Probing M-theory with tetrahedron instantons

The duality between type IIA superstring theory and M-theory enables us to lift bound states of D0-branes and n parallel D6-branes to M-theory compactified on an n-centered multi-Taub-NUT space TN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{TN} $$\end{document}n. Accordingly, the rank n K-theoretic Donaldson-Thomas invariants of ℂ3 are connected with the index of M-theory on ℂ3 × TN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{TN} $$\end{document}n. In this paper, we extend this connection by considering intersecting D6-branes. In the presence of a suitable Neveu-Schwarz B-field, the system preserves two supercharges. This system is T-dual to the configuration of tetrahedron instantons which we introduced in [1]. We conjecture a closed-form expression for the K-theoretic tetrahedron instanton partition function, which is the generating function of the D0-D6 partition functions. We find that the tetrahedron instanton partition function coincides with the partition function of the magnificent four model for special values of the parameters, leading us to conjecture that our system of intersecting D6-branes can be obtained from the annihilation of D8-branes and anti-D8-branes. Remarkably, the K-theoretic tetrahedron instanton partition function allows an interpretation in terms of the index of M-theory on a noncompact Calabi-Yau fivefold which is related to a superposition of Kaluza-Klein monopoles. The dimensional reduction of the system allows us to express the cohomological tetrahedron instanton partition function in terms of the MacMahon function, generalizing the correspondence between Gromov-Witten invariants and Donaldson-Thomas invariants for Calabi-Yau threefolds.


