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Partial chord diagrams and matrix models

In this article, the enumeration of partial chord diagrams is discussed via matrix model techniques. In addition to the basic data such as the number of backbones and chords, we also consider the Euler characteristic, the backbone spectrum, the boundary point spectrum, and the boundary length spectrum. Furthermore, we consider the boundary length and point spectrum that unifies the last two types of spectra. We introduce matrix models that encode generating functions of partial chord diagrams filtered by each of these spectra. Using these matrix models, we derive partial differential equations - obtained independently by cut-and-join arguments in an earlier work - for the corresponding generating functions.


Introduction
A partial chord diagram is a special kind of graph, which is specified as follows. The graph consists of a number of line segments (which are called backbones) arranged along the real line (hence they come with an ordering), with a number of vertices on each. A number of semi-circles (called chords) arranged in the upper half plane is attached at a subset of the vertices of the line segments, in such a way that no two chords have endpoints at the same vertex. The vertices which are not attached to chord ends are called the marked points. A chord diagram is by definition a partial chord diagram with no marked points. Partial chord diagrams occur in many branches of mathematics, including topology [14,30], geometry [9,10,3] and representation theory [16].
To each partial chord diagram c one can associate canonically a two dimensional surface with boundary Σ c , see Figure 1. Moreover, as discussed in [56,12,2,7], the notion of a fatgraph [42,43,44,45] is a useful concept when studying partial chord diagrams. A fatgraph is a graph together with a cyclic ordering on each collection of half-edges incident on a common vertex. A partial linear chord diagram c has a natural fatgraph structure induced from its presentation in the plane. The partial chord diagram c is characterized by various topological data, and we will consider the following five types of data, introduced in [2] and [7].
• The number of chords k in c and the number of backbones b in c.
• Euler characteristic χ and genus g.
Let χ and g denote respectively the Euler characteristic and genus of Σ c , which are related as follows Denoting by n the number of boundary components of Σ c , the Euler relation can be written as As a unification of the boundary length spectrum and the boundary point spectrum, we consider the boundary length and point spectrum introduced in [7]. Let us here recall its definition.
• Boundary length and point spectrum. The data {g, k, l; {b i }, {n i i i }} stores more detailed information on the distribution of marked points on each boundary component. One can determine the previous two kinds of spectra from the boundary length and point spectrum by forgetting the partitions of marked points on the boundary cycles. It is known that the enumeration of chord diagrams is intimately related to matrix models and cut-and-join equations [4,5,6,20,38]. In this paper, the enumeration of partial chord diagrams labeled by the boundary length and point spectrum with the genus filtration is studied using matrix model techniques. Let where Λ P and Λ L are external matrices [29] of rank N , and the normalization factor Vol N is defined in (2.4). In this matrix integral representation, the counting parameter u (i1,...,iK ) is identified with the trace of the corresponding product of external matrices (1.14) In Theorem 2.13 we show that This matrix integral representation provides a new, matrix model proof of the cut-and-join equation found by combinatorial means in [7]. The cut-and-join equation can be written as where M is the second order partial differential operator in variables u i i i (see Theorem 3.11 for details). This cut-and-join equation can be regarded as the evolution equation in the variable y, and its formal solution reads (1.17) Expanding the operator e yM around y = 0, one determines the number of connected partial chord diagrams N g,k,l ({b i }, {n i i i }) iteratively from this formal solution. The cut-and-join equation is a powerful method to systematically count partial chord diagrams of a given length and point spectrum.
In this work we also generalize the above analysis to non-oriented analogues of partial chord diagrams. By non-oriented partial chord diagrams we mean diagrams with all chords decorated by a binary variable, which indicates if they are twisted or not. When associating the surface Σ c to a non-oriented partial chord diagram, twisted bands are associated along the twisted chords as indicated in Figure 3. This construction leads to 2 k orientable or non-orientable surfaces associated to one particular partial chord diagram with k chords, if we consider all possible assignments of twisting or untwisting of k bands. In the non-oriented case the Euler characteristic is defined as follows.
