PREPRINT
國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
www.math.ntu.edu.tw/ ~ mathlib/preprint/2013- 01.pdf
Geometric Engineering of (Framed) BPS States
Wu-Yen Chuang, Duiliu-Emanuel Diaconescu, Jan Manschot, Gregory W. Moore, and Yan Soibelman
January 29, 2013
WU-YEN CHUANG1, DUILIU-EMANUEL DIACONESCU2, JAN MANSCHOT3,4, GREGORY W. MOORE5, YAN SOIBELMAN6
Abstract. BPS quivers for N = 2 SU (N ) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds.
While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N , relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with “exotic” SU (2)R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson- Thomas invariants for framed quiver representations.
Contents
1. Introduction 1
1.1. A (short) summary for mathematicians 6
1.2. BPS categories and mirror symmetry 9
2. Geometric engineering, exceptional collections, and quivers 11 2.1. Exceptional collections and fractional branes 14
2.2. Orbifold quivers 18
2.3. Field theory limit A 19
2.4. Stability conditions 28
2.5. BPS degeneracies and Donaldson-Thomas invariants 31
3. Field theory limit B 34
4. Large radius stability and the weak coupling BPS spectrum 40
4.1. Large radius stability 40
4.2. An example: SU (2) gauge theory 43
4.3. Limit weak coupling spectrum and absence of walls 47 4.4. SU (3) spectrum with magnetic charges (1, 1) 55
5. The SU (3) quiver at weak coupling 59
5.1. General considerations 59
5.2. W -bosons 60
5.3. Moduli spaces and stability chambers for magnetic charges (1, m) 62
5.4. Comparison with large radius spectrum 65
6. Strong coupling chamber for the SU (N ) quiver 67
1
6.1. A mutation of the SU (3) quiver 72
6.2. A deceptive chamber 73
7. Line defects and framed BPS states 75
7.1. Geometric construction of magnetic line defects 75
7.2. Framed stability conditions 78
7.3. Framed BPS states, Donaldson-Thomas invariants, and wallcrossing 79 7.4. A recursion formula for unframed BPS states 83
7.5. Absence of exotics I 86
8. BPS states and cohomological Hall algebras 90
8.1. Cohomological Hall algebras 90
8.2. Framing and SL(2,C)spin action 92
8.3. Absence of exotics II 94
Appendix A. Exceptional collections and quivers for XN 95
Appendix B. Motives for pedestrians 101
Appendix C. Kronecker modules 103
C.1. Harder-Narasimhan filtrations 104
C.2. Application to representations of the SU (3) quiver 106
Appendix D. Background material on extensions 107
Appendix E. Classifications of fixed points 110
References 115
1. Introduction
The BPS spectrum of four dimensionalN = 2 gauge theories has been a constant subject of research since the discovery of the Seiberg-Witten solution. An incom- plete sampling of references includes [125, 126, 61, 22, 101, 23, 112, 113, 127, 65, 85, 131]. Very recent intense activity in this field was motivated by the connection [67] between wallcrossing on the Coulomb branch and the Kontsevich-Soibelman formula [106]. An incomplete sampling of references includes [67, 70, 38, 68, 37, 2, 3, 39, 69, 34, 36, 35, 137]. For recent reviews see [115, 33].
On the other hand, it has been known for a while that manyN = 2 gauge theories are obtained in geometric engineering as a low energy limit of string theory dynam- ics in the presence of Calabi-Yau singularities [5, 96, 101, 99, 98]. This leads imme- diately to a close connection between the gauge theory BPS spectrum and the BPS spectrum of string theory in the presence of such singularities. The latter consists of supersymmetric D-brane bound states wrapping exceptional cycles, and hence can in principle be analyzed using derived category methods [104, 56, 57, 8, 128, 12, 7].
In principle geometric engineering is expected to provide a microscopic string the- ory derivation for the BPS quivers found in [47, 48, 37, 3] by low energy methods.
Indeed the BPS quivers constructed in loc. cit. for SU (N ) gauge theories were first derived by Fiol in [62] using fractional branes on quotient singularities. It is quite remarkable that this construction was confirmed ten years later by completely dif- ferent low energy methods. A similar approach, employing a more geometric point of view has been subsequently employed in [10, 54] for SU (2) gauge theories. Their results are again in agreement with the low energy constructions.
The goal of the present work is to proceed to a systematic study of the gauge theory BPS spectrum via categorical and geometric methods. Special emphasis is
placed on higher rank gauge theories, where the BPS spectrum is not completely known on the entire Coulomb branch, many problems being at the moment open.
In order to keep the paper to be of reasonable length, only pure SU (N ) gauge theories will be considered in this paper. In this case the local toric threefolds are resolved AN−1quotient singularities fibered overP1, such that the singularity type does not jump at any points on the base. Their derived categories are equivalent by tilting to derived categories of modules over the path algebra of a quiver with potential determined by an exceptional collection of line bundles. Physically, these quivers encode the quantum mechanical effective action of a collection of fractional branes on the toric threefold. Taking the field theory limit amounts to a truncation of the fractional brane quiver, omitting the branes which become very heavy in this limit together with the adjacent arrows. The resulting quiver for pure SU (N ) gauge theory is of the form
qN−1• dcN−1 //
N−1 // •pN−1 rN−2
sN−2
ssqN−2• cN−2 //d
N−2 //
bN−2
OO
•pN−1 aN−2
OO
... ...
qi+1• cdi+1 //
i+1 // •pi+1 ri
si
ssqi• ci //
di //
bi
OO
•pi ai
OO
... ...
q2• c2 //
d2 // •p2 r1
s1
ssq1• c1 //
d1 //
b1
OO
•p1 a1
OO
with a potential
W =
N∑−2 i=1
[ri(aici− ci+1bi) + si(aidi− di+1bi)] .
