UT-Komaba/13-3, CALT-68-2923

Yuto Ito^{1}^{1}1,
Kazunobu Maruyoshi^{2}^{2}2,
and
Takuya Okuda^{3}^{3}3,

University of Tokyo, Komaba
Meguro-ku, Tokyo 153-8902, Japan
California Institute of Technology
452-48, Pasadena, California 91125, USA

## 1 Introduction

In this paper we make an observation on the two schemes used in the literature for instanton counting in asymptotically locally Euclidean (ALE) spaces. The first scheme uses the enumeration of torus fixed points in the moduli space of instantons on [1], and keeps only contributions that are -invariant in the sense we will explain. The second is based on such enumeration of fixed points in the moduli space of instantons on the resolved -ALE space. Since this space is the minimal resolution of the orbifold , one naturally expects that the results of the two counting schemes are simply related. In fact, in the examples studied in the literature, the two schemes produce identical results.

We point out that in general the two schemes lead to different results. Our experience shows that the difference appears when there are a sufficient number of fundamental/anti-fundamental/bifundamental hypermultiplets and when the sectors with non-zero values of the first Chern class are considered. As far as we know, the appearance of the difference has not been noticed in the literature.

We illustrate our observation by calculating the instanton partition functions of the gauge theory with flavors, i.e., fundamental and anti-fundamental hypermultiplets.
In §2, we apply the counting of orbifolded instantons [2, 3]
and obtain the instanton partition function.
We then consider instanton counting in the resolved spaces in §3.
In §3.1 we first focus on the resolved -ALE space since the instanton counting scheme for this space has been rigorously established [4, 5, 6, 7, 8, 9].^{1}^{1}1See [10] for a review and references on instanton counting in toric spaces.
In §3.2 we analyze the resolved spaces with general by applying the physically motivated method developed in [11].
In §4 we propose simple relations between the instanton partition functions that result from the two schemes.

Our study of instanton counting in ALE spaces was motivated by a version of the AGT correspondence [12]. It was found in [13] that the norm of the Whittaker vector in the super Virasoro algebra is identical to the instanton partition function of the pure theory on . The super Virasoro algebra with a generic central charge is realized by super Liouville theory. The correspondence between super Liouville theory and theories on the -ALE space has been extended in many directions; in particular the gauge theory calculations were performed in various settings in [8, 14, 9, 15, 16, 17, 18, 19, 20, 11, 21, 22]. It was noticed in [8, 9] that the instanton partition function computed on the resolved space has a structure that naturally matches two copies of Liouville theory. This was studied in detail in [18]. We discuss implications of our observation for the correspondence with 2d theories in §5. We also note there that the results in this paper have useful applications for the computation of ’t Hooft line operators.

In the appendix we collect the details of the calculations that support our proposals.

## 2 Counting orbifolded instantons

In this section, we apply the counting scheme based on orbifolded instantons to the gauge theory with fundamental and anti-fundamental hypermultiplets. This scheme is based on [2] and was developed in [3]. (See also [23].) We denote the scalar vevs by , anti-fundamental masses by , and fundamental masses by . We set .

The asymptotic boundary of the space is the lens space , which has non-contractible torsion 1-cycles. Thus the gauge field can have a nontrivial holonomy

(2.1) |

along the generator of , where and . We use the notation .

The instanton partition function on can be obtained by summing the -invariant contributions of the torus fixed points in the moduli space of instantons on . The -action depends on . We can label the fixed points by the same -tuples of Young diagrams used in the instanton counting calculations on , which we review in §A.1.

Although we work with the singular orbifold, it is useful to keep track of the first Chern class of the gauge bundle that becomes well-defined once the space gets resolved. We decompose it as

(2.2) |

where is the flat line bundle with holonomy . Let us introduce the notation

(2.3) |

The precise range of depends on .

The combined data (, ) uniquely determines . The map is given as follows. Given the holonomy data we assign the additive charge mod to the box in the -th column and the -the row of the Young diagram . Denote by the number of elements with value in . Denote also by the number of boxes with charge mod in all the diagrams in . Each contributes to the partition function with the first Chern class given by

(2.4) |

For a given pair , there are infinitely many -tuples of Young diagrams that satisfy the relation (2.4). We denote the set of such by .