• Euler characteristic χ. The Euler characteristic of the two dimensional surface Σ c is defined by the formula where h is the number of cross-caps. The Euler relation holds With this setup, the enumeration of non-oriented partial chord diagrams can be considered analogously to the orientable case. Let N h,k,l ({b i }, {n i i i }) denote the number of connected non-oriented partial chord diagrams with the cross-cap number h, k chords, the backbone spectrum {b i }, l marked points, and the boundary length and point spectrum n i i i . The generating function F (x, y; We also define the generating function of the numbers of connected and disconnected non-oriented partial chord diagrams In Theorem 4.5 we show that this generating function can be expressed as a real symmetric matrix integral with two external symmetric matrices Ω P and Ω L where the normalization factor Vol N (R) is defined in (4.8), and H N (R) is the space of real symmetric matrices of rank N . The parameter u (i1,...,iK ) is identified with a trace of the external matrices via the formula (1.23) Using this matrix integral representation of the generating function, one can again prove the cut-and-join equation, established independently by combinatorial arguments in [7] ∂ ∂y where M is a second order partial differential operator in the variables u i i i . The details of the differential operator M and the matrix model derivation of the cutand-join equation are presented in Theorem 4.10.
1.1. Motivation: RNA chains. One important motivation to study partial chord diagrams in this and the preceding work [2,7] is a complicated problem of RNA structure prediction in molecular biology, which we now shortly review. An RNA molecule is a linear polymer, referred to as the backbone, that consists of four types of nucleotides: adenine, cytosine, guanine, and uracil, denoted respectively A, C, G, and U. The backbone is endowed with an orientation from 5'-end to 3'-end, and the primary sequence is the sequence of nucleotides read with respect to this orientation. Between nucleotides hydrogen bonds are formed, resulting in the so-called Watson-Click pairs involving A − U or G − C nucleotides; in addition Wobble pairs U − G can be formed. The set of base pairs formed by such hydrogen bonds is referred to as the secondary structure. 1 Prediction of the secondary structure from the primary sequence is an outstanding problem that was initiated by the pioneering work of Michael Waterman [57] and has been studied intensively for last three decades.

H type
Kissing hairpin Topologically, we can represent the base pairings for a given RNA structure by a partial chord diagram as follows. The backbone is represented as a disjoint union of horizontal straight line segments (arranged along the real line in the plane), one for each backbone component, and each nucleotide is represented as a marked point on this union of line segments. The base pairs are represented by chords in the upper-half plane attached at two marked points corresponding to the bonded pair of nucleotides.
Note that a partial chord diagram has genus zero if no two of its chords cross each other. If however such crossings exist, then the structure is referred to as a pseudoknot, and its genus is non-zero. Considerable number of pseudoknot structures have been observed, e.g. tRNAs, RNAseP [31], telomerase RNA [53] and ribosomal RNAs [28]. According to the online database "RNA-strand" half of the known structures form pseudoknots [13]. There are various kinds of pseudoknots classified by the topology of the RNA [12], referred to as e.g. H-type [1], kissing hairpin [17,51], etc.
In recent years, a combinatorial description of RNA structures in terms of linear chord diagrams has been developed in a series of works [41,56,55,12,11,8,4,5,2,49,46]. However, a large class of reasonable energy-based models that predict the secondary structure including pseudoknots are NP complete [32,1], and a fully satisfactory energy model for RNA, including pseudoknot structures, has not been established yet.
In the search of a realistic energy function for RNA structures with pseudoknots, the boundary length and point spectrum should provide a useful tool that includes more detailed information about the location of marked points. In standard algorithms developed by Waterman [58], Nussinov et al. [40], Zucker and Stiegler Figure 5. Partial chord diagrams unveil the difference in the topological structure of RNA molecules.