This is the same as the quiver found in [62], and is mutation equivalent to the quivers found in [37, 3] by different methods. This approach can be extended to gauge theories with flavors allowing the AN−1 singularity to jump at special points on the base.
In order to set the stage, geometric engineering and the field theory limit of Calabi-Yau compactifications is carefully reviewed in Section 2. Special emphasis is placed on categorical constructions, in particular exceptional collections of line bundles on toric Calabi-Yau threefolds. In particular an explicit construction of such collections is provided for toric Calabi-Yau threefolds XN engineering pure SU (N ) gauge theory. Not surprisingly, it is then shown that the associated frac- tional brane quiver is the same as the one obtained in [62] by orbifold methods. As opposed to the construction in loc. cit., the geometric approach provides a large radius limit presentation of fractional branes in terms of derived objects on XN. The main outcome of Section 2 is a conjectural categorical description of gauge theory BPS states in terms of a triangulated subcategoryG ⊂ Db(XN). As shown by detailed A-model computations in Section 2.3, G is a truncation of Db(XN) generated by fractional branes with finite central charges in the field theory limit.
It is perhaps worth noting that this conclusion involves certain delicate cancella- tions between tree level and world-sheet instanton contributions which were never spelled out in the literature.
According to [56, 57, 8] supersymmetric D-brane configurations on XN are iden- tified with Π-stable objects in the derived category Db(XN), or in rigorous mathe- matical formulation, Bridgeland stable objects [28]. Therefore one is naturally led to conjecture that gauge theory BPS states will be constructed in terms of Bridge- land stable objects in Db(XN) which belong toG. However it is important to note that agreement of the low energy constructions with [37, 2, 3] requires a stronger statement. Namely, that gauge theory BPS states must be constructed in terms of an intrinsic stability condition onG. Mathematically, these two statements are not equivalent since in general a stability condition on the ambient derived category does not automatically induce one on the subcategoryG. It is however shown in Section 2.4 that this does hold for quivery or algebraic stability conditions, anal- ogous to those constructed in [27, 15]. The above statement fails for geometric large radius limit stability conditions, such as (ω, B)-stability, which is analyzed in Section 4. Section 2 concludes with a detailed comparison of gauge theoretic BPS indices and the motivic Donaldson-Thomas invariants constructed in [106].
In particular it is shown that the protected spin characters defined in [68] corre- spond mathematically to a χy-genus type specialization of the motivic invariants.
In contrast, the unprotected spin characters introduced in [52, 54] are related to virtual Poincar´e or Hodge polynomials associated to the motivic invariants. This is explained in Section 2.5, together with a summary of positivity conjectures for gauge theory BPS states states formulated in [68].
We note here that different mathematical constructions of categories and stability conditions for BPS states is carried out by Bridgeland and Smith in [31, 32], and,
as part of a more general framework, by Kontsevich and Soibelman in [107]. The connection between their work and this paper will be explained in Section 1.2.
Section 3 consists of a detailed analysis of the field theory limit in terms of the local mirror geometry for SU (2) gauge theory. The results confirm the conclusions of Section 2.3 and also set the stage for the absence of walls conjecture formulated in the next section.
Section 4 is focused on large radius supersymmetric D-brane configurations on XN and their relation to gauge theory BPS states. Motivated by the SU (2) example in Section 4.2, we are led to conjecture a precise relation between large radius and gauge theory BPS states, called the absence of walls conjecture. As explained in the beginning of Section 3, for general N the complex structure moduli space of the local mirror to XN is parameterized by N complex coordinates zi, 0≤ i ≤ N − 1.
The large complex structure limit point (LCS) lies at the intersection of the N boundary divisors zi = 0, 0 ≤ i ≤ N − 1. On the other hand, the scaling region defining the field theory limit is centered at the intersection between the divisor z0= 0 and the discriminant ∆N, as sketched below.
•
•
LCS
Field theory scaling region
9
z0=0
z0
∆N
Possible marginal stability walls between LCS and field theory region
Figure 1. Schematic representation of the complex structure moduli space for general N≥ 2.
In principle there could exist marginal stability walls between the LCS limit point and the field theory scaling region as sketched in Figure 1. Therefore a correspon- dence between large radius BPS states and gauge theory BPS states is not expected on general grounds. We conjecture that for all charges γ ∈ Γ which support BPS states of finite mass in the field theory limit it is possible to choose a path connect- ing the two regions in the moduli space which avoids all such walls. This implies a one-to-one correspondence between BPS states in these two limits, which was first observed for SU (2) gauge theory in [54].
Section 4.3 contains a precise mathematical formulation of this conjecture em- ploying the notion of limit weak coupling BPS spectrum. Intuitively, the limit spectrum should be thought of as an extreme weak coupling limit of the BPS spec- trum where all instanton and subleading polynomial corrections to the N = 2 prepotential are turned off. Then the absence of walls conjecture implies that the limit weak coupling spectrum is identified with a certain limit of the large radius BPS spectrum. As a first test of this conjecture we next show that all large radius supersymmetric D-branes in this limit, with charges in the gauge theory lattice Γ≃ K0(G), actually belong to the triangulated subcategory G. This is a nontrivial
result, and an important categorical test of the field theory limit of Calabi-Yau compactifications.