Recall that the instanton partition function for takes the form

(2.5) |

where in each term is the product of the weights of the equivariant -action. The associated equivariant parameters are , and . The group is the complexified maximal torus of , where is the gauge group, is the flavor group, and is the Lorentz group. The explicit expression for can be found in §A.1. In the case of the product must be restricted to -invariant weights, i.e., the weights that have a vanishing charge. The Coulomb vev has charge , and have charges . The instanton partition function with holonomy and the first Chern class on is given as

(2.6) |

where . The factor is defined in the same way as , except that the products in (A.3-A.5) are restricted to the invariant weights. We present the explicit calculation up to several orders of in §A.2.

## 3 Instanton counting in the resolved spaces

In this section we review and apply the second scheme for instanton counting in -ALE spaces. Physically, the idea can be summarized as follows. Upon performing the orbifolding by and the minimal resolution, the maximal torus of the Lorentz group descends to an isometry of the resolved space. The resolution also produces homologically non-trivial submanifolds each isomorphic to . The submanifolds intersect with each other at their north and south poles. The torus action has fixed points precisely at the poles. The instanton partition function on the resolved space is obtained by gluing the instanton contributions from the fixed points, taking into account also the bulk contributions to the fluctuation determinant.

The case is mathematically more rigorous; the Poincaré polynomials were computed in [7], while [8, 9] adapted the method for the calculation of the instanton partition functions with the vanishing first Chern classes. For general , we use the method proposed in [11]. When specialized to , the latter method reproduces the results from the first one.

### 3.1 -ALE space

Here we consider the case and denote the first Chern class by . The supersymmetric saddle point configurations in the path integral are abelian and can be diagonalized. Such configurations can be partially classified by the first Chern class of each factor in the unbroken gauge group. We parametrize the first Chern class of the -th subgroup by . The normalization is such that .

The -ALE space has a by which the orbifold singularity are blown up. As mentioned above, the instanton partition function on this space is obtained by intertwining the contributions from two fixed points at the north and south poles of the , multiplied by the so-called -factor which will be introduced shortly. Since each fixed point has a neighborhood locally isomorphic to , its contribution is simply the instanton partition function on [1]. The weights and of the torus action on the local invariant coordinates at the north and south poles are given by

(3.1) |

Furthermore, at these poles the scalar vev gets shifted to (see [11] for an explanation)

(3.2) |

Let us introduce the -factors as follows. First we define

where , , and . Similarly, we define

With these definitions, the -factors for the bifundamental and (anti-)fundamental hypermultiplets are given by

while the -factors for the adjoint hypermultiplet and the vector multiplet are given by

(3.3) | ||||

We then define the total -factor of gauge theory with by taking a product over all multiplets:

(3.4) |

### 3.2 -ALE space

We now turn to the general case. The saddle point configurations are again abelian, and the gauge bundle decomposes into line bundles: . In the resolved space, it is natural to use by Poincaré duality the exceptional divisors () as a basis of the second (co)homology. We expand with the coefficients taking values in . The basis is dual to the basis we used in §2 [2]. The coefficients are related as , where is the Cartan matrix: (), (), and the other elements vanish. We use the notation and .

Let us review the method proposed in [11]. The total partition function on the resolved ALE space splits into the classical, one-loop, and instanton parts. The one-loop part is the fluctuation determinant in the topologically trivial background, and should be universal in all topological sectors once the asymptotic boundary condition is fixed by . Assuming that at least some sectors have the same partition functions as computed by the orbifold method in §2, we can compute the one-loop determinant by restricting to the -invariant factors of the one-loop factor by using the orbifold method explained in §2.

We expect that the total partition function for fixed precisely factorizes into the contributions from the fixed points in the ALE space, each of which can be written as the total partition function on [4]. Such factorization is expected because the total fluctuation determinant for each saddle point configuration should be calculable by the localization formula for the equivariant index of appropriate differential operators in the non-compact case. Examples include the blow-up of [24], and the -ALE space [9] above. Explicitly, for the resolved space we expect the relation

(3.7) |

Here the index labels the fixed points and the equivariant parameters of the torus action and shifts of the vevs are

(3.8) |

We also defined set as

(3.9) | ||||

The relation (3.7) specialized to implies the equality (3.5), with the arguments shifted as in (3.8), and with the -factors given as the ratio of the -orbifolded one-loop factor and the product of two one-loop factors on . The resulting -factors are precisely those given in (3.4). The authors of [11] proposed that this can be generalized to arbitrary ; one can obtain the -factors by computing the ratio of the -orbifolded one-loop factor and the product of one-loop factors on .

In order to write down the explicit one-loop contributions, let us introduce the functions^{2}^{2}2Our convention for agrees with [25, 24, 12] and differs from the one in [11].

where . The symbol for denotes the integer that satisfies and mod . The definition of the one-loop factor on a non-compact space requires a choice, and we choose here to work with the following expressions:

The expression for agrees with [24, 12], and the one for the ALE space is obtained by the orbifold method. Then, the -factor can be defined as

where and .^{3}^{3}3Our convention for agrees with [11]. In order to compare with §3.1, we set and .

