PPreprint typeset in JHEP style - HYPER VERSION
Notes On U(1) Instanton Counting On A l − ALESpaces
Haitao Liu
Department of Mathematics and Statistics,University of New Brunswick, Fredericton, Canada, E3B 5A3andDepartment of Applied Mathematics,Hebei University of Technology, Tianjin, P.R.China, 300130Email: [email protected]
Abstract:
In this note, we investigate the detailed relationship between the orbifoldpartition counting and the (l-quotient, l-core) pair counting. We show that the orbifoldpartition counting is exactly the same as the (l-quotient, l-core) pair counting.
Keywords: A l − ALE space, U(1) Instantons, Orbifold partitions, Quotient and Corepartition, ADHM description, Equivariant cohomology, l-quotient, l-core. a r X i v : . [ h e p - t h ] N ov ontents
1. Introduction 12. Orbifold partitions 23. Instantons on A l − ALE space 4 C A l − ALE space 5
4. Regular and fractional instantons 6
5. Conclusion 8A. Colored partitions , charged partitions and blended colored partitions 8B. Quotients and cores for Young diagrams 9C. Blended partition, l-quotient and l-core 11
1. Introduction
In recent years, people have made much progress on connecting the four-dimensional su-persymmetric gauge theories with the two dimensional conformal theories, e.g. [1, 2, 3].One of the first examples was found by Nakajima [4]. Further, in [5], Witten and Vafadiscussed this in four-dimensional supersymmetric gauge theory based on the ALE spaceby using the twisting method. Nakajima’s analysis shows that the partition functions ofN=4 four-dimensional U(N) gauge theories on the A l − ALE space is related to the affinecharacter of (cid:99) su ( l ) N . Since the boundary of A l − ALE space is the lens space S / Z l , so theU(N) gauge theory can approach some non-trivial flat connection at infinity [6, 7]. Suchflat connections are labeled by the N-dimensional representation of Z k λ ∈ Hom ( Z , U ( N )) , which can be decomposed into irreducible representations R a of Z l as λ = (cid:88) a R a N a , – 1 –here N a are the integers satisfying (cid:88) a N a dim R a = N. Thus once we choose boundary condition λ at infinity we get a vector-valued partitionfunction whose components are of the form Z λ ( v, τ ) = χ (cid:99) su ( l ) N ˆ λ ( v, q ) . (1.1)Witten and Vafa showed that the partition function of N=4 U(N) super-Yang-Mills theoryon the ALE spaces is related to the generating function of the Euler number of the instantonmoduli spaces [5]. Later, in [8], the authors employed equivariant cohomology techniquesto calculate the partition function explicitly. Recently Dijkgraaf and Su(cid:32)lkowski introducethe orbifold partitions and show how to get the affine character of (cid:99) su ( l ) for the U(1) gaugetheory [7].In this note, we investigate the detailed relationship between the orbifold partitioncounting and the (l-quotient, l-core) pair counting. In section 2, we review the definitionof orbifold partitions introduced in [7]. In section 3, we review the basic structures of U(1)instantons on A l − ALE spaces. In section 4, we investigate the detailed relation betweenthe orbifold partition counting and the (l-quotient, l-core) pair counting and show that theorbifold partition counting is exactly the same as the (l-quotient, l-core) pair counting.
