Blobbed topological recursion for the quartic melonic tensor model
Random tensor models are generalizations of random matrix models which admit expansions. In this article we show that the topological recursion, a modern approach to matrix models which solves the loop equations at all orders, is also satisfied in some tensor models. While it is obvious in some tensor models which are matrix models in disguise, it is far from clear that it can be applied to others. Here we focus on melonic interactions for which the models are best understood, and further restrict to the quartic case. Then Hubbard-Stratonovich transformation maps the tensor model to a multi-matrix model with multi-trace interactions. We study this matrix model and show that after substracting the leading order, it satisfies the blobbed topological recursion. It is a new extension of the topological recursion, recently introduced by Borot and further studied by Borot and Shadrin. Here it applies straightforwardly, yet with a novelty as our model displays a disconnected spectral curve, which is the union of several spectral curves of the Gaussian Unitary Ensemble. Finally, we propose a way to evaluate expectations of tensorial observables using the correlation functions computed from the blobbed topological recursion.
Random tensor models are generalizations of random matrix models. They were introduced as a way to generalize the successful relationship between random matrices and two-dimensional quantum gravity (see the classics (1) and the more modern (2)) to higher dimensions. Indeed, random matrix models are integrals over matrices of size which admit some expansions. At each order in the expansion, the matrix integrals give rise to Feynman diagrams which are ribbon graphs, or equivalently combinatorial maps, and represent discrete surfaces of genus . The original motivation behind tensor models for a tensor with indices is that their Feynman diagrams represent gluings of -dimensional building blocks.
The original proposals of tensor models (3); (4); (5) were plagued with difficulties. The Feynman diagrams represent gluings of simplices which can be quite singular (6). Moreover, no tensor models were known to admit a expansion (where is the range of each index) for twenty years. A reason which explains the absence of major development in tensor models for such a long time might have been the impossibility to generalize random matrix techniques. For instance, tensors cannot be diagonalized which implies no reduction to eigenvalues and therefore no orthogonal polynomials. As a consequence, methods to solve matrix models, such as the one considered in the present article, were not applicable as they rely on the existence of a expansion.
Tensor models have come a long way in the past six years. We now have at our disposal a family of building blocks known as bubbles (7) which generalize the unitary-invariant observables (matrix traces) of matrix models. Matrix models use those invariant observables as potential: , in the case of single-trace interactions, where the term of order represents a -gon to be used to form surfaces.
Similarly, bubbles are unitary-invariant polynomials in the tensor entries, which also represent simplicial building blocks: instead of -gons which are gluings of triangles, one can have arbitrary colored gluings of -simplices with a boundary. Using those polynomials as potential in the tensor integrals gives Feynman diagrams which are colored triangulations and which are classified according to some expansion. The fact that colored gluings of simplices emerges in modern tensor models is the reason why expansions exist and was the missing ingredient in the original tensor models (see Gurau’s original expansion articles for the combinatorics of colored triangulations (8)).
In contrast with matrix models for which the expansion is the genus expansion for any choice of potential, the expansions in tensor models depend on the choice of bubbles (9). Gurau’s expansion, with respect to Gurau’s degree, exists for all tensor models but leads to trivial large limits when the bubbles used as potential do not fall in the class of melonic bubbles (7). In the case of melonic bubbles, Gurau’s degree is the only way to get a expansion and it leads to very strong result, see Gurau’s universality theorem (12) and its extension to almost-melonic bubbles (10).
For tensor models whose potential consist in non-melonic bubbles, it is then possible to recourse to other expansions, which depend on the choice of bubbles (9); (10). Some models for instance resemble matrix models and admit a genus-like expansion (11). Recent efforts in tensor models have thus been organized in two directions:
Finding expansions and large limits for tensor models with non-melonic potentials,
Solving tensor models to all order in their expansions.
The first aspect is new compared to matrix models, but the second aspect is a typical problem in matrix models. Therefore, one can try and extend methods developed for matrix integrals to tensor integrals in order to solve tensor models at all orders in their expansions. Doing so however requires a good understanding of the large limit first, i.e. the first non-trivial order of the expansion.
