We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two differe
For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $Csubset X$ is an embedded curve and $Dsubset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.
We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, omega, tau)$ with nonempty and compact real part $L={rm Fix}(tau)$. For given $Lambdain (0, +infty]$ and $minNcup{0}$ we show the equivalence of the following two claims: (i) $sharp(Lcapphi^H_1(L))ge m$ for any Hamiltonian function $Hin C_0^infty([0, 1]times M)$ with Hofers norm $|H|<Lambda$; (ii) $sharp {cal P}(H,tau)ge m$ for every $Hin C^infty_0(R/Ztimes M)$ satisfying $H(t,x)=H(-t,tau(x));forall (t,x)inmathbb{R}times M$ and with Hofers norm $|H|<2Lambda$, where ${cal P}(H, tau)$ is the set of all $1$-periodic solutions of $dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=tau(x(t));forall tinR$ (which are also called brake orbits sometimes). Suppose that $(M, omega)$ is geometrical bounded for some $Jin{cal J}(M,omega)$ with $tau^ast J=-J$ and has a rationality index $r_omega>0$ or $r_omega=+infty$. Using Hofers method we prove that if the Hamiltonian $H$ in (ii) above has Hofers norm $|H|<r_omega$ then $sharp(Lcapphi^H_1(L))gesharp {cal P}_0(H,tau)ge {rm Cuplength}_{F}(L)$ for $F=Z_2$, and further for $F=Z$ if $L$ is orientable, where ${cal P}_0(H,tau)$ consists of all contractible solutions in ${cal P}(H,tau)$.
We revisit Atiyah and Botts study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Lius non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled On the Yang-Mills stratification for surfaces) will appear in the Proceedings of the AMS.
We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.
We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety).
Let $(M,g)$ be a closed Riemannian manifold, $L: TMrightarrow mathbb R$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$, we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q_0$ to $Q_1$ and satisfying the conormal boundary conditions. We introduce the Ma~ne critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.
We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].
We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-Kahler case
We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds.
