Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories by Artan Sheshmani* Abstract. This article provides the summary of 1. Introduction [GSY17a] and [GSY17b] where the authors studied the In recent years, there has been extensive math- enumerative geometry of “nested Hilbert schemes” ematical progress in enumerative geometry of sur- of points and curves on algebraic surfaces and faces deeply related to physical structures, e.g. their connections to threefold theories, and in around Gopakumar-Vafa invariants (GV); Gromov- particular relevant Donaldson-Thomas, Vafa-Witten Witten (GW), Donaldson-Thomas (DT), as well as and Seiberg-Witten theories. Pandharipande-Thomas (PT) invariants of surfaces; and their “motivic lifts”. There is also a tremen- dous energy in the study of mirror symmetry of sur- Contents faces from the mathematics side, especially Homo- logical Mirror Symmetry. On the other hand, phys- 1 Introduction . 25 ical dualities in Gauge and String theory, such as Acknowledgment . 26 Montonen-Olive duality and heterotic/Type II du- 2 Nested Hilbert Schemes on Surfaces . 26 ality have also been a rich source of spectacular 2.1 Special Cases . 27 predictions about enumerative geometry of moduli 2.2 Nested Hilbert Scheme of Points . 27 spaces on surfaces. For instance an extensive re- 3 Nested Hilbert Schemes and DT Theory of search activity carried out during the past years was Local Surfaces . 28 to prove the modularity properties of GW or DT in- References . 30 variants as suggested by the heterotic/Type II dual- ity. The first prediction of this type, the Yau-Zaslow conjecture [YZ96], was proven by Klemm-Maulik- Pandharipande-Scheidegger [KMPS10]. Further recent developments in this particularly fruitful direction * Center for Mathematical Sciences and Applications, Har- include: Pandharipande-Thomas proof [PT16] of the vard University, Department of Mathematics, 20 Garden Street, Room 207, Cambridge, MA, 02139 Katz-Klemm-Vafa conjecture [KKV99] for K3 sur- Centre for Quantum Geometry of Moduli Spaces, Aarhus faces, Maulik-Pandharipande proof of modularity of University, Department of Mathematics, Ny Munkegade 118, GW invariants for K3 fibered threefolds [MP13], building 1530, 319, 8000 Aarhus C, Denmark Gholampour-Sheshmani-Toda proof of modularity of National Research University Higher School of Economics, PT and Gholampour-Sheshmani proof of modular- Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow, Russia, 119048 ity of DT invariants of stable sheaves on K3 fibra- E-mail: [email protected] tions [GST17, GS18] (moreover, the generalizations DECEMBER 2019 NOTICES OF THE ICCM 25 of the latter in [BCDDQS16]), and finally Gholampur- Their main exciting realization was when L = KS, the Sheshmani-Thomas [GS14] proof of modular prop- canonical bundle of S. In this case the above DT in- erty of the counting of curves on surfaces deforming variants of X recover some well known invariants freely (in a fixed linear system) in ambient CY three- in physics, the Vafa-Witten (VW) invariants [VW94], folds. which are known to have modularity property, fol- SU(2)-Seiberg-Witten (SW) / DT correspondence. lowing the work of Vafa-Witten [VW94] (and verified The introduction of other versions of gauge theo- by Tanaka-Thomas [TT17), TT18(2)]). On the other ries in dimensions 6 started numerous exciting de- hand the authors also showed that in some cases the velopments in physical mathematics and mathemat- nested Hilbert scheme invariants also have modular ical physics involving enumerative geometry, mirror property [GSY17a, Theorem 7]. Therefore, we were symmetry, and related physics of 4 dimensional man- able to show that in some cases, the SW invariants ifolds, realized as a complex two dimensional sub- can be described in terms of modular forms. variety which could exhibit deformations inside am- bient higher dimensional target varieties, such as CY threefolds. The SW/DT correspondence is one of Acknowledgment such platforms to find interesting structural sym- The author was partially supported by NSF metries between theory of surfaces and theory of DMS-1607871, NSF DMS-1306313 and Laboratory of threefolds: having fixed a spin structure on a com- Mirror Symmetry NRU HSE, RF Government grant, ag. plex surface S the SW invariant of S roughly counts No 14.641.31.0001. The author would like to further with sign the number of points in the parametriz- sincerely thank the Center for Mathematical Sciences ing space (the moduli space) of solutions to SW and Applications at Harvard University, the center equations defined on S [D96, Section 1]. The fo- for Quantum Geometry of Moduli Spaces at Aarhus cus of SW theory in physics is the study of mod- University, and the Laboratory of Mirror Symmetry in uli space of vacua