Progress in identifying and solving large tensor models with non-melonic potentials is very recent and still limited. Some strategies and applications have been presented in the review (9) and in (11); (13); (14).
In this article, we will be interested in solving some tensor models at all orders in their expansions and we will therefore focus on cases which are better understood like the melonic case. One of the preferred strategies to do so in matrix models is by using Schwinger-Dyson equations, also known as loop equations in that context. Schwinger-Dyson equations are derived from the matrix integrals and form a sequence of equations which relate the expectation values of the unitary-invariant polynomials for different values of .
Loop equations can be rearranged as equations on the generating functions of products of traces, , known as the -point functions. Solving the loop equations at all orders in the expansion is then a difficult problem. A fascinating development known as the topological recursion has been put forward for the past ten years as it provides a recursive solution to the loop equations in a beautiful mathematical way (2); (15); (16).
The topological recursion was developed in the context of matrix models and enumeration of combinatorial maps by Eynard (17). It has since been found to be applicable to a much larger set of situations, leading to fascinating results in combinatorics, enumerative geometry (18); (19), quantum topology (20) and theoretical physics (15); (16). The existence of a solution to a system of equations satisfying the topological recursion furthermore comes with additional, interesting properties, such as symplectic invariance and integrability.
Tensor models are based on tensor integrals which generalize matrix integrals. It is thus not unexpected that tensor models also give rise to Schwinger-Dyson equations which generalize those of matrix models (21). They form a system of equations which relate the expectation values of unitary invariant polynomials labeled by different bubbles. Just like the Schwinger-Dyson equations of matrix models can be recast as a system of differential constraints forming (half) a Virasoro algebra, those of tensor models also form an algebra. A set of generators is formed by bubbles with a marked simplex and the commutator is related to the gluing of bubbles forming larger bubbles.
The Schwinger-Dyson equations of tensor models have been used to solve melonic tensor models in the large limit (22), i.e. to calculate the large expectation of polynomials for melonic bubbles in tensor models with melonic potentials. In this case, the Schwinger-Dyson equations simplify drastically and allow to rederive Gurau’s universality theorem. Also in the case of melonic potentials, the Schwinger-Dyson equations have been solved to next-to-leading and next-to-next-to-leading orders and in the double-scaling limit (23). A non-melonic example of Schinwger-Dyson equations, which combine melonic bubbles and other bubbles which are reminiscent of matrix models, has been given in (11). In fact, the equations then reduce at large to those of matrix models with double-trace interactions (24).
In spite of this progress, solving the Schwinger-Dyson equations for tensor models remains extremely challenging. Even for one of the simplest non-trivial models involving melonic bubbles with four simplices, going beyond a few finite orders in the expansion is still out of reach. However, we know some tensor models can be completely solved at all order in the expansion and in fact satisfy the topological recursion. This is because those tensor models are matrix models in disguise. From the tensor with four indices, one can built a matrix with multi-indices so that and then write a matrix model for . Such a tensor model obviously satisfy the topological recursion since its Schwinger-Dyson equations are loop equations of a matrix model. Although quite trivial, this statement can be seen as a proof of principle that tensor models can be solved using modern matrix model techniques.
Instead of facing directly the Schwinger-Dyson equations of tensor models, most of the recent progress in tensor models made use of a correspondence between tensor models and some multi-matrix models with multi-trace interactions. The correspondence is described through a bijection between the Feynman graphs of both sides in (13). In the case of quartic interactions, it is in fact the Hubbard-Stratonovich transformation which transforms the tensor model into a matrix model (25). This transformation was used in (26) to find the double-scaling limit with melonic quartic bubbles.
The correspondence between tensor models and matrix models offers the opportunity to study tensor models indirectly, by using matrix model techniques on the corresponding matrix models. The matrix models arising in the correspondence are however quite complicated as they involve multiple matrices which do not commute. Nevertheless, the matrix model obtained by the Hubbard-Stratonovich transformation applied to the quartic melonic tensor model is simpler and can be used to test the use of matrix model techniques in that context.