Introduction
One of the most striking consequences of string duality is the discovery of an elevendimensional interacting quantum theory known as M-theory [2,3], whose low-energy limit is the eleven-dimensional supergravity [4].Despite the lack of an intrinsic formulation of Mtheory in terms of its fundamental degrees of freedom, there has been diverse and convincing evidence of its existence.Especially, ten-dimensional type IIA superstring theory at finite string coupling is conjectured to be equivalent to M-theory compactified on a circle.All objects in type IIA superstring theory, including fundamental strings, D-branes, and NS5branes, can be naturally lifted in M-theory.
Over the past few decades, the conjectured duality between type IIA superstring theory and M-theory has led to many interesting results by comparing the computations of BPS quantities from two viewpoints.One prominent setup is the system of D0-branes probing n parallel D6-branes in type IIA superstring theory on S 1 t × C 3 × R 3 [5][6][7].This configuration is supersymmetric if a suitable Neveu-Schwarz B-field is turned on [8].The generating function of D0-D6 partition functions with varying number of D0-branes can be interpreted as the instanton partition function of the (6 + 1)-dimensional noncommutative U(n) super Yang-Mills theory on S 1 t × C 3 [9,10], and can be computed exactly [11].Mathematically, this instanton partition function computes rank n K-theoretic Donaldson-Thomas invariants of C 3 [12][13][14].Upon lifting to M-theory, the D0-branes become Kaluza-Klein (KK) modes of the graviton, and the D6-branes become KK monopoles, which can be described geometrically by an n-centered multi-Taub-NUT space TN n [2,[15][16][17][18].This means that the instanton partition function is related to the partition function of M-theory on S1 t × C 3 × TN n , or equivalently the index of M-theory on the noncompact Calabi-Yau fivefold C 3 × TN n .This sharp prediction has been confirmed for n = 1 in [10] and later for n ≥ 2 in [19]. 1  We notice that on both sides of this beautiful relation, there are four preserved supercharges, but two of them are not used in performing the exact computations.This observation opens an opportunity to extend the setup by breaking half of the supercharges while still keeping the ability to analyze exactly.We introduce additional D6-branes that intersect with the original D6-branes in type IIA superstring theory, producing defects of real codimension two in the worldvolume theory of the original stack of D6-branes.The whole system involves D0-branes and four stacks of intersecting D6-branes on S 1 t × C 4 × R.This is in fact T-dual to the brane configuration that realizes tetrahedron instantons, which we introduced in [1].In the presence of a suitable constant Neveu-Schwarz B-field, two supercharges are preserved.In [1], we have defined and calculated the cohomological/Ktheoretic/elliptic tetrahedron instanton partition function.The result is expressed as a statistical sum over a collection of random plane partitions.In this paper, we conjecture that the K-theoretic tetrahedron instanton partition function Z ⃗ n admits an elegant closed-form expression in terms of the plethystic exponential (3.15, 3.16, 3.17), where all the notations will be explained in sections 2 and 3 of the paper.Interestingly, it only depends on the Ω-deformation parameters q a and the instanton counting parameter p, but is independent of the Coulomb parameters associated with the positions of D6-branes.
It is fascinating to connect our system with the magnificent four model, which can be constructed using D0-branes probing pairs of D8-branes and anti-D8-branes [23,24].The partition function Z MF of the magnificent four model is given by a sum over random solid partitions.From the mathematical viewpoint, Z MF computes the K-theoretic version of the equivariant integral over the Hilbert scheme of points on C 4 and higher rank analogues thereof, or equivalently Donaldson-Thomas invariants of the noncompact Calabi-Yau fourfold C 4 [25][26][27][28].A simple closed-form expression for Z MF was conjectured in [23,24].We find that (1.1) coincides with certain specialization of Z MF .This matching strongly hints that the annihilation of D8-branes and anti-D8-branes can leave behind a system of intersecting D6-branes.
Although our system seems to be a special case of the magnificent four model from the perspective of partition functions, it is worthy studying independently.One important reason is that we have very little control over the lift of D8-branes and anti-D8-branes to M-theory, and it is utterly bewildering to provide a non-perturbative interpretation of Z MF .It was suspected in [23] that the magnificent four model is related to the compactification of a mysterious thirteen-dimensional theory M 13 on a Taub-NUT space: However, M 13 is not needed in our case since the M-theory lift of D6-branes is better understood.Indeed, we find a decomposition of Z ⃗ n , making it possible to interpret Z ⃗ n in terms of the index of M-theory on X ⃗ n , where X ⃗ n is obtained by coalescing four copies of Clearly, X ⃗ n is not a product of two lower-dimensional manifolds, and the compactification of M-theory on X ⃗ n preserves two instead of four supercharges.We also study the dimensional reduction of our system by shrinking the radius of S 1 t to zero.Accordingly, the K-theoretic tetrahedron instanton partition function is reduced to its cohomological counterpart.We find that the result can be expressed in terms of the MacMahon function.This allows us to identify the tetrahedron instanton partition function with certain gluing of A-model topological string partition functions, generalizing the celebrated Gromov-Witten/Donaldson-Thomas correspondence for Calabi-Yau 3-folds [5,6].The rest of the paper is organized as follows.In section 2, we describe our D0-D6 brane system in type IIA superstring theory.In section 3, we give the definition and the exact expression of the K-theoretic tetrahedron instanon partition function, both in an instanton-expansion form and in a closed form.In section 4, we discuss the connection of our system with the magnificent four model.In section 5, we explain the definition and the result of the index of M-theory.In section 6, we explore the relation between the Ktheoretic tetrahedron instanon partition function and the index of M-theory.In section 7, we consider the dimensional reduction of the system and discuss the relation with A-model topological strings, generalizing the correspondence between Donaldson-Thomas invariants and Gromov-Witten invariants.We end in section 8 with a summary and a discussion of interesting open questions.
Shortly after submitting this paper to arXiv, we received a paper by Fasola and Monavari [29] where a rigorous proof of our conjecture (1.1) was provided.They also proposed a mathematical definition of the moduli space of tetrahedron instantons in the language of Quot scheme of a singular variety.

