BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters
BBPS/CFT CORRESPONDENCE III:GAUGE ORIGAMIPARTITION FUNCTIONAND QQ -CHARACTERS NIKITA NEKRASOV
Abstract.
We study generalized gauge theories engineered by taking the low energylimit of the Dp branes wrapping X × T p − , with X a possibly singular surface in aCalabi-Yau fourfold Z . For toric Z and X the partition function can be computedby localization, making it a statistical mechanical model, called the gauge origami.The random variables are the ensembles of Young diagrams. The building block ofthe gauge origami is associated with a tetrahedron, whose edges are colored by vectorspaces. We show the properly normalized partition function is an entire function ofthe Coulomb moduli, for generic values of the Ω -background parameters. The orbifoldversion of the theory defines the qq -character operators, with and without the surfacedefects. The analytic properties are the consequence of a relative compactness of themoduli spaces M ( (cid:126)n, k ) of crossed and spiked instantons, demonstrated in "BPS/CFTcorrespondence II: instantons at crossroads, moduli and compactness theorem". Contents Introduction
Acknowledgments
Review of notations
Sets and partitions
Four and Six
Finite groups and quiver varieties
The local model data
Partition function of spiked instantons
The configuration space
The statistical weight
The perturbative prefactor
The main result
Orbi folding
Abelian orbifolds
The abelian × ALE case
The ALE × ALE case
The main fact
Conclusions and outlook a r X i v : . [ h e p - t h ] O c t NIKITA NEKRASOV Introduction
This paper is a continuation of the series [Ne2, Ne3]. There we proposed a set ofobservables in quiver N = 2 supersymmetric gauge theories. These observables are use-ful in organizing the non-perturbative Dyson-Schwinger equations. The latter relatedifferent instanton sectors contributions to the expectation values of gauge invariantchiral ring observables. We also introduced the geometric setting to which these ob-servables belong in a natural way. Namely, we defined the moduli spaces M X,G of whatmight be called supersymmetric gauge fields in the generalized gauge theories, whosespace-time X contains several, possibly intersecting, components:(1) X = (cid:91) A X A . The gauge groups G | X A = G A on different components may be different. The inter-sections X A ∪ X B lead to the matter fields charged under the product group G A × G B (bi-fundamental multiplets). In this paper we shall be studying the integrals over themoduli space M X,G , which we shall compute using equivariant localization.1.1.
Acknowledgments.
I am grateful to H. Nakajima for patient explanations aboutthe quiver varieties and to the anonymous referee for very useful comments on the man-uscript. The results of this paper were reported at various seminars [Ne4],[Ne5]. Theresearch was carried out at the IITP RAS with the support of the Russian Foundationfor Sciences (project No. 14-50-00150).2.
Review of notations
Sets and partitions.
Sequences.
For two sets X and S let X S = Maps( S, X ) denote the set of mapsfrom S to X . For a map f : S → X we sometimes use the notation(2) ( x s ) s ∈ S , with x s = f ( s ) ∈ X . For example, a sequence ( a n ) , n ∈ N would be denoted as ( a n ) n ∈ N or ( a n ) n ≥ , if the context is clear.2.1.2. Non-negative integers. are denoted by Z ≥ = N ∪ { } .2.1.3. Finite sets.
Let [ n ] denote the set { , , . . . , n } for n ∈ N . For a finite set X wedenote by X the number of its elements. Thus, for finite X and S (3) X S = ( X ) S PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 3
Partitions.
There are lots of sums over partitions in this paper. Let Λ denotethe set of all partitions. An element λ ∈ Λ is a non-increasing sequence λ = ( λ ≥ λ ≥ . . . ≥ λ (cid:96) ( λ ) > λ (cid:96) ( λ )+1 = λ (cid:96) ( λ )+2 = . . . = 0) of integers, with a finite number of positiveterms, sometimes called the parts of λ . The number (cid:96) ( λ ) of positive terms is called the length of the partition λ , the sum(4) (cid:96) ( λ ) (cid:88) i =1 λ i = | λ | is called its size . We also identify the partitions λ with the finite subsets of N = N × N ,as follows:(5) λ = { (cid:3) | (cid:3) = ( i, j ) , i, j ≥ , ≤ j ≤ λ i } The size | λ | of the partition λ is the number of elements λ of the corresponding finiteset. Not every finite subset of N corresponds to a partition, only those, for which thecomplement N \ λ is preserved by the action of the semi-group Z ≥ × Z ≥ on N bytranslations. Equivalently, the partitions are in one-to-one correspondence with finitecodimension monomial ideals in the ring of polynomials in two variables: λ ↔ I λ , I λ ⊂ C [ x, y ] , I λ = ∪ (cid:96) λ +1 i =1 C [ x, y ] x i − y λ i .We denote by Λ [ k ] the set of partitions of k , i.e. the set of all λ ∈ Λ , such that | λ | = k . We have:(6) Λ = (cid:71) k ≥ Λ [ k ] The celebrated Euler formula:(7) ∞ (cid:88) k =0 Λ [ k ] q k = ∞ (cid:89) n =1 − q n Four and Six.
Let denote the set [4] and let denote the set of -elementsubsets of (we write ab instead of { a, b } to avoid the clutter):(8) = { , , , } , = (cid:32) (cid:33) = { , , , , , } In (8) we exhibit the lexicographic order of the sets and which is used below insome formulas. For example, < < < . For A ∈ we denote by ¯ A = \ A itscomplement. Let denote the quotient / ∼ where A ∼ ¯ A . Identify = { , , } ⊂ bychoosing a representative A = a , a ∈ [3] . Define the map: ϕ : → by(9) ϕ ( A ) = inf ¯ A ∈ , so that(10) ϕ (12) = 3 , ϕ (13) = ϕ (14) = 2 , ϕ (24) = ϕ (23) = ϕ (34) = 1 . We also define the following map ε : → Z : write A = { a, b } , a < b ∈ , write ¯ A = { c, d } , c < d ∈ , then ε ( A ) = ε abcd . Thus,(11) ε (12) = ε (34) = ε (14) = ε (23) = +1 , ε (13) = ε (24) = − . NIKITA NEKRASOV
It may seem surprising that ε takes values +1 four times and − only two times, butin fact it is natural, since ε ( A ) = ε ( ¯ A ) , therefore ε is defined on . Since a two-valuedfunction on a set of odd cardinality cannot split it equally, more classes are bound tobe good rather then bad (assuming the values +1 and − are identified with “good”and “bad”).It is useful to view as the set of faces (or vertices) of the tetrahedron, while is theset of edges. The edge ab connects the vertices a and b . Alternatively the edge ab isthe common boundary of the faces a and b .2.3. Finite groups and quiver varieties. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)
Abelian (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) groups.
We denote by Γ ab a finite abelian group. It is well-known thatany such Γ ab is a product of cyclic groups whose orders are powers of primes:(12) Γ ab = d (cid:17) κ =1 (cid:16) Z /p l κ κ Z (cid:17) , l κ ∈ N , p κ − primes An element of Γ ab is a string t = ( t , . . . , t d ) of integers defined modulo lattice t κ ∼ t κ + p l κ κ Z . All irreducible representations L (cid:36) of Γ ab are complex one-dimensional,labeled by a string of integers(13) ν = (cid:16) n , . . . , n d (cid:17) ∈ Γ ∨ ab , n κ ∈ Z also defined modulo lattice n κ ∼ n κ + p l κ κ Z :(14) T L ν ( t ) = exp 2 π √− (cid:88) κ t κ n κ p l κ κ We set ν = to label the trivial representation with all n κ = 0 ,(15) T L ( t ) ≡ The set Γ ∨ ab is also an abelian group, isomorphic to Γ ab , with multiplication given by thetensor product of irreducible representations. We shall be using the addition symbolfor the group law on Γ ∨ ab :(16) L ν + ν = L ν ⊗ L ν , L ∗ ν = L − ν Let(17) δ Γ ∨ ab : Γ ∨ ab → { , } PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 5 be the indicator function of the trivial representation:(18) δ Γ ∨ ab ( ) = 1 , δ Γ ∨ ab ( ν ) = 0 , ν (cid:54) = (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Nonabelian (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) subgroups (cid:58)(cid:58) of (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) SU (2) . Let γ denote the affine Dynkin diagram of type D , or E , respectively (see the Fig. ):’ Fig.
Vert γ be the set of vertices of γ , Edge γ be the set of oriented edges (we pick anyorientation). For the edge e ∈ Edge γ let s ( e ) , t ( e ) ∈ Vert γ denote its source and target,respectively.Let Γ γ ⊂ SU (2) denote the corresponding non-abelian finite subgroup. For γ = ˜ E , , the group Γ γ is the binary tetrahedral, octahedral, icosahedral group, respectively.In this correspondence i ∈ Vert γ labels the irreducible representations R i ∈ Γ ∨ γ of Γ γ . The edges Edge γ show up in the tensor products: let denote the defining two-dimensional representation of SU (2) . Then:(19) ⊗ R i = (cid:77) e ∈ s − ( i ) R t ( e ) ⊕ (cid:77) e ∈ t − ( i ) R s ( e ) where is viewed as the representation of Γ γ ⊂ SU (2) . The dimensions dim R i are indi-cated on the corresponding nodes in the picture, the vector of dimensions is annihilatedby the affine Cartan matrix = 2 − incidence matrix of γ , cf. (19):(20) R i = (cid:88) e ∈ s − ( i ) dim R t ( e ) + (cid:88) e ∈ t − ( i ) dim R s ( e ) The trivial representation is colored pink on Fig. 1.2.3.3. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)
Walks (cid:58)(cid:58)(cid:58) on (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) quivers. Let i s , i t ∈ Vert γ . A path p connecting i s (the source of p )to i t (the target of p ) of length (cid:96) p on the quiver γ is the ordered sequence of pairs p i = ( e i , σ i ) , i = 1 , . . . , (cid:96) p where e i ∈ Edge γ , σ i = ± , and(1) the source of p : if σ = 1 , then s ( e ) = i s , otherwise t ( e ) = i s (2) the end-point of p : if σ (cid:96) p = 1 , then t ( e (cid:96) p ) = i t , otherwise s ( e (cid:96) p ) = i t (3) concatenation: if σ i = 1 , σ i +1 = 1 , then t ( e i ) = s ( e i +1 ) , if σ i = 1 , σ i +1 = − , then t ( e i ) = t ( e i +1 ) , if σ i = − , σ i +1 = 1 , then s ( e i ) = s ( e i +1 ) , if σ i = − , σ i +1 = − ,then s ( e i ) = t ( e i +1 ) NIKITA NEKRASOV
Let us denote the set of all paths on γ connecting i s to i t by P i t i s [ γ ] . There is an obviousassociative concatenation map:(21) (cid:63) : P i i [ γ ] × P i i [ γ ] −→ P i i [ γ ] ,p × ˜ p (cid:55)→ ˜ p (cid:63) p , ( ˜ p (cid:63) p ) i = (cid:40) p i , ≤ i ≤ (cid:96) p (cid:48) ˜ p i − (cid:96) p , (cid:96) p < i ≤ (cid:96) p + (cid:96) ˜ p = (cid:96) ˜ p(cid:63)p and the inversion map(22) − : P i t i s [ γ ] −→ P i s i t [ γ ] , p (cid:55)→ ¯ p, ¯ p i = (cid:16) e (cid:96) p +1 − i , − σ (cid:96) p +1 − i (cid:17) , ≤ i ≤ (cid:96) p (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Nakajima (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) varieties.