In order to carry out further tests, the large radius BPS spectrum of SU (3) theory is then investigated in Section 4.4. The geometrical setup determines a Cartan subalgebra of SU (3) together with a set of simple roots {α1, α2}. We determine the degeneracy of states with magnetic charge α1+ α2. The results show that one can find BPS states with arbitrarily high spin at weak coupling.
Section 5 presents some exact weak coupling results for BPS degeneracies in SU (3) gauge theories with magnetic changes (1, m) with m≥ 1. Explicit formulas are derived both for m = 1 by a direct analysis of the moduli spaces of stable quiver representations. It is also shown that for any m≥ 1 the BPS degeneracies vanish in a specific chamber in the moduli space of stability conditions. This yields exact results by wallcrossing, explicit formulas being written only for m = 2. It should be noted at this point that the above results are not in agreement with those obtained in [65] by monodromy arguments. The weak coupling spectrum found in [65] is only a subset of the BPS states found here by quiver computations. In addition, it is explicitly shown that there exist BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. This is also in agreement with the semiclassical analysis of [71, 133] based on counting zero modes of a Dirac operator on the monopole moduli space. Finally, these results are shown to be in agreement with their large radius counterparts in Section 5.4, confirming the predictions of the absence of walls conjecture.
Section 6 exhibits a strong coupling chamber for SU (N ) gauge theories where the BPS spectrum is in agreement with previous results [3, 69]. In contrast with loc. cit., here this chamber is obtained by a direct analysis of the spectrum of stable quiver representations. As a corollary, a deceptive adjacent chamber is found in Section 6.2 where the BPS spectrum coincides with the one generated in [65]
by monodromy transformations. However, the disposition of the central charges in the complex plane shows that the deceptive chamber cannot be a weak coupling chamber, hence justifying its name.
Building on the geometric methods developed so far, framed quiver models are constructed in Section 7 for framed BPS states corresponding to simple magnetic line defects. From a geometric point of view, such line defects are engineered by D4-branes wrapping smooth noncompact divisors in the toric threefold XN. This framework leads to a rigorous mathematical construction of such states in terms of weak stability conditions1 for framed quiver representations depending on an extra real parameter δ related to the phase of the line defect [93, 68, 123, 122]. The wallcrossing theory of [106] is shown to be applicable to such situations, resulting in a mathematical derivation of the framed wallcrossing formula of [68].
Moreover, in Section 7.4, a detailed analysis of the chamber structure on the δ- line leads to a complete recursive algorithm, determining the BPS spectrum at any point on the Coulomb branch in terms of the noncommutative Donaldson- Thomas invariants defined in [130]. It should be emphasized that this argument solely relies on wallcrossing on the δ-line, and is therefore valid at any fixed point on the Coulomb branch where this particular quiver description is valid. As an application, we show in Section 7.5 that the recursion formula implies the absence of exotics conjecture for framed and unframed BPS states first articulated in [68].
1The meaning of “weak stability conditions ” is explained in [132].
Note that rigorous positivity results are obtained in a similar context in [46]
by proving a purity result for the cohomology of the sheaf of vanishing cycles.
It is interesting to note that the the technical conditions used in [46] are not in general satisfied in gauge theory examples. Hence we are led to conjecture that such positivity results will hold under more general conditions, not yet understood from a mathematical point of view.
Finally, Section 8 addresses the same issues from the perspective of cohomological Hall algebras, introduced by Kontsevich and Soibelman in [108] as well as their framed stability conditions introduced in [105]. A geometric construction is outlined in this context for the action of the spin SL(2,C) group on the space of BPS states. Moreover, absence of exotics is conjectured to follow in this formalism from a hypothetical Atiyah-Bott fixed point theorem for the cohomology with rapid decay at infinity defined in [108].
1.1. A (short) summary for mathematicians. In this section we summarize the main results of this work for a mathematical audience. Recent physics results on BPS states [67, 70, 38, 68, 37, 2, 3, 39, 69, 34, 36, 35, 137] point towards a general conjectural correspondence assigning to an N = 2 supersymmetric gauge theory
(i) a triangulated CY3 categoryG, and
(ii) a map ϱ : C → Stab(G) from the universal cover C of the gauge theory Coulomb branch to the moduli space of Bridgeland stability conditions on G.
The central claim is then:
(G.1) The BPS spectrum of the gauge theory at any point a∈ C is determined by the motivic Donaldson-Thomas invariants [106] of ϱ(a)-semistable objects ofC.
Since supersymmetric quantum field theories do not admit a rigorous mathemat- ical construction, a natural question is whether the above correspondence can be converted into a rigorous mathematical statement. One answer to this question is presented in [31, 32, 107] (building on the main ideas of [70].) The present paper proposes a different approach to this problem based instead on geometric engineer- ing of gauge theories [5, 96, 101, 99, 98]. As explained in Subsection 1.2 below, geometric engineering and the construction of [31, 32, 107] are related by mirror symmetry, modulo certain subtle issues concerning the field theory limit.
Very briefly, geometric engineering is a physics construction assigning anN = 2 gauge theory to a certain toric Calabi-Yau threefold with singularities. It is not known whether any gauge theory can be obtained this way, but a large class of such theories admit such a geometric construction. For example SU (N ) gauge theories with Nf ≤ 2N fundamental hypermultiplets and quiver gauge theories with gauge group∏
iSU (Ni) belong to this class, as shown in [98].
Accepting geometric engineering as a black box, the present paper identifies the category G with a triangulated subcategory of the derived category Db(X).