D0-D6 brane system
We start by describing the D0-D6 brane system in type IIA superstring theory.This is T-dual to the string theory configuration of tetrahedron instantons [1].We will mostly follow the notations and conventions used there.The ten-dimensional spacetime is S 1 t × C 4 × R. The coordinates on S 1 t and R are taken to be x 0 and x 9 , respectively.We denote the set of coordinate labels of four complex planes by 4 = {1, 2, 3, 4} , and the complex coordinate on C a ⊂ C 4 by For each we define the complex three-plane (2.4) The complex plane that is complementary to We introduce k D0-branes along S 1 t and n A D6 A -branes along S 1 t × C 3 A for all A ∈ 4 ∨ .We will use the notation to capture the number of D6-branes of different stacks.In total, there are four stacks of intersecting D6-branes.A summary of the configuration is in Table 1.
The presence of the D0-branes and the D6-branes breaks the ten-dimensional Lorentz group SO (1,9) down to a∈4 U (1) a , where U (1) a rotates the complex plane C a .We turn on a constant Neveu-Schwarz B-field along C 4 [8,30], (2.7) As analyzed in [1], when 2π 3 Strings the original string theory vacuum is unstable and supersymmetry is completely broken, but the effect of tachyon condensation restores a part of supersymmetry.After rolling to the true vacuum, there are two unbroken supercharges.
We can have two different ways to understand this setup.The first one is the viewpoint of conventional field theory.Without loss of generality, we can take the physical spacetime to be S 1 t × C 3 123 .The bound states of D0-branes and D6 123 -branes realize noncommutative instantons in the (6 + 1)-dimensional super Yang-Mills theory, while the effect of the other D6-branes is to produce defects of real codimension two.Alternatively, we can adopt the interpretation of generalized field theory, which can be constructed by merging several ordinary field theories across defects [31][32][33].The spacetime of a generalized field theory is generally not a manifold, but a union of several intersecting components.In our case, the spacetime is (2.9) The fields and the gauge groups on different components can be different, and the matter fields living on the intersection In [1], we described this generalized field theory in the framework of noncommutative field theory.
We are interested in the low-energy effective theory T ⃗ n,k on the worldline of D0-branes.The field content is summarized in Table 2.The quantization of D0-D0 strings gives rise to an N = 4 U(k) vector multiplet and three massless N = 4 chiral multiplets Φ a , a ∈ A in the adjoint representation of U(k).There is a superpotential Here the U (n A ) symmetry, which is supported by the D6 A -branes, appears as a global symmetry from the perspective of D0-brane quantum mechanics.Notice that the N = 4 supersymmetry preserved by the D0-branes and the D6 A -branes is different from that by the D0-branes and the D6 B -branes for A ̸ = B ∈ 4 ∨ , and only an N = 2 supersymmetry is shared [1].Hence, T ⃗ n,k is an N = 2 supersymmetric quantum mechanics.In the language of N = 2 superspace, the Lagrangian is given by where D t is the covariant time derivative, and the parameter r is positive when the condition (2.8) is satisfied.The corresponding quiver diagram is presented in Figure 1.After integrating out the auxiliary fields, we can obtain the scalar potential which leads to the moduli space M ⃗ n,k of the supersymmetric ground states where ) and the U(k) symmetry acts on B a in the adjoint representation and In [1], M ⃗ n,k is called the moduli space of tetrahedron instantons with instanton number k.
When ⃗ n = (n, 0, 0, 0), all D6-branes are parallel, and the system preserves four instead of two supercharges.It provides a string theory realization of K-theoretic Donaldson-Thomas theory on C 3 [5][6][7].The moduli space M (n,0,0,0),k coincides with the moduli space of torsion free sheaves This moduli space is also isomorphic to a Quot scheme Quot k C 3 (O ⊕n ), which reduces to the Hilbert scheme Hilb k C 3 of k points on C 3 when n = 1 [34].For the general case of M ⃗ n,k , we sketched a geometric interpretation in the framework of Quot schemes in [1].A careful mathematical treatment was given recently in [29].

