BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem
BBPS/CFT CORRESPONDENCE II :INSTANTONS AT CROSSROADS,MODULI AND COMPACTNESS THEOREM
NIKITA NEKRASOV
Abstract.
Gieseker-Nakajima moduli spaces M k ( n ) parametrize the charge k non-commutative U ( n ) instantons on R and framed rank n torsion free sheaves E on CP with ch ( E ) = k . They also serve as local models of the moduli spaces of instantonson general four-manifolds. We study the generalization of gauge theory in which thefour dimensional spacetime is a stratified space X immersed into a Calabi-Yau four-fold Z . The local model M k ( (cid:126)n ) of the corresponding instanton moduli space is themoduli space of charge k (noncommutative) instantons on origami spacetimes. There, X is modelled on a union of (up to six) coordinate complex planes C intersecting in Z modelled on C . The instantons are shared by the collection of four dimensionalgauge theories sewn along two dimensional defect surfaces and defect points. We alsodefine several quiver versions M γ k ( (cid:126) n ) of M k ( (cid:126)n ) , motivated by the considerations ofsewn gauge theories on orbifolds C / Γ .The geometry of the spaces M γ k ( (cid:126) n ) , more specifically the compactness of the set oftorus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwingeridentities recently found to be satisfied by the correlation functions of qq -charactersviewed as local gauge invariant operators in the N = 2 quiver gauge theories.The cohomological and K-theoretic operations defined using M k ( (cid:126)n ) and their quiverversions as correspondences provide the geometric counterpart of the qq -characters,line and surface defects. Contents
1. Introduction 21.1. Organization of the paper 41.2. Acknowledgements 52. Gauge and string theory motivations 52.1. Generalized gauge theory 52.2. Gauge origami 62.3. Symmetries, twisting, equivariance 62.4. Gauge theories on stacks of D-branes 83. Spiked instantons 123.1. Generalized ADHM equations 123.2. Holomorphic equations 163.3. The moduli spaces M ∗ k ( (cid:126)n ) a r X i v : . [ h e p - t h ] A ug NIKITA NEKRASOV fine print U versus P U × U (1) × ADE 5811. Spiked instantons on orbifolds and defects 6012. Conclusions and future directions 61References 621.
Introduction
Recently we introduced a set of observables in quiver N = 2 supersymmetric gaugetheories which are useful in organizing the non-perturbative Dyson-Schwinger equa-tions, relating contributions of different instanton sectors to the expectation valuesof gauge invariant chiral ring observables. In this paper we shall provide the naturalgeometric setting for these observables. We also explain the gauge and string theorymotivations for these considerations. PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 3 (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)
Notations . In our story we explore moduli spaces, which parametrize, roughly speak-ing, the sheaves E supported on a union of coordinate complex two-planes ≈ C inside C . The C with a set of C ’s inside is a local model ( Z loc , X loc of a Calabi-Yau fourfold Z containing a possibly singular surface X ⊂ Z :(1) Z loc = C , X loc = (cid:91) A ∈ C A , supp( E ) = (cid:91) A ∈ n A C A We denote by the set of complex coordinates in C :(2) = { , , , } , a ∈ ↔ z a ∈ C and by(3) = (cid:32) (cid:33) = (cid:40) { , } , { , } , { , } , { , } , { , } , { , } (cid:41) the set of two-element subsets of , i.e. the set of coordinate two-planes in C .We shall sometimes denote the elements of by the pairs ab = ba ↔ { a, b } ∈ . Wealso define, for A ∈ , ¯ A = \ A , and(4) (cid:15) ( A ) = ε abcd , A = { a, b } , ¯ A = { c, d } , a < b, c < d so that, e.g.
12 = 34 , (cid:15) (23) = ε = 1 , (cid:15) (24) = ε = − .The two-plane C A ⊂ C corresponding to A ∈ is defined by the equations: z ¯ a = 0 ,for all ¯ a ∈ ¯ A .We denote by the quotient / Z where Z acts by the involution A (cid:55)→ ¯ A . Theelements a ∈ are the unordered pairs ( A, ¯ A ) .We can visualize the sets , , , using the tetrahedron:Figure 1: Tetrahedron with the sets and of vertices and edges,and the set = { red , green , orange } of crossed edgesOur story will involve four complex parameters q a ∈ C × , a ∈ , obeying(5) (cid:89) a ∈ q a = 1 We shall also use the additive variables e a ∈ C , a ∈ , obeying(6) (cid:88) a ∈ e a = 0 NIKITA NEKRASOV
Define the lattice(7) Z e = Z e + Z e + Z e ⊂ C which is the image of the projection:(8) Z → C , ( i, j, k, l ) (cid:55)→ e i + e j + e k + e l We shall use the following functions on :(9) p A = (cid:89) a ∈ A (1 − q a ) ,q A = (cid:89) a ∈ A q a , e A = (cid:88) a ∈ A e a = − e ¯ A . In what follows we denote by [ n ] , for n ∈ Z > , the set { , , , . . . , n } ⊂ Z > .Let S be a finite set, and ( V s ) s ∈ S a collection of vector spaces. We use the notation(10) (cid:88) s ∈ S V s for the vector space which consists of all linear combinations(11) (cid:88) s ∈ S ψ s , ψ s ∈ V s . Organization of the paper.
We review the gauge and string theory motivationin the section . The moduli space M k ( (cid:126)n ) of spiked instantons is introduced in thesection . The symmetries of spiked instantons are studied in the section . The mod-uli space of ordinary U ( n ) instantons on (noncommutative) R is reviewed in section . The section discusses in more detail two particular cases of spiked instantons,the crossed instantons and the folded instantons. The crossed instantons live on twofour-dimensional manifolds transversely intersecting in the eight-dimensional ambientmanifold (a Calabi-Yau fourfold), the folded ones live on two four-dimensional mani-folds intersecting transversely in the six dimensional ambient manifold. The section constructs the spiked instantons out of the ordinary ones, and studies the toric spikedinstantons in some detail. The section is the main result of this paper: the com-pactness theorem. In section we enter the theory of integration over the spiked andcrossed instantons, and relate the analyticity of the partition functions to the com-pactness theorem. The section discusses the ADE -quiver generalizations of crossedinstantons. The section describes the spiked instantons on cyclic orbifolds, and theassociated compactness theorem. The section is devoted to future directions andopen questions. PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 5
Figure 2: The origami wolrdvolume X = (cid:83) A X A Acknowledgements.
Research was supported in part by the NSF grant PHY1404446. I am grateful to A. Okounkov, V. Pestun and S. Shatashvili for discus-sions. I would also like to thank Alex DiRe, Saebyeok Jeong, Xinyu Zhang, NaveenPrabhakar and Olexander Tsymbalyuk for their feedback and for painfully checkingsome of the predictions of the compactness theorem proven in this paper.The constructions of this paper were reported in 2014-2016 in a series of lectures atthe Simons Center for Geometry and Physics http://scgp.stonybrook.edu/video_portal/video.php?id=2202 , at the Institute for Advanced Studies at Hebrew Univer-sity , at the Center for Mathemat-ical Sciences and Applications at Harvard University http://cmsa.fas.harvard.edu/nikita-nekrasov-crossed-instantons-qq-character/ and at the String-Math-2015conference in Sanya, China.2.
Gauge and string theory motivations
Generalized gauge theory.
We study the moduli spaces M X,G of what might becalled supersymmetric gauge fields in the generalized gauge theories, whose space-time X contains several, possibly intersecting, components: see Fig. 2. We call such X the origami worldvolume . The gauge groups G | X A = G A on different components may bedifferent. The intersections X A ∪ X B lead to the bi-fundamental matter fields chargedunder G A × G B . The arrangement is motivated by the string theory considerations,where the open string Hilbert space, in the presence of several D -branes, splits intosectors labelled by the boundary conditions. It is well-known [34, 10] that some featuresof the open string theory are captured by the noncommutative gauge theory. In fact, thetheories we shall study descend from the maximally supersymmetric Yang-Mills theory,which is twisted and deformed. One can view the fields of this theory as describing thedeformations of the four dimensional stratified manifolds X = ( X A , n A ) , i.e. singular, ingeneral, spaces, which can be represented as unions X = ∪ A X A of manifolds with certainconditions on closures and intersections, endowed with multiplicities, i.e. the strata X A are allowed to have different multiplicity n A . The local gauge group G A is simply U ( n A ) . The particular twist of the super-Yang-Mills theory we study corresponds to NIKITA NEKRASOV
X (cid:44) → Z × E , where E is a two torus T , a cylinder R × S , or a plane R , while Z is aspecial holonomy eight dimensional manifold, e.g. the Calabi-Yau fourfold.2.2. Gauge origami.
Now suppose Z has non-trivial isometries (it ought to be non-compact). It is natural, in this case, to deform the problem to take into account thesymmetries of Z . The partition function of the theory of stratified multiple X ’s localizesonto the set of fixed points, which are the configurations X = ( X A , n A ) where X A ’s areinvariant under the isometries of Z . For example, when Z is toric, with the threedimensional torus T acting by isometries, preserving the holomorphic top form, thenat each vertex z ∈ Z T pass at most six strata X A , A ∈ .We are interested in integrals over the moduli spaces M X ,G . We shall view M X ,G asthe “space, defined by some equations modulo symmetry”. More formally, M X ,G is thequotient of a set of zeroes of some G g -equivariant section s : M → V of G g -equivariantvector bundle V → M over some smooth space (vector space in our case) with G g -action, with some Lie group G g . If M is compact the integral over M X ,G of a closeddifferential form can be represented by the G g -equivariant integral over M of the pull-back of the corresponding form times the Euler class of V . In the non-compact case oneuses equivariant cohomology (mass deformation, in the physics language) with respectto both G g and some global symmetry group H , and Mathai-Quillen representatives ofthe Euler class.The resulting partition functions(12) Z X ,G ( ξ ) ∼ (cid:90) H equiv M X ,G ∼ G g ) (cid:90) Lie( G g ) (cid:90) ( G g × H ) equiv M Euler( V ) are functions on the Lie algebra of H C , ξ ∈ Lie( H C ) . The analytic properties of Z X ,G ( ξ ) reflect some of the geometric and topological features of M X ,G . They are the mainfocus of this paper.The equivariant localization expresses Z X ,G as the sum over the fixed points of H -action, which are typically labelled by multiple partitions, i.e. collections of Youngdiagrams. The resulting statistical mechanical model is called the gauge origami andis studied in detail in the companion paper [29].2.3. Symmetries, twisting, equivariance.
The partition functions Z X,G ( ξ ) are analyticfunctions of ξ ∈ Lie( H C ) , with possible singularities. Given ξ ∈ Lie( H C ) , the closure ofthe subgroup exp tξ , t ∈ C defines a torus T ξ . The partition function Z X,G ( ξ ) can becomputed, by Atiyah-Bott fixed point formula, as a sum over the T ξ -fixed points. Eventhough the moduli space M X,G may be noncompact (it is noncompact for noncompact X ), the fixed point set, for suitable ξ , may still be compact, so that the integrals over M X,G of the equivariant differential forms converge. The set M T ξ X,G of T ξ -fixed pointsmay have several connected components:(13) M T ξ X,G = (cid:91) f (cid:18) M T ξ X,G (cid:19) f PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 7
The contributions Z f of (cid:18) M T ξ X,G (cid:19) f are rational functions on Lie( H C ) , they have poles.In the nice situations the component (cid:18) M T ξ X,G (cid:19) f has a normal bundle in M X,G (or in theambient smooth variety, as in the case of the obstructed theory), N f , which inheritsan action of T ξ , and decomposes into the sum of complex line bundles (real rank twobundles) L f ,w , with w going through the set of T ξ -weights. The fixed point formulastates(14) Z f = (cid:90) (cid:18) M T ξX,G (cid:19) f Euler ξ (Obs f ) (cid:81) w (cid:16) w ( ξ ) + c ( L f ,w ) (cid:17) The poles in Z f occur when the Lie algebra element ξ crosses the hyperplane w ( ξ ) = 0 for some w occuring in the decomposition of N f . Geometrically this means that the ξ belongs to a subalgebra of Lie T ⊂ Lie( H C ) which fixes not only (cid:18) M T ξ X,G (cid:19) f , but also (atleast infinitesimally, at the linearized order) a two-dimensional surface passing through f , in the direction of L f ,w .We shall be interested in the analytic properties of Z X,G and one of the questionswe shall be concerned with is whether the poles in Z f are cancelled by the poles in thecontribution of some other component (cid:18) M T ξ X,G (cid:19) f (cid:48) of the fixed point set. More precisely,once ξ → ξ c where ξ c belongs to the hyperplane w ( ξ ) = 0 defined relative to the weightdecomposition of N f , the component of the fixed point set may enhance,(15) (cid:18) M T ξ X,G (cid:19) f ⊂ (cid:18) M T ξc X,G (cid:19) f (cid:48)(cid:48) , reaching out to the other component (cid:18) M T ξ X,G (cid:19) f (cid:48) (16) (cid:18) M T ξ X,G (cid:19) f (cid:48) ∩ (cid:18) M T ξc X,G (cid:19) f (cid:48)(cid:48) (cid:54) = ∅ If the enhanced component (cid:18) M T ξc X,G (cid:19) f (cid:48)(cid:48) is compact, then the pole at ξ = ξ c in Z f will becancelled by the pole in Z f (cid:48) .So the issue in question is the compactness of the fixed point set for the torusgenerated by the non-generic infinitesimal symmetries ξ c .In our case we shall choose a class of subgroups H ⊂ H . We shall show that the set of H -fixed points is compact. It means that for generic choice of (cid:98) ξ ∈ Lie( H C ) the partitionfunction Z X,G ( (cid:98) ξ + x ) as a function of x ∈ Lie( H C ) ⊥ ⊂ Lie( H ) has no singularities.The procedure of restricting the symmetry group of the physical system to a subgroupis well-known to physicists under the name of twisting [38]. It is used in the contextof topological field theories, which are obtained from the supersymmetric field theorieshaving an R -symmetry group H R such that the group of rotations G rot of flat spacetime NIKITA NEKRASOV can be embedded nontrivially into the direct product(17) G rot −→ G rot × H R . We shall encounter a lot of instances of the procedure analogous to (17) in what follows.2.4.
Gauge theories on stacks of D-branes.
The maximally supersymmetric Yang-Mills theory in p + 1 -dimensions models [39] the low energy behavior of a stack ofparallel Dp -branes. This description can be made p -blind by turning on a backgroundconstant B -field. In the strong B -field the “non-abelian Born-Infeld/Yang-Mills” theorydescription of the low energy physics of the open strings connecting the Dp -branescrosses over to the noncommutative Yang-Mills description [34]. In this paper we shalluse the noncommutative Yang-Mills to study the dynamics of intersecting stacks of Dp -branes.2.4.1. The Matrix models.