Define the Nakajima varieties M γ ( v , w ) associated with aquiver γ and two dimension vectors v , w ∈ Z Vert γ ≥ [Na1, Na2, Na3].Let γ be as before. To each vertex i ∈ Vert γ we associate two Hermitian vectorspaces W i , V i of dimensions w i , v i , respectively. Let(23) U γ ( v ) = (cid:17) i ∈ Vert γ U ( v i ) be the group of unitary transformations of V = ( V i ) i ∈ Vert γ . First, form the Hermitianvector space:(24) H γ ( v , w ) = T ∗ (cid:77) e ∈ Edge γ Hom( V s ( e ) , V t ( e ) ) (cid:77) (cid:77) i ∈ Vert γ Hom( W i , V i ) = (cid:40) (cid:16) B e , ˜ B e (cid:17) e ∈ Edge γ , ( I i , J i ) i ∈ Vert γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I i : W i → V i , J i : V i → W i , B e : V s ( e ) → V t ( e ) , ˜ B e : V t ( e ) → V s ( e ) (cid:41) which is acted upon by U γ ( v ) via:(25) ( u ω ) ω ∈ Vert γ · (cid:32)(cid:16) B e , ˜ B e (cid:17) e ∈ Edge γ , ( I i , J i ) i ∈ Vert γ (cid:33) = (cid:32)(cid:16) u t ( e ) B e u − s ( e ) , u s ( e ) ˜ B e u − t ( e ) (cid:17) e ∈ Edge γ , (cid:16) u i I i , J i u − i (cid:17) i ∈ Vert γ (cid:33) For a path p ∈ P i s i t [ γ ] define its holonomy B p : V i s → V i t in the obvious way:(26) B p = ← (cid:96)p (cid:89) i =1 (cid:40) B e i , σ i = +1˜ B e i , σ i = − This definition is compatible with the path multiplication:(27) B p B p = B p (cid:63)p PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 7
The action (25) preserves the hyper-Kähler structure of H γ ( v , w ) , with the three sym-plectic forms (cid:36) I,J,K given by:(28) (cid:36) I = (cid:88) e ∈ Edge γ Tr V t ( e ) (cid:16) dB e ∧ dB † e − d ˜ B † e ∧ d ˜ B e (cid:17) + (cid:88) i ∈ Vert γ Tr W i (cid:16) dJ i ∧ dJ † i − dI † i ∧ dI i (cid:17) ,(cid:36) J + √− (cid:36) K = (cid:88) i ∈ Vert γ Tr W i ( dJ i ∧ dI i ) + (cid:88) e ∈ Edge γ Tr V t ( e ) (cid:16) dB e ∧ d ˜ B e (cid:17) Then perform the hyper-Kähler reduction with respect to the action (25):(29) M γ ( v , w ) = (cid:126)µ − ( (cid:126)ζ ) /U γ ( v ) where (cid:126)µ = ( µ I, i , µ J, i , µ K, i ) i ∈ Vert γ ,(30) µ I, i = I i I † i − J † i J i + (cid:88) e ∈ t − ( i ) (cid:16) B e B † e − ˜ B † e ˜ B e (cid:17) + (cid:88) e ∈ s − ( i ) (cid:16) ˜ B e ˜ B † e − B † e B e (cid:17) ,µ J, i + √− µ K, i = I i J i + (cid:88) e ∈ t − ( i ) B e ˜ B e − (cid:88) e ∈ s − ( i ) ˜ B e B e , and we take (this is not the most general definition)(31) (cid:126)ζ = ( ζ i V i , , i ∈ Vert γ with all ζ i > . Stability . Instead of solving three equations (cid:126)µ = (cid:126)ζ one can actually solve only µ C ≡ µ J + √− µ K = 0 , and then take a quotient of the set of stable points in µ − C (0) bythe action of(32) G γ ( v ) = (cid:17) i ∈ Vert γ GL ( v i ; C ) so that(33) M γ ( v , w ) = µ − C (0) stable /G γ ( v ) The stable points are the G γ ( v ) -orbits of ( B e , ˜ B e , I , J ) s.t. the path algebra of γ repre-sented by the products of B e and ˜ B e acting on the image (cid:76) i ∈ Vert γ I i W i generates all of (cid:76) i ∈ Vert γ V i :(34) V i = (cid:88) i (cid:48) ∈ Vert γ (cid:88) p ∈P ii (cid:48) [ γ ] B p I i (cid:48) W i (cid:48) . In other words: any collection V (cid:48) = ( V (cid:48) i ) i ∈ Vert γ ⊂ V of vector subspaces V (cid:48) i ⊂ V i , obeying:(35) S1) I i W i ⊂ V (cid:48) i , for all i ∈ Vert γ , S2) B e ( V (cid:48) s ( e ) ) ⊂ V (cid:48) t ( e ) , ˜ B e ( V (cid:48) t ( e ) ) ⊂ V (cid:48) s ( e ) , for all e ∈ Edge γ must coincide with V : V (cid:48) i = V i for all i ∈ Vert γ . NIKITA NEKRASOV
A simple proof of the equivalence of (29) and (33) can be found along the lines of thearguments of the section 3.4 and [Ne3]: in one direction, any solution to µ I, i = ζ i · V i isstable. Indeed, V (cid:48) ⊂ V as in (35), and let P i denote the orthogonal projection V i → V (cid:48) ⊥ i .By (35) we have:(36) P i I i = 0 , P t ( e ) B e (1 − P s ( e ) ) = 0 , P s ( e ) ˜ B e (1 − P t ( e ) ) = 0 Define b e = P t ( e ) B e P s ( e ) , ˜ b e = P s ( e ) ˜ B e P t ( e ) , b (cid:48) e = (1 − P t ( e ) ) B e P s ( e ) , ˜ b (cid:48) e = (1 − P s ( e ) ) ˜ B e P t ( e ) . Then(37) ζ i dim (cid:16) V i /V (cid:48) i (cid:17) = Tr V i ( P i µ i P i ) = Tr ( V (cid:48) i ) ⊥ − j † i j i + (cid:88) e ∈ t − ( i ) (cid:16) b e b † e − ˜ b † e ˜ b e − ˜ b (cid:48) † e ˜ b (cid:48) e (cid:17) + (cid:88) e ∈ s − ( i ) (cid:16) ˜ b e ˜ b † e − b † e b e − b (cid:48) † e b (cid:48) e (cid:17) , hence, after obvious cancellations,(38) ≤ (cid:88) i ∈ Vert γ ζ i dim (cid:16) V i /V (cid:48) i (cid:17) = − (cid:88) i ∈ Vert γ Tr V (cid:48)⊥ i j † i j i + (cid:88) e ∈ Edge γ Tr V (cid:48)⊥ t ( e ) ˜ b (cid:48) † e ˜ b (cid:48) e + Tr V (cid:48)⊥ s ( e ) b (cid:48) † e b (cid:48) e ≤ which implies V (cid:48) i = V i for all i ∈ Vert γ . Conversely, given a stable solution ( B e , ˜ B e , I i , J i ) to µ C = 0 equations, run the gradient flow of the function:(39) f = 12 (cid:88) i ∈ Vert γ Tr V i (cid:16) µ I, i − ζ i V i (cid:17) which goes along the × i GL ( V i ) orbits. The end-point of the flow is either at f = 0 which would establish the rest of the equations in (29), or at the higher critical point.There, the End ( V i ) -matrices h i = µ I, i − ζ i V i solve:(40) h t ( e ) B e = B e h s ( e ) , h s ( e ) ˜ B e = ˜ B e h t ( e ) , h i I i = 0 , J i h i = 0 Therefore V (cid:48) i = ker h i obeys both S1) and S2) conditions of (35), therefore h i = 0 forall i ∈ Vert γ .2.3.5. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Framing (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) symmetries (cid:58)(cid:58)(cid:58) of (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Nakajima (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) varieties.
The Nakajima variety M γ ( v , w ) hasa symmetry group(41) U γ ( w ) = (cid:17) i ∈ Vert γ U ( w i ) acting in an obvious way on the operators ( I i , J i ) . The maximal torus T γ ( w ) ⊂ U γ ( w ) fixed point locus is the union(42) M γ ( v , w ) T γ ( w ) = (cid:71) v = (cid:80) i ∈ Vert γ ,α ∈ [ w i ] v i ,α (cid:17) i ∈ Vert γ (cid:17) α ∈ [ w i ] M γ ( v i ,α , δ i ) PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 9 where v i ,α ∈ Z Vert γ ≥ for each ( i , α ) , i.e.(43) v i ,α = (cid:16) v i ,α ˜ i (cid:17) ˜ i ∈ Vert γ , and(44) δ i = ( δ i , j ) j ∈ Vert γ We define the fundamental Nakajima variety(45) M i γ ( v ) = M γ ( v , δ i ) The Eq. (42) explains the importance of the fundamental Nakajima varieties.2.3.6. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)
Nakajima-Young (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) varieties.
