Let Fl_n+1 be the variety of complete flags in 𝔸^n+1 and let Ω^2_β(Fl_n+1) be the space of based maps f:ℙ^1→Fl_n+1 in the class f_*[ℙ^1]=β. We show that under a mild positivity condition on β, the class of Ω^2_β(Fl_n+1) in K_0(Var), the Grothendieck group of varieties, is given by [Ω^2_β(Fl_n+1)] = [GL_n×𝔸^a]. The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).

Let 𝒳 be a local orbifold Calabi-Yau threefold whose coarse space X has transverse A_N singularities along a smooth non-compact curve. We define orbifold Gopakumar-Vafa invariants, and we prove they are integers and satisfy a finiteness property. We compute our invariants for local orbifold K3 surfaces, where we prove an orbifold version of the classical Yau-Zaslow formula: we show that for an orbifold K3 surface 𝒮, the genus zero Gopakumar-Vafa invariant of 𝒮 in a class of square 2n is the Euler characteristic of the Hilbert scheme of n+1 points on the singular surface S.

We give a quiver description of the space of based algebraic maps from ℙ^1 to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's “naive algebro-geometric homotopy category” whose topological realization (after specializing to ℂ) recovers the classical homotopy equivalences Ω^2(Sp/U) ≃ BO×ℤ and Ω^2(O/U) ≃ BSp×ℤ.

A banana manifold is a Calabi-Yau threefold fibered by Abelian surfaces whose singular fibers contain banana configurations: three rational curves meeting each other in two points. A nano-manifold is a Calabi-Yau threefold X with very small Hodge numbers: h(1,1)(X) + h(2,1)(X) <= 6. We construct four rigid banana nano-manifolds (X) over tilde (N), N is an element of {5, 6, 8, 9}, each with Hodge numbers given by (h(1,1), h(2,1)) = (4, 0). We compute the Donaldson-Thomas partition function for banana curve classes and show that the associated genus g Gromov-Witten potential is a genus 2 meromorphic Siegel modular form of weight 2g-2 for a certain discrete subgroup P-N* subset of Sp(4)(R). We also compute the weight 4 modular form whose pth Fourier coefficient is given by the trace of the action of Frobenius on H-et(3)((X) over tilde (N), Q(l)) for almost all prime p. We observe that it is the unique weight 4 cusp form on Gamma(0)(N).

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves." A basic example is given by X-ban := Bl(Delta)(Sx(P1) S), the blowup along the diagonal of the fibered product of a generic rational elliptic surface S -> P-1 with itself. In this paper, we give a closed formula for the Donaldson-Thomas partition function of the banana manifold X-ban restricted to the 3-dimensional lattice Gamma of curve classes supported in the fibers of X-ban -> P-1. It is given by Z(Gamma)(X-ban) = Pi(d1,d2,d3 >= 0) Pi(k) (1 - p(k)Q(1)(d1)Q(2)(d2)Q(3)(d3))(-12c(parallel to(d) under bar parallel to,k)), where parallel to(d) under bar parallel to = 2d(1)d(2) + 2d(2)d(3) + 2d(3)d(1) - d(1)(2) - d(2)(2) - d(3)(2) and the coefficients c(a, k) have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely related to the equivariant elliptic genera of Hilb (C-2). In an appendix with S. Pietromonaco, it is shown that the corresponding genus g Gromov-Witten potential F-g is a genus 2 Siegel modular form of weight 2g-2 for g >= 2; namely, it is the Skoruppa-Maass lift of a multiple of an Eisenstein series, specifically 12 vertical bar B-2g vertical bar/2g(2g - 2)!E-2g(tau).

A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves."A basic example is given by X ban := Bl ∆ (S × P 1 S), the blowup along the diagonal of the fibered product of a generic rational elliptic surface S → P 1 with itself.In this paper, we give a closed formula for the Donaldson-Thomas partition function of the banana manifold X ban restricted to the 3-dimensional lattice Γ of curve classes supported in the fibers of X ban → P 1 .It is given by k) ,and the coefficients c(a, k) have a generating function given by an explicit ratio of theta functions.This formula has interesting properties and is closely related to the equivariant elliptic genera of Hilb C 2 .In an appendix with S. Pietromonaco, it is shown that the corresponding genus g Gromov-Witten potential F g is a genus 2 Siegel modular form of weight 2g-2 for g 2; namely, it is the Skoruppa-Maass lift of a multiple of an Eisenstein series, specifically 12|B 2g | 2g(2g -2)! E 2g (τ ) .

We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) X which is equipped with an involution & imath; preserving the holomorphic volume form. We define integers n g,h ( beta ) which give a virtual count of the number of genus g curves C on X , in the class beta E H 2 ( X ), which are invariant under & imath; , and whose quotient C/& imath; has genus h . We give two definitions of n g,h ( beta ) which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our invariants and give evidence for our conjecture in several cases. In particular, we compute our invariants when X = S x C , where S is an Abelian surface with & imath; ( a ) = - a or a K 3 surface with a symplectic involution (a Nikulin K 3 surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms.