Recall the dimensional reductions of the maximally super-symmetric Yang-Mills theory down to , and dimensions [4], [18], [9].We take the gauge group to be G g = U ( N ) for some large N . Following [23] we shallview the model of [18](18) G g (cid:90) R | ⊗ Lie G g DX m Dθ α exp − (cid:88) m Let us start in the -dimensional case.Let Σ be the worldsheet of our theory, with the local complex coordinates z, ¯ z . Thetheory has a gauge field A = A z dz + A ¯ z d ¯ z , Hermitian adjoint scalars X m , and adjoint fermions, which split into right ψ a + , and left χ ˙ a − ones. Here m , a and ˙ a are the indices of the v , s and c representations of the global symmetry group Spin(8) , respectively. The formalism that we shall adopt singles out a particular spinoramong c . The isotropy subgroup Spin(7) of that spinor has the following significance.The representation c , under Spin(7) , decomposes as ⊕ , the being the invariantsubspace. Accordingly, we split (cid:16) χ ˙ a − (cid:17) ˙ a ∈ c −→ (cid:16) χ i − (cid:17) i ∈ ⊕ η − . The representations s and v become the spinor of Spin(7) . We shall also need the auxiliary fields h i , whichare the worldsheet scalars, transform in the adjoint of the gauge group, and in therepresentation of Spin(7) . The theory, in this formalism, has one supercharge δ + ,which squares to the chiral translation on the worldsheet(19) δ = D ++ = ¯ ∂ ¯ z + A ¯ z PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 9 (the conjugate derivative D −− = ∂ z + A z ) which acts on the fields of the model as follows:(20) δ + X m = ψ m + , δ + ψ m + = D ++ X m = ¯ ∂ ¯ z X m + [ A ¯ z , X m ] δ + χ i − = h i , δ + h i = D ++ χ i − = ¯ ∂ ¯ z χ i − + [ A ¯ z , χ i − ] δ + A z = η − , δ + η − = F z ¯ z The Lagrangian(21) L = δ + (cid:90) Σ Tr (cid:16) ψ m + D −− X m + χ i − (cid:16) i ℘ imn [ X m , X n ] − h i (cid:17) + η − F z ¯ z (cid:17) becomes that of the standard N = 8 supersymmetric Yang-Mills once the auxil-iary fields h i are eliminated by their equations of motion. Here ℘ imn is the matrixof the projection ℘ : Λ v = LieSpin(8) −→ onto the orthogonal complement to LieSpin(7) ⊂ LieSpin(8) = LieSpin(7) ⊕ .Upon the dimensional reduction to dimensions, the gauge field A becomes a com-plex scalar σ = A ¯ z and its conjugate ¯ σ = A z .2.4.3. (0 , -formalism, SU(4) -instantons. In this formalism we have two supercharges Q + , ¯ Q + , obeying(22) Q = ¯ Q = 0 , Q + ¯ Q + + ¯ Q + Q + = D ++ so that δ + = Q + + ¯ Q + . We split Hermitian adjoint scalars X m into complex adjointscalars Z a , a ∈ , and their conjugates ¯ Z ¯ a , and the same for the fermions ψ m + −→ ψ a + , ¯ ψ ¯ a + .The χ i − ’s split as ⊕ : ( χ i − ) i ∈ → (cid:16) χ A, − = ε A ¯ A ¯ χ ¯ A, − (cid:17) A ∈ ⊕ χ − . This splitting breaksthe symmetry group Spin(8) × Spin(2) ⊂ Spin(10) of (18) down to SU(4) .The Lagrangian (21), in this formalism, reads as follows:(23) L = δ + Ψ , Ψ = (cid:90) Σ Tr (cid:16) ψ a + D −− ¯ Z ¯ a + ¯ ψ ¯ a + D −− Z a + η − F z ¯ z (cid:17) ++ i (cid:90) Σ Tr (cid:18) χ ab, − (cid:18) [ Z a , Z b ] + 12 ε abcd [ ¯ Z ¯ c , ¯ Z ¯ d ] (cid:19) + χ − µ (cid:19) −− (cid:90) Σ Tr (cid:16) χ − h + χ ab, − h ab (cid:17) where(24) µ = (cid:88) a ∈ [ Z a , ¯ Z ¯ a ] The supersymmetric (for flat Σ ) solutions of (23) are the covariantly holomorphicmatrices, solving the equations(25) D ¯ z Z a = 0 , D ¯ z ¯ Z ¯ a = 0 , a ∈ , µ = 0 and(26) [ Z a , Z b ] + 12 ε abcd [ ¯ Z ¯ c , ¯ Z ¯ d ] = 0 , { a, b } ∈ NIKITA NEKRASOV For finite dimensional C N these equations imply that all matrices commute and can besimultaneously diagonalized.2.4.4. Noncommutative gauge theory. We now wish to consider a generalization of themodel [18] in which the finite dimensional vector space C N is replaced by a Hilbertspace H . In order to keep the action (21) finite the combination(27) ℘ imn [ X m , X n ] could be deformed to(28) ℘ imn [ X m , X n ] − i ϑ i · H for some constants ϑ i . One possibility to have a finite action configuration (after h i ’sare integrated out) is to have the operators X m obey the Heisenberg algebra:(29) [ X m , X n ] = i ϑ mn · H with the c -number valued matrix ϑ mn = − ϑ mn obeying(30) ϑ i = (cid:88) m Now let us assume all θ a equal to ζ > . Take H = L ( R ) to be the Fock spacerepresentation of the algebra(35) [ c , c ] = [ c † , c † ] = 0[ c i , c † j ] = ζδ ij , i, j = 1 , Define:(36) (cid:98) z = 1 √ (cid:16) N ⊗ c † + N ⊗ c † + N ⊗ c † (cid:17)(cid:98) z = 1 √ (cid:16) N ⊗ c † + N ⊗ c † + N ⊗ c † (cid:17)(cid:98) z = 1 √ (cid:16) N ⊗ c † + N ⊗ c † + N ⊗ c † (cid:17)(cid:98) z = 1 √ (cid:16) N ⊗ c † + N ⊗ c † + N ⊗ c † (cid:17) These operators obey:(37) [ (cid:98) z a , (cid:98) z b ] = [ (cid:98) z a † , (cid:98) z b † ] = 0 (cid:88) a ∈ [ (cid:98) z a , (cid:98) z a † ] = − ζ N ⊗ H where(38) N = (cid:77) A ∈ N A The solution (37) describes six stacks of D -branes spanning the coordinate two-planes C ⊂ C , with n A = dim N A branes spanning the two-plane C A . This is a generalizationof the “piercing string” and “fluxon” solutions of [14, 13].We can easily produce more general solutions of the BPS equations. Take six so-lutions (cid:98) C A , (cid:98) C A of non-commutative instanton equations in R , viewed as operators in N A ⊗ H , obeying:(39) [ (cid:98) C A , (cid:98) C A ] = 0 , [ (cid:98) C A , (cid:98) C † A ] + [ (cid:98) C A , (cid:98) C † A ] = ζ Define operators in H = (cid:76) A ∈ N A ⊗ H :(40) (cid:98) Z a = 1 √ (cid:77) A (cid:51) a (cid:98) C h A ( a ) † A where h { a,b } ( a ) = 1 for a < b , and h { a,b } ( a ) = 2 for a > b . These operators satisfy thehigher dimensional analogues of the noncommutative instanton equations (26), (25), NIKITA NEKRASOV [28]:(41) [ (cid:98) Z a , (cid:98) Z b ] + 12 (cid:88) c,d ε abcd [ (cid:98) Z d † , (cid:98) Z c † ] = 0 , a, b, c, d ∈ (cid:88) a ∈ [ (cid:98) Z a , (cid:98) Z a † ] = − ζ N ⊗ H The aim of the next section is to produce the (almost) finite-dimensional model of themoduli space of finite action solutions to (41). Some of these solutions are of the form(40). . . . Recently the field theory description of two stacks of intersecting D branes in IIBstring theory sharing a common -dimensional worldvolume was explored in [6, 22].The theories exibit unusual holographic and renormalization properties. . . . The string theory of two stacks of transversely intersecting D branes in IIB theoryhas been recently studied in [36, 35], albeit in the ζ = 0 case. None of the beauty (tothe trained eye) of the picture presented below seem to survive in this limit.3. Spiked instantons We are going to work with the collections of vector spaces and linear maps betweenthem. The vector spaces will be labelled by the coordinate complex two-planes in thefour dimensional complex vector space C .3.1. Generalized ADHM equations. We start by fixing seven Hermitian vector spaces: K and N A , A ∈ . Let k = dim C ( K ) , n A = dim C ( N A ) . Consider the vector space A k ( (cid:126)n ) of linear maps ( B , I , J ) (42) B = ( B a ) a ∈ , B a : K → K , I = ( I A ) A ∈ , I A : N A → K , J = ( J A ) A ∈ , J A : K → N A . PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 13 Figure 3: Seven vector spaces and maps between themThe vector spaces and the maps are conveniently summarized by the tetrahedrondiagram on Fig. 3. The choice of the matrices can be motivated by the string theoryconsiderations. Namely, consider k D ( − -branes in the vicinity of the six stacks of D -branes (some of these stacks could be D -branes) spanning the coordinate two-planes C A ⊂ C . The number of branes spanning C A is n A .Figure 4: Open string sectors: fields B , I , J Then the open strings stretched between the D ( − and D ( − ’s produce, uponquantization, the matrices B a , B † a , together with their superpartners, and some auxiliaryfields, which enter the effective Lagrangian in such a way so as to impose the following3.1.1. KK equations. Define, for A = { a, b } , a < b ,(43) µ A = [ B a , B b ] + I A J A , and(44) s A = µ A + (cid:15) ( A ) µ † ¯ A : K → K , A ∈ NIKITA NEKRASOV obeying(45) s † A = (cid:15) ( A ) s ¯ A Define the real moment map (46) µ = (cid:88) a ∈ [ B a , B † a ] + (cid:88) A ∈ (cid:16) I A I † A − J † A J A (cid:17) The symmetry (45) allows to view the collection (cid:126)s = ( s A ) A ∈ ⊕ µ as the U ( K ) -equivariantmap(47) (cid:126)s : A k ( (cid:126)n ) −→ Lie U ( K ) ∗ ⊗ R , as a sort of an octonionic version of the hyperkähler moment map [15]. . . . Likewise, the open strings stretched between the D ( − and D ’s produce, uponquantization, the matrices I A , J A , I † A , J † A , together with their superpartners, and someauxiliary fields,Figure 5: Open string sectors: mutiplets of the equations σ ¯ aA , s A which enter the effective Lagrangian in such a way so as to impose the following3.1.2. KN equations. For each pair ( ¯ a, A ) , where A ∈ , and ¯ a ∈ ¯ A , define(48) σ ¯ aA = B ¯ a I A + ε ¯ a ¯ bA B † ¯ b J † A : N A → K where ¯ b ∈ ¯ A , and ¯ b (cid:54) = ¯ a . PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 15 NN equations. Now, for each A ∈ define(49) Υ A = J ¯ A I A − (cid:15) ( A ) I † ¯ A J † A : N A → N ¯ A which obey(50) Υ † A = − Υ ¯ A . Because of the symmetry (50) the collection of the maps ( Υ A ) A ∈ takes values in thereal vector space of dimension(51) (cid:88) A ∈ n A n ¯ A The equations (50) result from integrating out the open strings connecting the twostacks of D -branes which intersect only at a point, the origin in C . . . . For each pair A (cid:48) , A (cid:48)(cid:48) ∈ , such that A (cid:48) ∩ A (cid:48)(cid:48) = { a } , and i ≥ , define(52) Υ A (cid:48) ,A (cid:48)(cid:48) ; i = J A (cid:48) B i − a I A (cid:48)(cid:48) . These equations result (conjecturally) from integrating out the − strings connectingthe neighbouring stacks C A (cid:48) and C A (cid:48)(cid:48) , intersecting along a real two-dimensional plane C a . . . . Finally, for each A = { a (cid:48) , a (cid:48)(cid:48) } ∈ , e.g. a (cid:48) < a (cid:48)(cid:48) , and i, j ≥ , define(53) Υ A ; i,j = J A B i − a (cid:48) B j − a (cid:48)(cid:48) I A A very useful identity. Let us compute(54) (cid:88) A ∈ Tr s A s † A + (cid:88) A ∈ , ¯ a ∈ ¯ A Tr σ ¯ aA σ † ¯ aA + (cid:88) A ∈ Tr Υ A Υ † A =2 (cid:88) A ∈ Tr µ A µ † A + (cid:88) A ∈ ε ( A ) (cid:16) Tr µ A µ ¯ A + Tr µ † A µ † ¯ A (cid:17) + (cid:88) A ∈ ,a ∈ ¯ A Tr (cid:16) B † a B a Π IA + B a B † a Π JA (cid:17) + 2 (cid:88) A ∈ Tr Π JA Π I ¯ A + − (cid:88) A ∈ , ¯ A = { ¯ a, ¯ b } (cid:15) ( A ) Tr ([ B ¯ a , B ¯ b ] I A J A + c.c. ) − (cid:88) A ∈ (cid:15) ( A ) Tr ( I A J A I ¯ A J ¯ A + c.c. ) == 2 (cid:88) A ∈ (cid:16) (cid:107) µ A (cid:107) + (cid:107) J ¯ A I A (cid:107) (cid:17) + (cid:88) A ∈ ,a ∈ ¯ A (cid:107) B a I A (cid:107) + (cid:107) J A B a (cid:107) where(55) Π JA = J † A J A , Π IA = I A I † A NIKITA NEKRASOV Holomorphic equations. Using the identity (54) it is easy to show that the equa-tions(56) s A = 0 , Υ A = 0 , A ∈ σ ¯ aA = 0 , ¯ a ∈ ¯ A , which are not holomorphic in the variables B , I , J , imply stronger holomorphic equa-tions: for each A ∈ ,(57) µ A = 0 , J ¯ A I A = 0 B ¯ a I A = 0 , J A B ¯ a = 0 , ¯ a ∈ ¯ A, The moduli spaces M ∗ k ( (cid:126)n ) . Define M ik ( (cid:126)n ) to be the U ( k ) -quotient of the space ofsolutions to (56) (which imply, by the above argument, (57)), the additional equations(58) Υ A (cid:48) ,A (cid:48)(cid:48) ; j = 0 , ≤ j ≤ i, for all A (cid:48) , A (cid:48)(cid:48) ∈ with A (cid:48) ∩ A (cid:48)(cid:48) = 1 , and the “moment map” equation(59) µ = ζ · K The group U ( k ) acts by:(60) ( B , I , J ) (cid:55)→ ( g − B g, g − I , J g ) , g ∈ U ( k ) It is clear that, as a set(61) M ∞ k ( (cid:126)n ) ⊂ . . . ⊂ M ik ( (cid:126)n ) ⊂ M i − k ( (cid:126)n ) ⊂ . . . ⊂ M k ( (cid:126)n ) ⊂ M k ( (cid:126)n ) and that the sequence stabilizes at i ≥ k (use the fact that a k × k matrix obeys thedegree k polynomial equation).3.4. Stability. Imposing (59) with ζ > and dividing by U ( k ) is equivalent to imposingthe stability condition and dividing by the action (60) with g ∈ GL ( k ) ≡ GL ( k, C ) . Notethat we deal with the equations (57) when talking about the GL ( k ) symmetry. Thestability condition reads:(62) Any subspace K (cid:48) ⊂ K , such that I A ( N A ) ⊂ K (cid:48) , for all A ∈ and B a ( K (cid:48) ) ⊂ K (cid:48) , for all a ∈ coincides with all of K , K (cid:48) = K in other words , (cid:88) A ∈ C [ B , B , B , B ] I A ( N A ) = K The proof is standard. In one direction, let us prove (62) holds given that the GL ( k ) -orbit of the tuple ( B , I , J ) of matrices crosses the locus µ = ζ K . Indeed, assume there PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 17 is K (cid:48) which is B -invariant, and contains the image of I A ’s. Let K (cid:48)(cid:48) be the orthogo-nal complement K (cid:48)(cid:48) = ( K (cid:48) ) ⊥ . Let P (cid:48) , P (cid:48)(cid:48) be the orthogonal projections onto K (cid:48) , K (cid:48)(cid:48) ,respectively:(63) K = P (cid:48) + P (cid:48)(cid:48) , P (cid:48) P (cid:48)(cid:48) = P (cid:48)(cid:48) P (cid:48) = 0 , ( P (cid:48) ) = ( P (cid:48) ) † = P (cid:48) , ( P (cid:48)(cid:48) ) = ( P (cid:48)(cid:48) ) † = P (cid:48)(cid:48) Since the images of I ’s are in K (cid:48) , we have:(64) P (cid:48)(cid:48) I A = 0 , A ∈ Since B preserve K (cid:48) , we have:(65) P (cid:48)(cid:48) B a P (cid:48) = 0 , a ∈ Define(66) b a = P (cid:48)(cid:48) B a P (cid:48)(cid:48) , b † a = P (cid:48)(cid:48) B † a P (cid:48)(cid:48) , a ∈ ,j A = J A P (cid:48)(cid:48) , P (cid:48)(cid:48) J † A = j † A , A ∈ Thus:(67) ζP (cid:48)(cid:48) = P (cid:48)(cid:48) µP (cid:48)(cid:48) = (cid:88) a ∈ [ b a , b † a ] − (cid:88) a ∈ P (cid:48)(cid:48) B † a P (cid:48) B a P (cid:48)(cid:48) − (cid:88) A ∈ j † A j A Now, taking the trace of both sides of (67) we arrive at the conclusion K (cid:48)(cid:48) = 0 :(68) ≤ ζ dim K (cid:48)(cid:48) = − (cid:88) a ∈ (cid:107) P (cid:48) B a P (cid:48)(cid:48) (cid:107) − (cid:88) A ∈ (cid:107) j A (cid:107) ≤ ⇒ dim K (cid:48)(cid:48) = 0 Conversely, assume (62) holds. Let(69) f = 12 Tr ( µ − ζ K ) Consider the gradient flow, generated by f with respect to the flat Kähler metric(70) ds = (cid:107) d B (cid:107) + (cid:107) d I (cid:107) + (cid:107) d J (cid:107) The function f decreases along the gradient trajectory. Moreover, the trajectory be-longs to the GL ( k ) -orbit. Eventually, the trajectory stops at a critical point of f . Eitherit is the absolute minimum, i.e. the solution to (59), or the higher critical point, where(71) (cid:104) ξ, ∇ µ (cid:105) = 0 , ξ = µ − ζ K (cid:54) = 0 The one-parametric subgroup (exp tξ ) t ∈ C ⊂ GL ( K ) , preserves ( B , I , J ) ,(72) [ B a , ξ ] = 0 , a ∈ ,ξI A = 0 , J A ξ = 0 , A ∈ Define K (cid:48) = ker ξ . The Eq. (72) implies K (cid:48) is B -invariant, and contains the image of I .Therefore, by (62), K (cid:48) = K , ξ ≡ , i.e. (59) is satisfied. (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Notation . We denote by [ B , I , J ] the GL ( k ) -orbit (cid:16) g − B a g, g − I A , J A g (cid:17) a ∈ ,A ∈ ,g ∈ GL ( k ) . NIKITA NEKRASOV The symmetries of spiked instantons The moduli spaces M ∗ k ( (cid:126)n ) are acted on by a group H = H (cid:126)n of symmetries, definedbelow. The symmetry of M ∗ k ( (cid:126)n ) will be used in several ways. First, we shall be study-ing H -equivariant integration theory of the spiked instanton moduli, in cohomologyand equivariant K -theory. Second, the shall define new moduli spaces by studyingthe Γ -fixed loci (cid:16) M ∗ k ( (cid:126)n ) (cid:17) Γ in M ∗ k ( (cid:126)n ) , for subgroups Γ ⊂ H . These moduli spaces havethe commutant C Γ ( H ) as the symmetry group. Finally, the connected components M ∗ ,γ k ( (cid:126) n ) ⊂ (cid:16) M ∗ k ( (cid:126)n ) (cid:17) Γ can be defined using only the quiver of Γ , not the group Γ . Thedefinition can be then generalized to define more general quiver spiked instantons.Their symmetry H γ generalizes the commutant C Γ ( H ) .4.1. Framing and spatial rotations. First of all, we can act by a collection h = ( h A ) A ∈ of unitary matrices h A ∈ U ( n A ) , defined up to an overall U (1) multiple:(73) h · [ B a , I A , J A ] = (cid:104) B a , I A h A , h − A J A (cid:105) We call the symmetry (73) the framing rotation.Secondly, we can multiply the matrices B a by the phases B a (cid:55)→ q a B a , as long as theirproduct is equal to :(74) (cid:89) a ∈ q a = 1 and we supplement this transformation with the transformation J A (cid:55)→ q A J A :(75) q · [ B a , I A , J A ] = [ q a B a , I A , q A J A ] We can view q as the diagonal matrix(76) q = diag ( q , q , q , q ) ∈ U (1) e ⊂ SU (4) which belongs to the maximal torus U (1) e of the group SU (4) of rotations of C preserving some supersymmetry. We call (75) the spatial rotations.The group(77) H = P (cid:17) A ∈ U ( n A ) × U (1) e is the symmetry of the moduli space of spiked instantons for generic ζ and (cid:126)n . The com-plexification H C preserves the holomorphic equations (57) and the stability condition(62).The center Z H of H is the eight dimensional torus(78) Z H = U (1) x × U (1) e The maximal torus T H of H is the torus(79) T H = (cid:18)(cid:16) × A ∈ T A (cid:17) (cid:30) U (1) (cid:19) × U (1) e PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 19 where(80) T A ⊂ U ( n A ) is the group of diagonal n A × n A unitary matrices, the maximal torus of U ( n A ) , T A ≈ U (1) n A . In the Eq. (79) we divide by the U (1) embedded diagonally into the productof all (cid:80) A n A U (1) ’s.4.1.1. Coulomb parameters. Let ( a , e ) ∈ Lie( T H ) ⊗ C ,(81) e = ( e , e , e , e ) , e a ∈ C , (cid:88) a ∈ e a = 0 , a = ( a A ) A ∈ , a A = diag (cid:16) a A, , . . . , a A,n A (cid:17) ∈ Lie( T A ) ⊗ C The eigenvalues a A,α ∈ C are defined modulo the overall shift a A,α (cid:55)→ a A,α + x , x ∈ C .The integrals (12) which we define below are meromorphic functions of ( e , a ) .4.1.2. Symmetry enhancements. Sometimes the symmetry of the spiked ADHM equa-tions enhances. First of all, if all I = J = 0 (for N A = 0 , for all A ), then the q -transformations can be generalized to the action of the full SU(4) = Spin(6) :(82) B a (cid:55)→ (cid:88) c ∈ g a ¯ c B c , gg † = 1 , det( g ) = 1 In the case of less punitive restrictions on N A ’s, e.g. in the crossed instanton case, thesymmetry enhances to SU(2) × U (1) × SU(2) , and, if ζ = 0 , to SU(2) . Let us assume,for definiteness, that only N and N are non-zero. Then the transformations:(83) ( B , B , B , B , I , J , I , J ) (cid:55)→(cid:55)→ (cid:16) uaB + ubB , − u ¯ bB + u ¯ aB , ¯ ucB + ¯ udB , − ¯ u ¯ dB + ¯ u ¯ cB , uI , uJ , ¯ uI , ¯ uJ (cid:17) , (cid:32) a b − ¯ b ¯ a (cid:33) ∈ SU(2) , (cid:32) c d − ¯ d ¯ c (cid:33) ∈ SU(2) , (cid:32) u 00 ¯ u (cid:33) ∈ U (1) ∆ ⊂ SU(2) ∆ a ¯ a + b ¯ b = c ¯ c + d ¯ d = u ¯ u = 1 preserve the crossed instanton equations (56). When ζ = 0 the U (1) ∆ symmetry en-hances to the full SU(2) ∆ , acting by:(84) ( B , B , B , B ) , ( I , J , I , J ) (cid:55)→(cid:55)→ (cid:16) uB − vB † , vB † + uB , ¯ uB − ¯ vB † , ¯ vB † + ¯ uB (cid:17) , (cid:16) uI − vJ † , uJ + vI † , ¯ uI − ¯ vJ † , ¯ uJ + ¯ vI † (cid:17) , (cid:32) u ¯ v − v ¯ u (cid:33) ∈ SU(2) ∆ , u ¯ u + v ¯ v = 1 The equation Υ ≡ J I − I † J † , the equations s = − s † , s = s † , the equations σ , , σ , as well as the equations σ , , σ , are SU(2) ∆ -invariant, while the equations s , µ, s = s † form a doublet. NIKITA NEKRASOV Subtori. In what follows we shall encounter the arrangement of hyperplanes H l in Lie( T H ) ⊗ C defined by the system of linear equations:(85) L i ( a , e ) = (cid:88) A ∈ (cid:88) α ∈ [ n A ] (cid:36) i ; A,α a A,α + (cid:88) a ∈ n i ; a e a = 0 with (cid:36) i ; A,α ∈ {− , , +1 } , n i ; a ∈ Z and the matrix (cid:36) i ; A,α of maximal rank. Such equa-tions (85) can be interpreted as defining a subtorus H = T L ⊂ T H : simply solve (85)for the subset of a A,α ’s for which the matrix (cid:36) i ; A,α is invertible. We shall not worryabout the integrality of the inverse matrix in this paper, by using the covering tori, ifnecessary.One of the reasons we need to look at the subtori T L is the following construction.4.3. Orbifolds, quivers, defects. In this section the global symmetry group H is equalto H = P (cid:17) A ∈ U ( n A ) × G rot where(1) G rot = U (1) e if there are at least two A (cid:48) (cid:54) = A (cid:48)(cid:48) ∈ with non-empty intersection with n A (cid:48) n A (cid:48)(cid:48) (cid:54) =0 , and to(2) G rot = SU(2) A × U (1) ∆ × SU(2) ¯ A otherwise, i.e. there is at most one pair A, ¯ A with n A n ¯ A (cid:54) = 0 .In all cases G rot ⊂ SU(4) , so that to every γ ∈ Γ one associates a unitary × matrix q = (cid:107) q ba ( γ ) (cid:107) a,b ∈ with unitdeterminant. In the first case this matrix is diagonal, in the second case it is a × block-diagonal matrix with unitary × blocks of inverse determinants.The symmetry of M ∗ k ( (cid:126)n ) can be used to define new moduli spaces. Suppose Γ ⊂ H is a discrete subgroup. Let H Γ ⊂ H be the maximal subgroup commuting with Γ , thecentralizer of Γ . Let Γ ∨ be the set of irreducible unitary representations ( R ω ) ω ∈ Γ ∨ of Γ , and (cid:126)k ∈ Z Γ ∨ ≥ . The representations N A , A ∈ of H decompose as representations of Γ (86) N A = (cid:77) ω ∈ Γ ∨ N A,ω ⊗ R ω Let (cid:126) n now denote the collection ( n A,ω ) A ∈ ,ω ∈ Γ ∨ of dimensions(87) n A,ω = dim N A,ω of multiplicity spaces. The vector k = ( k ω ) ω ∈ Γ ∨ defines a representation of Γ :(88) γ ∈ Γ (cid:55)→ g γ ∈ U ( K ) , K = (cid:77) ω ∈ Γ ∨ K ω ⊗ R ω , k ω = dim K ω PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 21 We call the components k ω of the vector k fractional instanton charges. The modulispace of Γ -folded spiked instantons of charge (cid:126)k is the component M ∗ , Γ k ( (cid:126) n ) set of Γ -fixedpoints (cid:16) M ∗ k ( (cid:126)n ) (cid:17) Γ . The representation (88) enters the realization of the Γ -fixed locus inthe space of matrices ( B , I , J ) :(89) γ · ( B , I , J ) = (cid:16) g γ B g − γ , g γ I , J g − γ (cid:17) , g γ ∈ U ( K ) where(90) γ · ( B , I , J ) ≡ (cid:16) q ba ( γ ) B b , I A h A ( γ ) − , q A ( γ ) h A ( γ ) J A (cid:17) ,γ ∈ Γ (cid:55)→ ( h A ( γ )) A ∈ × (cid:107) q ba ( γ ) (cid:107) a,b ∈ ∈ H is the defining representation of Γ , with q A ( γ ) given by (9) in the case (1), and bythe projection to U (1) ∆ in the case (2). The equations (89) are invariant under thesubgroup(91) (cid:17) ω ∈ Γ ∨ U ( K ω ) ⊂ U ( K ) of unitary transformations of K commuting with Γ . The holomorphic equations (57)restricted onto the locus of Γ -equivariant i.e. obeying (89) matrices B , I , J becomethe holomorphic equations defining M Γ k ( (cid:126) n ) . The stability condition (62) can be fur-ther refined, analogously to the refinement of the real moment map equation µ = (cid:80) ω ∈ Γ ∨ ζ ω K ω ⊗ R ω . We shall work in the chamber where all ζ ω > .The moduli spaces M Γ k ( (cid:126) n ) in the case (1) parametrize the spiked instantons in thepresence of U (1) e -invariant surface operators, while in the case (2) they parametrize theinstantons in supersymmetric quiver gauge theories on the ALE spaces, with additionaldefect.The commutant H Γ acts on M Γ k ( (cid:126) n ) , so that the partition functions we study aremeromorphic functions on Lie (cid:16) H Γ (cid:17) ⊗ C .Note that if Γ has trivial projection to G rot then the moduli space of Γ -folded instan-tons is simply the product of the moduli spaces of spiked instantons for N ω ’s. In whatfollows we assume the projection to G rot to be non-trivial.4.3.1. Subtori for Γ -folds. Let us now describe the maximal torus T H Γ of the Γ -commutantas T L . In other words, the choice of a discrete subgroup Γ ⊂ H defines the hyperplanes L i ( a , e ) = 0 in Lie( T H ) .In the case (1) the G rot -part of Γ is abelian, i.e. it is a product of cyclic groups (if Γ is finite) or it is a torus itself. In either case there is no restriction on the e -parameters.The framing part of Γ reduces P ( × A U ( n A )) to P (cid:0) × A,ω U ( n A,ω ) (cid:1) which means that someof the eigenvalues a A,α , viewed as the generators of Lie U ( n A ) , must coincide, moreprecisely to be of multiplicity dim R ω . The minimal case, when Γ is abelian, imposesno restrictions on ( a , e ) , so that T H Γ = T H .In the case (2) the G rot -part of Γ need not be abelian. Let us assume, for definiteness,that A = 12 , ¯ A = 34 . If the image of Γ in SU (2) is non-abelian, then e = e . Likewiseif the image of Γ in SU (2) is non-abelian then e = e . The non-abelian discrete NIKITA NEKRASOV subgroups of SU (2) have irreducible representations of dimensions and higher, up to . Thus the corresponding a A,α eigenvalues will have the multiplicity up to .4.3.2. Subtori for sewing. Let us specify the integral data for the subtori, i.e. theexplicit solutions to the constraints (85). Let e = ( e a ) a ∈ , e a ∈ Z (cid:54) =0 , be a -tuple ofnon-zero integers, with no common divisors except for ± , which sum up to zero:(92) (cid:88) a ∈ e a = 0 . Such a collection e defines a split = + (cid:113) − , where ± being the set of A = { a, b } suchthat ± e a e b > .For A = { a, b } ∈ − , i.e. e a e b < , let p A = gcd( | e a | , | e b | ) > . Let us also fix for such A ∈ − a partition ν A = (cid:0) ν A,ι (cid:1) of size n A , whose parts do not exceed e A : ≤ ν A,ι ≤ p A .Let (cid:96) A = (cid:96) ( ν A ) be its length.Given ν A we partition the set [ n A ] as the union of nonintersecting subsets(93) [ n A ] = (cid:91) ι [ n A ] ι , n A ] ι = ν A,ι , [ n A ] ι (cid:48) ∩ [ n A ] ι (cid:48)(cid:48) = ∅ for ι (cid:48) (cid:54) = ι (cid:48)(cid:48) . Fix a map c A,ι : [ n A ] ι → Z obeying, for any a (cid:48) , a (cid:48)(cid:48) ∈ [ n A ] ι , a (cid:48) (cid:54) = a (cid:48)(cid:48) :(94) c A,ι ( a (cid:48) ) − c A,ι ( a (cid:48)(cid:48) ) (cid:54) = 0 (mod p A ) When p A = 1 the condition (94) is empty.For A = { a, b } ∈ + , let us fix a map c A : [ n A ] → Z , obeying, for any a (cid:48) , a (cid:48)(cid:48) ∈ [ n A ] ,(95) c A ( a (cid:48) ) − c A ( a (cid:48)(cid:48) ) / ∈ Z > e a + Z > e b Note that (95) does not forbid the situation where c A ( a (cid:48) ) = c A ( a (cid:48)(cid:48) ) for some a (cid:48) , a (cid:48)(cid:48) ∈ [ n A ] .To make the notation uniform we assign to such A , ν A = ( n A ) , (cid:96) A = 1 .The final piece of data is the choice ι A ∈ [ (cid:96) A ] for each A ∈ − . Define the set λ A = [ (cid:96) A ] \{ ι A } , of cardinality (cid:96) A − .Now, we associate to the data(96) L = ( ν, λ, c, e ) , the torus(97) H = T L = (cid:17) A ∈ U (1) (cid:96) A − × U (1) Note that only A ∈ − contribute to the product in (97). This torus is embedded into T H as follows: the element(98) ( e i ξ , e i t ) ≡ (cid:17) A ∈ − (cid:16) e i ξ A,i (cid:17) i ∈ λ A × e i t ∈ T L PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 23 is mapped to(99) (cid:34) (cid:17) A ∈ − diag − A × (cid:17) A ∈ + diag + A (cid:35) × diag (cid:16) e i e a t (cid:17) a ∈ ∈ P (cid:17) A ∈ U ( n A ) × SU(4) where diag ± A ∈ U ( n A ) are the diagonal matrices with the eigenvalues(100) Eigen (cid:0) diag − A (cid:1) = (cid:110) e i c A,ιA ( α ) t | α ∈ [ n A ] ι A (cid:111) ∪ (cid:26) e i ( ξ A,ι + c A,i ( α ) t ) | ι ∈ λ A , α ∈ [ n A ] ι (cid:27) (101) Eigen (cid:16) diag + A (cid:17) = (cid:110) e i c A ( α ) t | α ∈ [ n A ] (cid:111) Thus, the torus T L corresponds to the solution of the Eqs. (85) with(102) e a = e a u, a ∈ , a A,α = c A ( α ) u, A ∈ + , α ∈ [ n A ] a A,α = c A,ι A ( α ) u, A ∈ − , α ∈ [ n A ] ι A a A,α = ξ A,ι + c A,ι ( α ) u, A ∈ − , α ∈ [ n A ] ι , ι ∈ λ A In other words, the Ω -background parameters are maximally rationally dependent (theworst way to insult the rotational parameters), the framing of the spaces N A , A ∈ + is completely locked with space rotations (spin-color locking), while the framing of thespaces N A , A ∈ − is locked partially.4.4. Our goal: compactness theorem. Our goal is to establish the compactness ofthe fixed point sets M k ( (cid:126)n ) T L and (cid:16) M Γ k ( (cid:126) n ) (cid:17) T H Γ . Before we attack this problem we shalldiscuss a little bit the ordinary instantons, then look at a few examples of the particulartypes of spiked instantons: the crossed and the folded instantons, and then proceedwith the analysis of the general case. The reader interested only in the compactnesstheorem can skip the next two sections at the first reading.5. Ordinary instantons In this section we discuss the relations between the ordinary four dimensional U ( n ) instantons and the spiked instantons.5.1. ADHM construction and its fine print . In the simplest case only one of six vectorspaces is non-zero, e.g.