The Nakajima varieties M γ ( v , w ) with the choice(31) have a holomorphic C × -symmetry (its compact subgroup U (1) acts by an isome-try): u ∈ C × acts via(46) u · (cid:16) B e , ˜ B e , I i , J i (cid:17) = (cid:16) u B e , u ˜ B e , u I i , u J i (cid:17) Define Nakajima-Young variety Y i γ ( µ ) to be the connected component of the fixed pointset:(47) M i γ ( v ) C × = (cid:71) µ ∈ Λ i γ [ v ] Y i γ ( µ ) with(48) Λ i γ [ v ] = π (cid:16) M i γ ( v ) C × (cid:17) denoting the set of connected components. We define the sets Λ i γ for i ∈ Vert γ :(49) Λ i γ = (cid:71) v ∈ Z Vert γ ≥ Λ i γ [ v ] For µ ∈ Λ i γ [ v ] we define:(50) | µ | = v ∈ Z Vert γ ≥ Each Nakajima-Young variety Y i γ ( µ ) carries a set of vector bundles:(51) V ij ,n ( µ ) −→ Y i γ ( µ ) where j ∈ Vert γ , n ≥ , and(52) V ij ,n ( µ ) = (cid:88) p ∈P ji [ γ ] , (cid:96) p = n B p I ( C ) . The stability condition (34) implies, for any j ∈ Vert γ :(53) V j = ∞ (cid:77) n =0 V ij ,n It is easy to show that J ≡ on all Y i γ ( v ; µ ) , and V ij , = C δ i , j . Let us clarify the origin ofthe direct sum decomposition (53). The C × -invariance of the G γ ( v ) -orbit of ( B e , ˜ B e , I , J ) means that the transformation (46) can be compensated by an element ( g j ( u )) j ∈ Vert γ :(54) g t ( e ) ( u ) B e g s ( e ) ( u ) − = u B e , g s ( e ) ( s ) ˜ B e g − t ( e ) ( u ) = u ˜ B e , g i ( u ) I = u I , Jg i ( u ) − = u J Then(55) V ij ,n = Ker (cid:16) g j ( u ) − u n +1 (cid:17) ⊂ V j are obviously mutually orthogonal for different n ’s. The ranks ν ij ,n ( µ ) = rk V ij ,n ( µ ) areimportant local invariants of Y i γ ( µ ) . By definition:(56) ∞ (cid:88) n =0 ν ij ,n ( µ ) = v j The K -theory class of the tangent bundle T Y i γ ( µ ) to Y i γ ( µ ) can be expressed in terms ofthose of V ij ,n :(57) (cid:104) T Y i γ ( µ ) (cid:105) = (cid:104) V ii , (cid:105) + (cid:88) n ≥ ,e ∈ Edge γ (cid:104) Hom (cid:16) V i t ( e ) ,n , V i s ( e ) ,n +1 (cid:17) ⊕ Hom (cid:16) V i s ( e ) ,n , V i t ( e ) ,n +1 (cid:17)(cid:105) − (cid:88) n ≥ , j ∈ Vert γ (cid:104) Hom (cid:16) V ij ,n , V ij ,n (cid:17) ⊕ Hom (cid:16) V ij ,n , V ij ,n +2 (cid:17)(cid:105) .Remark. In the case of γ = (cid:98) A , where Edge γ = { e } , Vert γ = { } , s ( e ) = t ( e ) = 0 , thefundamental Nakajima variety is the Hilbert scheme of v points on C , a.k.a. themoduli space of noncommutative U (1) instantons on R , while the Nakajima-Youngvarieties are the connected components of the so-called graded Hilbert scheme of v points.2.4. The local model data.
To specify the basic local model data we fix:(1) The string(58) ¯ ε = ( ε a ) a ∈ of complex numbers which sum to zero:(59) ε + ε + ε + ε = 0 (2) The string ¯ n of non-negative integers n A ≥ , A ∈ . Let(60) N = (cid:71) A ∈ [ n A ] ≈ { ( A, α ) | A ∈ , α ∈ [ n A ] } . PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 11 (3) The string ¯ a ∈ C N of (cid:88) A ∈ n A complex numbers a A,α ∈ C , α = 1 , . . . , n A , also denoted as(61) a A = (cid:0) a A,α (cid:1) α ∈ [ n A ] ≡ (cid:16) a A, , . . . , a A,n A (cid:17) ∈ C n A . (4) The fugacity(62) q ∈ C , | q | < .We also use the notations: for any a ∈ ,(63) q a ( β ) = e βε a , P a ( β ) = 1 − q a ( β ) , q ∗ a ( β ) = e − βε a , P ∗ a ( β ) = 1 − q ∗ a ( β ) , and for any S ⊂ (64) q S ( β ) = (cid:89) a ∈ S q a ( β ) , q ∗ S ( β ) = (cid:89) a ∈ S q ∗ a ( β ) , P S ( β ) = (cid:89) a ∈ S P a ( β ) , P ∗ S ( β ) = (cid:89) a ∈ S P ∗ a ( β ) We shall often skip the argument β in the notations for q a , P ∗ S , etc. The notation (64),in particular, implies (cf. (59))(65) q = q ∅ = 1 , P = P P P P = P ∗ , q ¯ A = q ∗ A and(66) P ∗ S = ( − | S | q ∗ S P S We shall also encounter the relation(67) P = P + P ∗ in what follows.2.4.1. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Geometry (cid:58)(cid:58)(cid:58) of (cid:58)(cid:58)(cid:58)(cid:58) the (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) local (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) model (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) data. The meaning of the parameters a , ¯ ε is thefollowing. Define the gauge group G A corresponding to the stratum X A ≈ C A of thesingular toric surface X to be(68) G A = U ( n A ) Let T A ⊂ G A denote its maximal torus. Let U (1) ε ⊂ SU (4) be the maximal torus ofthe (4 , -volume preserving unitary symmetries of Z = C . The U (1) ε -action pre-serves X . The Lie algebra Lie T A ⊗ C is parametrized by diagonal matrices a A =diag (cid:16) a A, , . . . , a A,n A (cid:17) with complex entries a A,α ∈ C . The Lie algebra Lie U (1) ε ⊗ C is parametrized by diag( ε , ε , ε , ε ) with ε + ε + ε + ε = 0 . Let(69) T H = U (1) ε × (cid:17) A ∈ T A (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Additive (cid:58)(cid:58)(cid:58) to (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) multiplicative. Let I ± be two finite sets, I = I + (cid:113) I − . Let M be a spacewith an action of a Lie group G , and let E i , i ∈ I be a collection of G -equivariant vectorbundles over M . Let w i ∈ C . We combine them into the G × ( C × ) I -equivariant virtualbundle E = (cid:104)(cid:76) i ∈ I + E i (cid:9) (cid:76) i ∈ I − E i (cid:105) . Let(70) Ch β ( E i ) = (cid:88) α e βξ i,α be the refined Chern character (with ξ i,α equivariant Chern roots of E i ), so that in thenon-equivariant setting Ch β ( E ) = (cid:88) k ≥ β k ch k ( E ) , and define(71) f ( β ) = (cid:88) i ∈ I + e βw i Ch β ( E i ) − (cid:88) i ∈ I − e βw i Ch β ( E i ) To f ( β ) we associate the equivariant characteristic class, a rational function of w i ’s:(72) (cid:15) [ f ] = (cid:89) i ∈ I + c w i ( E + ,i ) (cid:89) i ∈ I − c w i ( E i ) − . where(73) c w ( E ) = rk E (cid:88) k =0 w k c rk E− k ( E ) is the usual G -equivariant Chern polynomial of E , evaluated at w ∈ H • C × ( pt ) = C ,equivalently, it is the top C × × G -equivariant Chern class of E . We define the ∗ -operationon the expressions f ( β ) :(74) f ∗ ( β ) ≡ (cid:88) i ∈ I + e − βw i Ch β ( E ∗ i ) − (cid:88) i ∈ I − e − βw i Ch β ( E ∗ i ) = f ( − β ) This definition is consistent with the notations (63).We have:(75) (cid:15) [ f ] = ( − f (0) (cid:15) [ f ∗ ] where(76) f (0) = (cid:88) i ∈ I + rk( E i ) − (cid:88) i ∈ I − rk( E i ) Therefore,(77) (cid:15) [ P S f ] = (cid:15) [ P S q ¯ S f ∗ ] ( − | S | , PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 13
The definition (72) is the generalization of the notation used in [Ne3], where we defined (cid:15) as a map from the space of Z -linear combinations of exponents to rational functions:(78) (cid:15) (cid:34) (cid:88) i ∈ I + e βw i − (cid:88) i ∈ I − e βw i (cid:35) = (cid:89) i ∈ I + w i · (cid:89) i ∈ I − w i − . Partition function of spiked instantons
In this section we define the statistical mechanical model. The random variablesare the strings of Young diagrams and the complex Boltzmann weights are rationalfunctions of the complex numbers (58), (61). The definition might look first a bitartificial. Its origin is geometric. Namely, in [Ne3] the moduli space of spiked instantons M ( k, (cid:126)n ) is introduced, with (cid:126)n = ( n A ) A ∈ . It has an action of the group H = × A ∈ U ( n A ) × U (1) ε . The fixed points of the maximal torus T H are in one-to-one correspondencewith the strings λ of partitions described below. The Boltzmann weight is simplythe localization contribution to the integral of over M ( k, (cid:126)n ) , multiplied by q k . Thiscontribution is the product of the weights of the T H -action on the virtual tangent spaceto M ( k, (cid:126)n ) , which is the difference of the kernel and the cokernel of the linearization ofthe equations defining M ( k, (cid:126)n ) at the fixed point. The kernel is always a complex vectorspace, henceforth it is naturally oriented and the product of weights is well-defined.The cokernel (the obstruction space) is only a real vector space, hence the product ofthe weights depends on the choice of its orientation. In what follows we specify thechoice of the orientation with the help of the choice of the order on and . Theresulting measure will not depend on this choice.3.1. The configuration space.