2. Orbifold partitions
In this section we briefly review the first type of orbifold partition counting in [7].We know that an ideal of functions I = { f ( x, y ) } ⊂ C [ x, y ] generated by a set ofmonomials x i y j for i, j ≥ λ in sucha way that a box ( m, n ) ∈ λ iff x m y n / ∈ I . For example, the following Young diagramshows the I which is generated by y , xy , x y, x . y y y xy x , For the A l − ALE space we will consider ideals of functions having definite transfor-mation properties under the action ( x, y ) → ( ωx, ¯ ωy ) , (2.1)where ω = e πi/l . All the monomials with the same transformation property form a periodicsub-lattice of Z . In particular, the set of invariant monomials is called the invariant sector;all others are called twisted sectors.In [7] the authors defined two type orbifold partitions. We only review the first type:– 2 – efinition 2.0.1 (Orbifold partition of the first type) It is an ordinary two dimen-sional partition, with some subset of its boxes distinguished; these distinguished boxes, aspoints in Z lattice, correspond to monomials with a definite transformation property underthe action ( x, y ) → ( ωx, ¯ ωy ) , where ω = e πi/l ; we define a weight of such a partition asthe number of these distinguished boxes. In [7] the authors also pointed out that this kind of orbifold partitions are related to statesof a Fermi sea.We define the generating functions of the generalized partitions of the first type forALE spaces of A l − type by Z lr,orbifold = (cid:88) first type orbifold partitions q , (2.2)where r = 0 , · · · , l − r specifies the power of ω in the action (2.1).The orbifold partitions of the first type can be identified with a blended partition [7].Let us review the definition of the blended partition [9, 10] . Consider a colored partition (cid:126)R = { R , · · · , R l − } with charges p i , i = 0 , · · · , l −
1. The blended partition K = ( K i ) i ∈ N is defined by the set of integers { p + K m − m | m ∈ N } = { k ( R i,m − m + p i ) + i | i = 0 , · · · , l − , m ∈ N } . (2.3)The total number of boxes of K is | K | = l (cid:88) i =1 (cid:18) l | R i | + l p i + ip i (cid:19) − ( l + 1) p − p , (2.4)where p = (cid:80) l − i =0 p i . In Appendix C, we’ll show that (cid:126)R is just the l-core of K, where K isplaced at point ( p, Z l − partitions of the first type are in one-to-onecorrespondence with the blended partition K obtained from k-colored partition (cid:126)R , suchthat • an orbifold Z l − partition has the same shape as the corresponding blended partition, • a weight of an orbifold partition (as given by the number of distinguished boxes itcontains) is specified by the total weight of a state of k fermions related to (cid:126)R ,the authors get the following formula about the number of distinguished boxes | K | − (cid:80) l − i =0 ip i + ( l +1) ln + ( r +1) r l = (cid:80) l − i =0 ( l | R ( i ) | + l p i ) − ( l +1) p − p + ( l +1) ln + ( r +1) r l , (2.5) The definition of blended partition can be found in Appendix A. The relationship between the blendedpartition and free fermions can be found in [7]. – 3 –here the total charge is p = nl + r, ≤ r ≤ l −
1. Then the partition function (2.2)becomes Z lr,orbifold = (cid:88) { R (0) , ··· ,R ( l − } q (cid:80) i | R ( i ) | (cid:88) { p i | (cid:80) l − i =0 p i = p = ln + r } q l p i − ( l +1) p − p
22 + ( l +1) ln r +1) r l = q l/ η ( q ) l (cid:88) n , ··· ,n l − q (cid:80) i ( n i − n i n i +1 )+ r + n r − r (2.6)= q l + r l − r η ( q ) χ (cid:99) su ( l ) r (0) , where η ( q ) = q (cid:81) ∞ n =1 (1 − q n ) is the Dedekind eta function and n , · · · , n l − ∈ N l − aredefined by ( p , · · · , p l − ) = n + n + rn − n + n n − n + n ... n − n l − + n l − n − n l − ∈ Z l . .