(103) N A = 0 , A (cid:54) = { , } . Let n = n . We shall now show that, set theoretically, the moduli space of spikedinstantons in this case is M k ( n ) , the ADHM moduli space (more precisely, its Gieseker-Nakajima generalization).Recall the ADHM construction of the U ( n ) framed instantons of charge k on (non-commutative) R [3, 24, 27]. It starts by fixing Hermitian vector spaces N and K ofdimensions n and k , respectively. Consider the space of quadruples ( B , B , I , J ) ,(104) I : N → K , J : K → N , B α : K → K , α = 1 , NIKITA NEKRASOV obeying(105) (cid:126)µ ≡ (cid:16) µ R , µ C + µ C † , i (cid:16) µ C − µ C † (cid:17) (cid:17) = ( ζ, , · K , where(106) µ C = [ B , B ] + I J, µ R = [ B , B † ] + [ B , B † ] + I I † − J † J Note that the number of equations (105) plus the number of symmetries is less thenthe number of variables. The moduli space M k ( n ) of solutions to (105) modulo the U ( k ) action(107) ( B , B , I , J ) (cid:55)→ ( g − B g, g − B g, g − I , Jg ) , g ∈ U ( k ) has the positive dimension(108) dim R M k ( n ) = 4 k ( n + k ) − k − k = 4 nk Again, the µ R -equation, with ζ > , can be replaced by the stability condition, andthe GL ( k ) -symmetry:(109) Any subspace K (cid:48) ⊂ K , such that I ( N ) ⊂ K (cid:48) , and B α ( K (cid:48) ) ⊂ K (cid:48) , for all α = 1 , K , K (cid:48) = K in other words , C [ B , B ] I ( N ) = K (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Notation. We denote by [ B , B , I , J ] the GL ( k ) -orbit (cid:16) g − B g, g − B g, g − I , Jg (cid:17) g ∈ GL ( k ) .5.2. Ordinary instantons from spiked instantons. Now, to show that the spiked in-stantons reduce to the ordinary instantons when (103) is obeyed, we need to show that B = B = 0 on the solutions of our equations (57). This is easy:(110) B f ( B , B ) I = f ( B , B ) B I = 0 where we used [ B , B ] = µ = 0 , [ B , B ] = µ = 0 , and B I = 0 . Therefore B acts byzero on all of K . The same argument proves the vanishing of B .5.3. One-instanton example. Let k = 1 . We can solve the equations (106) explic-itly. The matrices B , B are just complex numbers, e.g. b , b ∈ C . The pair I , J obeys I J = 0 , (cid:107) I (cid:107) −(cid:107) J (cid:107) = ζ . Assuming ζ > define the vectors w = J ( K ) ∈ N , w = √ ζ + (cid:107) J (cid:107) I † ( K ) ∈ N . They obey (cid:104) w , w (cid:105) = 0 , (cid:104) w , w (cid:105) = 1 . Dividing by the U (1) = U ( K ) symmetry we arrive at the conclusion:(111) M ( n ) = C × T ∗ CP n − The first factor parametrizes ( b , b ) , the base CP n − of the second factor is the spaceof w ’s obeying (cid:107) w (cid:107) = 1 modulo U (1) symmetry. PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 25 U versus P U . In describing the action of H in (73) specified to the case ofordinary instantons we use an element h of the group U ( n ) yet it is the group P U ( n ) = U ( n ) /U (1) = SU ( n ) / Z n which acts faithfully on M k ( n ) . Indeed, multiplying h by ascalar matrix(112) h → h ˜ u, ˜ u ∈ U (1) does not change the effect of the transformation (73) since it can be undone by the U ( k ) -transformation (107) with g = ˜ u − K ∈ U ( k ) .5.5. Tangent space. Let m ∈ M k ( n ) . Let ( B , B , I , J ) be the representative of m =[ B , B , I , J ] . Consider the nearby quadruple(113) ( B + δB , B + δB , I + δI , J + δJ ) Assuming it solves the ADHM equations to the linear order, the variations δB , δB , δI , δJ are subject to the linearized µ C equation:(114) d ( δB , δB , δI , δJ ) := [ B , δB ] + [ δB , B ] + ( δI ) J + I ( δJ ) = 0 and we identify the variations which differ by an infinitesimal GL ( K ) -transformationof ( B , B , I , J ) :(115) ( δB , δB , δI , δJ ) ∼ ( δB , δB , δI , δJ ) + d ( δσ ) ,d ( δσ ) := ( [ B , δσ ] , [ B , δσ ] , − δσ · I , J · δσ ) Since d ◦ d = 0 , the tangent space is the degree cohomology, T m M k ( n ) = ker d / im d = H T m M k ( n ) , of the complex(116) T m M k ( n ) =[0 → End( K ) −→ d −→ d End( K ) ⊗ C ⊕ Hom( N , K ) ⊕ Hom( K , N ) −→ d −→ d End( K ) ⊗ ∧ C → Fixed locus. In applications we will be interested in the fixed point set M k ( n ) T with T ⊂ H a commutative subgroup. The maximal torus T ⊂ H is the productof the maximal torus T n ⊂ P U ( n ) and the two dimensional torus U (1) × U (1) (cid:48) ⊂ SU (2) × U (1) (cid:48) . Let(117) a = i diag( a , . . . , a n ) , a α ∈ R be the generic element of Lie ( T n ) . It means that the numbers a α are defined up to thesimultaneous shift(118) a α ∼ a α + a, a ∈ R and we assume a α (cid:54) = a β , for α (cid:54) = β . Let(119) e = 12 ( e − e , e + e ) NIKITA NEKRASOV be the generic element of Lie (cid:16) U (1) × U (1) (cid:48) (cid:17) . The pair ( a , e ) generates an infini-tesimal transformation (73), (75) of the quadruple ( B , B , I , J ) :(120) δ a , e ( B , B , I , J ) = ( e B , e B , I a , ( e + e − a ) J ) For the U ( k ) -equivalence class f = [ B , B , I , J ] to be fixed under the infinitesimal trans-formation generated by the generic pair ( a , e ) there must exist an infinitesimal U ( k ) -transformation (107)(121) δ σ ( B , B , I , J ) = ( [ B , σ ] , [ B , σ ] , − σ I , Jσ ) undoing it: δ σ ( B , B , I , J ) + δ a , e ( B , B , I , J ) = 0 . In other words, there must exist theoperators (cid:98) d α , (cid:98) d , (cid:98) d ∈ End( K ) , such that:(122) σ = (cid:88) α ∈ [ n ] a α (cid:98) d α + e (cid:98) d + e (cid:98) d , (cid:88) α ∈ [ n ] (cid:98) d α = K obeys:(123) e a B a = [ σ , B a ] , a ∈ { , } I a = σ I , − ( e + e ) J + a J = Jσ , or, in the group form:(124) q a B a = g t B a g − t , q q h − t J = Jg − t , I h t = g t I where q a = e t e a , g t = e tσ , h t = e t a . Here t is an arbitrary complex number, the map t (cid:55)→ g t defines the representation T → GL ( k ) . The Eqs. (124) imply:(125) g t (cid:18) B i − B j − I (cid:19) = q i − q j − (cid:18) B i − B j − I (cid:19) h t The Eqs. (123) for generic ( a , e ) imply:(126) K = (cid:77) α ∈ [ n ] K α , (cid:98) d α | K β = δ α,β The eigenspace K α is generated by I α = I ( N α ) , where N α ⊂ N is the eigenline of a withthe eigenvalue a α :(127) K α = (cid:88) i,j ≥ B i − B j − I α . The subspace I α ⊂ K (it is one-dimensional for generic a ) obeys:(128) (cid:98) d β I α = δ α,β , (cid:98) d I α = (cid:98) d I α = 0 On K α the operators (cid:98) d , (cid:98) d have non-negative spectrum:(129) σ (cid:18) B i − B j − I α (cid:19) = ( a α + e ( i − 1) + e ( j − (cid:18) B i − B j − I α (cid:19) PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 27 Therefore, as long as a β − a α / ∈ e Z > + e Z > ,(130) J (cid:18) B i − B j − I α (cid:19) = 0 , as follows from the last Eq. in (123).Thus, we have shown that(131) J = 0 , [ B , B ] = 0 Define an ideal I ( α ) ⊂ C [ x, y ] in the ring of polynomials in two variables by:(132) P ( x, y ) ∈ I ( α ) ⇔ P ( B , B ) I α = 0 Define the partition λ ( α ) = (cid:18) λ ( α )1 ≥ λ ( α )2 ≥ . . . ≥ λ ( α ) (cid:96) λ ( α ) (cid:19) by(133) λ ( α ) i = min { j | B i − B j I α = 0 } Thus, K α = C [ z , z ] / I λ ( α ) . Here we denote by I λ ⊂ C [ x, y ] the ideal generated by themonomials x i − y λ i , i = 1 , , . . . , (cid:96) λ .Conversely, given the monomial ideal I λ ( α ) define the vector I α ⊂ K α to be the im-age of the polynomial in the quotient C [ x, y ] / I λ ( α ) . The operators B , B act bymultiplication by the coordinates x , y , respectively. Furthermore,(134) K α = (cid:96) λ (cid:77) i =1 λ i (cid:77) j =1 K α ; i,j where(135) (cid:98) d | K α ; i,j = i − , (cid:98) d | K α ; i,j = j − The map ( a , e ) (cid:55)→ σ ∈ End( K ) makes the space K a T -representation. Its characteris easy to compute:(136) K χ := Tr K g t = (cid:88) α ∈ [ n ] e t a α (cid:96) λ ( α ) (cid:88) i =1 q i − λ ( α ) i (cid:88) j =1 q j − The set of eigenvalues of σ is a union of n collections of centers of boxes of Youngdiagrams λ (1) , . . . , λ ( n ) NIKITA NEKRASOV Figure 6: Eigenvalues of σ The space N is a T -representation by definition:(137) N χ := Tr N h t = (cid:88) α ∈ [ n ] e t a α We also define:(138) K ∗ χ = Tr K g − t , N ∗ χ = Tr N h − t Tangent space at the fixed point. Finally, the tangent space T f M k ( n ) to themoduli space at the fixed point f is also a T -representation. Let us compute itscharacter. Let f = [ B , B , I , J ] . The quadruple ( B , B , I , J ) is fixed by the compositionof the T transformation ( e t a , e t e ) ∈ T C and the GL ( k ) -transformation e tσ ∈ GL ( k ) ,for any complex number t , cf. (123). Now take the nearby quadruple (cid:16) ˜ B = B + δB , ˜ B = B + δB , ˜ I = I + δI , ˜ J = J + δJ (cid:17) and act on it by the combination of the T C transformation ( e t a , e t e ) : (cid:16) ˜ B , ˜ B , ˜ I , ˜ J (cid:17) (cid:55)→ (cid:16) q ˜ B , q ˜ B , ˜ I , q q ˜ J (cid:17) and the GL ( k ) -transformation g t : (cid:16) ˜ B , ˜ B , ˜ I , ˜ J (cid:17) (cid:55)→ (cid:16) g − t ˜ B g t , g − t ˜ B g t , g − t ˜ I , ˜ Jg t (cid:17) , defining the T -action on ( δB , δB , δI , δJ ) :(139) e t · [ δB , δB , δI , δJ ] = (cid:104) q g − t δB g t , q g − t δB g t , g − t δI h t , q q h − t δJg t (cid:105) . PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 29 So the space T f M k ( n ) of variations ( δB , δB , δI , δJ ) is a T representation, with thecharacter:(140) Tr T f M k ( n ) ( h, q ) = ( q + q ) K χ K ∗ χ + N χ K ∗ χ + N ∗ χ K χ q q The first two terms on the right hand side account for δB , δB , the third term corre-sponds to the δI variations, and the last term accounts for the δJ variations.Now, the tangent space T f M k ( n ) is the degree cohomology H T f M k ( n ) of thecomplex (116), which has no H or H cohomology (for ζ > ). The character of T f M k ( n ) can be therefore computed by taking the alternating sum of the characters of T f M k ( n ) , T f M k ( n ) and T f M k ( n ) , giving:(141) Tr T f M k ( n ) ( h, q ) = N χ K ∗ χ + q N ∗ χ K χ − p K χ K ∗ χ Thus,(142) Tr T f M k ( n ) ( h, q ) = (cid:88) α,β ∈ [ n ] e t ( a α − a β ) T χ ( λ ( α ) , λ ( β ) ) where(143) T χ ( µ, λ ) = (cid:88) ( i,j ) ∈ λ q i − λ tj q µ i +1 − j + (cid:88) ( i,j ) ∈ µ q µ tj +1 − i q j − λ i We see that, as long as there is no rational relation between e and e , and a α − a β / ∈ e Z + e Z the weights which appear in the character of the tangent space are non-zero.In other words, the tangent space T f M k ( n ) does not contain trivial representations of T , i.e. f is an isolated fixed point.5.8. Smaller tori. Let ( Γ ∨ , d ) be a pair consisting of a finite or a countable set Γ ∨ (themeaning of the notation will become clear later), and a function d : Γ ∨ → N , which weshall call the dimension. We assign to each ω ∈ Γ ∨ a vector space(144) R ω = C d ( ω ) of the corresponding dimension.Let n be a d -partition of n ,(145) n = (cid:88) ω ∈ Γ ∨ n ω d ( ω ) , n ω ≥ with only a finite number of n ω > . Let(146) (cid:96) n = { ω | n ω > } We associate to n a decomposition of N into the direct sum of tensor products:(147) N = (cid:77) ω ∈ Γ ∨ N ω ⊗ R ω with n ω -dimensional complex vector spaces N ω .Define, for the d -partition n and a pair ( e , e ) of non-zero integers, the sub-torus(148) T n ; e ≈ T n × U (1) e ⊂ P U ( N ) × Spin(4) . NIKITA NEKRASOV Here U (1) e is embedded into U (1) × U (1) (cid:48) ⊂ U (2) ⊂ Spin(4) by(149) U (1) e : e i ϑ (cid:55)→ (cid:18) e i2 ( e + e ) ϑ , e i2 ( e − e ) ϑ (cid:19) , in other words, it acts on C by:(150) ( z , z ) (cid:55)→ (cid:16) e i e ϑ z , e i e ϑ z (cid:17) . The torus T n ⊂ T n is defined to be a quotient of the product of the maximal tori of U ( n ω ) by the overall center U (1) :(151) h = (cid:77) ω ∈ Γ ∨ h ω ⊗ R ω ∈ U ( N ) , (cid:16) h : . . . : h (cid:96) n (cid:17) = (cid:16) h u : . . . : h (cid:96) n u (cid:17) ∈ T n , where h ω h † ω = 1 , ω ∈ Γ ∨ , h ω = diag (cid:16) u ω, , . . . , u ω,n ω (cid:17) , and | u | = 1 .5.9. Fixed points of smaller tori. Let us start with n = 1 , so that T = T e . The T e -fixed points on M k (1) = Hilb [ k ] ( C ) are isolated for e e < and non-isolated for e e > , as we see from the T χ ( λ, λ ) character (143). Indeed, as soon as the partition λ has a box (cid:3) = ( i, j ) whose arm plus one-to-leg, or leg plus one-to-arm ratio is equalto e : e ,(152) e ( i − λ tj ) + e ( λ i + 1 − j ) = 0 , or e ( λ tj + 1 − i ) + e ( j − λ i ) = 0 then T λ M k (1) contains trivial T e -representations, i.e. λ is not an isolated fixed point.Geometrically, the fixed points of the T e -action for e e > are the ( e , e ) -gradedideals I in C [ x, y ] , i.e. the ideals which are invariant under the C × -action:(153) ( x, y ) → ( t e x, t e y ) For such an ideal I the quotient K = C [ x, y ] / I is also a graded vector space:(154) K = d K (cid:77) s =0 K s For general Γ ∨ and the general partition n the T n ; e -fixed point set is a finite union offinite product(155) M k ( n ) T n ; e = (cid:91) (cid:80) ω ∈ Γ ∨ ,α ∈ [ nω ] k α,ω = k (cid:17) ω ∈ Γ ∨ (cid:17) α ∈ [ n ω ] M k α,ω ( d ( ω )) T e of the T e -fixed point sets on the moduli spaces M k (cid:48) ( n (cid:48) ) . This is easy to show usingthe same methods as we employed so far. It suffices then to analyze the structure of M k ( n ) T e where the torus T e ≈ U (1) acts on the matrices ( B , B , I , J ) via:(156) e i t · [ B , B , I , J ] = (cid:104) e i e t B , e i e t B , I , e i( e + e ) t J (cid:105) PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 31 As usual, the GL ( K ) -equivalence class of the quadruple [ B , B , I , J ] is T e -invariant iffor every e t ∈ C × there is an operator g t ∈ GL ( K ) which undoes (156), i.e.(157) e i e t g − t B g t = B , e i e t g − t B g t = B g − t I = I , e i( e + e ) t Jg t = J The correspondence e t (cid:55)→ g t splits K as the sum of irreducible representations of T e (158) K = (cid:77) s ≥ K s ⊗ R s with K s being the multiplicity space of the charge s representation R s : e t (cid:55)→ e t s . Let(159) k s = dim K s . We have:(160) (cid:88) s k s = k The grade component is -dimensional:(161) K = I ( N ) . The operators B , B raise the grading by e and e , respectively:(162) B a : K s → K s + e a , a = 1 , The complex S and its cohomology P ± are also graded:(163) P ± = (cid:77) s P ± , s . The dimensions k s are constant throughout the connected component of the set of ( e , e ) -homogeneous ideals. In fact, for e = e the component is a smooth projectivevariety, [17]. See also [16], [20].5.10. Compactness of the fixed point set. The topology of the fixed point set M k ( n ) T depends on the choice of the torus T . In other words, it depends on how non-genericthe choice of ( a , e ) is.If there is no rational relation between a and e , e , more precisely, if for any α, β ∈ [ n ] and p, q ∈ Z (164) a α − a β + e p + e q = 0 = ⇒ α = β , p = q = 0 then the fixed points f are isolated, f ↔ (cid:16) λ ( α ) (cid:17) α ∈ [ n ] . Their total number, for fixed k isfinite, therefore the set of fixed points is compact.What if there is a rational relation between a α − a β and e , e ? That is for somenon-trivial α, β ∈ [ n ] and p, q ∈ Z ,(165) a α − a β + e p + e q = 0 . We shall assume all the rest of the parameters a γ , e , e generic. In particular weassume both e , e non-zero. There are three cases to consider: NIKITA NEKRASOV (1) α (cid:54) = β and pq > ;(2) α = β and pq > ;(3) pq < and no restriction on α, β ;In the case (1) the fixed locus is non-compact. It is parametrized by the value of theinvariant(166) J β B p − B q − I α We therefore must make sure, in what follows, that the eigenvalues ( a α ) α ∈ [ n ] of theinfinitesimal framing rotations and the parameters ( e , e ) of the spatial rotation donot land on the hyperplanes:(167) a α − a β + e p + e q (cid:54) = 0 , for all α (cid:54) = β , and integer p, q ≥ .In the case (2) the fixed points corresponding to the monomial ideals are isolated,since the weights in (143) have the form e p (cid:48) + e q (cid:48) with p (cid:48) q (cid:48) ≤ . We shall show belowthat the U (1) -fixed points in the case pq > correspond to the monomial ideals.In other words they are U (1) × U (1) -invariantFor fixed k the sizes of the Young diagrams λ ( α ) are bounded above, since(168) n (cid:88) α =1 | λ ( α ) | = k Since