in N = 2, D = 4 super Yang-Mills Higher School of Economics, Russian federation, for theory, and in particular certain dualities of the the great help and support. theory such as electric-magnetic duality (Montonen- Olive duality). In [GLSY17] Gukov-Liu-Sheshmani-Yau conjectured a relation between SW invariants (asso- 2. Nested Hilbert Schemes on ciate to SU(2) gauge invariant theory) of a projec- Surfaces tive surface S and DT invariants of a noncompact Hilbert schemes of points and curves on a non- threefold X obtained by total space of a line bun- singular surface S have been vastly studied. Their dle L on S. In a later work [GSY17a, GSY17b], which rich geometric structures have proved to have many will be elaborated below, Gholampour-Sheshmani- applications in mathematics and physics (see [N99] Yau proved the conjecture in [GLSY17, Equation for a survey). The current article is a survey of the 56]. articles [GSY17a] and [GSY17b], where the authors The key object which bridges the geometry of studied the enumerative geometry of “nested Hilbert complex surface to the ambient noncompact com- schemes” of points and curves on algebraic surfaces. plex threefold is the “nested Hilbert scheme of sur- Given the sequence face”. This space parametrizes a nested chain of configurations of curves and points in the surface. n := n1;n2;:::;nr;r ≥ 1 Let us assume that S is a projective simply con- nected complex surface and let L be a line bundle of nonnegative integers, and β := β1;:::;βr−1, a se- on S. In [GSY17a], Gholampour-Sheshmani-Yau an- 2 quence of classes in H (S;Z) such that βi ≥ 0, we de- alyzed the moduli space of stable compactly sup- note the corresponding “nested” Hilbert scheme by ported sheaves of modules on the noncompact three- [n] [n] Sβ . A closed point of Sβ corresponds to fold X. In physics terms these are D4-D2-D0 branes wrapping the zero section of X (i.e. sheaves are sup- (Z1;Z2;:::;Zr); (C1;:::;Cr−1) ported on S or a fat neighborhood of S in X). The au- thors showed in [GSY17a] that the DT invariants of where Zi ⊂ S is a 0-dimensional subscheme of length X satisfy an equation in terms of SW invariants of S ni, and Ci ⊂ S is a divisor with [Ci] = βi, and Zi+1 is a and certain correction terms governed by invariants subscheme of Zi [Ci for any i < r, or equivalently of nested Hilbert schemes: (1) IZi (−Ci) ⊂ IZi+1 : DT invariants of X = (SW invariants of S) · (Combinatorial coefficients) + Invariants of nested In [GSY17a] and [GSY17b], in order to define in- Hilbert scheme of S variants for the nested Hilbert schemes (see [GSY17a, 26 NOTICES OF THE ICCM VOLUME 7,NUMBER 2 [n] Definitions 2.13, 2.14]), the authors constructed a vir- where ωS is the rank n tautological vector bundle [n] vir [n] 1 tual fundamental class [Sβ ] and then integrate ap- over S associated to the canonical bundle ωS of S. propriate cohomology classes against it. More pre- 3. If β = 0 and n = n2 = n1 − 1, then it is known that [n+1;n] ∼ [n] [n] cisely, they constructed a natural perfect obstruction Sβ=0 = P(I ) is nonsingular, where I is the uni- [n] [n] theory over Sβ . This is done by studying the defor- versal ideal sheaf over S × S [L99, Section 1.2]. mation/obstruction theory of the maps of coherent Then, sheaves given by the natural inclusions (1) following [S[n+1;n]]vir = −[S[n+1;n]] \ c (O (1) ): Illusie. It turned out that this construction in partic- β=0 β=0 1 P ωS ular provides a uniform way of studying all known 4. If n = n = 0 and β 6= 0, then S[0;0] is the Hilbert obstruction theories of the Hilbert schemes of points 1 2 β [0;0] vir and curves, as well as the stable pair moduli spaces scheme of divisors in class β, and [Sβ ] coincides on S. The first main result of [GSY17a] is as fol- with virtual cycle used to define Poincaré invariants lows (see also [GSY17a, Proposition 2.5 and Corollary in [DKO07]. [0;n2] 2.6]): 5. If n1 = 0 and β 6= 0, then Sβ is the relative Hilbert scheme of points on the universal divisor over S[0;0], Theorem 1 ([GSY17a, Theorem 1]). Let S be a nonsin- β which as shown in [PT10], is a moduli space of sta- gular projective surface over C and ωS be its canonical [0;n ] [n] ble pairs and [S 2 ]vir is the same as the virtual fun- bundle. The nested Hilbert scheme S with r ≥ 2 car- β β damental class constructed in [KT14] in the context ries a natural virtual fundamental class of stable pair theory. If Pg(S) = 0 this class was used r−1 in [KT14] to define stable pair invariants. [n] vir [n] 1 [Sβ ] 2 Ad(Sβ ); d = n1 + nr + ∑ βi · (βi − c1(ωS)): 2 i=1 In certain cases, the authors constructed a re- [n1;n2] duced virtual fundamental class for Sβ by reducing the perfect obstruction theory leading to Theorem 1 2.1 Special Cases ([GSY17a, Propositions 2.10, 2.12]): In the simplest special case, i.e.