This program has been initiated in (27). It was shown that all eigenvalues fall into the potential well without spreading at large . This is in contrast with ordinary matrix models for which the Vandermonde determinant causes the eigenvalues to repel each other and prevent them from simultaneously falling in the potential well. However, in this matrix model, the Vandermonde determinant is a sub-leading correction at large . This result was in fact expected from the direct analysis of the Schwinger-Dyson equations of the quartic melonic tensor model at large (22).
It was further shown in (27) that after substracting the leading value of the eigenvalues and appropriately rescaling the corrections, another matrix model is obtained, involving matrices . The matrix is called the matrix of color . Although the model couples those matrices all together, it only does so at subleading orders in the expansion. At large , it reduces to a GUE for each color independently, with their expected semi-circle eigenvalue distributions. The matrix model action however contains corrections to the GUE at arbitrarily high order in the expansion. It is a natural question to ask whether the expansion of this matrix model satisfies the topological recursion. If so, it would give another example of a topological recursion for a tensor model, with a different origin than the matrix model with multi-indices mentioned above.
To answer that question, a quick look at the action which will be presented below shows that it is a multi-matrix model with matrices for and multi-trace interactions of the form . Interestingly and fortunately, the topological recursion for matrix models with multi-trace interactions was presented by Borot very recently (28). In that situation, it is necessary to supplement the ordinary topological recursion with an infinite set of “initial conditions” (which depend on the model under consideration) called the blobs. They give rise to the notion of blobbed topological recursion whose properties have been analyzed in details in (29) by Borot and Shadrin. The blobbed topological recursion splits the correlation functions into two parts, a polar part which is singular along a cut and is evaluated using the ordinary topological recursion, and the blobs which bring in a holomorphic part of the correlation functions.
We thus find that the quartic melonic tensor model, after some transformation to a matrix model, satisfies the blobbed topological recursion. It is a fairly straightforward application of (28) and (29), with one notable difference: the spectral curve is not connected. This novelty is a direct consequence of the multi-matrix aspect. Indeed, without the multi-trace interactions, the matrices of different colors would not be coupled. Then, correlation functions for any fixed color would be obtained from the ordinary topological recursion with a one-matrix spectral curve. Moreover, the connected correlators involving different colors would vanish. However, upon turning on the multi-trace interactions which couple the colors all together, it becomes necessary to package the fixed-color spectral curves into a global spectral curve which is the union of the fixed-color ones.
Once the correct spectral curve is found, the blobbed topological recursion can be applied fairly generically for all . However, we find that there are properties which depend on . In particular, the matrix model admits a topological expansion, i.e. an expansion in powers of for and a non-negative integer, so For other values of , the expansion is not topological and is actually an expansion in powers of (some orders of the expansion can vanish: for instance, at , it really is an expansion in ). Therefore, when expanding the correlation functions, we might distinguish the contribution which comes from the usual topological expansion in powers of and the others. This is not what we will do. Indeed, the non-topological orders come from the fact that the coupling constants of the model have expansions. We can thus perform all calculations of the correlation functions as in the topological cases, find them as functions of the coupling constants and eventually expand the latter. This is however very awkward and since it is obvious that it can be done in principle, we will instead focus on the topological case .
The organization of the paper is as follows. In Section I, we recall the definition of the quartic melonic tensor model and summarize the analysis of (27). It leads to the matrix model (11) which is the one the topological recursion is then applied to. In Section II, we define the correlation functions. A key point of our article is that we distinguish the local observables which are labeled by sets of colors from to and global observables obtained by packaging the local ones appropriately and which are defined globally on the spectral curve. The loop equations are then derived for the local observables. To find the spectral curve, we solve the loop equations in the planar limit for the 1-point and 2-point functions in Section III. In the same section, we further show that non-topological corrections do not have the singularities one expects for topological expansions.
The spectral curve is described explicitly in Section IV, which then offers the possibility of writing the loop equations on the spectral curve for the global observables. Those loop equations take exactly the same form as in the multi-matrix models of (28) except for the disconnected spectral curve and the fact that the correlation functions do not have a topological expansion for generic .