the number of collections of Young diagrams which obey (168) is finite, the setof points fixed by the action of the maximal torus T is compact. This set, as we justshowed, is in one-to-one correspondence with the collections(169) f ↔ (cid:16) λ ( α ) (cid:17) α ∈ [ n ] obeying (168).In the case (3) the fixed points are not isolated, but the fixed point set is compact.Let us show the T n ; e -fixed point set is compact. There are two cases:(1) e e > . In this case the minimal torus corresponds to Γ ∨ = { } , d (0) = n , n = 1 , i.e. for T n = 1 , T n ; e = T e . The corresponding Coulomb parametervanishes, a = 0 .We are going to demonstrate that for all T n ; e -fixed points on M k ( n ) the L -norm of ( B , B , I , J ) is bounded above by a constant which depends only on n , k , and ζ . We use the real moment map equation (46):(170) kζ = Tr K ( µ ) = (cid:107) I (cid:107) −(cid:107) J (cid:107) ζ Tr K σ = Tr K ( σ µ ) = e (cid:107) B (cid:107) + e (cid:107) B (cid:107) +( e + e ) (cid:107) J (cid:107) PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 33 where we used the Eqs. (123) with the specialization e = e , e = e :(171) e a B a = [ σ , B a ] , e a B † a = [ B † a , σ ] , a = 1 , , σ I , I † σ , ( e + e ) J = − Jσ , J † ( e + e ) = − σ J † , The Eqs. (171) imply, by the same arguments as before, that the spectrum of σ has the form:(172) s = e ( i − 1) + e ( j − , ( i, j ) ∈ Σ for a finite set Σ of pairs ( i, j ) of positive integers, and that J maps the eigen-vectors of σ in K to zero, unless the eigenvalue is equal to − e − e . Now, theeigenvalues of σ are of the form (172), which are never equal to − e − e . Thus, J | K = 0 , therefore B and B commute on K .Now, σ | I ( N ) = 0 , i.e. m = dim (im I ) ≤ n . Now, the vector spaces(173) K i,j = C · B i − B j − I ( N ) if non-zero, contribute dim K i,j ≤ n to k s with s = e ( i − 1) + e ( j − . It is clearthat(174) k s = dim (cid:88) e ( i − e ( j − s C (cid:18) B i − B j − I ( N ) (cid:19) ≤≤ (cid:88) e ( i − e ( j − s dim C (cid:18) B i − B j − I ( N ) (cid:19) ≤ n Coe ff t s (1 − t ke )(1 − t ke )(1 − t e )(1 − t e ) , since both i and j cannot be greater then k . The trace Tr K σ can be estimatedby(175) Tr K σ = (cid:88) s ∈ Σ s k s ≤ n ∞ (cid:88) s =0 s Coe ff t s (1 − t ke )(1 − t ke )(1 − t e )(1 − t e ) = n t ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =1 (1 − t ke )(1 − t ke )(1 − t e )(1 − t e ) == 12 k ( k − n ( e + e ) Thus, J = 0 , the norms (cid:107) B , (cid:107) of the operators B , are bounded above, whilethe norm of the operator I is fixed:(176) (cid:107) I (cid:107) = ζk , e e + e (cid:107) B (cid:107) + e e + e (cid:107) B (cid:107) ≤ ζ k ( k − n (2) e e < . In this case we take Γ ∨ = [ n ] , d ( ω ) = 1 for all ω ∈ [ n ] . The Coulombparameters are the generic n complex numbers a α ∈ C , α ∈ [ n ] , defined up toan overall shift. Below we further restrict the parameters a α to be real, so that NIKITA NEKRASOV they belong to the Lie algebra of the compact torus T n ; e . The equations (171)generalize to:(177) e a B a = [ σ , B a ] , e a B † a = [ B † a , σ ] , a = 1 , ,I a = σ I , a I † = I † σ , ( e + e − a ) J = − Jσ , J † ( e + e − a ) = − σ J † , The fixed point set M k ( n ) T n ; e splits:(178) M k ( n ) T n ; e = (cid:91) k + ... + k n = k (cid:17) α ∈ [ n ] M k α (1) T e The fixed points are isolated, these are our friends ( λ ( α ) ) α ∈ [ n ] , the n -tuples ofpartitions with the total size equal to k . Since it is a finite set, it is compact.Note that we couldn’t restrict the torus T n ; e any further in this case. Indeed,the crucial ingredient in arriving at (178) is vanishing of the J matrix for the T n ; e -invariant solutions of the ADHM equations. The argument below the Eq.(172) we used before would not work for e e < , since a α − ( e + e ) may beequal to a β + e ( i − 1) + e ( j − for some α, β ∈ [ n ] , i, j ≥ . In this case J mayhave a non-trivial matrix element, giving rise to a non-compact fixed locus.Now, insisting on the T n ; e -invariance with T n = U (1) n − means a α ’s in (177)are completely generic, in particular, for α (cid:54) = β , a α − a β / ∈ Z . This still leavesthe case α = β as a potential source of noncompactness. But this is the caseof the T e -action on M k (1) = Hilb [ k ] ( C ) . In this case J vanishes not because ofthe toric symmetry, but rather because of the stability condition [25]:(179) JI = Tr I J = Tr [ B , B ] = 0 ,J ( xB + yB ) l I = Tr ( xB + yB ) l [ B , B ] = Tr ( xB + yB ) l [ xB + yB , x (cid:48) B + y (cid:48) B ] = 0 , for any x, y, x (cid:48) , y (cid:48) , s . t . x (cid:48) y − xy (cid:48) = 1 ,Jf ( B , B ) B B g ( B , B ) I = Jf ( B , B ) B B g ( B , B ) I ++ ( Jf ( B , B ) I ) ( Jg ( B , B ) I ) = 0 , by induction= ⇒ J C [ B , B ] I = 0 = ⇒ J = 0 , by stability The compactness of M k ( n ) T n ; e is thus established.5.11. Ordinary instantons as the fixed set. Let us now consider the particular T x = U (1) symmetry of the spiked instanton equations,(180) ( I A , J A ) −→ (cid:16) e i ϑ A I A , e − i ϑ A J A (cid:17) where (cid:126)ϑ = ( ϑ , ϑ , ϑ , ϑ , ϑ , ϑ ) ∼ ( ϑ + ϑ, ϑ + ϑ, ϑ + ϑ, ϑ + ϑ, ϑ + ϑ, ϑ + ϑ ) for any ϑ .The T x -invariant configuration [ B , I , J ] defines a homomorphism of the covering torus ˜ T x ≈ U (1) −→ U ( k ) , via the compensating U ( k ) -transformation g ( (cid:126)ϑ ) obeying:(181) e i ϑ A I A = g ( (cid:126)ϑ ) I A , e − i ϑ A J A = J A g ( (cid:126)ϑ ) − , g ( (cid:126)ϑ ) B a g ( (cid:126)ϑ ) − = B a PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 35 The space K splits as the orthogonal direct sum(182) K = (cid:77) A ∈ K A , g ( (cid:126)ϑ ) | K A = e i ϑ A , K A = C [ B a , B b ] I A ( N A ) , for A = { a, b } This decomposition is preserved by the matrices B , I , J . Thus the solution is the directsum of the solutions of ADHM equations:(183) M ∗ k ( (cid:126)n ) T x = (cid:91) (cid:80) A ∈ k A = k (cid:17) A ∈ M k A ( n A ) Crossed and folded instantons Distorted shadows fellUpon the lighted ceiling:Shadows of crossed arms, of crossed legs-Of crossed destiny. † The next special case is where only two e.g. N A (cid:48) and N A (cid:48)(cid:48) out of six vector spaces N A are non-zero. There are two basic cases.6.1. Crossed instantons. Suppose A (cid:48) ∪ A (cid:48)(cid:48) = ∅ , e.g. A (cid:48) = { , } and A (cid:48)(cid:48) = { , } . In thiscase we define(184) M + k ( n, w ) = M k ( n = n, , , , , n = w ) We call the space M + k ( n, w ) the space of ( n, w ) -crossed instantons.The virtual dimension of the space M + k ( n, w ) is independent of k , it is equal to − nw .As a set, M + k ( n, w ) is stratified(185) M + k ( n, w ) = (cid:91) k (cid:48) + k (cid:48)(cid:48) ≥ k M + k (cid:48) ,k (cid:48)(cid:48) ; k ( n, w ) . The stratum(186) M + k (cid:48) ,k (cid:48)(cid:48) ; k ( n, w ) = (cid:40) [ B , I , J ] | dim K = k (cid:48) , dim K = k (cid:48)(cid:48) (cid:41) parametrizes the crossed instantons, whose ordinary instanton components have thecharges k (cid:48) and k (cid:48)(cid:48) , respectively: the crossed instanton [ B , I , J ] defines two ordinaryinstantons, [ B , B , I , J ] on C and [ B , B , I , J ] on C , of the charges(187) k (cid:48) = dim K , k (cid:48)(cid:48) = dim K † W inter night, f rom “ Dr.Zhivago (cid:48)(cid:48) , B.P asternak, English translation by Bernard G.Guerney NIKITA NEKRASOV One-instanton crossed example. When k = 1 the matrices B a are just complexnumbers b a ∈ C , a ∈ . The equations b I = b I = 0 and b I = b I = 0 implythat if ( b , b ) (cid:54) = 0 , then I = 0 , I (cid:54) = 0 , ( b , b ) = (0 , , K = K and the rest of thematrices define the ordinary charge U ( n ) instanton, parametrized by the space (111).Likewise, if ( b , b ) (cid:54) = 0 , then I = 0 , I (cid:54) = 0 , ( b , b ) = (0 , , K = K and the rest ofthe matrices define the ordinary charge U ( w ) instanton. Finally, if ( b , b , b , b ) = 0 ,then both I , I need not vanish. If, indeed, both I , I do not vanish, then J and J vanish, by the υ -equaitons, while I , I obey(188) (cid:107) I (cid:107) + (cid:107) I (cid:107) = ζ , which, modulo U (1) = U ( K ) symmetry, define a subset in CP n + w − , the complementto the pair of “linked” projective spaces, CP n − and CP w − , corresponding to thevanishing of I and I , respectively. The result is, then(189) M +1 ( n, w ) = C × T ∗ CP n − ∪ CP n + w − ∪ C × T ∗ CP w − , the first and second components intersect along (0 , × CP w − the second and thethird components intersect along (0 , × CP n − , where CP n − ∪ CP w − ⊂ CP n + w − arenon-intersecting CP n − ∩ CP w − = ∅ projective subspaces.Figure 7: Charge one crossed instanton moduli space: the planes represent the complexplane factors C and C , the girl represents the T ∗ CP w − factor, the man representsthe T ∗ CP n − factor, the orange ball is the CP n + w − component, the blue and greendots are the CP n − and CP w − loci of the intersections of components6.3. Folded instantons. In this case A (cid:48) ∩ A (cid:48)(cid:48) = { a } , e.g. A (cid:48) = { , } , A (cid:48)(cid:48) = { , } , a = 1 .We define:(190) M | _ k ( n, w ) = M kk ( n = n, n = w, , , , We call the space M | _ k ( n, w ) the space of ( n, w ) -folded instantons.There exists an analogue of the stratification (185) for the folded instantons. Again,the folded instanton data [ B , B , B , B , I , I , J , J ] defines two ordinary noncom-mutative instantons on R , one on C , [ B , B , I , J ] , another on C , [ B , B , I , J ] . PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 37 The stability implies that B vanishes. The spaces K = C [ B , B ] I ( N ) and K = C [ B , B ] I ( N ) generate all of K ,(191) K = K + K One-instanton folded example. When k = 1 , as before, the matrices B a are thecomplex numbers b a , a ∈ , except that b vanishes. Now, the equation b I = 0 im-plies that if b (cid:54) = 0 then I = 0 , and we have the charge one ordinary U ( n ) instanton on C . Likewise the equation b I = 0 implies that if b (cid:54) = 0 then I = 0 and we have thecharge one ordinary U ( w ) instaton on C . Finally, when both b = b = 0 , the remain-ing equations J I = J I = J I = J I = 0 , and (cid:107) I (cid:107) + (cid:107) I (cid:107) −(cid:107) J (cid:107) −(cid:107) J (cid:107) = ζ ,define the variety which is a product of a copy of C (parametrized by b ) and our friendthe union of three pieces: CP n + w − (this is the locus where J = J = 0 ), T ∗ CP n − (the locus where I = J = 0 ) and T ∗ CP w − (the locus where I = J = 0 ):(192) M | _ ( n, w ) = C × T ∗ CP w − ∪ C × CP n + w − ∪ C × T ∗ CP n − , the first and second components intersect along C × CP w − the second and the thirdcomponents intersect along C × CP n − , where CP n − ∪ CP w − ⊂ CP n + w − are non-intersecting CP n − ∩ CP w − = ∅ projective subspaces.6.5. Fixed point sets: butterflies and zippers. Let us now discuss the fixed point setsof toric symmetries of the crossed and folded instantons. The torus T n,w × U (1) e actson M + k ( n, w ) and M | _ k ( n, w ) :(193) ( B a , I A , J A ) (cid:55)→ (cid:16) e i t e a B a , I A e − i t a A , e i t e A e i t a A J A (cid:17) Here a A = diag (cid:16) a A, , . . . , a A,n A (cid:17) where the complex numbers a A,α , α ∈ [ n A ] are definedup to the overall shift(194) a A,α ∼ a A,α + y , with y ∈ C . Let e = ( e , e , e , e ) ,(195) (cid:88) a ∈ e a = 0 We assume e a (cid:54) = 0 for each a ∈ .6.5.1. Toric crossed instantons. The fixed point set M + k ( n, w ) T n,w × U (1) e is easy to de-scribe. The infinitesimal transformation generated by ( a , e ) is compensated by theinfinitesimal GL ( k ) transformation, generated by σ ∈ End( K ) . As in the previous sec-tion this makes K a representation of T n,w × U (1) e . The space K contains two subspaces, K and K , whose intersection K , = K ∩ K belongs to both P and P :(196) K , ⊂ P ∩ P NIKITA NEKRASOV Figure 8: The butterflyThe eigenvalues of σ on K have the form:(197) Eigen (cid:16) σ | K (cid:17) = (cid:40) a ,α + e ( i − 1) + e ( j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ∈ [ n ] , ( i, j ) ∈ λ (12 ,α ) (cid:41) The eigenvalues of σ on K have the form:(198) Eigen (cid:16) σ | K (cid:17) = (cid:40) a ,β + e ( i − 1) + e ( j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β ∈ [ w ] , ( i, j ) ∈ λ (34 ,β ) (cid:41) These two sets do not overlap when all the parameters a A,α , e a are generic. Therefore K ⊥ K and the T n,w × U (1) e -fixed points are isolated. These fixed points are,therefore, in one-to-one correspondence with the pairs(199) ( λ ; λ ) consisting of n - and w -tuples λ = (cid:16) λ (12 , , . . . , λ (12 ,n ) (cid:17) ; λ = (cid:16) λ (34 , , . . . , λ (34 ,w ) (cid:17) , of partitions, obeying(200) (cid:88) α ∈ [ n ] (cid:12)(cid:12)(cid:12)(cid:12) λ (12 ,α ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) β ∈ [ w ] (cid:12)(cid:12)(cid:12)(cid:12) λ (34 ,α ) (cid:12)(cid:12)(cid:12)(cid:12) = k Their number is finite, therefore the set M + k ( n ) T n,w × U (1) e of fixed points is compact.Now let us try to choose a sub-torus T (cid:48) ⊂ T n,w × U (1) e , restricted only by the con-dition that J = J = 0 for the T (cid:48) -invariant solutions of (57). We wish to prove thatthe set of T (cid:48) -fixed points is compact in this case as well. In the next sections we shalldescribe such tori in more detail. PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 39 Figure 9: (cid:4) = K , , a = a ,α , ν = a ,β , λ = λ (12 ,α ) , µ = λ (34 ,β ) We start by the observation that if K , (cid:54) = 0 then the two sets (197) and (198)of σ -eigenvalues must overlap. Therefore, for some ( α, β ) ∈ [ n ] × [ w ] , and for some ( i (cid:48) , j (cid:48) ) ∈ λ (12 ,α ) , ( i (cid:48)(cid:48) , j (cid:48)(cid:48) ) ∈ λ (34 ,β ) (201) a ,α + e ( i (cid:48) − 1) + e ( j (cid:48) − 1) = a ,β + e ( i (cid:48)(cid:48) − 1) + e ( j (cid:48)(cid:48) − Note that (201) is invariant under the shifts (194). Moreover, if (cf. (7))(202) a ,α (cid:48) − a ,α (cid:48)(cid:48) / ∈ Z e , α (cid:48) (cid:54) = α (cid:48)(cid:48) a ,β (cid:48) − a ,β (cid:48)(cid:48) / ∈ Z e , β (cid:48) (cid:54) = β (cid:48)(cid:48) and / ∈ Z e , then the condition (201) determines ( α, β ) and i (cid:48) j (cid:48) and i (cid:48)(cid:48) j (cid:48)(cid:48) uniquely, upto the shifts ( i (cid:48) , j (cid:48) , i (cid:48)(cid:48) , j (cid:48)(cid:48) ) (cid:55)→ ( i (cid:48) + k, j (cid:48) + k, i (cid:48)(cid:48) − k, j (cid:48)(cid:48) − k ) , k ∈ Z . The relation (201) definesthe codimension subtorus T (cid:48) ⊂ T n,w × U (1) e . Let us describe its fixed locus.If the condition (201) on a , e is obeyed, it does not imply that K , (cid:54) = 0 . However,if in addition to (201) also the condition (202) is obeyed, then the intersection K , is not more then one-dimensional. Let H = P α ; i (cid:48) j (cid:48) ⊂ P , H = P β ; i (cid:48)(cid:48) j (cid:48)(cid:48) ⊂ P be the one-dimensional eigenspaces of σ corresponding to the eigenvalue (201). If