Scholars@Duke
Mark A. Stern

Mark A. Stern

Professor of Mathematics
Mathematics
stern@math.duke.edu
Box 90320, Durham, NC 27708-0320
116 Physics Bldg, Durham, NC 27708
Office hours Monday: 2-3,  Tuesday : 10-11  

Selected Publications

ON THE BETTI NUMBERS OF FINITE VOLUME REAL- AND COMPLEX-HYPERBOLIC MANIFOLDS

Journal Article Journal of Differential Geometry · June 1, 2025 We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS22]. We also provide effective upper bounds for Bet ... Full text Cite

Harmonic Forms, Price Inequalities, and Benjamini–Schramm Convergence

Journal Article Journal of Geometric Analysis · January 1, 2025 We study Betti numbers of sequences of Riemannian manifolds which Benjamini–Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature ... Full text Cite

Lp-cohomology and the geometry of p-harmonic forms

Chapter · January 1, 2025 In this note we describe basic geometric properties of p-harmonic forms and p-coclosed forms and use them to reprove vanishing theorems of Pansu and new injectivity theorems for the Lp-cohomology of simply connected, pinched negatively curved ma ... Full text Cite

Instantons on multi-Taub-NUT Spaces II: Bow Construction

Journal Article Journal of Differential Geometry · June 30, 2024 Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperk\"ahler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of it ... Full text Open Access Link to item Cite

Price inequalities and Betti number growth on manifolds without conjugate points

Journal Article Communications in Analysis and Geometry · January 1, 2022 We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we ... Full text Cite

Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem

Journal Article Journal of Differential Geometry · December 9, 2021 We study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is the first in a series of papers prov ... Open Access Link to item Cite

Geometry of the Smallest 1-form Laplacian Eigenvalue on Hyperbolic Manifolds

Journal Article Geometric and Functional Analysis · December 1, 2018 We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equi ... Full text Open Access Cite

Asymptotic Hodge Theory of Vector Bundles

Journal Article Communications in Analysis and Geometry · December 1, 2015 Cite

$C^{\infty}$ Stability, Canonical Maps, and Discrete Dynamics

Journal Article · October 31, 2014 We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle $E$. We examine the relation between the distance of the chern classes of $E$ from the $(p,p)$ axis of the Hodge diamond and singularity ... Link to item Cite

Witten Spinors on Nonspin Manifolds

Journal Article Communications in Mathematical Physics · 2013 Motivated by Witten's spinor proof of the positive mass theorem, we analyze asymptotically constant harmonic spinors on complete asymptotically flat nonspin manifolds with nonnegative scalar curvature. © 2013 Springer-Verlag Berlin Heidelberg. ... Full text Cite

Target spaces from chiral gauge theories

Journal Article Journal of High Energy Physics · 2013 Chiral gauge theories in two dimensions with (0, 2) supersymmetry are central in the study of string compactifications. Remarkably little is known about generic (0, 2) theories. We consider theories with branches on which multiplets with a net gauge anomal ... Full text Cite

Geometry of Stable Yang Mills Connections

Chapter · 2012 This volume covers some of the most recent and important developments in geometry and theoretical physics today. ... Cite

Novel Branches of (0,2) Theories

Journal Article JHEP · 2012 We show that recently proposed linear sigma models with torsion can be obtained from unconventional branches of conventional gauge theories. This observation puts models with log interactions on firm footing. If non-anomalous multiplets are integrated out, ... Link to item Cite

Novel branches of (0, 2) theories

Journal Article Journal of High Energy Physics · January 1, 2012 We show that recently proposed linear sigma models with torsion can be obtained from unconventional branches of conventional gauge theories. This observation puts models with log interactions on firm footing. If non-anomalous multiplets are integrated out, ... Full text Cite

Geometry of minimal energy Yang-Mills connections

Journal Article Journal of Differential Geometry · 2010 We prove that energy minimizing Yang-Mills connections on compact homogeneous 4-manifolds are either instantons or split into a sum of instantons on passage to the adjoint bundle. We prove related results for Calabi-Yau 3-folds and for 3-dimensional manifo ... Cite

Spectral asymmetry for manifolds of special holonomy

Journal Article · February 12, 2009 We define a measure of spectral asymmetry for G_2 and Spin(7) manifolds. We show that this invariant can be computed in terms of characteristic classes and the covariant constant form defining the G_2 or Spin(7) structure. ... Link to item Cite

Fixed point theorems from a de Rham perspective

Journal Article Asian Journal of Mathematics · January 1, 2009 Full text Cite

B FIELDS FROM A LUDDITE PERSPECTIVE

Journal Article Quantum Theory and Symmetries · October 2004 Full text Cite

Mechanical D Branes and B Fields

Journal Article · October 2, 2003 We represent B fields and higher p-form potentials on a manifold M as connections on affine bundles over M. We realize D branes on M as special submanifolds of these affine bundles. We check the physical relevance of this representation by showing that the ... Link to item Cite