Tetrahedron instanton partition function
The D0-D6 partition function in the Ω-background is computed by the refined Witten index of the supersymmetric quantum mechanics T ⃗ n,k , with Here H ⃗ n,k is the Hilbert space of T ⃗ n,k on S 1 t , F is the fermion number operator, β is the circumference of S 1 t , Q and Q are two conserved supercharges, T A,α , α = 1, • • • , n A are the Cartan generators of U (n A ), and J a is the generator of the U (1) a symmetry of T ⃗ n,k originating from the symmetry rotating C a .The supercharges Q and Q commute with Thus, we can at most have three linearly independent combinations of J a that are Q-closed.as a contour integral [1], where the contour is specified by the Jeffrey-Kirwan residue prescription [35], and The genuine poles of the contour integral are classified by a collection of colored plane partitions where each π (A,α) is a plane partition, and |⃗ π| is the total number of boxes of ⃗ π.Every plane partition π (A,α) can be visualized as a set of boxes sitting in Z 3 + , so that there can be at most one box at (x, y, z), and a box can occupy (x, y, z) only if there are boxes in (x ′ , y, z) , (x, y ′ , z) , (x, y, z ′ ) for all 1 The K-theoretic tetrahedron instanton partition function Z ⃗ n is defined by packaging Z ⃗ n,k into a generating function, where the instanton counting parameter is taken to be −p for convenience.In terms of the characters we can express Z ⃗ n as an infinite sum over a collection of colored plane partitions [1], where the constant part that is independent of t A,α and q a is subtracted.The plethystic exponential operator PE is defined by and the dual operator ∨ acts on a character by We would like to point out that we define Z ⃗ n,k in such a way that makes formulas simple and puts four stacks of D6-branes on equal footing.This definition can be different from those used in [7,29,36] by a sign.However, the difference can be reconciled by a redefinition of p.
The expression (3.12) looks quite intricate, and appears to depend on all the parameters in the definition (3.1).In particular, each term in (3.12) depends on all the Coulomb parameters t A,α except for an overall factor associated with the center U(1) c of A∈4 ∨ U (n A ).However, we find that the dependence on all the Coulomb parameters t A,α disappears if we combine the contributions associated with all ⃗ π with fixed |⃗ π|.This means that Z ⃗ n does not depend on the positions of D6-branes at all.Inspired by [10,13,19,23,24], we propose that (3.12) has a simple closed-form expression where Here ∆ ⃗ n depends on ⃗ n and the Ω-deformation parameters, We have checked (3.15) up to k = 4 for various ⃗ n with n A ≤ 3. We believe that a rigorous proof of (3.15) can be given by extending the derivation in [37].We will not explore it in this paper, but it would be interesting to do so. 2hen ⃗ n = (n, 0, 0, 0), we can recover the previous result obtained in [10,13,36], where F n (q 1 , q 2 , q 3 , q 4 , p) = q 1 q 2 q 1 q 3 q 2 q 3 q 1 q 2 q 3 q n 2 4 p q which further reduces to the case when n = 1 as F 1 (q 1 , q 2 , q 3 , q 4 , p) = q 1 q 2 q 1 q 3 q 2 q 3 q 1 q 2 q 3 q 1 2 4 p q We would like to emphasize that F n has two classes of singularities: q a → 1, a = 1, 2, 3 and p → q . The first class is simply the standard thermodynamic limit in which D0-branes are allowed to move freely along the ath complex plane rather than being confined to its origin by the Ω-deformation.The second class is more interesting, and the existence of such singularities implies that extra flat directions appear in the finite coupling regime.The limit q 4 → 1 is smooth, and is the Calabi-Yau condition for C 123 .Thus there are five flat complex directions in total.An explanation for the singularity structure of F n can be given from the M-theory perspective.