aneigenbasis of N for a and the eigenbasis of N for a are chosen, then H and H are endowed with the basis vectors as well (act on the eigenvector of a correspondingto a ,α by B i (cid:48) − B j (cid:48) − I to get the basis vector of H ). NIKITA NEKRASOV The component ofthe fixed point setcorresponding to (201)is a copy of the complexprojective line: P ( H ⊕ H ) .It parametrizes rank onelinear relationsbetween H and H Figure 10: The component P ( H ⊕ H ) Let z be the coordinate on P ( H ⊕ H ) such that z = ∞ corresponds to the line H while z = 0 corresponds to the line H . For z (cid:54) = 0 , ∞ the linear spaces H and H coincide, z being the isomorphism. When z → the image ( p ( z = 0) , p ( z = 0)) isthe pair ( ˜ λ , λ ) , the image ( p ( z = ∞ ) , p ( z = ∞ )) is the pair ( λ , ˜ λ ) . Here ˜ λ isthe n -tuple of partitions which differs from λ in that the Young diagram of ˜ λ (12 ,α ) ( ˜ λ on the Fig. 10) is obtained by removing the ( i (cid:48) , j (cid:48) ) square from the Young diagram of λ (12 ,α ) . Similarly, the w -tuple ˜ λ ( ˜ µ on the Fig. 10) is obtained by modifying λ (34 ,β ) by removing the box ( i (cid:48)(cid:48) , j (cid:48)(cid:48) ) .In the next chapters we shall relax the condition (202). In other words, we shallconsider a subtorus in T n,w × U (1) e .6.5.2. Toric folded instantons. Now let us explore the folded instantons invariant underthe action of the maximal torus T n,w × U (1) e . It is easy to see that these are again thepairs ( λ , λ ) , with λ = ( λ (12 , , . . . , λ (12 ,n ) ) , λ = ( λ (13 , , . . . , λ (13 ,w ) ) . The spaces K and K do not intersect, K = K ⊕ K . In other words, the only T n,w × U (1) e -invariant folded instantons are the superpositions of the ordinary instantons on C and C , of the charges k and k , respectively, with k = k + k .Now let us consider the non-generic case, such that K , (cid:54) = ∅ . We call the cor-responding fixed point “the zipper”, see the Fig. 11. The codimension one subtorusfor which this is possible corresponds to the relation a ,α − a ,β / ∈ Z e between theparameters of the infinitesimal torus transformation.Figure 11: The zipper PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 41 The non-empty overlap K ∩ K implies the sets of eigenvalues of σ on K and K overlap, leading to(203) a ,α + e ( i (cid:48) − 1) + e ( j (cid:48) − 1) = a ,β + e ( i (cid:48)(cid:48) − 1) + e ( j (cid:48)(cid:48) − for some α ∈ [ n ] , β ∈ [ w ] , i (cid:48) , j (cid:48) , i (cid:48)(cid:48) , j (cid:48)(cid:48) . Unlike the Eq. (201) the Eq. (203) the integers i (cid:48) , i (cid:48)(cid:48) are not uniquely determined. Since the left hand side of (203) is the eigenvalueof σ | L , , while the right hand side is the eigenvalue of σ | L , , we conclude:(204) ( i (cid:48) , j (cid:48) ) ∈ λ (12 ,α ) , ( i (cid:48)(cid:48) , j (cid:48)(cid:48) ) ∈ λ (13 ,β ) , ( i (cid:48) , j (cid:48) + 1) / ∈ λ (12 ,α ) , ( i (cid:48)(cid:48) , j (cid:48)(cid:48) + 1) / ∈ λ (13 ,β ) i.e. j (cid:48) = λ (12 ,α ) ti (cid:48) , j (cid:48)(cid:48) = λ (13 ,β ) ti (cid:48)(cid:48) . The change ( i (cid:48) , i (cid:48)(cid:48) ) (cid:55)→ ( i (cid:48) ± , i (cid:48)(cid:48) ± maps the solutionof (203) to another solution of (203). Let l ≥ be the maximal integer such that j (cid:48) = λ (12 ,α ) i (cid:48) − l , j (cid:48)(cid:48) = λ (13 ,β ) i (cid:48)(cid:48) − l , and let l ≥ m ≥ be the maximal integer such that j (cid:48) = λ (12 ,α ) i (cid:48) − m , j (cid:48)(cid:48) = λ (13 ,β ) i (cid:48)(cid:48) − m , and e = B i (cid:48) − m − B j (cid:48) − I ,α ∈ K , , e = B i (cid:48)(cid:48) − m − B j (cid:48)(cid:48) − I ,β ∈ K , . In otherwords the vectors e and e are linearly dependent, e = ze . Consequenly, thearm-lengths a i (cid:48) − m,j (cid:48) = λ (12 ,α ) tj (cid:48) − i (cid:48) + m , a i (cid:48) − m,j (cid:48) = λ (13 ,β ) tj (cid:48)(cid:48) − i (cid:48)(cid:48) + m must be equal:(205) λ (12 ,α ) tj (cid:48) − i (cid:48) + m = a i (cid:48) − m,j (cid:48) = λ (13 ,β ) tj (cid:48)(cid:48) − i (cid:48)(cid:48) + m The component ofthe fixed point setcorresponding to (203)is a copy of the complexprojective line: P ( C e ⊕ C e ) .It parametrizes rank onelinear relationsbetween e and e Let z be the coordinate on P ( C e ⊕ C e ) such that z = ∞ corresponds to the line C e while z = 0 corresponds to the line C e . Then the image ( p ( z = 0) , p ( z = 0)) is the pair ( ˜ λ (12 ,α ) , λ (13 ,β ) ) , the image ( p ( z = ∞ ) , p ( z = ∞ )) is the pair ( λ (12 ,α ) , ˜ λ (13 ,β ) ) .Here we defined ˜ λ (12 ,α ) to be the partition whose Young diagram is obtained by re-moving the block of squares ( i (cid:48) − m, j (cid:48) ) . . . ( λ (12 ,α ) tj (cid:48) , j (cid:48) ) from the Young diagram of λ (12 ,α ) .Similarly, the Young diagram of ˜ λ (13 ,β ) is obtained by removing the block of squares ( i (cid:48)(cid:48) − m, j (cid:48)(cid:48) ) . . . ( λ (13 ,β ) tj (cid:48)(cid:48) , j (cid:48)(cid:48) ) from the Young diagram of λ (13 ,β ) .Note that the pair of Young diagrams λ (12 ,α ) , λ (34 ,β ) gives riseto several components ofthe fixed point set,isomorphic to CP ,e.g. the ones correspondingto the blocks of horizontal boxesof different length, NIKITA NEKRASOV see the pictures above on the left and on the right. But they actually belong to modulispaces of folded instantons of different charges (in computing the charge k we subtractthe length of the block from the sum of the sizes of Young diagrams). So despite thesimilarity in graphic design, these are pieces of different architectures.7. Reconstructing spiked instantons In this section we describe the sewing procedure, which produces a spiked instantonout of six ordinary noncommutative instantons. We then use the stitching to describethe spiked instantons invariant under the toric symmetry, i.e. the T L -fixed locus.7.1. The local K-spaces. For A = { a, b } ∈ we define:(206) K A = C [ B a , B b ] im( I A ) ⊂ K By definition, this is the minimal B a , B b -invariant subspace of K , containing the image im( I A ) = I A ( N A ) of N A .The equations (57), (58) for i ≥ k imply that(207) J B ( K A ) = 0 , B (cid:54) = A and(208) B b ( K A ) = 0 , b / ∈ A Toric spiked instantons. Now let us describe the spiked instantons, invariant un-der the torus action. The tori in question are the subgroups of H , the global symmetrygroup. We consider first the maximal torus T H (cf. (79)) and then its subtori T L forvarious choices of the L data.7.2.1. Maximal torus. First of all, let us consider the T H -fixed points. Let a = ( a A ) A ∈ be the collection of diagonal matrices a A = diag( a A, , . . . , a A,n A ) ∈ Lie U ( n A ) . The spikedinstanton [ B , I , J ] is T H -invariant iff for any a and e there exists an operator σ ∈ End( K ) ,such that:(209) e a B a = [ σ , B a ] , a ∈ ( e A − a A ) J A = − J A σ ,I A a A = σ I A , A ∈ Let N A,α , α ∈ [ n A ] , be the eigenspace of a A with the eigenvalue a A,α . Let I A,α = I A ( N A,α ) . We have (for A = { a, b } , a < b ):(210) K A = (cid:88) α ∈ [ n A ] ,i,j ≥ K i,jA,α , K i,jA,α = B i − a B j − b I A,α The eigenvalue of σ on K i,jA,α is equal to(211) σ | K i,jA,α = a A,α + e a ( i − 1) + e b ( j − On the other hand, Eq. (209) implies that the vector(212) ψ = J A ( K i,jA,α ) ∈ N A PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 43 is an eigen-vector of a A with the eigen-value:(213) a A ψ = ( a A,α + e a i + e b j ) ψ The T H -invariance means we are free to choose the parameters a A,α , e a in an arbitraryfashion. It means, that a A,α + e a i + e b j (cid:54) = a A,β for i, j ≥ , α, β ∈ [ n A ] . Therefore J A vanishes on all K i,jA,α subspaces, and therefore on all of K A . Therefore, all B a ’s com-mute with each other. Also, the eigenvalues (211) are different for different ( A, α ; i, j ) .Therefore, the spaces K i,jA,α are orthogonal to each other.Define, for A ∈ , α ∈ [ n A ] the partition λ ( A,α ) , by:(214) λ ( A,α ) i = sup (cid:26) j | j ≥ , K i,jA,α (cid:54) = 0 (cid:27) We have:(215) k A = (cid:88) α ∈ [ n A ] | λ ( A,α ) | The T H -fixed points are, therefore, in one-to-one correspondence with the collections(216) Λ = (cid:16) λ ( A,α ) (cid:17) A ∈ ,α ∈ [ n A ] of(217) n = (cid:88) A ∈ n A Young diagrams. In the companion paper [29] we shall be studying the statisticalmechanical model, where the random variables are the collections Λ , while the com-plexified Boltzman weights are the contributions of Λ to the gauge partition function,defined below.7.2.2. Subtori. Now fix the data L and consider the T L -invariant spiked instantons [ B , I , J ] . As usual, these come with the homomorphism T L → U ( K ) which associatesthe compensating U ( K ) -transformation g t,ξ for every ( e i ξ , e i t ) ∈ T L . Since K decomposesinto the direct sum of weight subspaces(218) K = (cid:77) w,n K w,n where(219) g t,ξ | K w,n = e i (cid:104) w,ξ (cid:105) +i nt where n ∈ Z , while w belongs to the weight lattice of (cid:16) A ∈ − U (1) (cid:96) A − .The relations:(220) e i e a t B a = g t,ξ B a g − t,ξ ,I A ( N A,α ) e i c A ( α ) t = g t,ξ I A ( N A,α ) , A ∈ + , α ∈ [ n A ] I A ( N A,ι,α ) e i c A,ι ( α ) t = g t,ξ I A ( N A,ι,α ) , A ∈ − , α ∈ [ n A ] ι NIKITA NEKRASOV imply:(221) I A ( N A,α ) ∈ K ,c A ( α ) , α ∈ [ n A ] , A ∈ + ,I A ( N A,ι A ,α ) ∈ K ,c A,ιA ( α ) , α ∈ [ n A ] ι A , A ∈ − ,I A ( N A,ι,α ) ∈ K (cid:36) A,ι ,c A,ι ( α ) , α ∈ [ n A ] ι , ι ∈ λ A , A ∈ − B a ( K w,n ) ⊂ K w,n + e a , a ∈ where (cid:36) A,ι is the fundamental weight, (cid:104) (cid:36) A,ι , ξ (cid:105) = ξ A,ι .Finally, the T L -invariance translates to(222) J A ( K w,n ) ∈ N A,α , A ∈ + , α ∈ [ n A ] ⇔ n = c A ( α ) − e A , w = 0 ,J A ( K w,n ) ∈ N A,ι A ,α , A ∈ − , α ∈ [ n A ] ι A ⇔ n = c A,ι A ( α ) − e A , w = 0 ,J A ( K w,n ) ∈ N A,ι,α , A ∈ − , α ∈ [ n A ] ι ⇔ n = c A,ι ( α ) − e A , w = (cid:36) A,ι . which imply, with our choice of T L , that J A = 0 . This is shown using the same argumentsas we used around the Eq. (213).7.2.3. K -spaces for toric instantons. Let A ∈ + . The local space K A is g t,ξ -invariant,and decomposes as(223) K A = (cid:77) n K nA with integer n , via(224) g t,ξ | K nA = e i nt where n ≥ c − A = inf α ∈ [ n A ] c A ( α ) , when e a , e b > and n ≤ c + A = sup α ∈ [ n A ] c A ( α ) when e a , e b < . For e a , e b > both operators B a , a ∈ A raise the grading. For e a , e b < both operators B a , a ∈ A lower the grading. Let k A,n = dim K nA . Since K A is finitedimensional, k A,n vanish for | n | > C A for some some constant C A ≤ k .Let A ∈ − . The local space K A is g t,ξ -invariant, and decomposes as(225) K A = (cid:77) n K nA ⊕ (cid:77) ι ∈ λ A ,n K nA,ι with(226) g t,ξ | K nA,ι = e i nt +i ξ A,ι for i ∈ λ A , and(227) g t,ξ | K nA = e i nt Since the eigenvalues of g t,ξ on K nA,ι for ι ∈ λ A differ from each other and from thoseon K n (cid:48) B for all B ∈ , n (cid:48) ∈ Z , the spaces K nA,ι are orthogonal to K n (cid:48) B and to each other:(228) K n (cid:48) A,ι (cid:48) ⊥ K n (cid:48)(cid:48) A,ι (cid:48)(cid:48) , ι (cid:48) (cid:54) = ι (cid:48)(cid:48) K n (cid:48) A,ι ⊥ K n (cid:48)(cid:48) A The action of B , I -operators respects the orthogonal decomposition (228). PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 45 We now prove that the spaces K nA and K nA,i have an additional U (1) -action. Indeed,let f a , f b be the two positive mutually prime integers, such that(229) e a f a + e b f b = 0 , so that e a = p ab f b , e b = − p ab f a (assuming e a > > e b ). Then the operator(230) E = B f a a B f b b commutes with g t,ξ , thanks to (220). Since all the eigenvalues of B a and B b vanish(again, thanks to (220)), the operators B a , B b , and E are nilpotent. By Jacobson-Morozov theorem, E can be included into the sl -triple, i.e. for each K nA , K nA,i there areoperators H , E ∨ , such that(231) [ E , E ∨ ] = H , [ H , E ] = 2 E , [ H , E ∨ ] = − E ∨ so that(232) K nA = (cid:77) h K n,hA , K nA,ι = (cid:77) h K n,hA,ι with h standing for the eigenvalue of H . Now, it is not difficult to prove that the ( n, h ) -grading is equivalent to the ( i, j ) -grading, with i, j ≥ :(233) i = i (cid:48) + ( h − h (cid:48) A ( α )) f a ,j = j (cid:48) + ( h − h (cid:48) A ( α )) f b ,n = c A ( α ) + p A ( f b ( i (cid:48) − − f a ( j (cid:48) − ,i (cid:48) , j (cid:48) ≥ , i (cid:48) ≤ f a and / or j (cid:48) ≤ f b h (cid:48) A ( α ) = inf Spec( H | K nA ) and α is uniquely determined by n mod p A = c A ( α ) mod p A . Thus,(234) K A = (cid:77) i,j ≥ K i,jA , K A,ι = (cid:77) i,j ≥ K i,jA,ι with(235) B a ( K i,jA ) ⊂ K i +1 ,jA , B b ( K i,jA ) ⊂ K i,j +1 A Now we are ready for the final push:8. The compactness theorem We now prove the compactness theorem which establishes the analyticity of thepartition function defined in the next chapter. To this end we estimate the norm of ( B , I , J ) whose U ( k ) -orbit is invariant with respect to the action of any minimal torus T L .Since J A ’s vanish, the real moment map equation reads as follows:(236) (cid:88) a ∈ [ B a , B † a ] + (cid:88) A ∈ I A I † A = ζ K NIKITA NEKRASOV The trace of this equation gives the norm of I A ’s:(237) (cid:88) A ∈ (cid:107) I A (cid:107) = ζk But we need to estimate the norms (cid:107) B a (cid:107) which drop out of trace. However, it is nottoo difficult to chase them down. We have:(238) (cid:88) a ∈ (cid:107) B a (cid:107) + (cid:88) A ∈ (cid:107) I A (cid:107) = (cid:88) a ∈ Tr K B a B † a + (cid:88) A ∈ Tr K I A I † A ≤ (cid:88) A ∈ Tr K A (cid:88) a ∈ B a B † a + (cid:88) C ∈ I C I † C = (cid:88) A ∈ ζk A + Tr K A (cid:88) a ∈ B † a B a = ζ (cid:88) A ∈ k A + (cid:88) A ∈ ,a ∈ A Tr K A B † a B a where we used the moment map equation (59), projected onto K A , and the Eq. (208).Define:(239) δ A,n = 1 ζ Tr K nA (cid:88) a ∈ B a B † a + (cid:88) A (cid:48) ∈ I A (cid:48) I † A (cid:48) δ A = 1 ζ Tr K A (cid:88) a ∈ B a B † a + (cid:88) A (cid:48) ∈ I A (cid:48) I † A (cid:48) = (cid:88) n δ A,n Now for A ∈ + use the decomposition (223), and (208) to show, that for e a (cid:54) = e b :(240) δ A,n = k A,n + 1 ζ (cid:88) a ∈ A Tr K n + eaA (cid:16) B a B † a (cid:17) ≤ k A,n + (cid:88) a ∈ A δ A,n + e a where we very conservatively estimated:(241) Tr K nA (cid:16) B a B † a (cid:17) ≤ δ A,n , for any a ∈ . This very conservative inequality can be used to show the boundenessof δ A,n .From now on let us assume e a > e b > . The case of negative e a , e b is treated analo-gously. First of all, let introduce the sequence of generalized Fibonacci numbers F p,qn , n ∈ Z , for positive integeres p > q > , by:(242) F p,qn = 0 , − p ≤ n ≤ F p,q = 1 ,F p,qn = F p,qn − p + F p,qn − q , n > PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 47 It is easy to write the formula for F p,qn in terms of the roots λ ω , ω = 0 , . . . , p − , of thecharacteristic equation(243) λ pω = λ p − qω + 1 , (244) F p,qn = (cid:88) ω f ω λ nω where the coefficients f ω are to be found from the linear equations F p,qn = δ n, , − p ≤ n ≤ .Now, we can estimate δ A,n by induction in n :(245) δ A,n ≤ (cid:88) n (cid:48) ≥ n k A,n (cid:48) F e a ,e b n (cid:48) +1 − n This leads to the following, also very conservative, bound on δ A :(246) δ A < k A F e a ,e b k A When e a = e b = e A / ≥ we can make a better estimate:(247) δ A,n = k A,n + 1 ζ (cid:88) a ∈ A Tr K n + eaA (cid:16) B a B † a (cid:17) ≤ k A,n + δ A,n + e A / which, by iteration, implies:(248) δ A,n ≤ k A,n + k A,n + e A / + k A,n + e A + . . . which in turns implies the upper bound on(249) δ A ≤ e A k A ( k A + 1) It remains to estimate δ A for A ∈ − . This is easy to do using the ( i, j ) -grading (234).Define:(250) δ A,n = 1 ζ (cid:88) i + j = n +2 Tr K i,jA (cid:88) a ∈ B a B † a + (cid:88) A (cid:48) ∈ I A (cid:48) I † A (cid:48) Then, using (59), projected onto K i,jA , and (235), we derive the estimate:(251) δ A,n ≤ δ A,n +1 + (cid:88) i + j = n +2 dim K i,jA from which we get the estimate:(252) δ A ≤ k A ( k A + 1) NIKITA NEKRASOV Integration over the spiked instantons The moduli spaces M ik ( (cid:126)n ) are not your favorite smooth varieties. They can bestratified by smooth varieties of various dimensions. Over these smooth componentsthe obstruction bundles keep track of the non-genericity of the equations we used todefine the spaces M ik ( (cid:126)n ) .In applications we need to compute the integrals over the spaces M ∞ k ( (cid:126)n ) , as well asto define and compute the equivariant indices of various twisted Dirac operators (forfive dimensional theories compactified on a circle).Mathematically one can take the so-called virtual approach [12], where the funda-mental cycle [ M ∞ k ( (cid:126)n )] is replaced by the virtual fundamental cycle [ M ∞ k ( (cid:126)n )] vir , whichis defined as the Euler class of the bundle of equations over the smooth variety of theoriginal variables ( B , I , J ) . There are two difficulties with this definition: i ) the spaceof ( B , I , J ) , being a vector space, is non-compact; ii ) the bundle of equations is infinitedimensional, unless we are in the situation with only the crossed or ordinary instantons.The problem i ) is solved by passing to the equivariant cohomology. The problem ii ) is cured by working with M ik ( (cid:126)n ) for large but finite i , and then regularize the limit i → ∞ by using the Γ -functions.Physically, the problem is solved by the considerations of the matrix integral (matrixquantum mechanics) of the system of k D ( − -branes ( k D -branes whose worldlineswrap S ) in the vicinity of six stacks of D branes ( D branes) wrapping coordinatetwo-planes C A (times a circle S ) in the I I B ( I I A ) background R × C ( R × S × C ).One can also define the elliptic genus by the study of the two-dimensional gaugetheory corresponding to the stack of k D -strings wrapping a T shared by six stacksof D branes in I I B string theory, wrapping T × C A .9.1. Cohomological field theory. Let us briefly recall the physical approach. For everyvariable, i.e. for every matrix element of the matrices B a , I A , J A , we introduce thefermionic variables Ψ B a , Ψ I A , Ψ J A with the same H × U ( K ) transformation properties.For every equation s A , µ, σ ¯ aA , Υ A , Υ A,B ; i we introduce a pair of fermion-boson variablesvalued in the dual space: ( χ A , h A ) , ( χ, µ ) , ( χ ¯ aA , h ¯ aA ) , ( ξ A , r A ) , ( (cid:36) A (cid:48) ,A (cid:48)(cid:48) ; i , y A (cid:48) ,A (cid:48)(cid:48) ; i ) .Finally, we need a triplet of variables ( σ , ¯ σ , η ) (two bosons and a fermion), valued inthe Lie algebra of U ( K ) .Our model has the fermionic symmetry acting by:(253) δB a = Ψ B a , δ Ψ B a = − [ σ , B a ] + e a B a δI A = Ψ I A , δ Ψ I A = − σ I A + I A a A δJ A = Ψ J A , δ Ψ J A = − a A J A + J A σ + e A J A PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 49 (cf. (209)) and(254) δχ A = h A , δh A = − [ σ , χ A ] + e A χ A ,δχ = h , δh = − [ σ , χ ] ,δχ ¯ aA = h ¯ aA , δh ¯ aA = − a A χ ¯ aA + χ ¯ aA σ − e ¯ a χ ¯ aA ,δξ A = r A δr A = a ¯ A ξ A − ξ A a A + e A ξ A ,δ(cid:36) A (cid:48) ,A (cid:48)(cid:48) ; i = y A (cid:48) ,A (cid:48)(cid:48) ; i ,δy A (cid:48) ,A (cid:48)(cid:48) ; i = − a A (cid:48) (cid:36) A (cid:48) ,A (cid:48)(cid:48) ; i + (cid:36) A (cid:48) ,A (cid:48)(cid:48) ; i a A (cid:48)(cid:48) + ( i − e a (cid:36) A (cid:48) ,A (cid:48)(cid:48) ; i , whenever A (cid:48) ∩ A (cid:48)(cid:48) = { a } and(255) δ ¯ σ = η, δη = [ σ , ¯ σ ] , δσ = 0 Now we can define the partition function(256) Z ik ( (cid:126)n ; e , a ) = (cid:90) e − S i Dσ Vol( U ( k )) D B D B † D Ψ B D Ψ B † . . . DχDh . . . D ¯ σ Dη i − (cid:89) j =1 Dχ A,B ; j Dh A,B ; j D ¯ χ A,B ; j D ¯ h A,B ; j where(257) S i = δ Ξ i , Ξ i = Ξ s + Ξ f + Ξ K + (cid:88) A Ξ NA + (cid:88) A (cid:48) ,A (cid:48)(cid:48) ; A (cid:48) ∩ A (cid:48)(cid:48) =1 i − (cid:88) j =1 Ξ NA (cid:48) A (cid:48)(cid:48) ,j , Ξ s = Tr K η [ σ , ¯ σ ] , Ξ f = Tr K (cid:88) a (cid:16) Ψ B a (cid:16) − [ ¯ σ , B † a ] + ¯ ε a B † a (cid:17) + c.c. (cid:17) ++ Tr K (cid:88) A (cid:16) ( − ¯ σ I A + I A ¯ a A ) Ψ I † A + c.c. (cid:17) + Tr K (cid:88) A (cid:16) Ψ J † A ( J A ¯ σ − ¯ a A J A + ¯ ε A J A ) + c.c. (cid:17) , Ξ K = Tr K χ ( − i ( µ − ζ K ) + h ) + (cid:88) A χ ¯ A ( − i s A + h A ) + (cid:88) ¯ a ∈ ¯ A χ † ¯ aA ( − i σ ¯ aA + h ¯ aA ) . Ξ NA = Tr N A (cid:16) ξ † A ( − i Υ A + r A ) + c.c. (cid:17) , Ξ NA (cid:48) ,A (cid:48)(cid:48) ; j = Tr N A (cid:16) (cid:36) † A (cid:48) ,A (cid:48)(cid:48) ; j (cid:16) − i Υ A (cid:48) ,A (cid:48)(cid:48) ; j + y A (cid:48) ,A (cid:48)(cid:48) ; j (cid:17) + c.c. (cid:17) Here ¯ ε a , ¯ a A are auxiliary elements of the Lie algebra of H , which can be chosen arbitrary,as long as the integral (256) converges.9.2. Localization and analyticity. The usual manipulations with the integral (256), forgeneric ( e , a ) , express it as a sum over the fixed points, which we enumerated in theEq. (216). Each fixed point contributes a homogeneous (degree zero) rational function NIKITA NEKRASOV of a A,α ’s, ≤ α ≤ n A and e a , times the product(258) (cid:89) A ∈ , ∈ ¯ A n A (cid:89) α =1 n ¯ A (cid:89) β =1 (cid:16) a A,α − a ¯ A,β + e ¯ A (cid:17) × (cid:89) A,B ∈ ,A ∩ B = { c } n A (cid:89) α =1 n B (cid:89) β =1 i − (cid:89) j =1 (cid:16) a A,α − a B,β + e c ( j − (cid:17) The compactness theorem of the previous chapter implies, among other things, thatthe partition functions Z ik ( (cid:126)n ; e , a ) , for i > k , have no singularities in(259) x A = 1 n A n A (cid:88) α =1 a A,α with fixed e a ’s and ˜ a A,α = a A,α − x A . In other words, they are polynomials of x A .10. Quiver crossed instantons Crossed quivers. For oriented graph γ let V γ denote the set of its vertices, and E γ the set of edges, with s, t : E γ → V γ the source and the target maps. Sometimeswe write γ = (cid:16) V γ , E γ , s, t (cid:17) . The crossed quiver is the data X = ( γ + , γ − , p ) , where γ ± aretwo oriented graphs, and p ∈ Z ≥ is a non-negative integer. Let Ξ p = Z /p Z be theadditive group with p elements, for p > and Z for p = 0 . Define V X = V γ + × V γ − × Ξ p .The group Ξ p acts on V X by translations of the third factor. The generator of Ξ p acts by ω = ( v + , v − , n ) (cid:55)→ ω + 1 ≡ ( v + , v − , n + 1) , with v ± ∈ V γ ± . We also define ± E X = E γ ± × V γ ∓ × Ξ p and the natural maps s, t : ± E X → V X , e.g. s ( e, u, n ) = ( s ( e ) , u, n ) for e ∈ E γ + , u ∈ V γ − , t ( e, v, n ) = ( v, t ( e ) , n ) for e ∈ E γ − , v ∈ V γ + etc. The group Ξ p also actson ± E X , so we shall write: η = ( e, u, n ) (cid:55)→ η ± ≡ ( e, u, n ± . The source and targetmaps are Ξ p -equivariant, i.e. s ( η ± 1) = s ( η ) ± .10.1.1. Paths and integrals. We shall use the notion of a path. Define the path p ± ω (cid:48) ,ω (cid:48)(cid:48) of length (cid:96) to be a sequence of pairs:(260) p ± ω (cid:48) ,ω (cid:48)(cid:48) = ( η , or ) , ( η , or ) , . . . , ( η (cid:96) , or (cid:96) ) with η j ∈ ± E X , or j ∈ {− , +1 } , (we call or j the orientation of the edge η j relative to p ± ω (cid:48) ,ω (cid:48)(cid:48) ) such that for any j = 1 , . . . , (cid:96) − either(261) s ( η j +1 ) = t ( η j ± , or j = or j +1 = 1 , or t ( η j +1 ) = t ( η j ) , or j = − or j +1 = 1 , or s ( η j +1 ) = s ( η j ) , − or j = or j +1 = 1 , or t ( η j +1 ) = s ( η j ∓ , − or j = − or j +1 = 1 , and also either s ( η ) = ω (cid:48) ( or = 1 ) or t ( η ± 1) = ω (cid:48) ( or = − ) and also either s ( η (cid:96) ) = ω (cid:48)(cid:48) ( or (cid:96) = − ) or t ( η (cid:96) ± 1) = ω (cid:48)(cid:48) ( or (cid:96) = 1 ). PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 51 For a function b : ± E X → R (a -chain) and a path p ± ω (cid:48) ,ω (cid:48)(cid:48) define the integral(262) (cid:90) p ± ω (cid:48) ,ω (cid:48)(cid:48) b = (cid:96) (cid:88) j =1 or j b ( η j ) The function b : ± E X → R is a coboundary, b = δc , of a -chain c : V X → R , iff b ( η ) = c ( t ( η ± − c ( s ( η )) . The integral of a coboundary obeys Stokes formula:(263) (cid:90) p ± ω (cid:48) ,ω (cid:48)(cid:48) δc = (cid:96) (cid:88) j =1 or j (cid:16) c ( t ( η j ± − c ( s ( η j )) (cid:17) = c ( ω (cid:48)(cid:48) ) − c ( ω (cid:48) ) Representations of crossed quivers. Fix four dimension vectors k , n + , n − , m : V X → Z ≥ . Let ( K ω , N ± ω , M ω ) ω ∈ V X be a collection of complex vector spaces whosedimensions are given by the components of the corresponding dimension vectors, e.g.dim K ω = k ω ≡ k ( ω ) . We view the spaces N ± ω , M ω as fixed, e.g. with some fixed basis,while the spaces K ω are varying, i.e. defined up to the automorphisms. We also fix adecomposition M ω = M (cid:48) ω ⊕ M (cid:48)(cid:48) ω as an additional refinement of our structure.10.1.3. Weight assignment for crossed quivers. For the crossed quiver γ and its rep-resentation let us fix the integral data: ( n ± ω , m (cid:48) ω , m (cid:48)(cid:48) ω ) ω ∈ V γ , t = ( t ( ω )) ω ∈ V X and e =( e ( η )) η ∈ E X , with integers t ( ω ) , e ( η ) ∈ Z , obeying t ( ω + 1) = t ( ω ) , e ( η + 1) = e ( η ) , and theintegral vectors n ± ω = (cid:16) n ± ω,α (cid:17) n ± ω α =1 ∈ Z n ± ω etc. The data ( n ± , e , m (cid:48) , m (cid:48)(cid:48) ) is defined up to theaction of the lattice Z V γ : a function f : V γ → Z shifts the data (289) by:(264) n ± ω,α (cid:55)→ n ± ω,α − f ( ω ) , e (cid:55)→ e + δf Crossed quiver instantons. Consider the vector superspace A γ k ( n ± , m ) of linearmaps ( B , I , J , Θ ) (265) bosons : B = ( B ± η , ˜ B ± η ) η ∈ ± E X , I = ( I + ω , I − ω ) ω ∈ V X , J = ( J + ω , J − ω ) ω ∈ V X ,B ± η : K s ( η ) → K t ( η ) ± , ˜ B ± η : K t ( η ) → K s ( η ) ± , η ∈ ± E X I ± ω : N ± ω → K ω ± , J ± ω : K ω → N ± ω ± , ω ∈ V X fermions : Θ = ( Θ (cid:48) ω , Θ (cid:48)(cid:48) ω ) ω ∈ V X , Θ (cid:48) ω ∈ Π Hom( K ω , M (cid:48) ω ) , Θ (cid:48)(cid:48) ω ∈ Π Hom( M (cid:48)(cid:48) ω − , K ω ) Let G γ k , G C γ k be the groups:(266) G γ k = (cid:17) ω ∈ VX U ( k ω ) , G C γ k = (cid:17) ω ∈ VX G L ( k ω , C ) , which act on A γ k ( n ± , m ) via:(267) ( g ω ) ω ∈ VX · ( B , I , J , Θ ) = (cid:16) g t ( η ) ± B ± η g − s ( η ) , g s ( η ) ± ˜ B ± η g − t ( η ) ; g ω ± I ± ω , J ± ω g − ω ; Θ (cid:48) ω g − ω , g ω Θ (cid:48)(cid:48) ω (cid:17) NIKITA NEKRASOV We impose the following analogues of the Eqs. (56):(268) µ ω = ζ ω K ω , ω ∈ V X , ζ ω > s e + ,e − ; n = 0 , e ± ∈ E γ ± , n ∈ Ξ p ˜ s e + ,e − ; n = 0 , e ± ∈ E γ ± , n ∈ Ξ p s ω = 0 , ω ∈ V X Σ η = 0 , η ∈ E X , ˜ Σ η = 0 , η ∈ E X , Υ ω = 0 , ω ∈ V X , where(269) Σ · = σ · + (cid:98) σ †· , ˜ Σ · = ˜ σ · − (cid:98) ˜ σ †· ,s ... = µ ... − (cid:16) µ ... (cid:17) † , ˜ s ... = µ ... + (cid:16) µ ... (cid:17) † , s ... = µ ... + (cid:16) µ ... (cid:17) † , Υ · = υ + ·− − ( υ −· +1 ) † with the linear maps(270) σ η : N ± s ( η ) → K t ( η ) , for η ∈ ∓ E X ˜ σ η : N ± t ( η ) → K s ( η ) , for η ∈ ∓ E X υ ± ω ∓ : N ∓ ω → N ± ω , and PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 53 (271) µ ω : K ω → K ω ,µ ω : K ω → K ω +2 ,µ ω : K ω → K ω − ,µ e + ,e − ; n : K s ( e + ) ,s ( e − ) ,n → K t ( e + ) ,t ( e − ) ,n ,µ e + ,e − ; n : K t ( e + ) ,t ( e − ) ,n → K s ( e + ) ,s ( e − ) ,n ,µ e + ,e − ; n : K s ( e + ) ,t ( e − ) ,n → K t ( e + ) ,s ( e − ) ,n ,µ e + ,e − ; n : K t ( e + ) ,s ( e − ) ,n → K s ( e + ) ,t ( e − ) ,n . The maps (270), for η ∈ ± E X , are given by, :(272) σ η = B ∓ η ± I ± s ( η ) , (cid:98) σ η = J ∓ s ( η ) ± ˜ B ± η , ˜ σ η = ˜ B ∓ η ± I ± t ( η ) , (cid:98) ˜ σ η = J ∓ t ( η ) ± B ± η , and(273) υ ± ω ∓ = J ± ω ∓ I ∓ ω The crossed quiver analogues (271) of real and complex moment maps are given by:for ω ∈ V γ ,(274) µ ω = I + ω − (cid:0) I + ω − (cid:1) † + I − ω +1 ( I − ω +1 ) † − ( J + ω ) † J + ω − ( J − ω ) † J − ω −− (cid:88) η ∈ s − ( ω ) ∩ + E X (cid:16) B + η (cid:17) † B + η − (cid:88) η ∈ t − ( ω ) ∩ + E X (cid:16) ˜ B + η (cid:17) † ˜ B + η −− (cid:88) η ∈ s − ( ω ) ∩ − E X (cid:16) B − η (cid:17) † B − η − (cid:88) η ∈ t − ( ω ) ∩ − E X (cid:16) ˜ B − η (cid:17) † ˜ B − η ++ (cid:88) η ∈ s − ( ω ) ∩ + E X ˜ B + η − (cid:16) ˜ B + η − (cid:17) † + (cid:88) η ∈ s − ( ω ) ∩ − E X ˜ B − η +1 (cid:16) ˜ B − η +1 (cid:17) † ++ (cid:88) η ∈ s − ( ω ) ∩ + E X B + η − (cid:16) B + η − (cid:17) † + (cid:88) η ∈ s − ( ω ) ∩ − E X B − η +1 (cid:16) B − η +1 (cid:17) † , and(275) µ ω = I + ω +1 J + ω + (cid:88) η ∈ + E X ∩ t − ( ω ) B + η +1 ˜ B + η − (cid:88) η ∈ + E X ∩ s − ( ω ) ˜ B + η +1 B + η NIKITA NEKRASOV (276) µ ω = I − ω − J − ω + (cid:88) η ∈ − E X ∩ t − ( ω ) B − η − ˜ B − η − (cid:88) η ∈ − E X ∩ s − ( ω ) ˜ B − η − B − η for e ± ∈ E γ ± , n ∈ Ξ p ,(277) µ e + ,e − ; n = B + e + ,t ( e − ) ,n − B − e − ,s ( e + ) ,n − B − e − ,t ( e + ) ,n +1 B + e + ,s ( e − ) ,n (278) µ e + ,e − ; n = ˜ B + e + ,s ( e − ) ,n − ˜ B − e − ,t ( e + ) ,n − ˜ B − e − ,s ( e + ) ,n +1 ˜ B + e + ,t ( e − ) ,n (279) µ e + ,e − ; n = ˜ B + e + ,t ( e − ) ,n − B − e − ,t ( e + ) ,n − B − e − ,s ( e + ) ,n +1 ˜ B + e + ,s ( e − ) ,n (280) µ e + ,e − ; n = B + e + ,s ( e − ) ,n − ˜ B − e − ,s ( e + ) ,n − ˜ B − e − ,t ( e + ) ,n +1 B + e + ,t ( e − ) ,n The moduli space of quiver crossed instantons M γ k ( n ± , m ) is the space of solutions to(268) modulo the action (266) of G γ k .The identity(281) (cid:88) η ∈ E γ Tr K t ( η ) (cid:16) σ η (cid:98) σ η (cid:17) − Tr K s ( η ) (cid:16) ˜ σ η (cid:98) ˜ σ η (cid:17) + (cid:88) ω ∈ V γ Tr N + ω (cid:0) υ + ω − υ − ω +1 (cid:1) + Tr K ω (cid:16) µ ω − µ ω (cid:17) ++ (cid:88) e ± ∈ E γ ± ,n ∈ Ξ p Tr K t ( e +) ,s ( e − ) ,n (cid:16) µ e + ,e − ; n µ e + ,e − ; n (cid:17) − Tr K t ( e +) ,t ( e − ) ,n (cid:16) µ e + ,e − ; n µ e + ,e − ; n (cid:17) = 0 PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 55 can be used to demonstrate, by the argument identical to that in (54), that the equa-tions (268) imply the holomorphic equations(282) σ η = 0 , ˜ σ η = 0 , (cid:98) σ η = 0 , (cid:98) ˜ σ η = 0 ,µ ω = 0 ,µ ω = 0 ,µ e + ,e − ; n = 0 ,µ e + ,e − ; n = 0 ,µ e + ,e − ; n = 0 ,µ e + ,e − ; n = 0 ,υ ± ω = 0 Thus, M γ k ( n ± , m ) is the space of stable solutions of (282) modulo the action (266) of G C γ k . Here, the stability condition is formulated as follows:(283) Any collection ( K (cid:48) ω ) ω ∈ V X of subspaces K (cid:48) ω ⊂ K ω , such that I ± ω ( N ± ω ) ⊂ K (cid:48) ω ± and B ± η ( K (cid:48) s ( η ) ) ⊂ K (cid:48) t ( η ) ± , for all η ∈ ± E X ˜ B ± η ( K (cid:48) t ( η ) ) ⊂ K (cid:48) s ( η ) ± , for all η ∈ ± E X coincides with all of K ω , K (cid:48) ω = K ω , in other words , K ω = K + ω + K − ω where K ± ω ⊂ K ω is the subspace, generated by acting with arbitrary (noncommutative)polynomials in B ± η , ˜ B ± η ’s with η ∈ ± E X on the image (cid:80) ω (cid:48) ∈ V X I ± ω (cid:48) ( N ± ω (cid:48) ) :(284) K ± ω = (cid:88) ω (cid:48) ∈ V X (cid:18) C (cid:104) B ± η , ˜ B ± η (cid:105) η ∈ ± E X I ± ω (cid:48) ( N ± ω (cid:48) ) (cid:19) ∩ K ω The space M γ k ( n ± , m ) is acted upon by the group(285) H γ = (cid:16) H γ ff × H γ edg (cid:17) / H γ ver × U (1) u , NIKITA NEKRASOV where(286) H γ ff = (cid:17) ω ∈ V γ U ( N + ω ) × U ( N − ω ) × U ( M (cid:48) ω ) × U ( M (cid:48)(cid:48) ω ) , H γ edg = (cid:17) η ∈ E γ + (cid:113) E γ − U (1) , H γ ver = (cid:17) ω ∈ V γ U (1) and the embedding of H γ ver into H γ ff × H γ edg is given by:(287) ( u ω ) ω ∈ V γ (cid:55)→ (cid:16) u ω · N + ω , u ω · N − ω , u ω · M (cid:48) ω , u ω · M (cid:48)(cid:48) ω (cid:17) ω ∈ V γ × (cid:16) u − s ( η ) ± u t ( η ) (cid:17) η ∈ ± E X The group (285) acts on M γ k ( n ± , m ) in the following fashion:(288) (cid:18) ( h + ω , h − ω , m (cid:48) ω , m (cid:48)(cid:48) ω ) ω ∈ V γ × (cid:16) u η (cid:17) η ∈ E γ × u (cid:19) · [ B , I , J , Θ ] == (cid:104) u ± u η g t ( η ) ± B ± η g − s ( η ) , u ± u − η g s ( η ) ± ˜ B ± η g − t ( η ) | η ∈ E γ u ± g ω ± I ± ω (cid:16) h ± ω (cid:17) − , h ± ω ± J ± ω g − ω u ± , m (cid:48) ω Θ (cid:48) ω , Θ (cid:48)(cid:48) ω ( m (cid:48)(cid:48) ω ) − | ω ∈ V γ (cid:21) , where we indicated the compensating G γ k -transformations. It is obvious from the Eq.