Non-commutativity and Supersymmetry

Journal Article Journal of High Energy Physics · March 6, 2002 Full text Cite

I. Non-commutativity and supersymmetry

Journal Article Journal of High Energy Physics · March 1, 2002 We study the extent to which the gauge symmetry of abelian Yang-Mills can be deformed under two conditions: first, that the deformation depend on a two-form scale. Second, that the deformation preserve supersymmetry. We show that (up to a single parameter) ... Cite

Pure hodge structure on the L2-cohomology of varieties with isolated singularities

Journal Article Journal fur die Reine und Angewandte Mathematik · 2001 Cite

Spinning bound states of two and three black holes

Journal Article Journal of High Energy Physics · January 1, 2001 Bound states of BPS particles in five-dimensional N = 2 supergravity are counted by a topological index. We compute this bound state index exactly for two and three black holes as a function of the SU(2)L angular momentum. The required regulator ... Full text Cite

Counting Yang-Mills dyons with index theorems

Journal Article Physical Review D - Particles, Fields, Gravitation and Cosmology · December 15, 2000 We count the supersymmetric bound states of many distinct BPS monopoles in N=4 Yang-Mills theories and in pure N=2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The ... Open Access Cite

The structure of the D0-D4 bound state

Journal Article Nuclear Physics B · July 3, 2000 We derive a set of equations for the wavefunction describing the marginal bound state of a single D0-brane with a single D4-brane. These are equations determining the vacuum of an N=8 Abelian gauge theory with a charged hypermultiplet. We then solve these ... Full text Cite

Invariance theorems for supersymmetric Yang-Mills theories

Journal Article Advances in Theoretical and Mathematical Physics · January 1, 2000 We consider quantum mechanical Yang-Mills theories with eight supercharges and a Spin(5) × SU(2)R flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a ... Full text Open Access Cite

Summing up instantons in three-dimensional Yang-Mills theories

Journal Article Advances in Theoretical and Mathematical Physics · January 1, 1999 We show that the four derivative terms in the effective action of three-dimensional N=8 Yang-Mills theory are determined by supersymmetry. These terms receive both perturbative and non-perturbative corrections. Using our technique for constraining the effe ... Full text Cite

Constraints from extended supersymmetry in quantum mechanics

Journal Article Nuclear Physics B · November 23, 1998 We consider quantum mechanical gauge theories with sixteen supersymmetries. The Hamiltonians or Lagrangians characterizing these theories can contain higher derivative terms. In the operator approach, we show that the free theory is essentially the unique ... Full text Cite

D-brane bound states redux

Journal Article Communications in Mathematical Physics · 1998 We study the existence of D-brane bound states at threshold in Type II string theories. In a number of situations, we can reduce the question of existence to quadrature, and the study of a particular limit of the propagator for the system of D-branes. This ... Cite

A comment on the spectrum of H-monopoles

Journal Article Physics Letters Section B Nuclear Elementary Particle and High Energy Physics · April 10, 1997 We consider the question of ground states in the supersymmetric system that arises in the search for the missing H-monopole states. By studying the effective theory near certain singularities in the five-brane moduli space, we find substantial evidence for ... Full text Cite

Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories

Journal Article Nuclear Physics Section B · December 25, 1995 We study the existence of monopole bound states saturating the BPS bound in N = 2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using ... Full text Open Access Cite

Lefschetz formulae for arithmetic varieties

Journal Article Inventiones Mathematicae · December 1, 1994 Full text Cite

Index theory for certain complete kähler manifolds

Journal Article Journal of Differential Geometry · January 1, 1993 Full text Cite

L2 -∂-cohomology of complex projective varieties

Journal Article Journal of the American Mathematical Society · January 1, 1991 Full text Cite

Eta invariants and Hermitian locally symmetric spaces

Journal Article Journal of Differential Geometry · January 1, 1990 Full text Cite

L²-cohomology of arithmetic varieties

Journal Article Annals of Mathematics · 1990 Link to item Cite

L2-index theorems on locally symmetric spaces

Journal Article Inventiones Mathematicae · June 1, 1989 Full text Cite

L²-cohomology of arithmetic varieties

Journal Article Proc Natl Acad Sci U.S.A. · August 1987 The L₂-cohomology of arithmetic quotients of bounded symmetric domains is studied. We establish the conjecture of Zucker equating the L₂-cohomology of these spaces to the intersection cohomology of their Baily-Borel compactifications. ... Link to item Cite