We then focus on the six-dimensional case in Section V. We derive the linear and quadratic loop equations in Sections V.2 and V.3, following (28); (29) which lead to the blobbed topological recursion in Section V.4. The most important equations are the split of correlation functions into polar and holomorphic parts (170), the recursion for the polar part (180) which takes the form of the ordinary topological recursion, and Borot’s formula (28) for the holomorphic part (182). We then apply the results of (29) to obtain a graphical expansion of the correlation functions at fixed number of points and genus and a graphical expansion of the blobs which are the completely holomorphic parts of the correlation functions.
I From the quartic melonic tensor model to a matrix model
Consider a tensor with entries , i.e. an –dimensional array of complex numbers where all indices range from to . We denote its complex conjugate with entries . Tensor models are based on tensorial invariants, i.e. polynomials in the tensor entries which are invariant under the natural action of on and . The simplest tensorial invariant is the quadratic contraction of with ,
The quartic melonic polynomial of color , denoted for , is defined by
where each sum runs from to . In other words, one can define the matrix to be the contraction of and on all their indices except those in position , i.e.
Then, the melonic polynomial of color simply is .
The quartic melonic tensor model has the partition function
where the are the coupling constants of the model. One can perform a Hubbard–Stratonovich transformation on this integral, (27)
Here is a Hermitian matrix and the matrix is defined by (3). Introducing the matrices that way, the integral over and becomes Gaussian and can thus be performed explicitly. The result is an integral over the matrices . It is convenient to introduce and rewrite the integral as
The matrices can be diagonalized simultaneously, in terms of the eigenvalues of . In (27), only the symmetric model was investigated where all coupling constants are equal, . The saddle point analysis at large (27) then reveals that the eigenvalues all fall into a potential well, without the usual spreading of the eigenvalue density. This is due to the fact that the Vandermonde determinant is here negligible at large . The eigenvalue density is thus where is the extremum of the potential where all the eigenvalues condensate,
That result was indirectly known from the analysis of the tensor model at large in (22), where a resolvent was proposed (in spite of the model not having eigenvalues since it is based on tensors which cannot be diagonalized) and found to be (it is the resolvent associated with the density ).
In the non-symmetric case the conclusion is quite the same as indeed the Vandermonde is still negligible at large . However the eigenvalue density for the matrix of color now depends on the color,
In (27) this result further led the authors to substract to the matrix the amount in order to study the fluctuations. Their idea can be extended to the non-symmetric case and lead to the same type of conclusion. Making the following change of variable,
the partition function rewrites
where and the partition function is following matrix model partition function
As was already the case in the symmetric model, the term of order at least 3 in the matrices , i.e. are subleading at large . To see that, we assume that the trace of is of order , we then use a power counting argument. The quadratic term of the action scales like , as in regular matrix models. Looking at the expansion of
we find that a typical term of order in (12) scales like
The Gaussian term thus scales like as expected in matrix models, but all higher order terms for are subleading.
Ii Loop equations for generic –matrix model with multi-trace couplings
The matrix model (11) is a multi–matrix model with matrices, coupling constants and some specific scalings with and multi-trace interactions which couple the matrices. It is not more difficult to write the loop equations for a generic –matrix model with interactions than for our model. We therefore derive the loop equations for the generic action
whose partition function is defined as a formal integral. It is a formal series in and the couplings .
ii.1 Observables and notations
If is a function of the matrices, its expectation is defined by
The observables of the model are the expectations of products of traces of powers of . Let and introduce
We will often have lists of elements, one for each color, like the list above. It is convenient to think of such data as an integer–valued vector
where with a in position . This provides a natural addition on lists of that type. We will moreover have lists labeled by such vectors. A typical example is the list of exponents of the matrices in the above observables. The list is for the color 1, and so on. It has elements of color . We denote such a list
This way, the observable defined above is simply written
The correlation functions are the generating functions of the above defined observables. They are therefore labeled by . They are obtained by multiplying with , where , and then summing over all values of .