The basic local model is a statistical ensemble. Therandom variables are the strings(79) λ = (cid:16) λ ( A,α ) (cid:17) A ∈ ,α ∈ [ n A ] ∈ Λ N of (cf. (60)) (cid:88) A ∈ n A = N partitions λ ( A,α ) ∈ Λ . In other words, the configuration space is(80) Λ N . Define, cf. (5):(81) N A ( β ) = n A (cid:88) α =1 e β a A,α , K A ( β ) = n A (cid:88) α =1 (cid:88) (cid:3) ∈ λ ( A,α ) e βc A,α ( (cid:3) ) , with(82) c A,α ( (cid:3) ) = a A,α + ε a ( i −
1) + ε b ( j − , for (cid:3) = ( i, j ) and (cf. (74))(83) T A = N A K ∗ A + q A N ∗ A K A − P A K A K ∗ A Let(84) k A = K A (0) = [ n A ] (cid:88) α =1 | λ ( A,α ) | and(85) | λ | = (cid:88) A ∈ k A It is well-known [Ne1, NY, AGT] that(86) T A = q A T ∗ A is a pure character, i.e.(87) T A = n A k A (cid:88) I =1 e t A,I where t A,I are integral linear combinations of a A,α , α ∈ [ n A ] , ε a , ε b , a, b ∈ A . Letus assume a A , ¯ ε are sufficiently generic, so that t A,I (cid:54) = 0 , t
A,I + ε ¯ a (cid:54) = 0 for any ¯ a ∈ ¯ A , I ∈ [2 n A k A ] .Define, finally,(88) K ( β ) = (cid:88) A ∈ K A ( β ) The statistical weight.
The complex Boltzmann weight of λ is given by the fol-lowing expression:(89) Z λ = q | λ | (cid:15) (cid:104) − T λ (cid:105) , where (cf. (9)):(90) T λ ( β ) = (cid:88) A ∈ P ϕ ( ¯ A ) T A + P ¯ A N A (cid:88) B (cid:54) = A K ∗ B − P (cid:88) A
Note that for generic a A , ¯ ε the measure (89) does not depend on the choice of theorder on or :(92) (cid:15) [ q ¯ a T A ] = (cid:15) (cid:104) q ∗ ¯ a T ∗ A (cid:105) = (cid:15) (cid:104) q ∗ ¯ a q ∗ A T A (cid:105) = (cid:15) [ q ¯ b T A ] ,(cid:15) P (cid:88) A
The partition function Z inst is the T H -equivariant integral of over the virtual fundamental cycle of the moduli spaceof spiked instantons [Ne3]. The latter is the space of solutions to certain quadraticmatrix equations, generalizing the ADHM equations [ADHM], on four complex k × k matrices B a , their Hermitian conjugates B † a , a ∈ , and twelve rectangular matrices I A , J A , of sizes n A × k and k × n A , A ∈ , and their Hermitian conjugates. The definition(90) stems from the equivariant localization. The strings of partitions λ are the T H -fixed points. The matrices ( B a , I A , J A ) of the construction [Ne3] obey, for such a fixedpoint:(94) [ B a , B b ] = 0 , a, b ∈ , J A = 0 , A ∈ the vectors(95) | i, j ; α ; ab (cid:105) = B i − a B j − b I ab ( N ab,α ) with α ∈ [ n ab ] , ≤ j ≤ λ ( ab,α ) forming the basis of the vector space K , N ab,α being theeigenspace of T ab action on the framing space N ab (see [Ne3] for the notations and moreexplanations).The equivariant weights of the matrices contribute(96) T + = (cid:88) a ∈ q a K K ∗ + (cid:88) A ∈ (cid:16) K ∗ N A + q A K N ∗ A (cid:17) with(97) K = (cid:88) A ∈ K A while the equivariant weights of the equations they obey, and the symmetries onedivides by, contribute (with the minus sign)(98) T − = (cid:88) c ∈ q c q K K ∗ + (cid:88) A ∈ (cid:88) ¯ a ∈ ¯ A q ¯ a K ∗ N A Moreover, the T + part is defined canonically by using the complex structure of thespace of matrices ( B a , I A , J A ) . The T − part is defined non-canonically, as the expression (98) does not respect the symmetry between q a ’s. The real (i.e. such that χ ∗ = χ )character T − + T ∗− is defined canonically. This subtlety has to do with the real, asopposed to complex, nature of the equations defining the spiked instantons [Ne3]. So, (cid:15) [ T − ] may have a sign ambiguity, as (cid:112) (cid:15) [ T − + T ∗− ] . Also, (cid:15) [ T − ] and (cid:15) [ T + ] separatelymay vanish, as some of the weights in (96) and (98) may vanish. It is easy to showthat formally (cid:15) [ T − − T + ] = (cid:15) [ − T λ ] . One simply uses (75) several times. The detailsof the choice of the sign will be clarified elsewhere (it uses the residue definition ofthe localization contribution, which was worked out in [MNS], it is similar to whatsometimes is referred to as the Jeffrey-Kirwan residue in the mathematical literature).The resulting measure factor(99) (cid:15) [ T − − T + ] = (cid:15) [ Obs λ ] (cid:15) [ Def λ ] = (cid:15) [ − T λ ] where Def λ , Obs λ are the T H -characters of ker D λ , coker D λ , respectively. Here D λ isthe linearization of the spiked instanton equations at the solution, corresponding to λ .The expressions N A , K A , T λ ( β ) etc. are the elements of the K-group K [ T H ] , i.e. theabelian group whose elements are the formal linear combinations(100) (cid:88) w ∈ T ∨ H n w L w where n w ∈ Z ,(101) L w are the irreducible representations of the torus T H , i.e. the elements of the lattice T ∨ H = Hom ( T H , U (1)) . We assign to the weight w = ( w A,α ) ⊕ ( w a ) a function of ( a , ¯ ε ) ,the character of T C H in the representation L w :(102) L w (cid:55)→ exp β (cid:88) A,α w A,α a A,α + (cid:88) a w a ε a Here w A,α ∈ Z , w a ∈ Z are defined up to a shift w a (cid:55)→ w a + w , w ∈ Z .3.2.2. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58) More (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) general (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) definition.
The definition (93) is fine as long as a and ¯ ε are generic.However, e.g. if for some ab ∈ the ratio ε a /ε b ∈ Q + is a positive rational number,or if for some α (cid:54) = β ∈ [ n A ] , a A,α = a A,β , the individual contributions Z λ to the formula(93) have apparent poles. Actually, the poles cancel. Let us give the presentation ofthe formula (93) which is applicable in these cases.(103) Z inst k = (cid:88) ( k A ) A ∈ , (cid:80) A k A = k (cid:90) (cid:16) A ∈ M kA ( n A ) S (cid:126)n,(cid:126)k ( a , ¯ (cid:15) ) where Gieseker-Nakajima moduli spaces M k ( n ) parametrize the charge k noncommu-tative U ( n ) instantons on R and framed rank n torsion free sheaves E on CP with PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 17 ch ( E ) = k , while S (cid:126)n,(cid:126)k ( a , ¯ (cid:15) ) is the equivariant characteristic class, given by (cf. (90)):(104) S (cid:126)n,(cid:126)k ( a , ¯ (cid:15) ) = (cid:89) A c m A (cid:16) T M k A ( n A ) (cid:17) ×× (cid:89) A (cid:54) = B ∈ (cid:89) α ∈ [ n A ] (cid:81) ¯ a ∈ ¯ A c a A,α + ε ¯ a ( K B ) c a A,α ( K B ) c a A,α − ε A ( K B ) × (cid:89) A
As an illustration, let us consider the case k = 1 . There are (cid:88) A n A possibilities, with(106) K B = δ A,B e β a A,α , A ∈ , α ∈ [ n A ] Thus, the -instanton partition function is given by:(107) Z inst1 = (cid:88) A ∈ (cid:88) α ∈ [ n A ] Z A,α , with(108) Z A,α = E ¯ A − E A E A (cid:89) α (cid:48) ∈ [ n A ] , α (cid:48) (cid:54) = α (cid:32) E ¯ A (cid:0) a A,α (cid:48) − a A,α (cid:1) (cid:0) a A,α (cid:48) − a A,α − ε A (cid:1) (cid:33) ×× (cid:89) B (cid:54) = A (cid:89) γ ∈ [ n B ] E ¯ B (cid:16) a B,γ − a A,α (cid:17) (cid:16) a B,γ − a A,α − ε B (cid:17) where(109) ε S = (cid:88) s ∈ S ε s = − ε ¯ S , E S = (cid:89) s ∈ S ε s The perturbative prefactor.
We introduce a common, i.e. λ -independent prefac-tor in the statistical weight. The so completed statistical weight is equal to:(110) W λ = Z pert (¯ a , ¯ ε ) Z λ with(111) Z pert (¯ a , ¯ ε ) = (cid:89) A ∈ Z pert ,AN =2 ∗ ( a A , ¯ ε ) × (cid:89) { a,b,c }⊂ Z pert ,a | bc fold ( a ab , a ac , ¯ ε ) × (cid:89) A ∈ ,A< ¯ A Z pert ,A cross ( a A , a ¯ A , ¯ ε ) where • for A = ab : define m A = ε ϕ ( A ) , and(112) Z pert ,AN =2 ∗ ( a A , ¯ ε ) = n A (cid:89) α,β =1 Γ (cid:16) a A,α − a A,β ; ε a , ε b (cid:17) × n A (cid:89) α,β =1 Γ − (cid:16) a A,α − a A,β + m A ; ε a , ε b (cid:17) with the Barnes double gamma functions Γ ( x ; ε a , ε b ) normalized in such a way so asto have simple zeroes on a quadrant of the integral lattice spanned by ε a , ε b :(113) Γ ( x ; ε a , ε b ) ∼ (cid:89) i,j ≥ ( x + ε a ( i −
1) + ε b ( j − , it is defined by the analytic continuation of the integral formula(114) Γ ( x ; ε a , ε b ) = exp − dds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 Γ ( s ) (cid:90) ∞ dtt t s e − tx (1 − e − tε a )(1 − e − tε b ) from the domain Re( x ) , Re( ε a ) , Re( ε b ) > . • (115) Z pert ,a | bc fold ( a ab , a ac , ¯ ε ) = n ab (cid:89) α =1 n ac (cid:89) β =1 Γ (cid:16) a ab,α − a ac,β + ε a + ε c ; ε a (cid:17) where Γ ( x ; ε a ) is essentially the ordinary Γ -function:(116) Γ ( x ; y ) ∼ ∞ (cid:89) i =1 ( x + y ( i − , Again, it can be defined by the analytic continuation of the integral(117) Γ ( x ; y ) = exp − dds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 Γ ( s ) (cid:90) ∞ dtt t s e − tx (1 − e − ty ) PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 19 from the domain
Re( x ) , Re( y ) > , giving(118) Γ ( x ; y ) = (cid:112) π/yy xy Γ (cid:16) xy (cid:17) , • (119) Z pert ,A cross ( a A , a ¯ A , ¯ ε ) = n A (cid:89) α =1 n ¯ A (cid:89) β =1 (cid:16) a A,α − a ¯ A,β + ε ¯ A (cid:17) Anomalies and other definitions of perturbative factors.