[61], etc., dynamic programming (DP) has been used to predict most likely secondary structures. Indeed, in famous algorithms such as [60,25], the (loop-based) energy in each configuration of RNA is considered. In these algorithms, the most probable secondary structure is determined as the minimum free energy configuration, and to make them more efficient the statistical mechanical ensemble (i.e. the partition function algorithm) is implemented [34]. The application of these algorithms, which include pseudoknot structures stratified by γ structures, was studied in [50,49]. Most of the energy functions essentially respect the boundary point and length spectra independently. In order to improve the energy model for RNA structure prediction with pseudoknots, it would be useful to explore energy parameters for more realistic and efficient energy function on the basis of the boundary length and point spectrum.
1.2. Plan of the paper. This paper is organized as follows. In Section 2 we construct Hermitian matrix models with external matrices, which encode generating functions of orientable partial chord diagrams labeled by the boundary point spectrum (in Subsection 2.1), the boundary length spectrum (in Subsection 2.2), and the boundary length and point spectrum (in Subsection 2.3). All these constructions are established by the correspondence between chord diagrams and Wick contractions via the Wick theorem. The matrix model encoding the boundary length and point spectrum is given in Theorem 2.13. In Section 3 we derive partial differential equations for matrix integrals found in Section 2. These partial differential equations coincide with the cut-and-join equations found combinatorially in [2,7]. The cut-and-join equation for partial chord diagrams labeled by the boundary length and point spectrum is determined in Theorem 3.11. Section 4 is devoted to the analysis of non-oriented analogues of the results obtained in Section 2 and 3. In Subsection 4.1 we find real symmetric matrix models with external matrices, that encode generating functions of both orientable and non-orientable partial chord diagrams. The non-oriented analogue of the matrix integral from Theorem 2.13 is given in Theorem 4.5. Non-oriented analogues of cut-and-join equations from Section 3 are determined in Theorem 4.10. In Appendix A we derive a partial differential equation from Proposition 4.7 for a real symmetric matrix integral with external matrices. In Appendix B we prove Lemma 4.9.

Enumerating partial linear chord diagrams via matrix models
The enumeration problem of partial chord diagrams with respect to the genus filtration has been reformulated in terms of matrix integrals. Matrix model techniques for enumeration of the RNA structures with pseudoknots have been developed in a series of papers [41,56,55], and independently in [4,5,6]. Subsequently the analysis involving boundary point and length spectra of partial linear chord diagrams has been conducted in [2,7]. In this section we develop a new perspective on this problem and construct a matrix model that enumerates partial chord diagrams labeled by the boundary length and point spectrum.
We introduce the generating function 2 for the numbers N g,k,l ({b i }, {n i }) The generating function for the numbers N k,b,l ({b i }, {n i }) of connected and disconnected partial chord diagrams arises in the usual way from the exponent In the following we rewrite the generating function Z P (x, y; {s i }; {t i }) as a Hermitian matrix integral. To this end, we consider first Gaussian averages over Hermitian matrices.
This quantity is called the Wick contraction. By definition, a multiple Wick contraction is a product of the Gaussian average of each Wick contracted pair.
It follows from the definition (2.3) that Gaussian averages of an odd number of matrix elements vanish. On the other hand, Gaussian averages of an even number of matrix elements are non-zero, and can be computed using the Wick theorem [15,43,37], as we now recall. Consider an ordered sequence Let P k denote a set of matchings by k Wick contractions among the 2k matrix elements in the above sequence. P k is isomorphic to the following quotient of groups Here the elements of the permutation group S 2k permute 2k matrix elements. The factors S k of G E act by permuting k Wick contractions and (S 2 ) k swaps matrix elements in each Wick contracted pair. The Wick theorem implies the following result.