Connection with the magnificent four model
The simple formula (3.16) is indicative of a connection with a closely related model, the so-called magnificent four model [23], which can be constructed by a system of k D0-branes probing N D8-branes and N anti-D8-branes on S 1 t × C 4 × R.This configuration preserves two supercharges when a suitable constant B-field is turned on [8].The low-energy effective theory is described by an N = 2 supersymmetric quantum mechanics T MF k .The partition function Z MF of the magnificent four model is the generating function of the refined Witten index Here where H MF k is the Hilbert space of T MF k on S 1 t , g is an element of the global symmetry group U (N ) × U (N ) ⊂ U ( N | N ) associated with D8-branes and anti-D8-branes, J a rotates the ath complex plane C a ⊂ C 4 , and (q 1 , q 2 , q 3 , q 4 ) ∈ U (1) 3 ⊂ SU (4) are the Ω-deformation parameters.The partition function Z MF also admits a closed-form expression [23,24], where s depends only on Coulomb branch parameters associated with U (N ) × U (N ).
Comparing (4.3) and (3.16), we immediately find that Z MF (q 1 , q 2 , q 3 , q 4 , s = ∆ ⃗ n , p) = Z ⃗ n (q 1 , q 2 , q 3 , q 4 , p) , (4.4) which generalizes the known agreement for ⃗ n = (n, 0, 0, 0) [24].This suggests that the annihilation of D8-branes and anti-D8-branes in the Ω-background happens not only when they coincide but also when they are separated by a suitable distance.Furthermore, a system of D6-branes will be produced after the annihilation.It is interesting that the configuration of D6-branes can be either parallel or intersecting, depending on the distance of the D8-branes and the anti-D8-branes when the annihilation happens.

The index of M-theory
In the previous sections, we take the perspective of type IIA superstring theory.In this section, we shift our attention to M-theory and introduce the index of M-theory on X, where X is a noncompact toric Calabi-Yau fivefold.
The eleven-dimensional Minkowski vacuum of M-theory preserves an eleven-dimensional Majorana spinor supercharge with 32 real components, among which only a part is preserved when M-theory lives in S 1 t × X.According to the branching rule for Spin(1, 10) ⊃ Spin (10) the eleven-dimensional supercharge can be decomposed as In general, only two supercharges Q and Q associated with representations 1 ±5 are unbroken.Meanwhile, from the branching rule only the time translation component H = −i∂ t of the eleven-dimensional momentum remains.It follows from the eleven-dimensional supersymmetry algebra that the effective one-dimensional theory on S 1 t enjoys an N = 2 supersymmetry algebra, and U(1) R is identified with the R-symmetry.We would obtain a local N = 2 supersymmetry in one dimension if X was a compact Calabi-Yau fivefold.Although the component fields of the one-dimensional gravity multiplet are not dynamical, they generate constraints and cannot be neglected [38].On the contrary, since X of interest are noncompact, the gravitational effects are turned off, and the low-energy dynamics is captured by an N = 2 supersymmetric quantum mechanics SQM (X) on S 1 t .
Furthermore, since X is a toric manifold, the U(1) 5 isometries lead to extra global symmetries in SQM (X) with generators J I , I = 1, • • • , 5. None of J I commutes with Q and Q, but there are four linearly independent combinations of J I that commute with Q and Q.This is consistent with the fact that only U(1) 4 ⊂ U(1) 5 preserves the nowhere-vanishing holomorphic top-form Ω of X.
We define the index of M-theory on X to be the refined Witten index of SQM (X), where H(X) is the Hilbert space of SQM (X) on S 1 t , F is the fermion number operator, and β is the circumference of S 1 t .The constraint on the fugacities v I decouples a U(1) ⊂ U(1) 5 symmetry that is not Q-closed.Equivalently, Z X can be viewed as the partition function of M-theory on a fiber bundle over S 1 t with fiber X.Based on the standard lore of Witten index, Z X only receives contributions from the ground states of SQM (X), which are constant modes along S 1 t .Furthermore, if Z X gets contributions only from supergravity fields, including the eleven-dimensional graviton g µν , a Majorana gravitino Ψ µ , and a three-form potential A µνρ , then a simple expression for Z X was derived by Nekrasov and Okounkov [13,14], where F X is computed by a geometric formula and the formula (5.7) reproduces the result obtained earlier by Nekrasov in [10], where we drop the subscript of F X when X = C 5 for simplicity.The same result was also derived in the framework of twisted eleven-dimensional supergravity in [39].It is evident that When X is a product manifold of the form Y × W, where Y and W are respectively a Calabi-Yau threefold and a Calabi-Yau twofolds, the holonomy group SU(5) is further reduced to SU(3) × SU(2), and we get two extra preserved supercharges associated with the representation (1, 1) in the decomposition and its conjugation.In total, the number of preserved supercharges is four, which matches that in type IIA superstring theory with all D6-branes are parallel.Nevertheless, these two supercharges are unnecessary in the definition and the computation of the index of M-theory.
It is worthy to note that for certain types of X, especially when X = Y × W, the contributions from other ingredients of M-theory cannot be neglected in order to get a sensible index of M-theory [20,22].It is beyond the scope of this paper to derive a complete formula of Z X for general X.