In [AGT] another nor-malization for the perturbative prefactor is used: the second line in (112) would read,in our notation, as (cid:89) ≤ α<β ≤ n A (cid:89) ¯ a ∈ ¯ A Γ − (cid:16) a A,α − a A,β + ε ¯ a ; ε a , ε b (cid:17) This normalization makes explicit the symmetry between ¯ a ∈ ¯ A , however the gaugeinvariance, i.e. the Weyl symmetry of U ( n A ) acting on a A,α is partly broken.Unlike the instanton partition function Z inst the perturbative factor does dependon the choice of the order on which is used in the definition of m A = ε ϕ ( A ) . Thisdependence will be analyzed elsewhere.3.3.2. Subtleties for tuned parameters.
When the equivariant parameters a , ¯ ε are ratio-nally dependent, the torus ˜ T H ⊂ T H they generate is strictly smaller than T H . Accord-ingly, the fixed points on the moduli space of spiked instantons need not be isolated, andthe formula (103) is used. It can be further localized to the set of torus-fixed pointson M k A ( n A ) , which are relatively well-understood in the case n A = 1 [I, IY, L]. Weshall encounter these complications when dealing with gauge theories on C / Γ spaces,or on the complex surfaces in the C / Γ (cid:48) × Γ (cid:48)(cid:48) spaces, with finite SU (2) subgroups Γ , Γ (cid:48) , Γ (cid:48)(cid:48) ⊂ SU (2) of D or E type.3.4. The main result.
Here is the main fact about the partition function of spikedinstantons: the compactness theorem proven in [Ne3] implies Z spiked ( a , ¯ ε ) defined by(120) Z spiked ( a , ¯ ε ) = (cid:88) λ W λ has no singularities in the variables:(121) x A = 1 n A n A (cid:88) α =1 a A,α with fixed(122) ˜ a A,α = a A,α − x A Remark . The reason we have to keep the majority of our variables fixed is the denom-inator Γ − in the perturbative prefactors Z pert , A N =2 ∗ . Without it the partition functionwould have been an entire function of all a A,α ’s. Orbi folding
In this section we discuss the partition function of the generalized gauge theoriesdefined on the orbifolds with respect to a discrete (finite) group Γ . Both the worldvol-ume and the transverse space of the theory may be subject to the orbifold projection.Geometrically, the action of Γ factors through the linear action in C , which we assumeto preserve the Calabi-Yau fourfold structure:(123) ρ geom : Γ −→ SU (4) This construction is motivated by the consideration of D-branes on C / Γ . As is ex-plained in [DM], the orbifold projection involves an action of Γ on the Chan-Patonspaces:(124) ρ CP : Γ −→ (cid:17) A ∈ U ( n A ) which amounts to the decomposition:(125) N A = (cid:77) $ ∈ Γ ∨ N A, $ ⊗ R $ The global symmetry group H is reduced to the Γ -centralizer: the subgroup H Γ ⊂ H which commutes with Γ . The particular cases of this construction are the quiver gaugetheories, the theories in the presence of special surface operators, possibly intersecting,and the theories on the ALE spaces.The parameters of the partition function of the orbifolded theory are ( (cid:101) a , (cid:101) ε ) ∈ Lie T H Γ ⊗ C , where (cid:101) a is in the Cartan subalgebra of the centralizer of the image ρ CP ( Γ ) in (124),while (cid:101) ε is in the Cartan subalgebra of the centralizer of the image of ρ geom ( Γ ) in SU (4) .In addition, the fugacity q of the original model fractionalizes:(126) q −→ q = ( q $ ) $ ∈ Γ Choices of discrete groups.
Since we want the action of Γ to admit the invariantcomplex two-planes supporting the strata ( X A , G A ) of the generalized gauge theories,at least for one A ∈ , the choice of Γ is reduced to the following three possibilities:(1) The abelian case:(127) Γ = Γ ab , represented in U (1) ε ⊂ SU (4) with the help of the homomorphism ρ geom =diag( ρ a ) a ∈ :(128) ρ a e π √− mκplκκ κ = exp π √− (cid:88) κ m κ n κa p l κ κ where n κa = 0 , . . . , p l κ κ − , and(129) (cid:88) a ∈ n κa = p l κ κ n κ n κ ∈ Z , PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 21
The Chan-Paton representation (124) amounts to the choice of multiplicities n A, ν :(130) ρ CP : (cid:18) e π √− m κ p − lκκ (cid:19) κ (cid:55)→ diag (cid:89) κ e π √− m κ n κ p − lκκ · n A, ν ν ∈ Γ ∨ ab ∈ U ( n A ) where ν = ( n κ ) κ ∈ Γ ∨ ab labels the irreducible (one-dimensional) representationsof Γ ab . The centralizer H Γ is equal to(131) H Γ = U (1) ε × (cid:17) ν ∈ Γ ∨ ab (cid:17) A ∈ U ( n A, ν ) , its maximal torus(132) T H Γ = U (1) ε × (cid:17) ν ∈ Γ ∨ ab (cid:17) A ∈ T n A, ν , with T n A, ν ⊂ U ( n A, ν ) the maximal torus of diagonal matrices.Define (cf. (60)):(133) N Γ ab = (cid:71) A ∈ , ν ∈ Γ ∨ ab [ n A, ν ] = { ( A, ν , α ) | A ∈ , ν ∈ Γ ∨ ab , α ∈ [ n A, ν ] } (2) The abelian × ALE case:(134) Γ = Γ ab × Γ γ represented in SU (4) with the help of the homomorphism(135) ρ geom = ρ = ρ l ρ r ρ = ρ l ρ − r ρ = ρ − l T ∈ U (2) , with(136) ρ α (cid:20) (cid:18) e π √− m κ p − lκκ (cid:19) κ × h (cid:21) = (cid:89) κ e π √− m κ ρ κα p − lκκ ∈ U (1) , α = l, r where ρ κα = 0 , . . . , p l κ κ − , and(137) ρ (cid:20) (cid:18) e π √− m κ p − lκκ (cid:19) κ × h (cid:21) = (cid:89) κ e − π √− m κ ρ κl p − lκκ T ( h ) with T the defining two-dimensional representation of SU (2) (cid:51) Γ γ .The irreducible representations of the group Γ = Γ ab × Γ γ are the tensorproducts:(138) R $ = L ν ⊗ R i , labelled by the pairs $ = ( ν , i ) , ν ∈ Γ ∨ ab , i ∈ Vert γ . With the choice (134) of Γ the only non-trivial Chan-Paton spaces are N and N . The choice of the Chan-Paton representation ρ CP in this case amounts to the choice of multiplicityspaces N , $ , N , $ , i.e. the dimension vectors(139) n $ = dim N , $ , w $ = dim N , $ , The centralizer(140) H Γ = U (1) γ × (cid:17) $ ∈ Γ ∨ U ( n $ ) × U ( w $ ) where U (1) γ ⊂ U (1) ε consists of the diagonal matrices of the form:(141) e √− ϑ e √− ϑ e − √− ( ϑ + ϑ ) · ∈ SU (4) , Define (cf. (60), (133)), for Γ = Γ ab × Γ γ :(142) N Γ = N + Γ (cid:116) N − Γ with(143) N + Γ = (cid:71) $ ∈ Γ ∨ [ n $ ] = (cid:110) ( $ , α ) | $ ∈ Γ ∨ , α ∈ [ n $ ] (cid:111) , N − Γ = (cid:71) i ∈ Vert γ N i , − Γ , N i , − Γ = (cid:71) ν ∈ Γ ∨ ab [ w i , ν ] = (cid:110) ( i , ν , β ) | ν ∈ Γ ∨ ab , β ∈ [ w i , ν ] (cid:111) (3) The ALE × ALE case:(144) Γ = Γ ab × Γ γ (cid:48) × Γ γ (cid:48)(cid:48) represented in SU (4) with the help of the homomorphism ρ geom [ t × h (cid:48) × h (cid:48)(cid:48) ] = (cid:32) ρ ( t ) · T ( h (cid:48) ) 00 ρ ( t ) − · T ( h (cid:48)(cid:48) ) (cid:33) ∈ SU (4) , with h (cid:48) ∈ Γ γ (cid:48) , h (cid:48)(cid:48) ∈ Γ γ (cid:48)(cid:48) , ρ ∈ Γ ∨ ab . The irreducible representations of Γ are thetensor products(145) R $ = L ν ⊗ R (cid:48) i (cid:48) ⊗ R (cid:48)(cid:48) i (cid:48)(cid:48) labelled by $ = ( ν , i (cid:48) , i (cid:48)(cid:48) ) , where $ ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) , and R (cid:48) , R (cid:48)(cid:48) are the irreps of Γ γ (cid:48) , Γ γ (cid:48)(cid:48) , respectively. Again, with the choice (144) of Γ theonly non-trivial Chan-Paton spaces are N and N . The choice of the Chan-Paton representation ρ CP in this case amounts to the choice of multiplicityspaces N , $ , N , $ , i.e. the dimension vectors(146) n $ = dim N , $ , w $ = dim N , $ , PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 23 for $ = ( ν , i (cid:48) , i (cid:48)(cid:48) ) ∈ Γ ∨ . The centralizer(147) H Γ = U (1) γ (cid:48) ,γ (cid:48)(cid:48) × (cid:17) $ ∈ Γ ∨ U ( n $ ) × U ( w $ ) where U (1) γ (cid:48) ,γ (cid:48)(cid:48) ⊂ U (1) ε consists of diagonal matrices of the form:(148) e √− ϑ · e −√− ϑ · ∈ SU (4) , Define (cf. (60), (133), (142)), for Γ = Γ ab × Γ γ (cid:48) × Γ γ (cid:48)(cid:48) :(149) N Γ = N + Γ (cid:116) N − Γ with N + Γ = (cid:70) i (cid:48) ∈ Vert γ (cid:48) N i (cid:48) , + Γ , N − Γ = (cid:70) i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) N i (cid:48)(cid:48) , − Γ , and(150) N i (cid:48) , + Γ = (cid:71) ν ∈ Γ ∨ ab , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) [ n ν , i (cid:48) , i (cid:48)(cid:48) ] = (cid:110) ( ν , i (cid:48) , i (cid:48)(cid:48) , α ) | ν ∈ Γ ∨ ab , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) , α ∈ [ n ν , i (cid:48) , i (cid:48)(cid:48) ] (cid:111) , N i (cid:48)(cid:48) , − Γ = (cid:71) ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) [ w ν , i (cid:48) , i (cid:48)(cid:48) ] = (cid:110) ( ν , i (cid:48) , i (cid:48)(cid:48) , β ) | ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , β ∈ [ w ν , i (cid:48) , i (cid:48)(cid:48) ] (cid:111) In what follows the expressions N A , K A etc. are promoted to N A , K A etc. which arevalued in K [ T H Γ ] ⊗ K [ Γ ] , i.e. they are the formal linear combinations:(151) (cid:88) w ∈ T ∨ H Γ , $ ∈ Γ ∨ n w, $ L w ⊗ R $ where L w are the characters of T H Γ , and R $ are the irreducible representations of Γ .Likewise, the “tangent space” character T A is promoted to T A ∈ K [ T H Γ ] ⊗ K [ Γ ] .4.0.2. Orbifold partition functions.