(2.6) 2.1.1. Chord diagrams and Wick contractions. Let c be a chord diagram. We now recall the explicit relation between a surface Σ c associated to a chord diagram c and k-matchings or Wick contractions in the Gaussian average. To illustrate this correspondence we depict chord ends on backbones in Σ c as trivalent vertices that consist of upright and horizontal line segments, see Figure 6. This correspondence is specified by the following four points C1-C4. This assignment encodes matrix multiplication of matrix elements corresponding to adjacent chord ends on the backbone.
which corresponds to a backbone with i chord ends, the following quantity is assigned to the bottom edge of the backbone Thus, a backbone with i chord ends corresponds to a single trace of the i'th power of M , namely N TrM i . C4: The Wick contraction between M αj βj and M α ′ j ′ β ′ j ′ corresponds to a band connecting two chord ends. Each Wick contraction imposes a constraint δ αj β ′ j ′ δ α ′ j ′ βj on matrix indices assigned to the edges of the chord ends matched by the Wick contraction. The above rules imply the following bijective correspondence between matchings by k Wick contractions in the Gaussian average on one hand, and chord diagrams that consist of b i backbones with i chord ends on the other hand, see Figure 7. The Wick contractions (2.6) in W C N ({b i }) replace all matrix elements M 's by products of δ's, and summing over matrix indices along a boundary cycle one finds a factor of N corresponding to each boundary cycle in a chord diagram. Therefore the overall N dependence following from the above rules amounts to assigning N b−k+n factor to the term W C N ({b i }), corresponding to a chord diagram with backbone spectrum {b i } and n boundary cycles. Combing the Wick theorem and this bijective correspondence between matchings by k Wick contractions in the Gaussian average W C N ({b i }) and the set of chord diagrams with backbone spectrum {b i }, the following proposition follows.
(2.9) 2.1.2. Partial chord diagrams and Wick contractions. We now generalize the above bijective correspondence to partial chord diagrams. Let c be a partial chord diagram. On the boundary cycles of the surface Σ c we add additional marked points, which correspond to those marked points on c which are not chord ends. These marked points are represented by external matrices Λ P of rank N in the Gaussian average. The rules P1-P5 below provide the correspondence between partial chord diagrams with backbone spectrum {b i } and matchings with k Wick contractions in the Gaussian average. just as in C2.
corresponding to a backbone which is an ordered sequence of v j chord ends and w j marked points, we assign to the bottom edge of this backbone. It follows that the trace is assigned to this backbone. P5: The Wick contraction between M αj βj and M α ′ j ′ β ′ j ′ corresponds to a band connecting two chord ends, and it is represented in the same way as specified in C4.
where we introduced the reverse Miwa times If there are n i boundary components containing i marked points, then one finds a trace factor (TrΛ i P ) ni in the corresponding term in the Gaussian average (2.14), see Figure 9. Therefore, for partial chord diagrams with the backbone spectrum {b i } and the boundary point spectrum Therefore, from Wick theorem and the above bijective correspondence between partial chord diagrams and matchings by Wick contractions, one finds the following proposition.
Proposition 2.5. The Gaussian average (2.14) is the generating function for the numbers N k,b,l ({b i }, {n i }) of partial chord diagrams with the backbone spectrum {b i } and the boundary point spectrum where the summation is constrained by Using this proposition, we consider the full generating function Z P N (y; Since the contribution from a partial chord diagram is invariant under permutations of its backbones, the full generating function can be rewritten as a sum over all backbone spectra {b i } of the terms It follows that Performing the summation over b i 's, one finds that the full generating function is given by the matrix integral This matrix integral and Z P (x, y; {s i }; {t i }) in equation (2.2) are identified by a change of variables. Since the reverse Miwa time for i = 0 yields r 0 = 1 automatically, we need to introduce the parameter t 0 by the following change of variables As a result, we find the main theorem in this subsection.
Theorem 2.6. The generating function (2.2) is given by the matrix integral (2.17), We introduce the following generating function of these numbers 3 denote the generating function of chord diagrams labeled by the boundary length spectrum In the same way as the generating function Z P (x, y; {s i }; {t i }) in (2.2), the generating function for the numbers N k,b ({b i }, {p i }) of connected and disconnected chord takes form  Figure 10. Figure 10. Decorating a chord diagram with new marked points for partitions.