This identification is based on a presentation of Db(X) in terms of an exceptional collection of line bundles {Lα} [6, 92, 21]. Any such collection determines a dual collection of objects {Pα} of Db(X) such that RHomX(Lα, Pβ) = Cδα,β. These are usually called fractional branes in the physics literature. Then the conjecture proposed in this paper is:
(G.2) There exists a subset{Lα′} ⊂ {Lα} such that the gauge theory category G is the triangulated subcategory of Db(X) generated by the fractional branes {Pβ′} satisfying RHomX(Lα′, Pβ′) = 0.
For illustration, this is explicitly shown in Sections 2.1 and 2.3 for pure SU (N ) gauge theory of arbitrary rank. More general models can be treated analogously, explicit statements being left for future work.
Granting the above statement, the results of [14, 24, 124] further identify G with a category of twisted complexes of modules over the path algebra of a quiver with potential (Q, W ). Moreover, a detailed analysis of geometric engineering as in Section 2.3 further yields an assignment of central charges zβ′ :C → C to the objects {Pβ′}. Therefore one obtains a well defined stability condition in Stab(G) for any point a ∈ C where the images zβ′(a) belong to a half-plane Hϕ. This defines a map ϱ(Q,W ) : C(Q,W ) → Stab(G) over a certain subspace C(Q,W ) ⊂ C.
We further conjecture that, using mutations, one can extend this map to a map ϱ : C → Stab(G), and moreover the image of ϱ is contained in the subspace of algebraic (or quivery) stability conditions in the terminology of [26, 27, 15].
The above construction also leads to a mathematical model for framed BPS states of simple magnetic line defects [68] in terms of moduli spaces of weakly stable framed quiver representations. This is explained in Section 7.
In this framework, one is naturally led to a series of conjectures, or at least ques- tions of mathematical interest. First note that four dimensional Lorentz invariance predicts the existence of a Lefschetz type SL(2,C)spin-action on the cohomology of the sheaf of vanishing cycles of the potential W on moduli spaces of stable quiver representations. In addition there is a second SL(2,C)R-action, encoding the R- symmetry of the gauge theory. The action of the maximal torusC×R⊂ SL(2, C)Ris determined by the Hodge structure on the above cohomology groups, as explained in Section 2.5.
Assuming the existence of the above actions a series of positivity conjectures are formulated in [68], and reviewed in Section 2.5. The strongest of these conjectures claims that the C×R-action is trivial, and the virtual Poincar´e polynomial of the vanishing cycle cohomology decomposes into a sum of irreducible SL(2,C)spin in- tegral spin characters with positive integral coefficients. This is called the no exotics conjecture.
Granting the existence of the SL(2,C)spin-action, in order to prove the no exotics conjecture it suffices to prove that all refined DT invariants belong to the subring generated by (xy)1/2, (xy)−1/2. This follows from the integrality result proven in [108]. Here we provide an alternative proof for pure SU (N ) gauge theory in Section 7.5 using a framed wallcrossing argument. Furthermore, as explained in the last paragraph of Section 7.5, physical arguments suggest that the no exotics conjecture should hold for refined DT invariants of toric Calabi-Yau threefolds in general. Again four dimensional Lorentz invariance predicts a Lefschetz type action on the moduli space of stable quiver representations. Moreover, there is also a C×R-action [52] corresponding to an U (1)R-symmetry. Combining all these statements, one is led to claim that a no exotics result will hold for toric Calabi-Yau threefolds, if one can prove that the motivic DT invariants belong to the subring generated by L1/2,L−1/2, as conjectured in [106]. For DT invariants defined in terms of algebraic stability conditions, this follows from the results of [108]. For geometric stability conditions, this follows from the results of [108] and the motivic
wallcrossing formula [106, 108]. Explicit computations in some examples have been carried out in [117, 118, 41].
It is important to note that some cases of the no exotics conjecture are proven in [59, 46] via purity results for the vanishing cycle cohomology. However, the proof relies on certain technical assumptions – such as compactness of the moduli space in [46] – which are not generically satisfied for gauge theory quivers. Physics arguments predict that similar results should hold in a much larger class of exam- ples of quivers with potential, although the mathematical reason for that is rather mysterious.
Finally, note that the above conjectures are formulated in the language of coho- mological Hall algebras [108] in Section 8. In particular a series of conjectures of [105] are generalized to moduli spaces of weakly stable framed quiver representa- tions.
In addition, geometric engineering also suggests an absence of walls conjecture stating an equivalence between refined DT invariants of large radius limit stable objects of Db(X) and refined DT invariants of gauge theory quiver representations.
The precise statement requires some preparation and is given in Section 4.3. As explained there it claims the existence of special paths in the complex K¨ahler moduli space of X avoiding certain marginal stability walls.
1.2. BPS categories and mirror symmetry. For completeness, we explain here a general framework emerging from string theory dualities, which ties together geometric engineering, N = 2 theories of class S, and the constructions of [31, 32, 107]. Our treatment will be rather sketchy with the details and is highly conjectural.
Our purpose here is merely to give a bird’s eye framework for relating several different approaches to the BPS spectrum ofN = 2 theories.
We will restrict ourselves to the gauge theories of class S introduced in [136, 66, 70]. These are in one-to-one correspondence with the following data
• a compact Riemann surface C with a collection of marked points {pi}
• a Hitchin system with gauge group G on C with prescribed singularities at the marked points{pi}.