(288) that the H γ ff × H γ edg -transformations which are in the image of H γ ver can be undoneby a G γ k -transformation.10.1.5. Compactness theorem in the crossed quiver case. Let us demonstrate the com-pactness of the set (cid:16) M γ k ( n ± , m ) (cid:17) T γ of T γ -fixed points in M γ k ( n ± , m ) , where T γ ⊂ H γ is asubtorus of the global symmetry group. The choice of T γ is restricted by the followingrequirement: it must contain a U (1) -subgroup, to be denoted by U (1) v , such that 1)the composition p ◦ i , where i : U (1) v (cid:44) → H γ is the embedding, and p : H γ → U (1) u isthe projection, is a non-trivial homomorphism, v (cid:55)→ v k , k (cid:54) = 0 , 2) the embedding into H γ is parametrized by the collection(289) v ∈ U (1) v (cid:55)→ (cid:18) v n + ω , v n − ω , v m (cid:48) ω , v m (cid:48)(cid:48) ω (cid:19) ω ∈ V γ × v e × v k The symmetry (264) reflects the quotient by H γ ver in (285).We shall impose an additional requirement on the data ( n ± V γ , m (cid:48) V γ , m (cid:48)(cid:48) V γ , e , k ) : for any ω (cid:48) , ω (cid:48)(cid:48) ∈ V γ and any α (cid:48) ∈ [ n ± ω (cid:48) ] , α (cid:48)(cid:48) ∈ [ n ± ω (cid:48)(cid:48) ] (290) n ± ω (cid:48) ,α (cid:48) − n ± ω (cid:48)(cid:48) ,α (cid:48)(cid:48) ± k ( (cid:96) + 1) + (cid:90) path ± ω (cid:48) ,ω (cid:48)(cid:48) e (cid:54) = 0 for any path p ± ω (cid:48) ,ω (cid:48)(cid:48) . Note (290) is invariant under (264). The requirement (290) can beslighlty weakened, namely one can allow (290) to fail for a single pair ( ω (cid:48) , α (cid:48) ) = ( ω (cid:48)(cid:48) , α (cid:48)(cid:48) ) .In what follows we insist on (290), though. PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 57 The proof goes as follows: Define the function δ on the Grassmanian of subspaces V ⊂ ⊕ ω K ω :(291) δ V = Tr V (cid:88) η ∈ E γ , ± B ± η (cid:16) B ± η (cid:17) † + ˜ B ± η (cid:16) ˜ B ± η (cid:17) † + (cid:88) ω, ± I ± ω (cid:16) I ± ω (cid:17) † + J ± ω (cid:16) J ± ω (cid:17) † The function δ is monotone: δ V (cid:48) ≤ δ V (cid:48)(cid:48) for V (cid:48) ⊂ V (cid:48)(cid:48) . We have:(292) (cid:107) B , I , J (cid:107) = (cid:88) ω ∈ V γ δ K ω Now, the T γ -invariance implies, by the usual arguments, that the spaces K ± ω , K ω , foreach ω ∈ V γ are T γ -representations,(293) K ± ω = (cid:77) w ∈ T ∨ γ K ± ω, w ⊗ R w where R w are the irreps of T γ .First, we need to prove that J ± ω = 0 for the T γ -invariant ( B , I , J ) . The equations (282)imply that J ± ω ( K ∓ ω ) = 0 . Let us restrict (293) onto U (1) v :(294) K ± ω = (cid:77) w ∈ Z K ± ω,w ⊗ R w where R w are the irreps of U (1) v : v (cid:55)→ v w . We have:(295) I ± ω ( N ± ω ) = n ± ω (cid:77) α =1 N ± ω,α ⊗ R n ± ω,α and(296) B ± η ( K ± s ( η ) ,w ) ⊂ K ± t ( η )+1 ,w + k + e ( η ) , ˜ B ± η ( K ± t ( η ) ,w ) ⊂ K ± s ( η )+1 ,w + k − e ( η ) Thus the weights s which occur in the decomposition (294) have the form:(297) n ± ω (cid:48) ,α (cid:48) ± k(cid:96) + (cid:90) p ± ω (cid:48) ,ω e for some ω (cid:48) ∈ V γ and some length (cid:96) path p ± ω (cid:48) ,ω .Now the U (1) v equivariance implies that J ± ω ( K ω,w ⊗ R w ) belongs to the eigenspaceof v ∈ U (1) v in N ± ω ± with the eigenvalue v w ∓ k . Since the eigenvalues of v ∈ U (1) v on N ± ω ± are given by: v n ± ω ± ,α the non-vanishing J ± ω means that for some ω (cid:48) , α (cid:48) theeigenvalue (297) coincides with n ω ± ,α ∓ k , which contradicts (290). Thus, J = 0 . NIKITA NEKRASOV Now, use the real moment map equation to deduce:(298) δ K ± ω,w = ζ ω dim( K ± ω,w ) ++ (cid:88) η ∈ s − ( ω ) ∩ ± E X Tr K ± t ( η ) ± ,w + k + e ( η ) B ± η (cid:16) B ± η (cid:17) † + (cid:88) η ∈ t − ( ω ) ∩ ± E X Tr K ± s ( η ) ± ,w + k − e ( η ) ˜ B ± η (cid:16) ˜ B ± η (cid:17) † ≤≤ ζ ω dim( K ± ω,w ) + (cid:88) η ∈ s − ( ω ) ∩ ± E X δ K ± t ( η ) ± ,w + k + e ( η ) + (cid:88) η ∈ t − ( ω ) ∩ ± E X δ K ± s ( η ) ± ,w + k − e ( η ) Now repeat the same estimate by pushing the arguments w (cid:48) of the δ K ω (cid:48) ,w (cid:48) ’s in the righthand side of (298) outside the domain where the corresponding K ω (cid:48) ,w (cid:48) spaces are non-trivial (this is possible because the total dimension of the K space is finite). In thisway we get an upper bound on δ K ω (cid:48) ,w (cid:48) ’s and the norms of B ’s, I ’s and J ’s, as promised.10.2. Orbifolds and defects: ADE × U (1) × ADE. The construction above can bemotivated by the following examples.Recall that the moduli space M + k ( n, w ) of crossed instantons has an SU(2) × U(1) × SU(2) symmetry.10.2.1. Space action. Let Γ be a discrete subgroup of SU(2) × U(1) ∆ × SU(2) ,(299) ι : Γ −→ G rot = SU(2) × U(1) ∆ × SU(2) , Framing action. Now let us endow the spaces N and N with the structureof Γ -module:(300) N = (cid:77) ω ∈ Γ ∨ N ω ⊗ R ω ,N = (cid:77) ω ∈ Γ ∨ W ω ⊗ R ω , in other words let us fix the homomorphisms(301) ρ A : Γ −→ U ( n A ) Let us denote by n , w the vectors of dimensions (dim N ,ω ) ω ∈ Γ ∨ , (dim N ,ω ) ω ∈ Γ ∨ , re-spectively.The data (300), (299) defines the embedding of the group Γ into H , the symmetrygroup of M + k ( n, w ) .10.2.3. New moduli spaces. The set of Γ -fixed points in M + k ( n, w ) splits into components(302) M + k ( n, w ) Γ = (cid:91) k M + ,γ Γ k ( n , w ) This is a particular case of the space M γ k ( n , w , . Indeed, let V γ = Γ ∨ , while E γ ± aredefined by decomposing(303) R ω ⊗ C = (cid:77) e ∈ s − ( ω ) ∩ E γ ± R t ( e ) (cid:77) e ∈ t − ( ω ) ∩ E γ ± R s ( e ) PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 59 The requirement that Γ preserves the U( k ) -orbit of ( B , I , J ) translates to the fact that Γ is unitary represented in K , so that(304) γ · ( B , I , J ) = ( g γ B g − γ , g γ I , J g − γ ) , γ (cid:55)→ g γ ∈ U( K ) Thus, we can decompose K into the irreps of Γ :(305) K = (cid:77) ω ∈ Γ ∨ K ω ⊗ R ω The operators B , I , J then become linear maps between the spaces K ω (cid:48) , N A,ω (cid:48)(cid:48) , whichcan be easily classified by unraveling the equivariance conditions (304).The components M + ,γ Γ k ( n , w ) can then be deformed by modifying the real momentmap equation to(306) µ = (cid:88) ω ∈ Γ ∨ ζ ω K ω In the particular case Γ ⊂ SU(2) the orbifold produces the moduli spaces of super-symmetric gauge configurations in the(307) G g = (cid:17) ω ∈ Γ ∨ U( n ω ) gauge theory in the presence of a point-like defect, the qq -character(308) X ( w ω ) ω ∈ Γ ∨ (cid:16) x, ν ω,β (cid:17) The gauge theory in question is the affine ADE quiver gauge theory.10.2.4. Odd dimensions and finite quivers. We can also obtain the moduli space ofsupersymmetric gauge field configurations in the quiver gauge theories built on finitequivers. The natural way to do that is to start with an affine quiver and send some ofthe gauge couplings to zero, i.e. by making some of dimensions k ω vanishing.Remarkably, this procedure produces the superspace, the odd variables originatingin the multiplet of the cohomological field theory. Let us explain this in more detail.Let us consider, for simplicity, the group Γ ⊂ SU (2) , so that E γ + has one element.The linear algebra data (cid:16) B ,ω , B ,ω , I ,ω , J ,ω , I ,ω , J ,ω , B e , ˜ B e (cid:17) ∈ A γ k ( n , w , is parametrized by the(309) (cid:88) ω ∈ Γ ∨ k ω ( k ω + n ω + w ω ) + 2 (cid:88) e ∈ E k t ( e ) k s ( e ) complex dimensional space. The Eqs. (268) plus the GL γ ( k ) -invariance remove(310) (cid:88) ω ∈ Γ ∨ ( n ω w ω + 2 k ω ( k ω + w ω )) + (cid:88) e ∈ E k s ( e ) k t ( e ) + k s ( e ) n t ( e ) + k t ( e ) n s ( e ) dimensions (this is half the number of equations (282)). The result is k -linear,(311) virtual dim M + ,γ k ( n , w ) = (cid:88) ω ∈ Γ ∨ ( k ω m ω − n ω w ω ) NIKITA NEKRASOV where(312) m ω = 2 n ω − (cid:88) e ∈ s − ( ω ) n t ( e ) − (cid:88) e ∈ t − ( ω ) n s ( e ) Now, if for all ω ∈ Γ ∨ the deficits m ω are non-negative, and at least for one vertexthe deficit is positive then the quiver is, in fact, a finite ADE Dynkin diagram. Inthis case we can add the odd variables taking values in the spaces Hom( K ω , M ω ) withthe complex vector space M ω of dimension m ω , and define the moduli space to bethe supermanifold which is the total space of the odd vector bundle Π Hom( K ω , M ω ) over the previously defined bosonic moduli space. In practice this means that theintegration over the “true” moduli space is the integral over the coarse moduli spaceof the equivariant Euler class of the vector bundle Hom( K ω , M ω ) . This is what thecohomological field theory applied to the affine quiver case with the subsequent setting k ω (cid:48) = 0 for some ω (cid:48) ∈ Γ ∨ would amount to. With the “compensator” vector bundle inplace the virtual dimension of the moduli space becomes k -independent. This is thetopological counterpart of the asymptotic conformal invariance of the gauge theory.11. Spiked instantons on orbifolds and defects Now let us go back to the general case of spiked instantons. Choose a discretesubgroup Γ of U (1) e , e.g. Γ ≈ Ξ p × Ξ p × Ξ p , Γ ∨ ≈ Γ . Let t a : Γ → U (1) , a ∈ be thehomomorphisms corresponding to the embedding U (1) e ⊂ SU (4) . We have:(313) (cid:89) a ∈ t a ( γ ) = 1 Let R a ∈ Γ ∨ , a ∈ be the corresponding one-dimensional representations of Γ , e.g R = R , , , R = R − , − , − . We shall use the additive notation, R a = R (cid:126)(cid:36) a so that R (cid:126)ω ⊗ R a = R (cid:126)ω + (cid:126)(cid:36) a . Fix the framing homomorphisms: ρ A, Γ → U ( n A ) :(314) N A = (cid:77) (cid:126)ω ∈ Γ ∨ N A, (cid:126)ω ⊗ R (cid:126)ω The set of Γ -fixed points in M k ( (cid:126)n ) splits into components(315) M k ( (cid:126)n ) Γ = (cid:91) k M γ Γ k ( (cid:126) n ) It describes the moduli spaces of spiked instantons in the presence of additional surfaceand point-like conical defects. The compactness theorem holds in this case. The proofis a simple extension of the proof of section with the spaces K A replaced by K A, (cid:126)ω ,where (cid:126)ω = ( ω , ω , ω ) , ω i ∈ Ξ p i :(316) K A, (cid:126)ω = (cid:88) (cid:126)ω (cid:48) ∈ Γ ∨ (cid:88) f ∈ C [ x,y ] f ( B a , B b ) I A ( N A, (cid:126)ω (cid:48) ) where the sum is over polynomials obeying:(317) f ( t a ( γ ) x, t b ( γ ) y ) = χ R (cid:126)ω − (cid:126)ω (cid:48) ( γ ) PS/CFT, INSTANTONS AT CROSSROADS, GAUGE ORIGAMI 61 for all γ ∈ Γ . As before B c, (cid:126)ω (cid:48) ( K A, (cid:126)ω ) = 0 whenever c / ∈ A . The vector k encodes thedimensions of the spaces(318) K (cid:126)ω = (cid:88) A ∈ K A, (cid:126)ω the operators B a have the block form:(319) B a ( K (cid:126)ω ) ⊂ K (cid:126)ω + (cid:126)(cid:36) a The norms (cid:107) B a (cid:107) are estimated with the help of the quantities(320) δ A, (cid:126)ω,n = Tr K A,(cid:126)ω (cid:88) a B a B † a + (cid:88) C I C I † C Conclusions and future directions In this paper we introduced several moduli spaces: M + , M | _ , M ∗ of matrices solvingquadratic equations modulo symmetries. These moduli spaces generalize the Gieseker-Nakajima partial compactification M of the ADHM moduli space of U ( n ) instantonson R . We gave some motivations for these constructions and proved the compactnesstheorem which we shall use in the next papers to establish useful identities on thecorrelation functions of supersymmetric gauge theories in four dimensions.In this concluding section we would like to make a few remarks.First of all, one can motivate the crossed instanton construction by starting thewith the ordinary ADHM construction and adding the co-fields [8, 7] which mirror theembedding of the N = 2 super-Yang-Mills vector multiplet into the N = 4 super-Yang-Mills vector multiplet [37].Secondly, we would like to find the crossed instanton analogue of the stable envelopesof [21].Third, it would be nice to generalize the spiked instanton construction to allow moregeneral orbifold groups Γ ⊂ SU (4) , and more general (Lagrangian?) subvarieties in C / Γ .Now, to the serious drawbacks of our constructions. The purpose of the ADHMconstruction, after all, is the construction of the solutions to the instanton equations F + A = 0 We didn’t find the analogue of the ADHM construction for the spiked instantons. Con-jecturally, the matrices [ B , I , J ] solving the Eqs. (56) are in one-to-one correspondencewith the finite action solutions to Eqs. (41).Finally, we have proposed a definition of quiver crossed instantons, which are definedfor quivers more general then the products of ADE Dynkin diagrams. It would be inter-esting to find the precise restrictions on these quivers compatible with the compactnesstheorem.In the forthcoming papers the compactness theorem will be used to derive the mainstatements of the theory of qq -characters [30]. While this paper was in preparation,the algebraic counterpart of our compactness theorem was studied in [19]. Various NIKITA NEKRASOV consequences of the compactness theorem will be studied in [29]. 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