In order to construct a Gaussian matrix integral which counts this type of chord diagrams we introduce another external matrix Λ L , which is an invertible rank N matrix that keeps track of new marked points. We introduce a new model model based on the following rules L1-L5, in which Wick contractions in the Gaussian average correspond bijectively to decorated chord diagrams.  Figure 11. Bijective correspondence between decorated chord diagrams and matchings of Wick contractions.
Repeating the same discussions as in the previous subsection, one finds that every chord diagram with the backbone spectrum {b i } corresponds to matchings with k = ib i /2 Wick contractions, which arise from the following Gaussian average where we introduced Miwa times  We also consider the full generating function for the numbers This full generating function is given by the sum of Gaussian averages (2.21), and in consequence by the following Hermitian matrix integral Comparing this matrix integral and the generating function Z L N (y; {s i }; {q i }) in equation (2.20), we arrive at the main theorem of this subsection. Under this specialization, the matrix integral (2.24) reduces to In Subsection 3.2 we derive the cut-and-join equation for this specialized model, and show the agreement with the cut-and-join equation found by combinatorial means in [2].
2.3. The boundary length and point spectrum and the unified model. So far we have discussed separately the enumeration of chord diagrams and partial chord diagrams labeled by the boundary point spectrum and the boundary length spectrum. In this subsection we consider a unification of these two kinds of spectra, which is referred to as the boundary length and point spectrum. This unified spectrum was introduced and analyzed by cut-and-join methods in [7]. In what follows we construct a matrix model that encodes this new spectrum, and in Subsection 3.3 we show how the cut-and-join equation found in [7] follows from this matrix model.
where l, k, and b obey The enumeration of partial chord diagrams decorated by the boundary length and point spectrum can also be expressed in terms of Gaussian averages over Hermitian matrices. To this end we again make use of extra marked points, just as in the previous section (concerning the length spectrum to mark the separation between marked points on the backbone, counted by the index i i i), see Figure 13. Indeed, the boundary length and point spectrum also encodes the length spectrum, simply as the number K of partitions of marked points on boundary cycles.
To represent the boundary length and point spectrum, we introduce two external matrices Λ P and Λ L . In order to faithfully represent the ordering between marked points and partitions on each boundary cycle, we assume that these two external matrices do not commute [Λ P , Λ L ] = 0.
The correspondence between partial chord diagrams with the backbone spectrum {b i } and matchings by Wick contractions in a Gaussian average is given by a combination of the previous rules C1, C2, P2, L2, P4, L4, and L5. We summarize this correspondence in Table 1.

A chord Wick contraction M M
Based on these rules, one finds a bijective correspondence between partial chord diagrams with the backbone spectrum {b i } and matchings by Wick contractions in the Gaussian average where in order to represent trace factors Λ P and Λ L we introduced the generalized Miwa times If a partial chord diagram c contains a boundary cycle labeled by i i i = (i 1 , . . . , i K ), one finds the following trace factor in the corresponding Wick contractions in Finally, combining Propositions 2.5 and 2.8, we obtain the key proposition.
Repeating the same combinatorics as in the previous subsections, we find the main theorem of this section.

Cut-and-join equations via matrix models
In Section 2 we discussed matrix models that enumerate partial chord diagrams filtered by the boundary point spectrum, the boundary length spectrum, and the boundary length and point spectrum. In this section we derive partial differential equations for these matrix models, and show that they agree with the cut-andjoin equations found in [2,7]. To derive these differential equations, it is useful to introduce the following matrix integral. By the following specializations of this matrix integral one finds matrix integrals discussed in Section 2 Z P N (y; where I N is the rank N identity matrix. The matrix integral (3.1) satisfies the following partial differential equation.