The definition of the partition function in the orb-ifold situation is the following. The random variable is a string λ of objects, whichnow involve both Young diagrams and connected components of the Nakajima-Youngvarieties, specifically:(1) In the abelian case the random variables are again the Young diagrams (parti-tions) λ ( A, ν ,α ) ∈ Λ , now labeled by triples: :(152) λ ab = (cid:16) λ ( A, ν ,α ) (cid:17) A ∈ , ν ∈ Γ ∨ ab ,α ∈ [ n A, ν ] ∈ Λ N Γ ab (2) In the abelian × ALE case the random variables are the collections of of twotypes of objects: Young diagrams as before, and the connected components of
Nakajima-Young varieties:(153) λ ab × ale = (cid:18) (cid:16) λ ( $ ,α ) (cid:17) $ ∈ Γ ∨ ,α ∈ [ n $ ] ; (cid:16) µ ( $ ,β ) (cid:17) $ ∈ Γ ∨ ,β ∈ [ w $ ] (cid:19) ∈ Λ N + Γ × (cid:17) i ∈ Vert γ (cid:16) Λ i γ (cid:17) N i , − Γ with λ ( $ ,α ) ∈ Λ , $ ∈ Γ ∨ , µ (( i , ν ) ,β ) ∈ Λ i γ , for i ∈ Vert γ , ν ∈ Γ ∨ ab .(3) Finally, in the ALE × ALE case the random variables are the collections ofconnected components of Nakajima-Young varieties:(154) λ ale × ale = (cid:18) (cid:16) µ ( $ ,α ) (cid:17) $ ∈ Γ ∨ ,α ∈ [ n $ ] ; (cid:16) ˜ µ ( $ ,β ) (cid:17) $ ∈ Γ ∨ ,β ∈ [ w $ ] (cid:19) with µ ( $ ,α ) ∈ Λ i (cid:48) γ (cid:48) ( v (cid:48) ) , ˜ µ ( $ ,β ) ∈ Λ i (cid:48)(cid:48) γ (cid:48)(cid:48) ( v (cid:48)(cid:48) ) , with $ = ( ν , i (cid:48) , i (cid:48)(cid:48) ) , ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) .We first describe the case of abelian orbifolds, and then proceed with the somewhatmore restricted case of the non-abelian orbifolds. In the latter case our formulas areless explicit.4.1. Abelian orbifolds.
We define the statistical model, which is parametrized by thefollowing generalization of the data of the section :(1) The string ¯ ε = ( ε a ) a ∈ of complex numbers ε a , a ∈ which sum up to zero, asin (59)(2) The string ρ geom = ( ρ a ) a ∈ of irreducible Γ ab -representations ρ a ∈ Γ ∨ ab obeying(155) (cid:88) a ∈ ρ a = 0 ∈ Γ ∨ ab In other words, ρ is a homomorphism Γ ab → U (1) ε ⊂ SU (4) , so that(156) (cid:3) SU (4) = (cid:77) a ∈ L ρ a We shall also use the notation ρ S with S ⊂ for the sum:(157) ρ S = (cid:88) s ∈ S ρ s . so that ρ ∅ = ρ = ρ and(158) ∧ • (cid:3) SU (4) = (cid:77) S ⊂ L ρ S (3) The string ¯ n of Γ ab -representations(159) N A = (cid:77) ν ∈ Γ ∨ ab N A, ν ⊗ L ν , A ∈ with the multiplicity spaces N A, ν ≈ C n A, ν of dimensions n A, ν = dim N A, ν . PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 25 (4) The string ¯ a = (cid:0) a A, ν ,α (cid:1) α ∈ [ n A, ν ] , ν ∈ Γ ∨ ab ∈ C N Γ ab of(160) (cid:88) A ∈ , ν ∈ Γ ∨ ab n A, ν = N Γ ab complex numbers a A, ν ,α , α ∈ [ n A, ν ] .The data (¯ a ; ¯ ε ) parametrizes the Cartan subalgebra of the centralizer H Γ ab .Define, for A = ( ab ) ∈ , a, b ∈ , a < b :(161) N A = (cid:88) ν ∈ Γ ∨ ab (cid:88) α ∈ [ n A, ν ] e β a A, ν ,α L ν = (cid:88) ν ∈ Γ ∨ ab N A, ν ( β ) L ν (5) The string q = ( q ν ) ν ∈ Γ of | Γ ab | = (cid:81) κ p l κ κ fugacities(162) q ν ∈ C , ν ∈ Γ ∨ ab obeying | q ν | < .Define, for S ⊂ :(163) P S, ν ( β ) = (cid:88) J ⊂ S (cid:89) a ∈ J (cid:16) − e βε a (cid:17) δ Γ ∨ − ν + (cid:88) a ∈ J ρ a , P S = (cid:88) ν ∈ Γ ∨ P S, ν L ρ S Define, for λ ∈ Λ N Γ ab , A ∈ , A = ( ab ) as before:(164) K A = (cid:88) ν ∈ Γ ab ∨ (cid:88) α ∈ [ n A,(cid:36) ] (cid:88) ( i,j ) ∈ λ ( A, ν ,α ) e β ( a A, ν ,α + ε a ( i − ε b ( j − ) L ν + ρ a ( i − ρ b ( j − = (cid:88) ν ∈ Γ ∨ ab K A, ν ( β ) L ν (cid:58)(cid:58)(cid:58)(cid:58) The (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) abelian (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) orbifold (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) model (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) statistical (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) weights.
Define, for A = { a, b } , a < b ∈ ,cf. (163)(165) T A = (cid:88) ν ∈ Γ ∨ ab T A, ν L ν = N A K ∗ A + q A K A N ∗ A ⊗ L ρ A − K A K ∗ A P A . The statistical weight of λ is given by the following expression:(166) Z Γ ab λ = (cid:89) ν ∈ Γ ∨ ab q k νν (cid:15) (cid:104) − T Γ ab λ (cid:105) where(167) k ν = (cid:88) A ∈ ,α ∈ [ n A, ν ] | λ ( A, ν ,α ) | , (168) T Γ ab λ = (cid:88) A ∈ , ν ∈ Γ ∨ ab P ϕ ( A ) , − ν T A, ν + (cid:88) ν (cid:48) ∈ Γ ∨ ab P ¯ A, ν N A, ν (cid:48) (cid:88) B (cid:54) = A K B, ν + ν (cid:48) ∗ − P , ν K A, ν (cid:48) (cid:88) B>A K B, ν + ν (cid:48) ∗ , and the full abelian orbifold partition function is defined as(169) Z Γ ab spiked (˜ a , ρ , ¯ (cid:15) ; ˜ q ) = Z pert , Γ ab (˜ a , ρ , ¯ ε ) (cid:88) λ Z Γ ab λ where the prefactor is given by the following formulas:4.1.2. (cid:58)(cid:58)(cid:58)(cid:58) The (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) abelian (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) orbifold (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) gauge (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) origami (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) perturbative (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) factors.
Define:(170) Z pert , Γ ab (˜ a , ρ , ¯ ε ) = (cid:89) A ∈ Z pert ,A, Γ ab N =2 ∗ ( a A , ρ a , ρ b , ¯ ε ) × (cid:89) { a,b,c }⊂ Z pert ,a | bc, Γ ab fold ( a ab , a ac , ρ a , ρ b , ρ c , ¯ ε ) × (cid:89) A ∈ ,A< ¯ A Z pert ,A, Γ ab cross ( a A , a ¯ A , ρ , ¯ ε ) where ( A = ab ):(171) Z pert ,A, Γ ab N =2 ∗ ( a A , ρ , ¯ ε ) = (cid:89) ν , ν (cid:48) ∈ Γ ∨ ab (cid:89) α ∈ [ n A, ν ] ,α (cid:48) ∈ [ n A, ν (cid:48) ] Γ , Γ ab (cid:32) a A, ν ,α − a A, ν (cid:48) ,α (cid:48) ; ε a , ε b ν − ν (cid:48) ; ρ a , ρ b (cid:33) Γ , Γ ab (cid:32) a A, ν ,α − a A, ν (cid:48) ,α (cid:48) + ε ϕ ( A ) ; ε a , ε b ν − ν (cid:48) + ρ ϕ ( A ) ; ρ a , ρ b (cid:33) where the projected double gamma(172) Γ , Γ ab (cid:32) x ; y (cid:48) , y (cid:48)(cid:48) ν ; ρ (cid:48) , ρ (cid:48)(cid:48) (cid:33) ∼ (cid:89) i,j ≥ ( x + y (cid:48) ( i −
1) + y (cid:48)(cid:48) ( j − δ Γ ∨ ab ( ν + ρ (cid:48) ( i − ρ (cid:48)(cid:48) ( j − can be easily expressed in terms of the ordinary Γ ’s, • (173) Z pert ,a | bc, Γ ab fold ( a ab , a ac , ρ, ¯ ε ) = (cid:89) ν , ν (cid:48) ∈ Γ ∨ ab (cid:89) α ∈ [ n ab, ν ] (cid:89) β ∈ [ n ac, ν (cid:48) ] Γ , Γ ab (cid:32) a ab, ν ,α − a ac, ν (cid:48) ,β + ε a + ε c ; ε a ν − ν (cid:48) + ρ a + ρ c ; ρ a (cid:33) where(174) Γ , Γ ab (cid:32) x ; y ν ; ρ (cid:33) ∼ ∞ (cid:89) i =1 ( x + y ( i − δ Γ ∨ ab ( ν + ρ ( i − can be easily expressed in terms of the ordinary gamma-functions, • PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 27 (175) Z pert ,A, Γ ab cross ( a A , a ¯ A , ρ, ¯ ε ) = (cid:89) ν , ν (cid:48) ∈ Γ ∨ ab (cid:89) α ∈ [ n A, ν ] (cid:89) β ∈ [ n B, ν (cid:48) ] (cid:16) a A, ν ,α − a ¯ A, ν (cid:48) ,β + ε ¯ A (cid:17) δ Γ ∨ ab ( ν − ν (cid:48) − ρ A ) The abelian × ALE case.