3. Instantons on A l − ALE space
In this section we’ll review the ADHM construction of instantons on the C and A k − ALEspace [11, 8, 12, 13]. C The moduli space of U(1) instantons with instanton number k on C has a very beautifuldescription–the ADHM description. Basically the moduli space is a U ( k ) quotient of ahypersurface on C k +2 k defined by the ADHM constraints[ B , B ] + IJ = 0 (3.1)[ B , B † ] + [ B , B † ] + II † − J † J = ξ k × k , (3.2)where B α , α = 1 , V to itself, I is a linear map from V to a 1-dimensional complex vector space W and J isa linear map from W to V . So B α , α = 1 , k × k matrices, I is a k × J is a 1 × k matrix. The U(k) action is defined by B α (cid:55)→ U B α U † , I (cid:55)→ U I, J (cid:55)→
J U † , (3.3)for U ∈ U ( k ). There is a U (1) action on C . It also induces a U (1) on the instantonmoduli space. According to [14], the fixed points of U (1) action on the instanton moduli– 4 –pace is in one-to-one correspondence with the Young diagram. The tangent space of theinstanton moduli space is T M k = V ∗ ⊗ V ⊗ ( Q − ∧ Q −
1) + W ∗ ⊗ V + V ∗ ⊗ W ⊗ ∧ Q, (3.4)where Q is a two-dimensional U (1) module. A l − ALE space
The A l − type ALE space is defined by the blowup of the quotient C / Γ where Γ is the Z l action: Γ : (cid:32) z z (cid:33) (cid:55)→ (cid:32) e πi/l e − πi/l (cid:33) (cid:32) z z (cid:33) . (3.5)In fact the U(1) instantons on C / Γ can be described by the set { ( Y, r ) } where Y is aYoung diagram with l types of boxes and r is an integer mod p . The integer r specifiesthe Z l representation R r under which the first box in Y transforms. We know that thereexists a U (1) action on the instantons on C . The specified r gives rise to an embeddingof Γ into U (1) : Γ : R a (cid:55)→ e πia/l R a , T (cid:55)→ e πia/l T , T (cid:55)→ e − πia/l T . (3.6)Under this action V and W have the following decomposition: V = l − (cid:88) a =0 V a ⊗ R a , dim V a = k a , (3.7) W = l − (cid:88) a =0 W a ⊗ R a , dim W a = N a . (3.8)Since we are dealing with the U(1) instantons, only one of N a is non-vanishing, and itdepends on the a specified. Thus the tangent space of the instanton moduli space is givenby the Γ − invariant component of the C result [8, 12] T M Y = (cid:0) V ∗ ⊗ V ⊗ ( Q − ∧ Q −
1) + W ∗ ⊗ V + V ∗ ⊗ W ⊗ ∧ Q (cid:1) Γ . (3.9)The dimension of the instanton moduli space isdim C M Y = ( k a k a +1 + k a k a − − k a + sk a N a ) (3.10)= − ˆ C ab k a k b + 2 k a N a , (3.11)where ˆ C ab = 2 δ ab − δ a,b +1 − δ a,b − is the extended A l − Cartan matrix.Now let us consider the tautological bundle [11]. We know the A l − singularity isresolved by replacing the singularity with l − P . This leads to new self-dual connections with non-trivial fluxes along the exceptional divisors. The U(1) bundlesΥ a , a = 0 , · · · , l − is the trivialbundle. These Υ a , a = 0 , · · · , l − (cid:90) c (Υ a ) ∧ c (Υ b ) = − C ab , (cid:90) C a c (Υ b ) = δ ba , (3.12)– 5 –here C ab is the inverse of the A l − Cartan matrix and C a is the a th exceptional divisor.The gauge bundle F Y is given by [11, 12] F Y = (cid:0) V ∗ ⊗ Υ ⊗ ( Q − ∧ Q −
1) + W ∗ ⊗ Υ (cid:1) Γ , (3.13)where Υ = (cid:80) l − a =0 Υ a R a is the tautological bundle.The Chern characters are given by [12] ch ( F Y ) = (cid:88) a u a ch (Υ a ) , (3.14) ch ( F Y ) = (cid:88) a u a ch (Υ a ) − Kl Ω , (3.15)where K = (cid:80) a k a , Ω is the normalized volume form of the manifold and u a := N a + k a +1 + k a − − k a = N a − ˆ C ab k b . (3.16)Further, the instanton number k ∈ k Z is defined by k = − (cid:90) M ch ( F Y ) = 12 (cid:88) a C aa u a + Kk = k + 12 (cid:88) a C aa N a , (3.17)where C aa = l ( l − a ) a .According to [5, 12], the partition function is given by Z ( q, z a ) = (cid:88) k,u a χ ( M k,u a ) q k e − z a u a , (3.18)where χ ( M k,u a ) is the Euler number of the instanton moduli space with first and secondChern characters u a and k respectively.