Proposition 3.2. The matrix integral Z N (y; {g i }; A; B) obeys a partial differential equation where the trace in the second term is defined, for rank N matrices X and Y , as Proof. By a shift M = X − y −1/2 BA, the matrix integral (3.1) can be rewritten as

(3.3)
Here H N is the space of shifted matrices X = M + y −1/2 BA with M ∈ H N . The invariance of this matrix integral under the infinitesimal scaling X αβ → (1 + ǫ)X αβ leads to a constraint equation where the first term N 2 comes from the measure factor, as dX → (1 + N 2 ǫ)dX.
Here we have defined the unnormalized average for an observable O(X) one finds that the constraint equation (3.4) yields It follows from (3.3) that the last two derivatives in the expression above can be replaced by 2y∂/∂y, so that the partial differential equation (3.2) is obtained.
Remark 3.3. In the above proof of the constraint equation (3.4) we considered the infinitesimal scaling X αβ → (1 + ǫ)X αβ . More generally, matrix integral (3.3) is invariant under infinitesimal shifts It is known that for the matrix integral without external matrices A and B this symmetry yields the Virasoro symmetry, and in particular the scaling X αβ → (1 + ǫ)X αβ is related to the Virasoro generator L Vir 0 [22,19]. This corollary implies the following theorem.
Theorem 3.5. Let L 0 and L 2 be the differential operators 6 (3.7) The matrix integral Z P N (y; {s i }; {r i }) obeys the cut-and-join equation The formal solution of this cut-and-join equation, which gives the matrix integral Z P N (y; {s i }; {r i }), is iteratively determined from the initial condition at y = 0, This theorem follows from the lemma below by rewriting the derivative Tr∂ 2 /∂Λ 2 P in the partial differential equation (3.6).
For a function f ({r i }) of the reverse Miwa times r i , the derivative Tr∂ 2 /∂Λ 2 P can be rewritten as Proof. Consider the derivative ∂/∂Λ Pβα of r i , Then the derivative Tr∂ 2 /∂Λ 2 P of the function f ({r i }) is re-expressed as This coincides with the right hand side of (3.10).
The cut-and-join equation for the rescaled matrix integral (2.18) yields where L is given by This cut-and-join equation agrees with the partial differential equation in Theorem 1 of [2], where it was proven combinatorially by the recursion relation for the number of partial chord diagrams. 7 This completes the proof of Theorem 2.6.
7 For the Grothendieck's dessin counting, a similar cut-and-join equation was found in [27].

3.2.
The boundary length spectrum. In Subsection 2.2 we showed that the matrix integral Z L N (y; {s i }; {q i }) in (2.24) enumerates chord diagrams labeled by the boundary length spectrum. By the specialization where the Miwa times q i are defined in equation (2.22). Obviously, for A = 0 the partial differential equation (3.2) does not hold. Instead we consider the matrix integral (2.26) obtained by the specialization s i = s The same matrix integral can be obtained by the specialization and thus Z N (y; (3.14) Then from (3.2) we obtain a partial differential equation for Z L N (y; s; {q i }). Corollary 3.7. The matrix integral Z L N (y; s; {q i }) obeys the partial differential equation This corollary implies the following theorem.
Theorem 3.8. Let K 0 and K 2 be the differential operators (3.16) The matrix integral Z L N (y; s; {q i }) obeys the cut-and-join equation The cut-and-join equation (3.17) was combinatorially proven in Theorem 2 of [2] for the generating function Z L (x, y; s; {q i }) in (2.19), and thus Theorem 2.9 for s i = s is reproved.
The claim of Theorem 3.8 is proven by rewriting the derivative Tr∂ 2 /∂Λ 2 L in the partial differential equation (3.15) using the following lemma. Tr Proof. By acting ∂/∂Λ L on the Miwa time q i one obtains Adopting this relation via the chain rule applied to the Λ L derivatives, one finds that This coincides with the right hand side of (3.19). where the generalized Miwa times u (i1,...,iK ) are defined in (2.31) From (3.2) we obtain a partial differential equation for Z N (y; {s i }; {u i i i }).
