we have, f (z) (z za)n 1 n 1 k=0 n k 1 1 (e z 1)n k e z n k 1;0 X n k 2 m+1 m=0 l=0 n 1 = X k=0 (z 1 (1 e z)(z za)n ( 1 )k za)n+k (n 1)! 1)! n kCl (m + 1 l)n k 1e (m+1)z + (B.24) (B.26) (B.27) (B.28) (B.29) (B.30) X ( 1 )l n kCl (m + 1 l)n k 1e (m+1)z (e z 1 1 e z n k 1;0 n k 1 1)n k n k 2 m+1 X X ( 1 )l Evaluating the above expression at both the poles z = za and zb, one can compute S1 as expressed in equation (B.25). Now let us discuss about evaluating the summation S2 as given in equation (B.23). With a weighting function f (z) = 1e ez=z2 , one can replace the above summation with the following contour integral, S2 = I e z=2dz (1 e z)(((q 1)m i q + z)(m + i q z))n Notice that we encounter the same n-th order poles in the contour integral as we had with S1. The residue computation for evaluating this contour integral needs to evaluate the e z=2 1 e z 2n(e ne z where, B(n) is the Eulerian polynomial of type-B in e 1)n+1 B(n) z, given by, B(n) = X( 1 )m k n+1Cm k(2k + 1)n e zm (B.31) (B.32) (B.33) Finally, using equation (B.26), (B.32) and (B.33), one can obtain, f (z) (z ( 1 )k za)n+k e z=2 za)n i (n 1)Ck (n + k (n 1)! 1)! 2n k 1(e z 1)n k n k 1 X( 1 )m l n kCm l (2l + 1)n k 1e (2m+1) z=2 (B.34) X m=0 k=0 n 1 X k=0 (z X n k 1 m m=0 l=0 Now using the above equation one can compute the residue and hence the integral (B.31). This nishes the computation of S2 as given in equation (B.23). One nds that S1 depends only linearly on x = e m noting that the di erence in A and B in (B.19) behaves as A f1 = (A B)O(x 2q +1) = O(xq=2) and f2 = (A B)O(x 2 ) = O(x 2q 1). Therefore in q+1 the scaling limit one can take A = B = 2q 2 1 x P e i j . Therefore evaluating (f1 + f2)N F1 + F2 in the scaling limit | where once again k 2 Zodd, F (n) = S1F1 + S2F2. Here F1;2 = F1;2(z = 0).44 F1 consists of terms with e kz where k 2 Z while F2 consists of terms e kz =2 where The fact that only these two type of summations contribute for any integer value of k, makes it easier to evaluate equation (B.18) in the scaling limit as, q 1 j=1 while S2 depends as px. Further B = O(xq 1) we nd that k=0 I(2n)=J 2n nX2 2(q 1)nx n k (mq)n+k (n)2 (2n 22+kn)(n 1Ck) (n + k) Y 1 m (B.35) q 1 m=1 44Since their z dependences were where taken into account in evaluating S1 and S2. which can be re-written as I(2n) = J 2 mq +O( ) n n 2 X k=0 or equivalently keep all orders in as J 2 k qJ 2 k 2(q 1)nx (n)2 (2n q 1 m=1 1 (B.36) I(2n) = J 2n ( 1 )q(n 1) m2nqn (n 1)! (n 1)! qn 1 n 2 + X k=0 (n + k 1)! 1)! qk (1 2k+2 nn)(m ) n k x q 1 Y m=1 1 : 2n + 3 (B.37) HJEP06(218)94 This multiplied with (B.9) N q 1 gives contribution of a circle diagram with n melons. B.3 Evaluating the subleading correction We end this appendix by presenting a technical result which we do not use in the main text of the paper, but record here anyway, just in case this result nds application subsequent work. The technical result we report here is the evaluation of the Feynman integral for diagram gure 7 (the gure is drawn for q = 4 but we present the evaluation in general), which is one of the diagrams that would contribute to the generalization of the results presented in this paper to subleading orders in 1 . We present the result for the Feynman diagram ignoring the symmetry factor (which can easily be independently evaluated). We evaluate the diagram of gure 7 as follows. In order to get the integrand of the diagram we rst multiply together all the propgators that make it up, keeping careful track of holonomy factors and making use of the fact that holonomies at any interaction vertex sum to zero. The integrand is the term in the big square bracket in (B.38) with 1 and 2 temporarily set to zero. The rst two lines on the r.h.s. of (B.38) are the n 2 factors on the in the diagram gure 8.45 The next four lines on the