We say that are the variables of color and there are of them. To emphasize the assignment of the variables to different colors, we introduce copies of , respectively denoted for convenience , so that a variable of color belongs to . The correlation functions are thus functions of the variables . These variables are packed into a list denoted . We thus write the correlation functions as
Note that it is symmetric under the permutations which preserve the colors of the variables, i.e. under the group which acts like on the variable of color , for . As a consequence, it turns out that depends on the variables of color as a set and not as a list.
are by definition formal series in and and the couplings . The correlations functions are also formal series in and . At each order of the expansion, they are holomorphic functions at in their variables.
We use similar notations for the cumulants, or connected correlation functions
Here are -dimensional vectors with integral coefficients and we say that is a partition of if and each and we write .
To recover the observables from their generating functions, one can extract the coefficients in front of . We will often have to extract the coefficient of the generating function in front of only a subset of variables. Assume that , with . One can then choose a splitting of the list of variables into a list and a list . For instance, the former consists of the variables, say, for and , while the latter consists of the variables for and . Notice that there is no ordering between the variables of a given color due to the symmetry under color–preserving permutations. Therefore any splitting can be used for each color as long as it has length on one side and on the other side.
Assume that we want to extract the coefficient of with respect to at order . The list of exponents can be stored as a list denoted . Then we will use the shorthand notation
where if .
In the matrix models literature (see for instance (28)), these coefficients are often expressed as integrals around the cut of the planar resolvent. As we will see later, in our model there are several cuts and we can use the same integral representation provided we integrate over the corresponding cuts. We use both notations depending on the context.
If the potential is the same for all matrices, and if is symmetric, then the correlation functions are invariant under color permutations. This means that if acts on by permuting its elements as follows , then
Here we abuse the notation a little bit and denote the list of variables where the sets have been permuted by .
The correlation functions can be described in two ways.
The -point correlation functions we have just described are functions over , i.e. one copy of for each variable and keeping track of the color assigned to each variable (in fact, as expected, those functions have cuts, so it should read minus some cut which depends on the color ). This will be the preferred choice to derive the loop equations and perform actual calculations. We call them local observables, as they require a choice of color assigned to each variable.
Equivalently, one can package them, at fixed , in a tensor where indicates the color of the -th variable, for . Either way, the essential point is that each variable is assigned a color.
As an alternative, and this will be the better choice to understand the structure of the model, we can package all -point functions into a single -point function by allowing the variables to live on any possible color. This means that the variable lives on the union of the copies of , or in fact where is the cut on the color , as we will see after solving for the 1-point functions at large , i.e.
Then the global -point function is defined as
These global observables will be the key objects satisfying the topological recursion, see Section V. Before that, we need to find the planar 1-point and 2-point functions using their local versions, which we do in Section III. This allows us to identify the spectral curve and show that it is disconnected in Section IV, leading to a precise definition of the global correlation functions in Equation (101). Then one can write the loop equations directly in terms of the global correlation functions, Equation (107)
ii.2 Exact resolvent equations
In regular matrix models, the disc equation is the loop equation obtained for all from
In the model we investigate, these equations are replaced by a family of the form
Moreover the form of the multi-matrix interaction of the action gives rise to unusual terms in the loop equations. We then provide a detailed derivation of the corresponding set of discs equations. Computing the above derivative explicitly gives
for all , and .
We turn those equations into an equation on generating functions by multiplying the equation for on matrix with – to emphasize that is the variable for the matrix , we further write . We then sum them over . There are three types of contributions. The first term gives . To rewrite the others, we use the standard trick
where importantly the last term is a polynomial. This way, one finds for the second contribution
with . The last contribution is similar, with the insertion of into the expectations,
The quantity can be rewritten as an extraction of coefficients in the generating function ,
We also introduce
The disc equations thus read, for all and ,
which is not convenient since it loses track of the variable . Therefore, we need to keep track of the part which contains the vector when considering decompositions . Since there is no ordering, we can also assume that is in and redefine the latter to be . In contrast with other for which cannot vanish, the new can be zero. This way, we introduce a family of “almost–partitions”, akin to a partition where a specific, labeled part can be empty. We denote
a list such that and while may be zero. Then,
and similarly for ,
We then have to extract some coefficients out of them.
The disc equation now reads in terms of connected generating functions