]>NUPHB114681114681S0550-3213(19)30167-110.1016/j.nuclphysb.2019.114681The AuthorsHigh Energy Physics – TheoryFig. 1Any level n operator with loop is obtained by the join operation of the level n − 1 operator K with K2.Fig. 1Generalized cut operation associated with higher order variation in tensor modelsH.Itoyamaabcitoyama@sci.osaka-cu.ac.jpR.Yoshiokac⁎yoshioka@sci.osaka-cu.ac.jpaNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, JapanNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP)Osaka City UniversityJapanNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City UniversitybDepartment of Mathematics and Physics, Graduate School of Science, Osaka City University, JapanDepartment of Mathematics and PhysicsGraduate School of ScienceOsaka City UniversityJapanDepartment of Mathematics and Physics, Graduate School of Science, Osaka City UniversitycOsaka City University Advanced Mathematical Institute (OCAMI), 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, JapanOsaka City University Advanced Mathematical Institute (OCAMI)3-3-138, SugimotoSumiyoshi-kuOsaka558-8585JapanOsaka City University Advanced Mathematical Institute (OCAMI), 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan⁎Corresponding author.Editor: Clay CórdovaAbstractThe cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian tensor model. We point out that, by successive choices of appropriate variations, the cut operation generalized this way can generate those operators which do not appear in the ring of the join operation, providing a tool to enumerate the operators by a level by level analysis recursively. We present a set of rules that control the emergence of such operators.1IntroductionThe tensor model has a long history of its research [1–6]. Recent interest has come from an intersection of thoughts on holography and randomness which are realized by several phenomena in quantum gravity in lower dimensions [7–26]. Some of the insights come from the Virasoro structure of the matrix models [27–34] and its combinatorics and finding its counterpart in tensor models in general is expected to pave the way to bring progress in this field. Recent references include [35–48].The Virasoro algebra has a natural extension named w1+∞ algebra whose roles at the 2-dimensional gravity and some integrable models have been well investigated. In particular, the constraints of w1+∞ type coming from the higher order contribution of the variation [31] have turned out to be algebraically independent and nontrivial in some of the matrix models such as the two-matrix model. Here we would like to discuss such higher order contributions arising from the change of the integration measure under the variation.The basic structure of the contributions from the action is the join operation defined by(1.1){K,K′}=∂K∂Aa1a2⋯ar∂K′∂A¯a1a2⋯ar, where K and K′ are arbitrary operators and the summation over the repeated indices is implied. When we choose appropriate keystone operators for K and K′, a block of independent operators called join pyramid is successively generated by the join operation. In other words, the join operation forms a ring whose elements are independent operators and the multiplication is given by (1.1). There are, however, operators which are not involved in the join pyramid and these can not be ignored because the cut operation which underlies the contribution from the variation of the integration measure generates these. These pieces of structure were discovered in [24]. (In contrast, the cut and join operation in one matrix model, namely, r=2, is depicted as going up and down at integer points on one-dimensional half-line.) The cut operation is defined by(1.2)ΔK=∂2K∂Aa1a2⋯ar∂A¯a1a2⋯ar, and corresponds to going up one stair (by one level) in the join pyramid. There is no systematic way to predict when a new operator appears and, in the situation of [24], one can try to discover this only by acting the cut operation on all operators in the join pyramid deep below in the level. In other words, in order to furnish a complete basis of operators, the operations of cut and join must be applied to certain keystone operators which are not given at first. The present work is an attempt to fix this point by proposing (and conjecturing) a systematic way of obtaining a keystone set of operators level by level, where the level of an operator is the number of fields that build the operator.Taking the above mentioned role of the cut and join operation into account, we expect that further thoughts on the cut operation will give a hint to organize the invariant operators in sets according to the variation of the action. Below we will investigate higher order contributions to the constraints from the variation of the integration measure.The enumeration and classification of invariants has been given by means of combinatorial considerations [41], and successfully polished for finite ranks of the gauge groups by means of representation theory [42–44]. As mentioned above, the virtue of the cut and join method is to organize the invariant operators in sets according to the variation of the action and this can potentially bring interesting information on dynamical questions when the action is non-Gaussian. This last point is explained in [24].This paper is organized as follows: In section 2, the higher order variations of the integration measure are considered. In section 3, we discuss the successive choices of the variations. In section 4, we check that our choice of the variation is correct up to the level 6 operators. In section 5, a procedure of generating the operators not included in the join pyramid is described.We owe most of our terminology to those seen in [22–24].2Higher order variationLet us consider the rank r=3 Aristotelian tensor model. Let A be a rank 3 tensor with its component Image 1 and be its conjugate A¯ with Image 2. Each index ai, i=1,2,3 runs over 1,⋯,Ni and is colored respectively in red, green and blue. (For interpretation of the colors in the text, the reader is referred to the web version of this article.) The shift of integration variables of the partition function is defined by A→A+δA and A¯→A¯+δA¯ with(2.1) for arbitrary K. As the line element is given by(2.2) its response under (2.1) is(2.3)dsA+δA2=(dA¯dA)(1+F)2(dAdA¯), where the matrix F of size (2N1N2N3)×(2N1N2N3) is defined by(2.4)F=(∂∂¯K∂¯∂¯K∂∂K∂∂¯K),∂=∂∂A,∂¯=∂∂A¯. The measure is, therefore, transformed