Fix Γ ab , γ of D or E type. Let Γ = Γ ab × Γ γ . Theirreps of Γ are labelled by the pairs $ = ( ν , i ) , ν ∈ Γ ∨ ab , i ∈ Vert γ . Fix the discretedata: the dimension vectors n = ( n $ ) $ ∈ Γ ∨ , w = ( w $ ) $ ∈ Γ ∨ , and the two characters ρ l , ρ r : Γ ab → U (1) . The orbifold gauge origami in this case depends on the followingcontinuous data:(1) Two complex numbers ε , ε , and ε = ε + ε .(2) Two sets of Coulomb parameters: a = ( a ν , i ,α ) ∈ C N + Γ , b = ( b ν , i ,β ) ∈ C N − Γ (176) a ν , i ,α ∈ C , α ∈ [ n ν , i ] , b ν , i ,β ∈ C , β ∈ [ w ν , i ] (3) The string q = ( q $ ) $ ∈ Γ ∨ of | Γ | fugacities:(177) q $ ∈ C , | q $ | < The geometric action Γ → SU (4) defines the following three representations:(178) C ≡ L ρ l + ρ r , C ≡ L ρ l − ρ r , C = L − ρ l ⊗ , which obey(179) C ⊗ C ⊗ Λ C = R , the trivial representation. Write P = P P , with:(180) P = 1 − q C − q C + e βε L ρ l , P = 1 − q − C + q − L − ρ l . Finally, define(181) N = (cid:88) $ ∈ Γ ∨ (cid:88) α ∈ [ n $ ] e β a $ ,α R $ , W = (cid:88) $ ∈ Γ ∨ (cid:88) ˜ α ∈ [ w $ ] e β b $ , ˜ α R $ • The random variables λ ab × ale in the Γ -orbifold gauge origami model were defined in(153). The statistical weight of λ ab × ale is an integral over the product of Nakajima-Young varieties:(182) X λ ab × ale = (cid:17) ν ∈ Γ ∨ ab (cid:17) i ∈ Vert γ (cid:17) ˜ α ∈ [ w ν , i ] Y i γ (cid:0) µ ν , i , ˜ α (cid:1) , µ ν , i , ˜ α ∈ Λ i γ Define(183) K = (cid:88) ν ∈ Γ ∨ ab , i ∈ Vert γ ,α ∈ [ n ν , i ] (cid:88) ( i,j ) ∈ λ ( i ,(cid:36),α ) e β a ν , i ,α q i − q j − L ν + ρ l ( i + j − ρ r ( i − j ) ⊗ R i , (184) T = N K ∗ + N ∗ K q L ρ l − P KK ∗ = (cid:88) $ ∈ Γ ∨ T , $ R $ so that, in particular(185) T , ν , = (cid:88) i ∈ Vert γ (cid:88) ν (cid:48) ∈ Γ ∨ ab (cid:16) N ν + ν (cid:48) , i K ∗ ν (cid:48) , i + q N ∗ ρ l + ν (cid:48) , i K ν + ν (cid:48) , i − K ν + ν (cid:48) , i K ∗ ν (cid:48) , i + q K ν + ν (cid:48) , i K ∗ ρ l + ρ r + ν (cid:48) , i + q K ν + ν (cid:48) , i K ∗ ρ l − ρ r + ν (cid:48) , i + q K ν + ν (cid:48) , i K ∗ ρ l + ν (cid:48) , i (cid:17) . Define:(186) V = (cid:88) i , ˜ i ∈ Vert γ (cid:88) ν ∈ Γ ∨ ab (cid:88) ˜ α ∈ [ w ν , i ] (cid:88) n ≥ e β b ν , i , ˜ α q − n Ch (cid:16) V i ˜ i ,n ( µ ν (cid:48) , i (cid:48) , ˜ α ) (cid:17) L ν − n ρ l ⊗ R ˜ i (187) T = W V ∗ + q − W ∗ V L − ρ l − P V V ∗ = (cid:88) $ ∈ Γ ∨ T , $ R $ We view N , W , K , V , T , T as the K [ T H Γ ] ⊗ K ( Γ ) -valued linear combinations of Cherncharacters of vector bundles over X λ ab × ale , as well as P = 1 − q C , P = 1 − q C .The measure (89) dressed with a partial perturbative contribution, the orbifold ver-sion of (175), is generalized to(188) Z pert Γ , cross z λ ab × ale = (cid:89) $ ∈ Γ ∨ q k $ (cid:90) X λ ab × ale (cid:15) (cid:104) − [ R ] T λ ab × ale + [ L R ] T λ ab × ale (cid:105) , where (cf. (57)):(189) [ R ] T λ ab × ale = [ R ] (cid:16) − q L ρ l N ∗ W + T + P N V ∗ + P T + P W K ∗ − P KV ∗ (cid:17) −− q − [ R ] (cid:88) ν ∈ Γ ∨ ab (cid:88) e ∈ Edge γ (cid:16) N ν + ρ l ,t ( e ) K ∗ ν ,s ( e ) − K ν + ρ l ,t ( e ) K ∗ ν ,s ( e ) − q K ν − ρ l ,t ( e ) K ∗ ν ,s ( e ) ++ q K ν − ρ r ,t ( e ) K ∗ ν ,s ( e ) + q K ν + ρ r ,t ( e ) K ∗ ν ,s ( e ) (cid:17) , [ L R ] T λ ab × ale = [ L R ] T = T X λ ab × ale and [ R ]( . . . ) denotes taking the Γ -invariant part in ( . . . ) , i.e. the contribution of thetrivial representation of Γ , while [ L R ] T (cf. (101)) denotes the T H Γ × Γ -invariantpart. Geometrically, the T H Γ × Γ -invariant [ L R ] T is the tangent space to the variety X λ ab × ale so its contribution is subtracted from the measure as the rest is being integrated PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 29 over X λ ab × ale (note that L R = L R as Γ ab action is contained in T H Γ ). Finally,(190) k ν , i = (cid:88) ν (cid:48) ∈ Γ ∨ ab (cid:88) α ∈ [ n ν (cid:48) , i ] (cid:88) ( i,j ) ∈ λ ( ν (cid:48) , i ,α ) δ Γ ∨ ab ( ν (cid:48) + ρ l ( i + j −
2) + ρ r ( i − j ) − ν ) ++ (cid:88) i (cid:48) ∈ Vert γ (cid:88) ˜ α ∈ [ w ν (cid:48) , i (cid:48) ] (cid:88) n ≥ δ Γ ∨ ab ( ν (cid:48) − n ρ l − ν ) ν i (cid:48) i ,n ( µ ν (cid:48) , i (cid:48) , ˜ α ) Of course, this formalism also applies to γ = (cid:98) A k . In this case the formulas (190),(188) reduce to the ε = ε limit of the abelian orbifold case of crossed instantons [Ne3].4.3. The ALE × ALE case.