4. Regular and fractional instantons
According to [8, 12, 13], the partition function of instantons on A l − -ALE space can befactorized into a product of contributions of regular and fractional instantons. We definethe r sector of the partition function Z ( q, z a = 0) by Z lr = Z reg Z lr,frac , (4.1)where the detailed definitions of Z reg and Z lr,frac can be found in the subsection 4.1 andsubsection 4.2. We’ll show that Z lr ≡ Z lr,orbifold . (4.2) In fact, we only use the r sector of this partition function. The detailed definitions of the r sector canbe found in next section. – 6 – .1 Regular instantons The regular instantons are instantons in the regular representation of Γ = Z l . They arefree to move on C / Γ. The moduli space is M regkl = ( C / Z l ) k /S k , (4.3)which is related to the Hilbert scheme of k-points on C / Γ via the Hilbert-Chow morphism.The first Chern class of the regular instanton moduli space is vanishing. Further, accordingto [8, 12, 13], the regular instantons correspond to the l-quotients (cid:126)R of Young diagramswhose definition can be found in Appendix B. Thus the instanton number k has followingrelationship with the number of boxes of the l-quotients (cid:126)R = ( R , R , · · · , R l − ) of Youngdiagram K [12]: k = l − (cid:88) i =0 | R i | . (4.4)Hence it is not hard to find that the partition function of the regular instantons is [13] Z reg = (cid:88) k q k χ ( M regkl ) = q l η ( q ) l , (4.5)where χ ( M regkl ) is the Euler number of M regkl . Further, the formula (4.5) is consistent withthe formula (2.6) and the Young diagram K is just the blended partition of (cid:126)R . In AppendixC, we show that the (cid:126)R is just the l-quotient of the blended partition K of (cid:126)R . According to [8, 12, 13], the fractional instantons correspond to the Young diagrams whichdo not have any boxes whose hook length (cid:96) ( s ) satisfies (cid:96) ( s ) = 0 mod l . In fact, the fractionalinstantons correspond to the l-cores of Young diagrams whose definition can be found inAppendix B. Following [13], we define the partition function Z lr,frac by Z lr,frac := (cid:88) Y ∈C ( l ) q N kl − r , (4.6)where C ( l ) is the subset of all Young diagrams consisting of l-cores whose definition can befound in Appendix B. According to [13], we have the following formula: Z lr,frac = (cid:88) { p i | (cid:80) i p i =0 } q (cid:80) l − i =0 p i + (cid:80) l − i = l − r p i (4.7)= (cid:88) { ˜ p i | (cid:80) i ˜ p i = p = ln + r } q l (cid:80) i ˜ p i − ( l +1) p − p
22 + ( l +1) ln r +1) r l , (4.8)where the definition of ˜ p i and the detailed derivation can be found in the equations (B.4,B.8, B.9) in Appendix B. According to the Proposition 2.28 in [13], the blended partition Comparing the q here with notations in theorem 4.6 in [13], our q is equal to t and q = t i = 1 , for i (cid:54) =0 mod l . – 7 – is uniquely determined by the pair (l-quotient, l-core). Hence counting the blendedpartitions is equivalent to counting the pair of (l-quotient, l-core). Furthermore, accordingto [7], the shape of blended partition is the same as the orbifold partitions. So we have Z lr,orbifold ≡ Z lr = Z reg Z lr,frac . (4.9)
5. Conclusion
In this note, we have investigated the detailed relationship between the orbifold partitionsof first type introduced in [7] and the pair (l-quotient, l-core) introduced in [13]. Wefind that orbifold partition counting presented in [7] is exactly the same as counting the(l-quotient, l-core) pair. According to [12, 13] the U(N) partition function factorizes into Z U ( N ) = Z NU (1) = (cid:0) Z U (1) ,reg Z U (1) ,frac (cid:1) N . (5.1)It would be interesting to see how to generalize the orbifold partition counting to the U(N)case. Acknowledgments
The author wants to thank Prof. Jack Gegenberg for his support of this work.
A. Colored partitions , charged partitions and blended colored partitions
In this Appendix, we shall review the definitions of colored partitions, charged partitionsand blended partitions in [9].