Relating two counting problems
So far we have discussed two counting problems, one in type IIA superstring theory and the other in M-theory.In this section, we explore the relation between them.

Parallel D6-branes
Let us start with the old examples in which all D6-branes are parallel.
According to the duality between type IIA superstring theory and M-theory, the D0branes are lifted to the KK modes of the graviton carrying momentum along the eleventh direction S 1 R of M-theory, where R is the radius of S 1 R .Meanwhile, the parallel D6-branes become KK monopoles.The geometry in the transverse direction is an n-centered multi-Taub-NUT space TN n , which approaches In the limit R → ∞, the metric on TN n reduces to that on the A n−1 -type ALE space C 2 /Z n , which is the resolution of the orbifold C 2 /Z n .The KK modes of the graviton and the KK monopoles carry electric and magnetic charges under A µ = G µ,10 , respectively.It is natural to expect that the instanton partition function multiplied by a perturbative contribution should coincide with the index of M-theory on C 3 × TN n , In the limit of infinite Taub-NUT radius R → ∞, TN n becomes C 2 /Z n , and correspondingly the U(1) factor in the gauge group U(n) decouples.Assuming that Z can be computed exactly in the supergravity limit, one has provided that the parameters on both sides are properly identified [10,13].
Notice that there are four independent parameters on both sides of (6.3).On the type IIA superstring theory side, three parameters encode U (1) 3 ⊂ U (1) 4 isometries of C 4 and which is equal to F . As shown in [24], the n terms in the sum correspond to n isolated fixed points of C 2 /Z n , with the local weights under the torus action determined by the equivariant index of Dirac operators on C 2 and C 2 /Z n , (6.10)

Intersecting D6-branes
Now we would like to generalize what we have reviewed to the case involving intersecting D6-branes.Since an honest calculation from the perspective of M-theory is not available, it is our aim to examine whether a matching of two counting results is possible.We first fix the perturbative contribution.If one assumes the equality of all D6-branes, regardless of the spatial directions A ∈ 4 ∨ in the worldvolume, then the only reasonable choice is Naively, the expression of F ⃗ n (q 1 , q 2 , q 3 , q 4 , p) given in (3.16) becomes singular not only when q a → 1 but also when p → ∆ Therefore, it appears that we have six flat complex directions, which is too many for an interpretation in terms of M-theory.We have observed that the tetrahedron instanton partition function coincides with a specialization of the partition function of the magnificent four model.In fact, it is difficult to provide a Mtheory reinterpretation of Z MF , partly because of the poor understanding of the M-theory lift of D8-and anti-D8-branes.It was speculated that a mysterious thirteen-dimensional theory should exist to reproduce the result of Z MF [23].

Relation with Gromov-Witten invariants
A fascinating correspondence between Donaldson-Thomas invariants and Gromov-Witten invariants was discovered in [5,6].According to this correspondence, the Donaldson-Thomas invariants of C 3 , which is computed by the partition function of D-instantons probing parallel D5-branes along C 3 in type IIB superstring theory, is equivalent to the A-model closed topological string theory on C 3 .To explore possible generalizations of this correspondence, we consider the dimensional reduction of our system.More precisely, we perform a T-duality of the brane configuration shown in Table 1 in type IIA superstring theory along S 1 t .The cohomological tetrahedron instanton partition function can be deduced from its K-theoretic counterpart by introducing q a = e βεa and taking the limit β → 0 with ε a and p fixed, The closed-form expression (1.1) of Z ⃗ n then leads to the following closed-form expression for Z coh ⃗ n , where In terms of the MacMahon function M 3 (p) [40], we have The MacMahon function appears in the all-genus A-model topological string partition function Z top C 3 of C 3 [41][42][43].In the equivariant form, where g s is the string coupling constant and F g is the free energy of genus g.Therefore, we can express (7.5) as a product of computes the equivalent volume of the moduli space M ⃗ n,k , which was interpreted geometrically as a Quot scheme in [1].Therefore, the identity (7.5) gives a correspondence between invariants that can be extracted from Z coh ⃗ n and Gromov-Witten invariants on C 3 A .