r.h.s. of (B.38) represent the second factor in gure 8. Lines 3{6 on the r.h.s. of (B.38) are the remaing factors (the propagators outside the square 45t1 in this term is the length of the straight line in these factors, while t2 is the length of the 3 (or more generally q 1) melonic lines in the part gure 8 that is enclosed in the square bracket. Really there are 1 di erent t1s and n 2 di erent t2. As t1 and t2 are dummy variables that we integrate over, we have used the same symbol for all of them. 1 X !1;!2= 1 " Z =2 =2 dt1e (m+i 1+i 1 )t1 sgn(t1) + tanh m +i 1 Z =2 Z =2 =2 =2 =2 =2 Z =2 =2 Z =2 Z =2 dt2e((q 1)m i 1 i 1 )t2 (sgn(t2)A1 B1) dt1e (m+i 1+i 1 )t1 sgn(t1) + tanh m +i 1 dt3e(m+i 2+i 2 )t3 sgn(t3) tanh m +i 2 dt4e((q 2)m i( 1+ 2) i ( 1+ 2) )t4 (sgn(t4)A1;2 dt5e( (q 1)m+i 2+i 2 )t5 (sgn(t5)A2 + B2) !n 2 2 2 (B.38) After evaluating the integrand we need to perform the integrals. Roughly speaking we must integrate all propagator lengths in the integrand above from 2 to 2 . However we need to do this subject to the constraint that as we go round either of the two circles in the diagram gure 7 we come back to the same time as we started out, modulo . This is where the parameters 1 and 2 in (B.38) come in. 1 couples to the sum of lengths of propagators in units of around the big circle in gure 7, while 2 multiplies the sum of the lengths of all the propagators as we go around the small circle | again in units of in gure 7. The constraint that these lengths evaluate to an integral multiple of can then be implemented by setting 1;2 = 2 !1;2 and then summing !i over all integral values, as we have done in (B.38). 46The third line in (B.38) is the straight line in this part of gure 8. The last and secondlast lines in (B.38) are, respectively, the blobs of q 1 and q 2 propagators in this part of gure 8. Finally the fourth line in (B.38) is the product of the two proagatogrs that run between the `q 1 blob' and the `q 2 blob'. The times in all these terms represent the lengths of the corresponding propagators. In order to proceed we perform the time integrals in an unconstrained manner. The result can be rearranged (according to its !i dependence) as a sum of four types of terms. 1. Terms containing ek(z1+z2) where k 2 Z 2. those with ekz1 =2 where k 2 Zodd 3. with ekz2 =2 where k 2 Zodd 4. and ek(z1+z2) =2 where k 2 Zodd; We deal with these four classes of terms spearately; for each class we explicitly perform the sum over !i (by reducing it to a contour integral as in the previous subsection) and expand the resultant expression in a Taylor series in x (again as in the previous subsection), keep only the terms that are linear in x. Combining together the results from each of the four classes we obtain our nal result q 1 n n 4 x(q 1) 2(q 1)n(2n +(n 1)23+k)(2n+k 2) (n+k 1) Y X (mq )k+1 1) (n) (1 + k) +O( 2) (the terms O( 2) that we have omitted to list in (B.39) are the terms with k = n k = n 2 which exist in the nal answer but the values of whose coe cients do not follow the uniform rule of the other terms). Note that (B.39) scales like 1 in coordinated large small J limit in which J 2 is 1 m (B.39) C The holonomy e ective action from the sigma model In this section we ask the following question: what is the contribution to Se (U ) | the e ective action for holonomies | resulting from integrating out the new light degrees of freedom discovered in the massless tensor model in early sections in this paper? In the bulk of this section we address this question at the technical level. At the end of the section we turn to a quick discussion of its physical import. Turning on holonomy is equivalent to putting appropriate boundary condition on fermion This translates into boundary condition on Vl, given by Vl U Vl + 2 , U 2 O(N ). This boundary