We then define the total -factor of theory with :

Then the instanton partition function for the sector with holonomy and the first Chern class on the -ALE spaces is

(3.10) | ||||

We give an explicit calculation for a few orders of in §A.3.

## 4 Proposed relations

For the theory with on the -ALE space, we propose the following relation between the instanton partition functions given in (2.6) and (3.5):

(4.1) | ||||

Here has repeated entries of and

(4.2) |

We recall that denotes the hypermultiplet masses, labels the holonomies at infinity, and parametrizes the first Chern class. We checked our proposal (4.1) for , , and all possible values of holonomies , up to . This is the main result of the paper.

We give examples of the calculations in §A.4. The relation (4.1) predicts that for the two expressions on the right hand side are equal. We also observe that the orbifold partition function is invariant under the sign flip of , , for all the terms we calculated although the two sides of the equality involve sums over different sets of Young diagrams. We expect this property to hold to all orders in .

We also investigated the case. When all of are simultaneously non-negative or non-positive, we found the following relations for the terms we calculated:

(4.3) | ||||

where

(4.4) |

Note the exchange of and in the last line of (4.3); this is immaterial in the case because the two parameters have the same charge . For other values of the first Chern class we have not found conclusively such simple relations. In §A.4, we summarize the calculations that we performed to check (4.3). We also checked, for the terms we computed, that the orbifold partition function (2.6) is invariant under the action of the Weyl group.

## 5 Discussion

Our proposals in the previous section immediately raise the question: are there a pair of distinct two-dimensional theories that naturally correspond to the two schemes? Obvious candidates for are the two theories discussed in [18], the first one involving two copies of Liouville theory and the second involving super Liouville theory. The pair of 2d theories naturally generalizes to [11]. We leave the study of this question for the future.

The two counting schemes we reviewed in §2 and §3 deal differently with the singularities in the instanton moduli space that appear when the exceptional cycles are blown down.
One might try to interpret the relations (4.1, 4.3) as an analog of wall-crossing formulas for equivariant Donaldson invariants (see for example [26] and the references therein).^{4}^{4}4We are very grateful to Y. Tachikawa for suggesting the possible relevance of wall-crossing.

Finally, we note that different instanton counting schemes for ALE spaces can be used to define different ’t Hooft line operators in four-dimensional gauge theories. The correspondence [27] between instantons on a multi-centered Taub-NUT(ALE) space and monopoles with Dirac singularities can be used to compute the expectation value of a ’t Hooft operator on various geometries [28, 29, 30]. Our findings can be adapted for the calculation of ’t Hooft operators, which precisely match the predictions from Liouville theory [31].

## Acknowledgements

We thank Giulio Bonelli, Yuji Tachikawa, Masato Taki, Alessandro Tanzini, and Futoshi Yagi for valuable discussions and comments. The research of Y.I. is supported in part by a JSPS Research Fellowship for Young Scientists. The research of K.M. is supported in part by a JSPS Postdoctoral Fellowship for Research Abroad. T.O. is supported in part by the Grant-in-Aid for Young Scientists (B) No. 23740168 and by the Grant-in-Aid for Scientific Research (B) No. 20340048.

## Appendix A Explicit calculations

### a.1 Instantons on

We review briefly the instanton partition function [1] for gauge theory with on . We denote an -tuple of Young diagrams by . The instanton number is given by the total number of boxes in the -tuple of Young diagrams . The contribution of the fixed point labeled by takes the form

(A.1) |

This includes contributions from the vector multiplet as well as the anti-fundamental and fundamental hypermultiplets; they are denoted by , and respectively. These are obtained by taking products of weights of the equivariant action , whose parameters are .

Let be a Young diagram where is the height of the -the column. We set when is larger than the width of the diagram . Let be its transpose. For a box in the -the column and the -th row, we define its arm-length and leg-length with respect to the diagram by , . We then define

(A.2) |

We set . The contribution from the vector multiplet is [32]

(A.3) |

Note that in is negative when the box is inside the diagram but outside the diagram . The contributions from the fundamental and the anti-fundamental hypermultiplets are given by

(A.4) | |||

(A.5) |

where for the box at the position [33].

Then the instanton partition function is denoted by

(A.6) |

where is the one-instanton factor and the coefficient is the sum of contributions (A.1) with . For example, the coefficient for and is calculated as