Corollary 3.10. The matrix integral Z N (y; {s i }; {u i i i }) obeys the partial differential equation This corollary implies the following main theorem of this section.
Theorem 3.11. Let M 0 and M 2 be the following differential operators with respect to parameters u i i i , where The formal solution of this cut-and-join equation, which gives the matrix integral Z N (y; {s i }; {u i i i }), is iteratively determined from the initial condition at y = 0, The partial differential equation (3.23) agrees with the cut-and-equation obtained combinatorially in Theorem 1.1 of [7]. Here we prove this theorem by rewriting the derivative in the second term of the partial differential equation (3.21), taking advantage of the following lemma.
Lemma 3.12. For a function h({u i i i }) of the generalized Miwa times u i i i , the derivative in the second term of the partial differential equation (3.21) can be rewritten as follows

25)
Proof. By the chain rule, the derivative on the left hand side of (3.25) is rewritten as follows Each of the coefficients yields and In this way one obtains the right hand side of (3.25).
As a corollary of Theorem 3.11, one finds the cut-and-join equation for the 1backbone generating function. 8

26)
obeys the cut-and-join equation The solution is iteratively determined by

Non-oriented analogues
In this section we consider the enumeration of both orientable and non-orientable (jointly called non-oriented) partial chord diagrams [2,7]. To this end we generalize the matrix models introduced in Section 2. In Subsection 4.1, matrix models for the boundary point spectrum, the boundary length spectrum, and the boundary length and point spectrum are introduced, based on the corresponding Gaussian matrix integrals over the space of rank N real symmetric matrices. Subsequently, in Subsection 4.2, we derive cut-and-join equations for the generating functions of non-oriented partial chord diagrams, using analogous methods as those discussed in Section 3.

4.1.
Non-oriented analogues of the matrix models. In this subsection we generalize matrix models found in Section 2, in order to enumerate both orientable and non-orientable partial chord diagrams [2,7].
and introduce generating functions and Generating functions of connected and disconnected partial chord diagrams are related by   9 The Gaussian matrix integral over the space of real symmetric matrix is also referred to as the Gaussian orthogonal ensemble [21,35].
This Wick contraction consists of two terms, which encode the corresponding fatgraph as follows. The first term 1 N δ αǫ δ βγ is the same as in the Hermitian matrix integral (2.5), and it can be identified with an untwisted band in the two dimensional surface Σ c associated to the partial chord diagram c. The second term 1 N δ αγ δ βǫ in (4.9) relates opposite matrix indices compared to the first term and can be identified with the twisted band in Σ c , see Figure 14. Hence, for the real symmetric Gaussian average, the correspondence rules C4, P5, L5 in Section 2 are replaced by the following rules [24,54,52,26,39,23].
N5: The Wick contraction between M αjβj and M α ′ j ′ β ′ j ′ corresponds to a band or a twisted band connecting two chord ends. Each Wick contraction imposes either the constraint δ αj β ′ j ′ δ α ′ j ′ βj or the constraint δ αj α ′ j ′ δ βjβ ′ j ′ for matrix indices assigned to edges of chord ends matched by Wick contractions. Figure 14. Wick contraction and the untwisted / twisted bands.
In order to construct matrix models that enumerate non-oriented partial chord diagrams, we introduce two external real symmetric matrices matrices Ω P and Ω L Ω P = Ω T P , Ω L = Ω T L , which take account of the fact that boundary cycles of non-oriented partial chord diagrams are not endowed with a specific orientation. To model the index structure i i i of the boundary length and point spectrum correctly, we assume these two matrices do not commute [Ω P , Ω L ] = 0.
Furthermore, we introduce corresponding generalized Miwa times This assignment implies the bijective correspondence (analogous to the orientable case discussed earlier) between non-oriented partial chord diagrams and Wick contractions, which is summarized in Table 2.
