Search

We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Nahm data and prove a Donaldson-Uhlenbeck-Yau type theorem to relate real and complex formulations. We use these results to construct a canonical Ehresmann connection on the family of non-singular $D_2$ ALF manifolds. In the complex formulation, we exhibit explicit relationships between these $D_2$ ALF manifolds and corresponding $A_1$ ALE manifolds. We conjecture analogous constructions and results for general orbifold quotients of $\mathbb{R}^{4-r} \times T^r$ with $2 \le r \le 4$. The case $r = 4$ produces K3 manifolds as hyperkahler quotients., Comment: 142 pages

We extend our recent study of K3 metrics near the $T^4/Z_2$ orbifold locus to the other torus orbifold loci. In particular, we provide several new constructions of K3 surfaces as hyper-K\"ahler quotients, which yield new formulae for K3 metrics. We then relate these to the construction of arXiv:1810.10540. As a corollary, we derive infinitely many constraints on the (as yet unknown) BPS spectra of the Minahan-Nemeschansky SCFTs with $E_n$ global symmetry. Specifically, we find linear combinations of $E_n$ characters (evaluated at different points) hiding within K3 metrics and we compute their second order Taylor expansions. We also find novel strong relationships between the BPS spectra of these SCFTs, as well as with that of the $SU(2)$ $N_f = 4$ SCFT. Finally, we provide a new derivation of the class S constructions of these SCFTs and state some experimental observations regarding their BPS spectra., Comment: 72 pages

The Attractor Conjecture for Calabi-Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. We provide a family of counterexamples to the Attractor Conjecture in all suitably high, odd dimensions conditional on the Zilber-Pink conjecture., Comment: Comments welcome! 24 pages

We provide an explicit construction of Ricci-flat K3 metrics. It employs the technology of D-geometry, which in the case of interest is equivalent to a hyper-K\"ahler quotient. We relate it to the construction of arXiv:1810.10540, and in particular show that it contains the solution to the BPS state counting problem (that of computing the BPS index of a heterotic little string theory compactified on $T^2$) discussed therein, which is the data needed for this second construction of K3 metrics., Comment: 85 pages. V2: corrected typos

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins' "theorem of the cube" construction of the ${\rm MString}$-orientation of elliptic cohomology., Comment: Revised based on referee suggestions. To appear in Advances in Mathematics

We relate a number of results in the theory of flat surfaces to BPS spectra of a class of 4d $\mathcal{N}=2$ supersymmetric quantum field theories arising from M5 branes wrapped on Riemann surfaces -- $A_1$ class S theories. In particular, we apply classic results of Eskin and Masur, which determine the asymptotic growth of geodesic counts at large length on flat surfaces, as well as more recent progress in the mathematics literature, to determine the large mass asymptotics of the BPS spectra of a wide class of such theories at generic points in the Coulomb branch.

Attractor black holes in type II string compactifications on $K3 \times T^2$ are in correspondence with equivalence classes of binary quadratic forms. The discriminant of the quadratic form governs the black hole entropy, and the count of attractor black holes at a given entropy is given by a class number. Here, we show this tantalizing relationship between attractors and arithmetic can be generalized to a rich family, connecting black holes in supergravity and string models with analogous equivalence classes of more general forms under the action of arithmetic groups. Many of the physical theories involved have played an earlier role in the study of "magical" supergravities, while their mathematical counterparts are directly related to geometry-of-numbers examples in the work of Bhargava et. al. This paper is dedicated to the memory of Peter Freund. The last section is devoted to some of M.G's personal reminiscences of Peter Freund., Comment: 22 pages; v2: minor changes, references added, reminiscences about Peter Freund added by first author; v3: two footnotes, one note and references added. Typos corrected

Certain six-dimensional (1,0) supersymmetric little string theories, when compactified on $T^3$, have moduli spaces of vacua given by smooth K3 surfaces. Using ideas of Gaiotto-Moore-Neitzke, we show that this provides a systematic procedure for determining the Ricci-flat metric on a smooth K3 surface in terms of BPS degeneracies of (compactified) little string theories., Comment: 36 pages. v2: added references. v3: updated joke

We consider asymptotics of certain BPS state counts in M-theory compactified on a K3 surface. Our investigation is parallel to (and was inspired by) recent work in the mathematics literature by Filip, who studied the asymptotic count of special Lagrangian fibrations of a marked K3 surface, with fibers of volume at most $V_*$, in a generic twistor family of K3 surfaces. We provide an alternate proof of Filip's results by adapting tools that Douglas and collaborators have used to count flux vacua and attractor black holes. We similarly relate BPS state counts in 4d ${\cal N}=2$ supersymmetric gauge theories to certain counting problems in billiard dynamics and provide a simple proof of an old result in this field., Comment: 11 pages, 1 figure

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy-theoretic construction of elliptic cohomology., Comment: Significant updates informed by the de Rham model in 1908.02868

We conjecture that the Fourier coefficients of a degree three Siegel form, $1/\sqrt{\chi_{18}}$, count the degeneracy of three-center BPS bound states in type II string theory compactified on $K3 \times T^2$. We provide evidence for our conjecture in the form of consistency with physical considerations of wall-crossing, holographic bounds, and the appearance of suitable counting functions (involving the inverse of the modular discriminant $\Delta$ and the inverse of the Igusa cusp form $\Phi_{10}$) in limits where the count degenerates to involve single-center or two-center objects.