Conclusions and open questions
In this paper, we study bound states of D0-branes and intersecting D6-branes in type IIA superstring theory.We compute the K-theoretic tetrahedron instanton partition function and conjecture a closed-form expression.It serves as an interpolation of the K-theoretic Donaldson-Thomas invariants of C 3 and the partition function of the magnificent four model.What is more, we find a decomposition of the partition function, making it possible to reinterpret the result non-perturbatively through the index of M-theory on a noncompact Calabi-Yau fivefold.After dimensional reducing the system, we discuss the connection between the cohomological tetrahedron instanton partition function and Gromov-Witten invariants.
There are several interesting open questions to be answered.It would be desirable to find the solution in the eleven-dimensional supergravity that describes the superposition of KK monopoles.In the limit of large radius of M-theory circle, the solution should lead to the Calabi-Yau fivefold X ⃗ n that we are interested in.Then one may try to evaluate the index of M-theory on X ⃗ n and test our conjectured expression (6.16).One may start with the simplest nontrivial situation in which there are two stacks of intersecting D6-branes in type IIA superstring theory.The corresponding M-theory is in S 1 t × C 2 × Y, where Y is a Calabi-Yau threefold obtained from merging two Taub-NUT spaces.The papers [22,[44][45][46] may be relevant for this direction.
In this work, we exclusively dealt with the case involving D0-branes and D6-branes in the flat spacetime.It is natural to ask what happens if the spacetime background is more general [47] or if there are D2-and D4-branes in type IIA superstring theory.In M-theory, these extra branes correspond to M2-branes and M5-branes, and the computation of the index of M-theory becomes more complicated [13,20,21].
From the mathematical viewpoint, an exciting problem is to provide a rigorous formulation of the connection between the invariants computed by the cohomological tetrahedron instanton partition function and Gromov-Witten invariants.This may also turn out to be useful for the study of Donaldson-Thomas/Pandharipande-Thomas correspondence for Calabi-Yau fourfolds [48].
In [49,50], the cohomological Hall algebra (COHA) of Calabi-Yau threefolds was constructed.Recently it was found that the moduli spaces of spiked instantons [32,51] also admits an action of COHA [52].As there are several resemblances between tetrahedron instantons and spiked instantons, it would be also interesting to investigate whether there is an action of certain generalization of COHA on moduli spaces of tetrahedron instantons.

. 10 )
which arises from the dimensional reduction of the ten-dimensional U (k) super Yang-Mills theory.The quantization of D0-D6 A and D6 A -D0 strings leads to a massless N = 4 chiral multiplet Φ A , transforming in the bifundamental representation of U (k) × U (n A ). )

Figure 1 .
Figure 1.The quiver diagram of the low-energy theory T ⃗ n,k on k D0-branes probing n A D6 Abranes for A ∈ 4 ∨ .Each solid line represents an N = 2 chiral superfield, while each dashed line represents an N = 2 Fermi superfield.

Table 1 .
Bound states of D0-branes and four stacks of intersecting D6-branes in type IIA superstring theory.Here − and • indicate the worldvolume and the transverse directions of the D-branes, respectively.
It is remarkable that we are able to decomposeF ⃗ n into A∈4 ∨ n A terms of the form F 1 , and F ⃗ n can be naturally combined with F pert e βε 2 , e βε 3 , e βε 4 , p .