Definition A.0.1 (Colored partition)
The colored partition (cid:126)R is the l-tuple of parti-tions: (cid:126)R = ( R , · · · , R l − ) , (A.1) where R k = ( R k, ≥ R k, · · · ≥ R k,n k > R k,n k +1 = 0 = · · · ) . (A.2) | (cid:126)R | := (cid:88) k,i R k,i (A.3) Definition A.0.2 (Charged partition)
The charged partition ( p, (cid:126)R ) is the set of non-increasing integers R i = R i + p , where (cid:126)R = ( R ≥ R ≥ · · · ) , (A.4) is a partition, and p ∈ Z . The limit R ∞ ≡ p is called the charge. Definition A.0.3 (Blending of colored partition)
Given a vector (cid:126)p = ( p , · · · , p l − ) ,with l − (cid:88) i =0 p i = p (A.5) and an N-tuple of partition (cid:126)R , we define the blended partition K, as follows: { p + K m − m | m ∈ N } = { k ( R i,m − m + p i ) + i | i = 0 , · · · , l − , m ∈ N } . (A.6)– 8 – igure 1: Young diagram and Maya diagram copied from [13]
B. Quotients and cores for Young diagrams
In this appendix, we shall review the definitions of quotients and cores for Young diagrams[13, 12, 15].
Definition B.0.4 (Maya diagram)
A Maya diagram is a sequence { µ ( k ) } k ∈ Z which con-sists of 0 or 1 and satisfies the following property: there exist N, M ∈ Z such that for all k > N (resp. k < N ), µ ( k ) = 1 (resp. µ ( k ) = 0 ). There is a one-to-one correspondence between the set of Maya diagrams and the set ofYoung diagrams. We can place the Young diagram Y on the (x, y)-plane by the followingway: the bottom-left corner of the Young diagram is at the origin (0,0) and a box in theYoung diagram is an unit square. We call an upper-right borderline of { x − axis } ∪ { y − axis } ∪ Y the extended borderline of Y and denote it by ∂Y . The line defined by y = x is called the medium . The Maya diagram { µ ( k ) } k ∈ Z corresponds to a Young digram Y isdefined as follows. We give the direction to the extended borderline ∂Y of Y , which goesfrom (0 , + ∞ ) to (+ ∞ ,
0) (see Figure 1). Then each edge of the extended borderline isnumbered by k ∈ Z , if we set the edge which is located at the next to the medium to be0. Next we encode a edge ↓ (resp. ↑ ) to 0 (resp. 1). By this way, we have a 0/1 sequence { µ ( k ) } k ∈ Z , where each µ Y ( k ) corresponds to a edge of ∂Y . Definition B.0.5 (Quotients)
For each ≤ i ≤ l − , we define µ Y ∗ i ( k ) := µ Y ( lk + i ) , k ∈ Z . (B.1) Then we have an p-tuple (cid:16) { µ Y ∗ ( k ) k ∈ Z } , · · · , { µ Y ∗ l − ( k ) k ∈ Z } (cid:17) of subsequences µ Y ( k ) k ∈ Z andeach { µ Y ∗ i ( k ) k ∈ Z is also a Maya diagram. The p-quotient for Y is the l-tuple of Young dia-grams (cid:126)Y := ( Y ∗ , · · · , Y ∗ l − ) corresponding to the Maya diagrams (cid:16) { µ Y ∗ ( k ) k ∈ Z } , · · · , { µ Y ∗ l − ( k ) k ∈ Z } (cid:17) . – 9 – emark B.0.6 Here sometime there does not exist a Young diagram corresponding to theMaya diagram { µ Y ∗ i ( k ) k ∈ Z } , if so, we denote this Young diagram as the empty set ∅ . We define p i ( Y ) ∈ Z ( i = 0 , · · · , l −
1) by the following condition: (cid:93) { µ Y ∗ i ( k ) = 1 | n < p i ( Y ) } = (cid:93) { µ Y ∗ i ( k ) = 0 | n ≥ p i ( Y ) } . (B.2) Remark B.0.7 l − (cid:88) i =0 p i ( Y ) = 0 . (B.3) Remark B.0.8
Notice that here we place the vertex of Young diagram to the origin. Ifwe place the vertex at ( p, , then we have ˜ p i = p + n,p + n, ... ,p l − m − + n,p l − m + n + 1 ,p l − m +1 + n + 1 , ... ,p l − + n + 1 , (B.4) where p = nl + m and ˜ p i is the new one satisfying (B.2) in the new Young diagram whosevertex is placing at ( p, . Hence we have l − (cid:88) i =0 ˜ p i = p. (B.5) Definition B.0.9 (l-core)