Fix Γ ab , and two quivers γ (cid:48) , γ (cid:48)(cid:48) of D or E type. In thissection Γ = Γ ab × Γ γ (cid:48) × Γ γ (cid:48)(cid:48) , with its irreps $ = ( ν , i (cid:48) , i (cid:48)(cid:48) ) , ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) .Fix the discrete data: the dimension vectors n = ( n $ ) $ ∈ Γ ∨ , w = ( w $ ) $ ∈ Γ ∨ , onecharacter ρ : Γ ab → U (1) , equivalently a representation L ρ ∈ Γ ∨ ab . The orbifold gaugeorigami in this case depends on the following continuous data:(1) A complex number ε ∈ C .(2) Two sets of Coulomb parameters (cf. (149)):(191) a = ( a $ ,α ) ∈ C N + Γ , b = ( b $ ,β ) ∈ C N − Γ , where a $ ,α ∈ C , α ∈ [ n $ ] , b $ ,β ∈ C , β ∈ [ w $ ] .(3) The string q = ( q $ ) $ ∈ Γ ∨ of | Γ | fugacities:(192) q $ ∈ C , | q $ | < The geometric action Γ → SU (4) defines the following two representations:(193) C ≡ L ρ ⊗ (cid:48) , C = L − ρ ⊗ (cid:48)(cid:48) Define:(194) P = 1 − q C + q L ρ , P = 1 − q − C + q − L − ρ • The random variables λ ale × ale in the Γ -orbifold gauge origami model were definedin (154). The statistical weight of λ ale × ale is given by the integral over the product ofNakajima-Young varieties(195) X λ ale × ale = (cid:17) ν ∈ Γ ∨ ab (cid:17) i (cid:48) ∈ Vert γ (cid:48) , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) (cid:17) α ∈ [ n ν , i (cid:48) , i (cid:48)(cid:48) ] Y i (cid:48) γ (cid:48) ( µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; α ) ) × (cid:17) β ∈ [ w ν , i (cid:48) , i (cid:48)(cid:48) ] Y i (cid:48)(cid:48) γ (cid:48)(cid:48) ( ˜ µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; β ) ) with µ ( $ ,α ) ∈ Λ i (cid:48) γ (cid:48) , ˜ µ ( $ ,β ) ∈ Λ i (cid:48)(cid:48) γ (cid:48)(cid:48) . Define(196) N = (cid:88) ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) (cid:88) α ∈ [ n ν , i (cid:48) , i (cid:48)(cid:48) ] e β a ν , i (cid:48) , i (cid:48)(cid:48) ; α L ν ⊗ R (cid:48) i (cid:48) ⊗ R (cid:48)(cid:48) i (cid:48)(cid:48) , (197) K = (cid:88) i , i (cid:48) ∈ Vert γ (cid:48) i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) ν ∈ Γ ab ∨ (cid:88) n ≥ α ∈ [ n ν , i , i (cid:48)(cid:48) ] e β a ν , i , i (cid:48)(cid:48) ; α q n Ch (cid:16) V i (cid:48) i ,n (cid:16) µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; α ) (cid:17)(cid:17) L ν + n ρ ⊗ R (cid:48) i (cid:48) ⊗ R (cid:48)(cid:48) i (cid:48)(cid:48) = (cid:88) $ ∈ Γ ∨ K $ R $ , (198) T = N K ∗ + N ∗ K q L ρ − P KK ∗ , and(199) W = (cid:88) ν ∈ Γ ∨ ab , i (cid:48) ∈ Vert γ (cid:48) , ˜ i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) (cid:88) ˜ α ∈ [ w ν , i (cid:48) , i (cid:48)(cid:48) ] e β b ν , i (cid:48) , i (cid:48)(cid:48) ; ˜ α L ν ⊗ R (cid:48) i (cid:48) ⊗ R (cid:48)(cid:48) i (cid:48)(cid:48) , (200) V = (cid:88) i (cid:48) ∈ Vert γ (cid:48) ˜ i , i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) ν ∈ Γ ∨ ab (cid:88) n ≥ α ∈ [ w ν , i , i (cid:48)(cid:48) ] e β b ν , i , ˜ i ; ˜ α q − n Ch (cid:16) ˜ V i (cid:48)(cid:48) ˜ i ,n (cid:16) ˜ µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; β ) (cid:17)(cid:17) L ν − n ρ ⊗ R (cid:48) i (cid:48) ⊗ R (cid:48)(cid:48) ˜ i = (cid:88) $ ∈ Γ ∨ V $ R $ , (201) T = W V ∗ + W ∗ V q − L − ρ − P V V ∗ , a K [ T H Γ ] ⊗ K ( Γ ) -valued linear combination of vector bundles over X λ ale × ale , where thevector bundles over X λ ale × ale denoted with some abuse of notation by V ii (cid:48) ,n (cid:16) µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; α ) (cid:17) , ˜ V ˜ i ˜ i (cid:48)(cid:48) ,n (cid:16) ˜ µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; β ) (cid:17) are the pullbacks of the bundles V ii (cid:48) ,n (cid:16) µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; α ) (cid:17) → Y i γ (cid:48) (cid:16) µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; α ) (cid:17) , V ˜ i ˜ i (cid:48)(cid:48) ,n (cid:16) ˜ µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; β ) (cid:17) → Y i γ (cid:48)(cid:48) (cid:16) ˜ µ ( ν , i (cid:48) , i (cid:48)(cid:48) ; β ) (cid:17) under the projections to the respective factors in(195).The measure (89) dressed with a partial perturbative contribution, the Γ -orbifoldversion of (175), is now generalized to(202) Z pert Γ , cross z λ ale × ale = (cid:89) $ ∈ Γ ∨ q k $ (cid:90) X λ ale × ale (cid:15) (cid:104) − [ R ] T λ ale × ale + [ L R ] T λ ale × ale (cid:105) , PS/CFT, QQ-CHARACTERS, GAUGE ORIGAMI 31 where (cf. (57)):(203) [ R ] T λ ale × ale = [ R ] (cid:16) − q L ρ N ∗ W + T + P N V ∗ + T + P W K ∗ − P KV ∗ (cid:17) −− [ R ] (cid:16) q − C N K ∗ + q C W V ∗ − q − C KK ∗ − q C V V ∗ (cid:17) −− (cid:88) ν ∈ Γ ∨ ab (cid:88) e (cid:48) ∈ Edge γ (cid:48) (cid:88) e (cid:48)(cid:48) ∈ Edge γ (cid:48)(cid:48) (cid:16) K ν ,t ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) K ∗ ν ,s ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) + K ν ,t ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) K ∗ ν ,s ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) (cid:17) − (cid:88) ν ∈ Γ ∨ ab (cid:88) e (cid:48) ∈ Edge γ (cid:48) (cid:88) e (cid:48)(cid:48) ∈ Edge γ (cid:48)(cid:48) (cid:16) V ν ,s ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) V ∗ ν ,t ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) + V ν ,t ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) V ∗ ν ,s ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) (cid:17) , [ L R ] T λ ale × ale = [ L R ] ( T + T ) = T X λ ale × ale and(204) k ν , i , ˜ i = (cid:88) ν (cid:48) ∈ Γ ∨ ab (cid:88) n ≥ (cid:88) i (cid:48) ∈ Vert γ (cid:48) (cid:88) α ∈ [ n ν (cid:48) , i (cid:48) , ˜ i ] δ Γ ∨ ab ( ν (cid:48) + n ρ − ν ) ν i (cid:48) i ,n ( µ ν (cid:48) , i (cid:48) , ˜ i ; α )++ (cid:88) ˜ i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) (cid:88) ˜ α ∈ [ w ν (cid:48) , i , ˜ i (cid:48)(cid:48) ] δ Γ ∨ ab ( ν (cid:48) − n ρ − ν ) ν ˜ i (cid:48)(cid:48) ˜ i ,n ( ˜ µ ν (cid:48) , i , ˜ i (cid:48)(cid:48) ; ˜ α ) To compute the measure (204) we use (19) to write:(205) [ R ] ( (cid:48) ⊗ (cid:48)(cid:48) ⊗ KK ∗ ) = (cid:88) ν ∈ Γ ∨ ab (cid:88) e (cid:48) ∈ Edge γ (cid:48) (cid:88) e (cid:48)(cid:48) ∈ Edge γ (cid:48)(cid:48) (cid:16) K ν ,s ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) K ∗ ν ,t ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) + K ν ,t ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) K ∗ ν ,s ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) + K ν ,s ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) K ∗ ν ,t ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) + K ν ,t ( e (cid:48) ) ,t ( e (cid:48)(cid:48) ) K ∗ ν ,s ( e (cid:48) ) ,s ( e (cid:48)(cid:48) ) (cid:17) and similarly for (cid:48) ⊗ (cid:48)(cid:48) ⊗ V V ∗ . To compute, e.g. the contribution [ R ] ( P W K ∗ ) to(204) we also use (19):(206) [ R ] ( P W K ∗ ) = (cid:88) ν ∈ Γ ∨ ab (cid:88) i (cid:48)(cid:48) ∈ Vert γ (cid:48)(cid:48) (cid:88) i (cid:48) ∈ Vert γ (cid:48) (cid:16) W ν , i (cid:48) , i (cid:48)(cid:48) K ∗ ν + ρ , i (cid:48) , i (cid:48)(cid:48) + q W ν , i (cid:48) , i (cid:48)(cid:48) K ∗ ν +2 ρ , i (cid:48) , i (cid:48)(cid:48) (cid:17) −− q (cid:88) e ∈ Edge γ (cid:48) (cid:16) W ν ,t ( e ) , i (cid:48)(cid:48) K ∗ ν + ρ ,s ( e ) , i (cid:48)(cid:48) + W ν ,s ( e ) , i (cid:48)(cid:48) K ∗ ν + ρ ,t ( e ) , i (cid:48)(cid:48) (cid:17) The main fact.
For all Γ , let us denote by(207) x A = 1 (cid:80) $ ∈ Γ ∨ n A, $ (cid:88) $ ∈ Γ ∨ (cid:88) α ∈ [ n A, $ ] a A, $ The partition function of the orbifold gauge origami, defined by (169) in the abelian case, by(208) Z Γ cross ( a , b ; ε , ε ; q ) = (cid:88) λ ab × ale Z pert Γ , cross z λ ab × ale in the abelian × ALE case,(209) Z Γ cross ( a , b ; ε ; q ) = (cid:88) λ ale × ale Z pert Γ , cross z λ ale × ale in the ALE × ALE case, has no singularities in the x A variables, with ˜ a A, $ = a A, $ − x A fixed. Again, this follows from the compactness theorem proven in [Ne3].5. Conclusions and outlook
The partition function of the gauge origami model, can be viewed as the expectationvalue in the N = 2 ∗ U ( n A ) theory on C A of an operator. In the crossed case, N A N B = 0 , A ∩ B (cid:54) = ∅ , this operator is the qq-character of the (cid:98) A -type [Ne2]. In the orbifoldedcrossed case this operator is the qq-character of the (cid:98) g γ -type. The orbifold partitionfunctions in the abelian case describe the (cid:98) A -type quiver gauge theories on the A -typeALE spaces in the presence of various surface defects invariant under the rotationalsymmetries of the maximal Ω -deformation. In the abelian × ALE case these partitionfunctions describe either the qq-characters of the (cid:98) D or (cid:98) E -type quiver gauge theories,possibly with the surface defects, or the (cid:98) A -type quiver gauge theory on the D or E -typeALE space, possibly with a novel type of surface defect (which collapses to a point-likedefect in the orbifold limit of the ALE space), and a qq-character. Finally, in the ALE × ALE case we are dealing with the (cid:98) D or (cid:98) E -type quiver gauge theories, on the D or E -type ALE space, with the qq-characters and novel surface defects.The physics of these defects will be discussed in the companion paper [Ne7].The regularity of these expectation values will be used in the forthcoming publi-cations [Ne8, SX] to derive the KZ and BPZ equations [BPZ, KZ] on the partitionfunctions of supersymmetric gauge theories with and without surface operators. References [ADHM] M. Atiyah, V. Drinfeld, N. Hitchin, Yu. Manin, Construction of Instantons, Phys. Lett.
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