A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order $N=2$ in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal S}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions $(2,A,B)$ and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in ${\cal S}_{LME}/K$ correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup $S \subset \operatorname{SL}(d_1, \mathbb{C}) \times \cdots \times \operatorname{SL}(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.) 24 (2) (2018), 1527–1548, arXiv:math/1603.05271 ], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

We give a general overview of the Donaldson-Thomas invariants of elliptic fibrations and their relation to Jacobi forms. We then focus on the specific case of where the fibration is S × E, the product of a K3 surface and an elliptic curve. Oberdieck and Pandharipande conjectured (Oberdieck and Pandharipande, K3 Surfaces and Their Moduli, Progress in Mathematics, vol. 315 (Birkhäuser/Springer, Cham, 2016), pp. 245–278, arXiv:math/1411.1514) that the partition function of the Gromov-Witten/Donaldson-Thomas invariants of S × E is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in S of square 2h − 2, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight − 10 and index h − 1. We calculate the Donaldson-Thomas partition function for primitive classes of square − 2 and of square 0, proving strong evidence for their conjecture. Our computation uses reduced Donaldson-Thomas invariants which are defined as the (Behrend function weighted) Euler characteristics of the quotient of the Hilbert scheme of curves in S × E by the action of E. Our technique is a mixture of motivic and toric methods (developed with Kool in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369)) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute both versions of the invariants: unweighted and Behrend function weighted Euler characteristics. Our Behrend function weighted computation requires us to assume Conjecture 18 in (Bryan and Kool, Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex (2016), arXiv:math/1608.07369).

We study a question which has natural interpretations both in quantum mechanics and in geometry. Let V_1,⋯ , V_n be complex vector spaces of dimension d_1,… ,d_n and let G= SL_d_1×⋯×SL_d_n . Geometrically, we ask: Given (d_1,… ,d_n) , when is the geometric invariant theory quotient ℙ(V_1⊗⋯⊗ V_n)//G non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space V_1⊗⋯⊗ V_n has a locally maximally entangled state, i.e., a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if R(d_1,⋯ ,d_n)⩾ 0 where R(d_1,⋯ ,d_n) = ∏ _id_i +∑ _k=1^n (-1)^k∑ _1⩽ i_1<⋯ <i_k⩽ n( (d_i_1,⋯ ,d_i_k) ) ^2. We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson–Thomas theory of toric Calabi–Yau threefolds, or the open string partition function of \({\mathbb {C}}^{3}\). We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson–Thomas invariants of elliptically fibered Calabi–Yau threefolds at the end of the paper.

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasi-modular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Gottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Gottsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.

We construct curve counting invariants for a Calabi-Yau threefold $Y$ equipped with a dominant birational morphism $\pi:Y \to X$. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when $\pi:Y\to Y$ is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in the case when $Y$ is a crepant resolution of $X$, the coarse space of a Calabi-Yau orbifold $\mathcal{X}$ satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold $\mathcal{X}$. Our methods include defining a new notion of stability for sheaves which depends on the morphism $\pi $. Our notion generalizes slope stability which is recovered in the case where $\pi $ is the identity on $Y$. Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence via the Hall algebra and Joyce's integration map.

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic Donaldson-Thomas theory.

Since the spectacular advances in quantum field theory and string theory of the early 90s, BPS states have been a central object of study in theoretical physics with deep mathematical implications. In a nutshell, BPS states are quantum states classified by short supersymmetry multiplets in supersymmetric theories. Such multiplets exhibit remarkable stability properties under quantum fluctuations, providing a reliable interpolation tool between strong and weak coupling in many theories of physical interest. This special property leads to striking results in string theory compactifications. A prominent example is the relation between the low energy spectrum of Seiberg-Witten theories and supersymmetric D-brane states in geometric engineering constructions, as pointed out by Kachru and Vafa [28], and also Katz, Klemm and Vafa [29]. This relation has led in particular to the connection between D-brane bound states and instanton sums discovered by Nekrasov [43]. From a mathematical point of view, supersymmetric D-brane states in string theory compactifications are loosely speaking cohomology classes on moduli spaces of stable bundles on Calabi-Yau threefolds. More generally, it has become increasingly clear in the past years due to work of Douglas and Aspinwall [3] that a complete mathematical construction must involve moduli spaces of stable objects in Calabi-Yau derived categories. Virtual numbers of such objects, Donaldson–Thomas (DT) invariants [17, 49], are identified with BPS indices, which count BPS states with a sign determined by their intrinsic spin quantum number. The special properties of the BPS spectrum of the underlying physical theories lead to deep predictions concerning such invariants, which very often crystalize into beautiful mathematical results. A particular long range mathematical program prompted by the physics connection is concerned with motivic and categorical generalizations of DT invariants. A first step in this direction is focused on refined DT invariants, a construction loosely analogous to promoting the Euler characteristic of a topological space to the Poincaré polynomial, or refining the Jones polynomial of a knot to a multi-variable polynomial coming from Khovanov–Rozansky theory. As pointed out by Gukov, Schwartz and Vafa [23] , and also more recently by Gaiotto, Moore and Nietzke [19], such constructions are motivated by a refinement of the BPS index in physics theories, which is promoted to a generating function which counts BPS states keeping track of their space-time spin quantum number. In the toric context, a refinement of topological string amplitudes was provided by Iqbal, Kozcaz and Vafa [27], employing the topological vertex formalism of Aganagic-KlemmMarino-Vafa [1] for topological string amplitudes. The resulting refined vertex was related to refined DT invariants by Dimofte and Gukov [16].

Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key example is affine three-space, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef-Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives of the Hilbert schemes of affine three-space via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Gottsche's formula for the Poincare polynomials of the Hilbert schemes of points on surfaces.