Using this correspondence, generating functions Z P (x, y; {s i }; {r i }), Z L (x, y; {s i }; {q i }), and Z(x, y; {s i }; {u i i i }) can be re-expressed in terms of matrix integrals. Repeating the same combinatorial arguments as for the orientable case in Section 2, we obtain the following three theorems. Table 2. The correspondence between partial chord diagrams and operator products in the real symmetric matrix integral.

Partial chord diagram
Gaussian average A chord end on a backbone Ω −1 L M A marked point on a backbone Ω P An underside of a backbone where r i are reverse Miwa times where q i are Miwa times   The cut-and-join equation that follows from this reduced model is derived in the next subsection.  4.2. Non-oriented analogues of cut-and-join equations. We derive now nonoriented analogues of cut-and-join equations discussed in Section 3. Analogously to the Hermitian matrix integral in (3.1), we introduce the following matrix integral.
Definition 4.6. Let U = U T and V = V T be rank N invertible symmetric matrices. We define a formal real symmetric matrix integral with parameters y, (4.21) The matrix integrals discussed in the previous subsection follow from this matrix integral by specializations where I N is the rank N identity matrix.
In Appendix A we prove the following proposition.
Proposition 4.7. The matrix integral Z N (y; {g i }; U ; V ) in (4.21) obeys the partial differential equation where A is a matrix such that From this proposition and by the specializations (4.22), (4.24), and (4.25) we find partial differential equations for the corresponding matrix integrals. For the specialization (4.23), because of U = 0 (and thus A = 0), the partial differential equation (4.26) cannot be reduced to a partial differential equation.
where Λ P and Λ L are matrices satisfying From this corollary we obtain non-oriented analogues of cut-and-join equations, by rewriting the derivatives with respect to the external matrices Λ P and Λ L in Corollary 4.8 in terms of Miwa times r i in (4.12), q i in (4.16), and u i i i in (4.10) as follows.

(4.31)
Then the derivatives with respect to Λ P and Λ L in Corollary 4.8 are rewritten as The proof of this lemma is given in Appendix B. By combining Corollary 4.8 with Lemma 4.9 one arrives at the following theorem.  where Assuming certain initial conditions at y = 0, one can iteratively determine the above matrix integrals by solving the cut-and-join equations  The cut-and-join equations (4.33) agree with those of [2,7]. Finally, from Theorem 4.10 we find non-oriented analogues of cut-and-join equations for 1-backbone generating functions. In this appendix we prove the Proposition 4.7, which states that the matrix integral obeys the partial differential equation where A is a matrix that satisfies U = A + A T . Proof. In order to differentiate the matrix integral Z N (y; {g i }; U ; V ) with respect to A we use the identities ∂U αβ ∂A γǫ = δ αγ δ βǫ + δ αǫ δ βγ , where X = M + y −1/2 V U . Using this shifted variable X one obtains Here H N (R) is the space of shifted matrices X = M + y −1/2 V U with M ∈ H N (R). It follows that 1 On the other hand, by differentiating the matrix integral Z N (y; {g i }; U ; V ) with respect to y one obtains the same expression as (A.2) times N/2, from which the partial differential equation (A.1) is obtained.
Appendix B. Proof of Lemma 4.9 In this appendix we prove the Lemma 4.9, which states that for functions f ({r i }), g({q i }), and h({u i i i }) of Miwa times r i in (4.12), q i in (4.16), and u i i i in (4.10), we find 1 4N Tr where L 0,1,2 , K 0,1,2 , M 0,1,2 , and M ∨ 2 are defined in (3.7), (3.16), (3.22), and in Lemma 4.9. Here the matrices Λ P and Λ L satisfy Ω P = Λ P +Λ T P and Ω L = Λ L +Λ T L . Proof. First we prove (B.1). Consider the derivative ∂/∂Λ P of the reverse Miwa time r i Using these relations the left hand side of (B.1) is rewritten as This agrees with the right hand side of (B.1). Second, we prove (B.2). Using the identity one finds that Then the left hand side of (B.2) yields This agrees with the right hand side of (B.2).   Then one obtains the right hand side of (B.3).