Non-self-dual Yang-Mills connections with quadrupole symmetry

We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.Research partially supported by NSF Grant DMS-8806731Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918     

Holonomy and Skyrme's Model

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L²_loc connection; the local developing maps for such connections need not be continuous.The first author was partially supported by NSF grant DMS-0204651.The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero. The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248. The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.     

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves. Research of the first author was partially supported by NSF grant DMS-0204824 Research of the second author was partially supported by NSF grant DMS-0505835     

On algebraic equations satisfied by hypergeometric correlators in WZW models. I

It is proven that integral expressions for conformal correlators insl(2) WZW model found in [SV] satisfy certain natural algebraic equations. This implies that the above integrals really take their values in spaces of conformal blocks.The second author was supported in part by the NSF grant DMS-9202280. The third author was supported in part by the NSF grant DMS-9203939     

A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models

We study families of dependent site percolation models on the triangular lattice and hexagonal lattice that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of Aizenman-Burchard [3]) as independent site percolation on .The work was conducted while this author was at Department of Physics, New York University, New York, NY 10003, USA. Research partially supported by the U.S. NSF under grants DMS-98-02310 and DMS-01-02587.Research partially supported by the U.S. NSF under grants DMS-98-03267 and DMS-01-04278.Research partially supported by FAPERJ grant E-26/151.905/2000 and CNPq.     

Frobenius Modules and Hodge Asymptotics

We exhibit a direct correspondence between the potential defining the H^1,1 small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules. E. Cattani was partially supported by NSF Grant DMS-0099707     

On the Averages of Characteristic Polynomials From Classical Groups

We provide an elementary and self-contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to Weyl and Littlewood. The first author was supported in part by the NSF grant FRG DMS-0354662. The second author was supported in part by the NSF postdoctoral fellowship and by the NSF grant DMS-0501245.     

Grad ø perturbations of massless Gaussian fields

We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.Research partially supported by NSF Grant DMS-8802912Research partially supported by NSF Grant DMS-8601978 and DMS-8806731     

Rotation Sets of Billiards with One Obstacle

We investigate the rotation sets of billiards on the m-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.The first author was partially supported by NSF grant DMS 0456748.The second author was partially supported by NSF grant DMS 0456526.The third author was partially supported by NSF grant DMS 0457168.     

Derivation of the Euler Equations from Quantum Dynamics

We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian.©2003 B. Nachtergaele and H.-T. Yau. This article may be reproduced in its entirety for non-commercial purposes.Research partially supported by NSF # DMS-0070774.Research partially supported by NSF # DMS-0072098, the Veblen Fund from the Institute for Advanced Study and a Fellowship from the MacArthur Foundation.     

Classification of singular Sobolev connections by their holonomy

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.Research partially supported by NSF Grant DMS-8701813Research partially supported by NSF Grant INT-8511481     

Spin Gromov-Witten Invariants

We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r^th Gelfand-Dickey hierarchy (or, equivalently, the A_r−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397     

Computer calculation of Witten's 3-manifold invariant

Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.The first author is supported by NSF grant DMS-8805684, an Alfred P. Sloan Research Fellowship, and a Presidential Young Investigators award. The second author is supported by NSF grant DMS-8902153     

A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation

We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in hyperbolic coordinates to the study of an ODE. Similar arguments extend to higher dimensions and other long range type nonlinear problems. Mathematics Subject Classifications (2000): 35L15, 74J30, 76B15 ★ Part of this work was done while H.L. was a Member of the Institute for Advanced Study, Princeton, supported by the NSF grant DMS-0111298 to the Institute. H.L. was also partially supported by the NSF Grant DMS-0200226. † Also a member of the Institute of Advanced Study, Princeton. Supported in part by NSF grant DMS-0100490.     

On a Penrose Inequality with Charge

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfieswhere m is the total mass, is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality for charged black holes.The research of the first author was supported in part by NSF Grant DMS-0205545.The research of the second author was supported in part by NSF Grant DMS-0222387.     

Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S². It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given. The first author was partially supported by NSF grant DMS-0204651 The second author was partially supported by EPSRC grant GR/R66982/01     

Topological Calculation of the Phase of the Determinant of a Non Self-Adjoint Elliptic Operator

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.The first author was partially supported by the Alfred P. Sloan foundation.The second author was partially supported by the NSF grant DMS-0204421.