condition is equivalent to the computation of the partition function Z = e Se (U) = Tr e H U^ ; where H is the Hamiltonian of the quantum mechanical system (1.3) and U is the quantum mechanical operator that implements left rotations on the sigma model by the O(N )q 1 group rotation U . The partition function (1.2) is the product of q 1 factors, associated with the sigma models on the q 1 gauge groups. It follows that the e ective action Se (U ) that follows from this computation takes the form In the rest of this section we compute the functions S(Ui) Let us rst note that the Hilbert H space on which any one of the factors of q 1 distinct factors the sigma model (1.3) acts is given as follows. The Hamiltonian acts on (C.4) HJEP06(218)94 the Hilbert space H H = X R~i Ri R~i: The sum Ri runs over all genuine (as opposed to spinorial) representations of O(N ). R~i denotes the vector space on which O(N ) acts in the ith representation. The space R~i transforms in the representation Ri Ri under O(N )L as an O(N ) rotation on the rst R~i but as identity on the second R~i. The Hamiltonian corresponding to action (1.3) is diagonal under the decomposition (C.3); the energy of the O(N )R; the operator U^ acts ith factor of the Hilbert space is 2JACN2(qRi2) . Representations of O(N ) are conveniently labeled by the highest weights (h1; h2; h3 : : :), the charges under rotations in mutually orthogonal two planes. Let h = Pi hi. At leading order in the large N limit the dimensionality of the representation Ri depends only on h and is given by Moreover the Casimir C2(Ri) of representations of O(N ) also depends only on h at leading order in the large N limit and is given by Let Ri (U ) denote the character in the Ri representation of O(N ) and let d(Ri) = N h C2(Ri) = N h: n(U ) = Ri (U ); X { 56 { where n^ denotes the collection of all representations of O(N ) with h = n. In other words n(U ) is the sum over the characters of all representations with h = n. Note that all representations with h = n can be constructed | and can be constructed exactly once | from the direct products of n vectors of O(N ) (this is true when N n as we assume).47 Let Pn denote the projector onto representations with h = n Pn [f (U ))] = Ri (U ) Ri (U 0)f (U 0): Z dU 0 X Ri2n^ n(U ) = Pn [(Tr U )n] : where U on the r.h.s. of (C.6) represents the group element in the vector representation It follows immediately from all the facts and de nitions presented above that Using (C.6), (C.8) can be rewritten in the (perhaps deceptively) elegant form It follows that of O(N ). Finally we de ne Note that (C.6) (C.8) (C.10) HJEP06(218)94 z = e 2ANJq 3 : e S(Ui) = X1 (zN )n n=0 Z dU e S(U) = 1: This is an immediate consequence of the fact that the vacuum is the only representation in the spectrum of the group sigma model that is a singlet under O(N )L. It follows that the partition function generated by S(U ) by itself is trivial. However S(U ) is only one piece of the e ective action for U in the massless tensor model (1.1); we get other contributions to the e ective action by integrating out the fermionic elds themselves (as was explicitly done earlier in this paper for the case of massive fermions). When put together with other contributions the e ective action (C.9) could have a signi cant impact on the partition function, especially at temperatures scaled to ensure that the matter contribution to the e ective action | like the contribution of the sigma model considered in this section | is of order N 2. Open Access. 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Sayantan Choudhury, Anshuman Dey, Indranil Halder, Lavneet Janagal, Shiraz Minwalla, Rohan R. Poojary. Notes on melonic O(N)q−1 tensor models, Journal of High Energy Physics, 2018, 94, DOI: 10.1007/JHEP06(2018)094