as(2.5)[dA][dA¯]⟶det(1+F)[dA][dA¯], where(2.6)det(1+F)=1+∑n=1∞(−1)nntrFn+12∑n,m(−1)n+mnmtrFntrFm+⋯.The cut operator Δ (1.2) corresponds to(2.7)trF=2ΔK. We are interested in the higher order contribution at the response of the measure under the general variation and the gauge-invariant operators which are contained in det(1+F). We use the pictorial representation of the operators as follows: The tensor A (resp. A¯) are denoted by a white circle (resp. a black dot) and the contractions of indices are denoted by colored lines connecting between the white circles and the black dots. For example,(2.8)The connected operators come from trFn. The number of A in the operator under consideration is called level of the operator. In the case of n=2 and higher, the generalized cut operation trFn raises the level and therefore it can be used as the procedure which generates the higher level operators, while the usual cut operation (1.2) lowers the level of the operators by one. In the next section, we see that all connected operators at each level are included in trFn if K is appropriately chosen.3Choice of KIn this section, we seek for the appropriate choice of the variation K to construct all operators. Now let us choose temporarily(3.1) where(3.2)The operator (3.1) is the linear combination of the level 2 operators and all operators in trFn are level k=n. Although trFn consists of ∂∂¯K≤2, ∂∂K≤2 and ∂¯∂¯K≤2, it turns out that only ∂∂¯K≤2 is necessary below. Pictorially,(3.3)(3.4)(3.5) where(3.6)(3.7)(3.8)In the subsections in what follows, we will show that all operators at the first few levels denoted generically by k are included in trFn.3.1Level k=1The only connected operator is Image 14. In the case of n=1, trF is the cut operation itself as mentioned above and we have(3.9)trF=4(N1+N2+N3+(N1N2+N2N3+N3N1))K1. Conversely, we can obtain K1 as the form of, for example,(3.10) Here the trace “Tr” denotes the contraction of all indices. In the pictorial representation, it corresponds to connecting the two open lines with the same color on the both sides.3.2Level k=2All connected operators are listed in the appendix A2 of [24]. Similarly to the case of level k=1, trF2 contains(3.11) The operators Image 17 and Image 18 are also obtained in a similar way.3.3Level k=3All connected operators are listed in the appendix A3 of [24]. At n=3, trF3 contains not only(3.12)(3.13) but also(3.14)The last one K3W cannot be obtained by the join operation. Hereafter, and in [22–24] such operators are called secondary operators. In the original procedure of [22–24], we had to act the original cut operation (1.2) on all of the level k=4 operators in order to discover the secondary operator K3W.3.4Level k=4The independent operators are listed in the appendix A4 of [24]. At n=4, trF4 contains(3.15)(3.16)(3.17)(3.18)(3.19)(3.20)(3.21)(3.22) The two operators K4C and Image 30 denoted by ✓ are secondary operators. We adopt ✓ notation to indicate secondary.3.5Level k=5At level k=5, KXXV, KXXVI and KXXVIII are still missing even with the generalized cut operation of this paper by the choice (3.1). In order to resolve this, let us replace (3.1) by(3.23) In this case, we have, in addition,(3.24) where(3.25)(3.26)(3.27)The subscripts Image 36, Image 37 and Image 38 denote the color which acts trivially. Eq. (3.23) is the linear combination of operators whose levels are greater than or equal to 2. The levels of the operators in trFn are not always equal to n in such case. To be more specific, operators of level k must be included in trFn for some n≤k.Then, one can observe11KXXVIII is equivalent to KXXVI except for the replacement of the coloring.(3.28)(3.29) In addition, KXIV is the secondary operator,(3.30) which we can generate in the generalized cut operation already with (3.1).We now arrive at a conjecture: in order to predict all connected operators at the higher levels, all we need to do is to add the new secondary operator at each lower level to K successively. Then all connected operators at a given level k are included in trFn(n∃≤k).(3.31)4Examination at level 6At level 5, K4C and Image 43 (also Image 44 and Image 45, of course) appear as the new secondary operators. Hence, we choose(4.1) We then have(4.2) where(4.3)(4.4)and(4.5) where(4.6)(4.7)(4.8)We checked by direct inspection that all operators at level 6 are included in trFn(n∃≤6) with K≤4. We plan to elaborate upon this in the future. In particular, we found 10 independent secondary operators at level 6 up to the coloring,(4.9)(4.10)(4.11)(4.12)(4.13)(4.14)(4.15)(4.16)(4.17)(4.18)The secondary operators can be constructed by the appropriate product of the objects Image 64, Image 65, Image 66, Image 67 and so on and the trace “Tr”.5Construction of the secondary operatorsIn the previous section, we have seen that, up to level 6, all operators appear as the constituents of trFn(n≤6). In particular, the secondary operators are constructed by the trace of the product of the ingredients Image 64, Image 65, Image 66, Image 67 and so on. Then a natural question arises as to what combinations of these ingredients the secondary operators consist of. Unfortunately, we do not have an complete answer. However, there may be some rules as to the correspondence between a “word” and each of the secondary operators.The join operation {K,K2} is the following operation in the pictorial representation: one of the white circles (resp. the black dots) in K (resp. K2) are removed and then the open lines with the same color are connected with each other. Thus if an operator can be split into two sub-diagrams by cutting one line per each color, it appears in the join operation pyramid. From this fact, the following corollary follows at once: Since the operators including the loop Image 68 can always be split into two diagrams as shown in Fig. 1, they are obtained by the join operation.Since the existence of the loops in a diagram means that the operator can be obtained by the join operation, the diagram that includes Image 69, Image 70 or Image 71 does not correspond to the secondary operators by construction.Moreover, each of the ingredients Image 64, Image 65 and Image 66 can not be repeated if these have the same color because loops are always generated in such cases. For example, Image 72 generates one green-blue loop,(5.1)This restriction on the repeated use of the ingredients with the same coloring is extended to the objects with subscript Image 74, Image 75, Image 76, such as Image 67. 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