Let H denote the total space of the Hitchin system and π : H → B the Hitchin map. The targetB of the Hitchin map is an affine linear space and the fibers of π are Prym varieties. We will denote by ∆⊂ B the discriminant of the map π.
The connection with M-theory is based on the spectral cover construction of the Hitchin system. Let D =∑
ipi denote the divisor of marked points on C, and SD the total space of the line bundle KC(D) on C. Let also S = SD\ ∪iKC(D)pi be the complement of the union of fibers of KC(D) at the marked points. Note that S is isomorphic to the complement of the union of fibers Tp∗iC in the total space of the cotangent bundle T∗C. In particular S is naturally a holomorphic symplectic surface.
If the Hitchin system has simple regular singularities at the marked points, the total spaceH is identified with a moduli space of pairs (¯Σ, ¯F ) where ¯Σ⊂ SD is a compact effective divisor in SD and ¯F a torsion free sheaf on Σ. At generic points in the moduli space ¯Σ is reduced and irreducible and ¯F is a rank one torsion free sheaf. For physics reasons, it is more convenient to think of the data ( ¯Σ, ¯F ) as a non-compact curve Σ⊂ S and a torsion free sheaf F on Σ with prescribed behavior at “infinity” i.e. at the points of intersection with the fibers KC(D)pi ⊂ SD. In
the following we will assume such a spectral cover construction to hold even if the Hitchin system has irregular singularities.
The holomorphic symplectic surface S can be used to construct an M-theory backgroundR3,1× S × R3. The data (Σ, F ) determines a supersymmetric M five- brane configuration with world-volume of the formR3,1× Σ. Now the connection with [31, 32, 107] can be explained employing M-theory/IIB duality. Suppose two out of the three transverse directions are compactified on a rectangular torus such that the M-theory background becomesR3,1× S × SM1 × S1A× R. Then a standard chain of string dualities shows that such a configuration is dual to a IIB background on a Calabi-Yau threefold Y .
The construction of Y for Hitchin systems with no singularities, i.e. no marked points pihas been carried out in [49]. More precisely, according to [49], any Hitchin systemH → B of ADE type determines naturally a family Y → B of Calabi-Yau threefolds such that
• For any b ∈ B \∆, Ybis smooth and isomorphic to the total space of a conic bundle over the holomorphic symplectic surface S with discriminant Σ.
• For any point b ∈ B \ ∆ the intermediate Jacobian J(Yb) is isogenous to the Prym π−1(b).
The family is defined over the entire base B, and Yb is isomorphic to the total space of a singular conic bundle over S at points b ∈ ∆. Furthermore note that by construction all fibers Yb, b ∈ B \ ∆ are equipped with a natural symplectic structure.
The duality argument sketched above leads to the conjecture that one can con- struct a familyY → B with analogous properties for Hitchin systems H → B with prescribed singularities at marked points. Since string duality preserves the spec- trum of BPS states, one is further led to the following conjecture, which provides a string theoretic framework for the constructions of [31, 32, 107].
(F.1) For any b ∈ B \ ∆, let F(Yb) be the Fukaya category of Yb generated by compact lagrangian cycles. Let eB denote the universal cover of B \ ∆. Then for any point ˜b ∈ eB over b ∈ B \ ∆ there is a unique point σ˜b ∈ Stab(F(Yb)) in the moduli space of Bridgeland stability conditions onF(Yb) such that the gauge theory BPS spectrum at the point ˜b is determined by the motivic DT invariants of moduli spaces of σ˜b-semistable objects in F(Yb).
Furthermore, there is a natural equivalence of triangulated A∞-categories of all categories F(Yb), b ∈ B \ ∆ with a fixed triangulated A∞-category F. Hence one obtains a map ϱ : eB → Stab(F) as predicted in the first paragraph of Section 1.1, withG ≃ F.
The construction of the familyY → B was carried out in [107], where the case of arbitrary irregular singularities was considered. Loc. cit. generalizes the results of [49] to a wide class of non-compact Calabi-Yau threefolds. It also gives a mathe- matically precise meaning to Conjecture (F.1) above and relates the DT-invariants of Fukaya categories from Conjecture (F.1) to the geometry of the corresponding Hitchin integrable system.
In order to explain the relation with the geometric engineering of the present paper, recall that any toric Calabi-Yau threefold X is related by local mirror sym- metry [86, 119] to a familyZ of non-compact Calabi-Yau threefolds. As explained
in more detail in Section 3, the mirror family is a family of hypersurfaces of the form
Pα(v, w) = xy,
where (v, w, x, y) ∈ (C×)2× C2, and Pα is a polynomial function depending on some complex parameters α. Each such hypersurface is a conic bundle Zα over (C×)2 with discriminant Pα(z, w) = 0. Homological mirror symmetry predicts an equivalence of triangulated A∞-categories
(1.1) Db(X)≃ Fuk(Zα)
for any smooth Zα in the family, where Fuk(Zα) is the Fukaya category of Zα. In local mirror variables, the field theory limit is presented as a degeneration of the familyZ. Referring the reader to Section 3 for more details, the parameters α are written in the form α = α(u, ϵ) for another set of parameters u to be identified with the Coulomb branch variables of the field theory, u∈ B. Then one takes the limit ϵ → 0 obtaining a family of threefolds Z0 over a parameter space B. Note that this degeneration has been studied explicitly in the physics literature [99, 98], but some geometric aspects would deserve a more detailed analysis. To conclude, string duality arguments predict the following conjecture:
(F.2) The limit family Z0→ B is the same as the family of threefolds Y → B in (F.1). Moreover the equivalence (1.1) restricts to an equivalence
(1.2) G ≃ F,
whereG ⊂ Db(X) is the category defined in (G.2)
Note that this conjecture predicts an interesting class of examples of homological mirror symmetry. The category G is defined algebraically as the subcategory of Db(X) spanned by a subset of fractional branes, whileF is obtained from Fuk(Zα) by degeneration. Hence it is natural to ask whether the categoryG can be obtained directly by constructing the mirror of the threefold familyY → B.