We show that BPS jumping loci -- loci in the moduli space of string compactifications where the number of BPS states jumps in an upper semi-continuous manner -- naturally appear as Fourier coefficients of (vector space-valued) automorphic forms. For the case of $T^2$ compactification, the jumping loci are governed by a modular form studied by Hirzebruch and Zagier, while the jumping loci in K3 compactification appear in a story developed by Oda and Kudla-Millson in arithmetic geometry. We also comment on some curious related automorphy in the physics of black hole attractors and flux vacua., Comment: 22 pages. Comments welcome!

We define a natural counting function for BPS black holes in $K3 \times T^2$ compactification of type II string theory, and observe that it is given by a weight 3/2 mock modular form discovered by Zagier. This hints at tantalizing relations connecting black holes, string theory, and number theory., Comment: 6 pages, essay awarded third prize in the Gravity Research Foundation 2017 essay competition

We consider $M$-theory compactified on $T^4 \times T^2$ and describe the count of spinning $1/8$-BPS states. This refines the classic count of Maldacena-Moore-Strominger in the physics literature and the recent mathematical work of Bryan-Oberdieck-Pandharipande-Yin, which studied reduced Donaldson-Thomas invariants of abelian surfaces and threefolds. As in previous work on $K3 \times T^2$ compactification, we track angular momenta under both the $SU(2)_L$ and $SU(2)_R$ factors in the 5d little group, providing predictions for the relevant motivic curve counts.

We study BPS jumping loci, or the subloci in moduli spaces of supersymmetric string vacua where BPS states come into existence discontinuously. This phenomenon is distinct from wall-crossing. We argue that these loci should be thought of as special cycles in the sense of Noether-Lefschetz loci or special Shimura subvarieties, which are indeed examples of BPS jumping loci for certain string compactifications. We use the Hodge-elliptic genus as an informative tool, suggesting that our work can be extended to understand the jumping behavior of motivic Donaldson-Thomas invariants., Comment: 24 pages. Comments welcome!

We propose that Borcherds' Fake Monster Lie algebra is a broken symmetry of heterotic string theory compactified on $T^7 \times T^2$. As evidence, we study the fully flavored counting function for BPS instantons contributing to a certain loop amplitude. The result is controlled by $\Phi_{12}$, an automorphic form for $O(2, 26, \mathbb{Z})$. The degeneracies it encodes in its Fourier coefficients are graded dimensions of a second-quantized Fock space for this large symmetry algebra. This construction provides a concrete realization of Harvey and Moore's proposed relationship between Generalized Kac-Moody symmetries and supersymmetric string vacua., Comment: Comments welcome!

We present a general conjecture on the divisibility of a certain expression in terms of Kostka numbers and their close variants. This conjecture is closely related to a variant of the period-index problem of noncommutative algebra, with partial implications in both directions. We present a description of the connection between these two problems via Schubert calculus as motivation and evidence for the conjecture before turning to a proof of the conjecture in a family of cases., Comment: 14 pages; comments welcome!

We perform a refined count of BPS states in the compactification of M-theory on $K3 \times T^2$, keeping track of the information provided by both the $SU(2)_L$ and $SU(2)_R$ angular momenta in the $SO(4)$ little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $K3 \times T^2$, simultaneously refining Katz, Klemm, and Pandharipande's motivic Donaldson-Thomas counts on $K3$ and Oberdieck-Pandharipande's Gromov-Witten counts on $K3 \times T^2$. This provides the first full answer for motivic curve counts of a compact Calabi-Yau threefold. Along the way, we develop a Hodge-elliptic genus for Calabi-Yau manifolds -- a new counting function for BPS states that interpolates between the Hodge polynomial and the elliptic genus of a Calabi-Yau., Comment: 21 pages. Comments welcome!

We study the integral Hodge conjecture in complex codimension $2$ and $3$ for approximations to the classifying space of groups of type A. In degree two, we prove a conjecture of Ben Antieau, extending his two counterexamples to a general family of groups. In particular, when reduced to a finite field, these spaces extend Antieau's counterexamples to the integral $\ell$-adic Tate conjecture from the cases $\ell = 2, 3$ to all primes $\ell$., Comment: Comments welcome!

We show that under mild hypotheses on a proper algebraic space $X$, the functors of taking its symmetric powers and its \'{e}tale realisation commute up to weak equivalence. We conclude an effective version of the Dold-Thom theorem for the \'{e}tale site and discuss the stabilisation results for the natural morphisms of \'{e}tale homotopy groups $\pi_k \mathrm{Sym}^n X \to \pi_k \mathrm{Sym}^{n+1} X$ in the context of the Weil conjectures.