A Young diagram is called l-core if its l-quotient (cid:126)Y is empty.Denote C ( l ) as the subset of all Young diagrams consists of l-cores. Definition B.0.10
Let Y ( l ) be the Young diagram obtained by removing as many hooks oflength l as possible from Y. It is called the l-core of Y. Proposition B.0.11
For any l ≥ , a Young diagram Y is uniquely determined by itsl-core Y ( l ) and l-quotient (cid:126)Y . Then we have | Y | = | Y ( l ) | + l | (cid:126)Y | . (B.6) Proposition B.0.12
For Y ∈ C ( l ) which determined by ( p , p , · · · , p l − ) and (cid:80) i p i = 0 ,then the number of boxes on y = x − nl + j N nl − j ( Y ) is (cid:88) k ∈ Z N kl − j ( Y ) = 12 l − (cid:88) i =0 p i ( Y ) + l − (cid:88) i = l − j p i ( Y ) . (B.7)– 10 –f we place the Young diagram at ( p, p = ln + r (0 ≤ r ≤ l − p i by ˜ p i in the formula (B.7), then we have the following formula (cid:88) k ∈ Z N kl − j ( Y ) = 12 l − j − (cid:88) i =0 (˜ p i − n ) + l − (cid:88) i = l − j (˜ p i − n − + l − (cid:88) i = l − j (˜ p i − n − l − j − (cid:88) i =0 (˜ p i − n ˜ p i + n ) + l − (cid:88) i = l − j (cid:2) ˜ p i − n + 1)˜ p i + ( n + 1) (cid:3) + l − (cid:88) i = l − j (˜ p i − n − l − (cid:88) i =0 ˜ p i − n l − (cid:88) i =0 ˜ p i + n l n + 1) j − ( n + 1) j = 12 l − (cid:88) i =0 ˜ p i − n ( nl + r ) + n l − j
2= 12 l − (cid:88) i =0 ˜ p i − nr − n l − j . (B.8)If j = r , then we have (cid:88) k ∈ Z N kl − r ( Y ) = l − (cid:88) i =0 ˜ p i − nr − n l − r (cid:80) l − i =0 ( l ˜ p i ) − ( l +1) p − p + ( l +1) ln + ( r +1) r l , (B.9)where p = ln + r . C. Blended partition, l-quotient and l-core
Suppose we have a blended partition K as follows: { p + K m − m | m ∈ N } = { l ( R i,m − m + p i ) + i | i = 0 , · · · , l − , m ∈ N } . (C.1)According to the l-quotient’s definition (B.0.5), it is easy to find that the right hand sideof the formula (C.1) tells us the l-quotient of K is (cid:126)R = ( R , · · · , R ) and the positions of 0are ( R i,m − m + p i ,
0) in Maya diagram corresponding to the Young diagram R i . Thus itis not hard to find that R i is placed at ( p i ,
0) position. Further, the left hand side tells usthat the positions of 0 of the Maya diagram of K are ( p + K m − m, p, p = p + p + p = 0. Table 1 showsthe Maya diagrams corresponding to this blended partition. In Table 1, the central verticalline is the position of the origin. It is not hard to see that the diagrams R , R , R are– 11 – igure 2: The blended partition copied from [7].
K 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 R R R Table 1:
Maya diagrams with vanishing total charge p=0 exactly same as the left Young diagrams in Figure 2. It also shows the correct position ofthe vertices of R , R , R . Now let us examine the situation of p (cid:54) = 0. For simplicity, letus assume p = 1, so that the Maya diagram is changed to one shown in Table 2. It showsK 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 R R R Table 2:
Maya diagrams with p = 1 that p = − p , p does not change; which means the R is placed at ( − , R , R , R to a single Young diagram by using the formula (C.1), it is easily tosee that this blended Young diagram is exactly same as the K here.– 12 –he 3-core of the Young diagram K in figure 2 is as follows: b bb , (C.2)where b stands for black color. References [1] N. A. Nekrasov and S. L. Shatashvili,
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