Acknowledgements. We are very grateful to Paul Aspinwall for collaboration and very helpful discussions at an early stage of the project. We thank Arend Bayer, Tom Bridgeland, Clay Cordova, Davide Gaiotto, Dmitry Galakhov, Zheng Hua, Amir Kashani-Poor, Albrecht Klemm, Maxim Kontsevich, Pietro Longi, Dav- esh Maulik, Andy Neitzke, Andy Royston, and Balazs Szendr¨oi for very helpful discussions and correspondence. W.Y.C. was supported by NSC grant 101-2628- M-002-003-MY4 and a fellowship from the Kenda Foundation. D.E.D was partially supported by NSF grant PHY-0854757-2009. J.M. thanks the Junior Program of the Hausdorff Research Institute for hospitality. G.M. was partially supported by DOE grant DE-FG02-96ER40959 and by a grant from the Simons Foundation (♯ 227381). D.E.D also thanks Max Planck Institute, Bonn, the Simons Center for Geometry and Physics, and the Mathematics Department of National Taiwan University for hospitality during the completion of this work. GM also gratefully acknowledges partial support from the Institute for Advanced Study and the Am- brose Monell Foundation. The research of Y.S. was partially supported by NSF grant. He thanks IHES for excellent research conditions.
2. Geometric engineering, exceptional collections, and quivers This section contains a detailed construction of a discrete family XN, N ≥ 2 of toric Calabi-Yau threefolds employed in geometric engineering [5, 96, 101, 99, 98]
of pure SU (N ) gauge theories with eight supercharges. Physical aspects of this correspondence will be discussed in Section 2.3.
Let Ya be the total space of the rank two bundleOP1(a)⊕ OP1(−2 − a) over P1, where a ∈ Z. For any N ∈ Z, N ≥ 2, there is a fiberwise ZN-action on Ya with weights±1 on the two summands. The quotient Ya/ZN is a singular toric threefold with a line of quotient AN singularities which admits a smooth Calabi-Yau toric resolution XN. For concreteness, let a = 0 in the following2. Then XN is defined by the toric data
(2.1)
x0 x1 x2 x3 . . . xN−1 xN y1 y2
C×(1) 1 −2 1 0 . . . 0 0 0 0
C×(2) 0 1 −2 1 . . . 0 0 0 0
...
C×(N−1) 0 0 0 0 . . . −2 1 0 0
C×(N ) −2 0 0 0 . . . 0 0 1 1
with disallowed locus
(2.2) ∪
0≤i,j≤N
|i−j|≥2
{xi = xj= 0} ∪ {y1= y2= 0}.
The toric fan of XN is the cone in R3 over the planar polytope in Fig. (2.a) embedded in the coordinate hyperplane ⃗r· ⃗e3= 1.
AA AA AA AA A
QQ QQ Q
SS SS SS
•
•
•
• •
•
(2.a)
AA AA AA AA A
•
• •
•
(2.b)
Figure 2. The toric polytope for X3 and the singular threefold Y0/Z3. The polytope (2.a) for XN is similar, except it will contain N− 1 inner points on the vertical axis.
Note that the toric data of the singular threefold Y0/ZN is the same, the disallowed locus being
{y1= y2} = 0.
The toric fan of the singular threefold is represented in Fig (1.b).
2Different values of a will lead to different Calabi-Yau threefolds, and the category of branes on these 3-folds will depend nontrivially on a. It is expected, however, that the field theoretic subcategories of interest in this paper will in fact be a-independent. Whether this is really so is left to future investigation.
As expected, there is a natural toric projection π : XN → P1, its fibers being isomorphic to the canonical resolution of the two dimensional AN singularity. The divisor class of the fiber is H = (y1) = (y2). The inner points of the polyhedron correspond to the N − 1 irreducible compact toric divisors Si ⊂ XN determined by xi = 0, i = 1, . . . , N− 1. Each of them is isomorphic to a Hirzebruch surface, Si≃ F2i, i = 1, . . . , N− 1.
For completeness, we recall that a Hirzebruch surfaceFm, m∈ Z is a holomorphic P1-bundle overP1. It has two canonical sections Σ−, Σ+and the homology H2(Fm) is generated by Σ−, Σ+, C, where C is the fiber class. The intersection form is
Σ2−=−m, Σ2+= m, Σ±· C = 1, C2= 0 and there is a relation
Σ+= Σ−+ mC.
The canonical bundle is
KFm =−c1(Fm) =−Σ−− Σ+− 2C,
and ∫
Fm
c2(Fm) = 4.
In addition XN contains two noncompact toric divisors S0, SN determined by x0 = 0 and xN = 0 respectively. The first, S0 is isomorphic to C × P1 and the second, SN, is isomorphic to the total space of the line bundle OP1(−2N − 2).
Note that Si and Si+1 intersect transversely along a (2i,−2i − 2) rational curve Σi, i = 0, . . . , N − 1, which is a common section of both surfaces over P1. All other intersections are empty. Note also that the equations
y1= 0, xi= 0, i = 1, . . . , N
determine a fiber Ciin each divisor Si, a compact rational curve for i = 1, . . . , N−1, and a complex line for i = 0, N . These curve classes satisfy the relations
(2.3) Σi= Σi−1+ 2iCi, i = 1, . . . , N− 1, which follow for example from [80, Prop. 2.9. Ch. V].
The rational Picard group of XN is generated by N divisors classes D1, . . . , DN−1, H, one for each factor of the torus (C×)N. This is so because for eachC× factor we can associate a canonical associated line bundle to the principal torus bundle over the quotient. From the weights of the action on homogeneous coordinates in (2.1) we see that a section of Dican be taken to be xi0xi1−1· · · xi−1, 1≤ i ≤ N − 1. The canonical toric divisors are equivalent to a linear combination of the generators D1, . . . , DN−1, H with coefficients determined by the columns of the charge matrix in (2.1). In particular
(2.4) Si=−
N∑−1 j=1
Ci,jDj, i = 1, . . . , N− 1, SN = DN−1,
where Ci,j is the Cartan matrix of SU (N ) normalized to have +2 on the diagonal.
One can obviously invert these relations, obtaining Di = −∑N−1
i=1 C−1ij Sj, i = 1, . . . , N−1, where the coefficients C−1ij are fractional. Alternatively, relations (2.4)
can be recursively inverted starting with DN−1 = SN. This yields the integral linear relations
(2.5) Di=
N∑−i j=1
jSi+j, i = 1, . . . , N− 1,
which will be used in the construction of an exceptional collection on XN. Note that this equation involves SN, hence is compatible with Di = −∑N−1
i=1 C−1ij Sj. Moreover note the following intersection numbers
(2.6) (Ci· Dj)XN = δij, (Ci· Sj)XN =−Cij i, j = 1, . . . , N− 1.
For the construction of line defects in Section 7.1 it is important to note that each class Di contains a smooth irreducible surface given by
(2.7) [
a1,iy2i1 + a2,iy22i](
xi0xi1−1· · · xi−1)
+ bixi+1x2i+2· · · xNN−i= 0,
with a1,i, a2,i, bi∈ C, i = 1, . . . , N−1, generic coefficients. This follows from the fact that the global holomorphic sections of the line bundlesOX(Si) are homogeneous polynomials in the toric coordinates (x0, . . . , xN, y1, y2) with (C×)N charge vector equal to the xi column of the charge matrix (2.1). Then using equations (2.5) one computes the charges of the sections ofOX(Di), 1≤ i ≤ N −1. Smoothness follows from the observation that the homogeneous toric coordinates in equation (2.7) are naturally divided into two groups, (xk)1≤k≤i−1 and (xl)i+1≤l≤N. According to equation (2.2), no two variables xk, xl with 1 ≤ k ≤ i and i + 2 ≤ xl ≤ N are allowed to vanish simultaneously. Since y1, y2 are also not allowed to vanish simultaneously, a straightforward computation shows that the divisors (2.7) are smooth and irreducible for generic coefficients a1,i, a2,i, bi ∈ C. Abusing notation, the same notation will be used for the divisor classes Di and a generic smooth irreducible representative in each class. The distinction will be clear from the context.
2.1. Exceptional collections and fractional branes. Adopting the definition of [6], a full strong exceptional collection of line bundles on a toric threefold X is a finite set{Lα} of line bundles which generate Db(X) and satisfy
ExtkX(Lα,Lβ) = 0
for all k > 0, and all α, β. Given such a collection the direct sum T = ⊕αLα is a tilting object in the derived category Db(X) as defined in [14, 24, 124]. Then the results of loc. cit. imply that the functor RHom(T , • ) determines an equivalence of the derived category Db(X) with the derived category of modules over the finitely generated algebra EndX(T )op.
Full strong exceptional collections of line bundles on toric Calabi-Yau threefolds can be constructed [92, 21] using the dimer models introduced in [63, 64, 77, 78]. A different construction for the threefolds XN, N≥ 2, exploiting the fibration struc- ture XN → P1 is presented in Appendix A. The resulting exceptional collection consists of the line bundles
(2.8) Li=OXN(Di), Mi=OXN(Di+ H), i = 1, . . . , N,
where Di, i = 1, . . . , N − 1 are the divisor classes given in (2.5) and DN = 0. So LN =OXN. Therefore there is an equivalence of derived categories
(2.9) Db(XN)≃ Db(End(T )op− mod), E7→ RHomXN(T, E),
where T =(
⊕Ni=1Li
)⊕(
⊕Ni=1Mi
), and End(T ) is the endomorphism algebra of T . According to Appendix A, this algebra is isomorphic to the path algebra of the quiver (A.4) with the quadratic relations given in equation(A.5). Reversing the arrows yields the periodic quiverQ below
(2.10) ...
rN
sN
vv
...
qN• cN //
dN //
bN
•pN rN−1
sN−1
ss
aN
qN−1• cdN−1 //
N−1 //
bN−1
OO
•pN−1 aN−1
OO
... ...
qi+1• ci+1 //
di+1 // •pi ri
si
rrqi• ci //
di //
bi
OO
•pi ai
OO
... ...
q2• c2 //
d2 // •p2
r1
s1
rrq1• c1 //
d1 //
b1
OO
•p1 a1
OO
sN rN
...
bN
OO
...
aN
OO
where the vertices pi, qicorrespond to the line bundles Li, Mi, i = 1, . . . , N respec- tively. At the same time, the relations (A.5) are derived from the cubic potential
(2.11) W =
N∑−1 i=1
[ri(aici− ci+1bi) + si(aidi− di+1bi)]
+ rN(aNcN − c1bN) + sN(aNdN − d1bN).
The resulting quiver with potential (Q, W) has a dual interpretation [9, 83, 84], as the Ext1 quiver of a collection of fractional branes (Pi, Qi)1≤i≤N. The latter are objects of Db(XN) corresponding to the simple (Q, W)-modules associated to the vertices (ui, vi)1≤i≤N under the equivalence (2.9). The simple module associated to a particular node is the representation consisting of a dimension 1 vector space assigned to the given node and trivial vector spaces otherwise. They are uniquely determined by the orthogonality conditions3
(2.12) RHomXN(Li, Pj) = δi,jC, RHomXN(Li, Qj) = 0 RHomXN(Mi, Pj) = 0, RHomXN(Mi, Qj) = δi,jC,
1 ≤ i, j ≤ N, where C denotes the one term complex of vector spaces with C in degree zero. As shown in Appendix A, the following collection of objects satisfy conditions (2.12).
(2.13) Pi= Fi[1], Qi= Fi(−H)[2], i = 1, . . . , N− 1, PN = FN, QN = FN(−H)[1]
where
(2.14)
Fi=OSi(−Σi−1), i = 1, . . . , N− 1, FN =OS, S =
N∑−1 i=1
Si. For future reference we note here that
(2.15) ch0(Pi) = 0, ch1(Pi) =−Si, ch2(Pi) =−(i + 1)Ci, χ(Pi) = 0 ch0(Qi) = 0, ch1(Qi) = Si, ch2(Qi) = iCi, χ(Qi) = 0
for 1≤ i ≤ N − 1, respectively
(2.16)
ch0(PN) = 0, ch1(PN) = S, ch2(PN) = Σ0+
∑N i=1
(i + 1)Ci, χ(PN) = 1 ch0(QN) = 0, ch1(QN) =−S
ch2(QN) =−Σ0−
∑N i=1
iCi, χ(QN) =−1,
where Si, 1 ≤ i ≤ N − 1, will also stand for a degree 2 cohomology class via pushforward, and similarly for Ci, Σ0. For completeness recall that the holomorphic
3Here RHomXN( , ) denotes the right derived functor of global HomXN( , ), which assigns to a pair of sheaves E, F the linear space of global sheaf morphisms E→ F . For any pair (E, F ), RHomXN(E, F ) is a finite complex of vector spaces whose cohomology groups are isomorphic to the global extension groups ExtkXN(E, F ). See [72] for abstract definition and properties.
Euler character χ(E) of an object E of Db(XN) with compact support is defined as
(2.17) χ(E) =∑
k∈Z
(−1)kdim HomDb(XN)(OXN, E[k]).
Since E is compactly supported and bounded, this is a finite sum and all vector spaces HomDb(XN)(OXN, E[k]), k ∈ Z, are finite dimensional. For E = F [p], with F a sheaf with compact support and p∈ Z, this definition agrees with the standard definition of the holomorphic Euler character of F up to sign,
(2.18) χ(E) = (−1)pχ(F ) = (−1)p∑
k∈Z
Hk(XN, F ).
Here Hk(XN, F ) are the ˇCech cohomology groups of F . Since F has compact support on XN, the ˇCech cohomology groups are finite dimensional and vanish for k < 0 and k > dim supp(F ). Furthermore note the Riemann-Roch formula
(2.19) χ(F ) =
∫
XN
ch(F )Td(XN).
The quiver Q is then identified with the Ext1-quiver of the collection of frac- tional branes (Pi, Qi)1≤i≤N. The nodes pi, qi correspond to the objects Pi, Qi, i = 1, . . . , N respectively while the arrows between any two nodes are in one-to-one correspondence with basis elements of the Ext1-space between the associated ob- jects. Moreover note that the equivalence (2.9) relates the objects (Pi, Qi) to the simple quiver representations supported respectively at each of the nodes (pi, qi), 1 ≤ i ≤ N. In contrast, the line bundles Li, Mi are related to the projective modules canonically associated to the nodes pi, qi respectively.
The potential (2.11) is related to the A∞-structure on the triangulated subcat- egoryF ⊂ Db(XN) generated by the fractional branes (Pi, Qi)1≤i≤N, as explained below. Consider an object in this category of the form
⊕N i=1
(Vi⊗ Pi)⊕
⊕N i=1
(Wi⊗ Qi),
where Vi, Wi, i = 1, . . . , N are finite dimensional vector spaces. This object is identified by the equivalence (2.9) to a representation ρ of (Q, W) assigning the vector spaces Vi, Wi to the nodes pi, qi, i = 1, . . . , N respectively, and the zero map to all arrows. Physically, this is a collection of fractional branes on XN. The space of open string zero modes between such a collection of fractional branes is isomorphic to the extension group Ext1F(ρ, ρ). The latter is in turn isomorphic to the linear space
(2.20)
Vρ=
⊕N i=1
Hom(Wi, Vi)⊕2⊕
N⊕−1 i=1
(Hom(Vi, Vi+1)⊕ Hom(Wi, Wi+1)⊕ Hom(Vi+1, Wi)⊕2).
Using canonical projective resolutions for simple modules as in Appendix D, one can construct a cyclic A∞ structure on F. The cyclic A∞ structure determines in particular a holomorphic superpotential Wρ on the above extension space as explained in detail in [82, 11, 9, 88]. This is the tree level superpotential in the
