String Topology, J-holomorphic Curves, and Symplectic Geometry

Differential geometry is the study of multi-dimensional curved spaces called "manifolds". The surfaces of a sphere or a torus (doughnut) are examples of 2-dimensional manifolds. Often one puts extra geometric structures on manifolds. "Symplectic manifolds" are manifolds with a "symplectic structure". They are part of the mathematical foundations of mechanics -- the basic physics of how objects move -- and quantum mechanics.

If M is a symplectic manifold, a "Lagrangian submanifold" N in M is a manifold N inside M, with half the dimension of M, compatible with the symplectic structure in a certain way. They are central objects in symplectic geometry.

Topology studies the "shape" of spaces including manifolds, focussing on qualitative properties that are unchanged by stretching or bending the space. The "number of holes" (genus) of a 2-dimensional manifold is a topological invariant. The sphere has no holes, and the torus has one hole, so they are different as topological spaces.

One important topological invariant of a space M is its "homology" H(M). Introduced by Henri Poincare in the 19th century, H(M) is an algebraic object which measures things like the "number of holes" in M.

If M is a space, the "loop space" LM is the space of all loops (circles) inside M. For example, if M is the 3-dimensional space we live in, then (embedded) loops are knots. Loop spaces are important in topology, and also String Theory in Physics, which models elementary particles not as points but as tiny "loops of string".

In 1999, Chas and Sullivan discovered that if M is a manifold, then the homology H(LM) of the loop space LM carries some extra algebraic structures coming from intersections of families of loops in M with each other -- basically, H(M) has a multiplication on it. This was unexpected and exciting, and has grown into a subject called "String Topology".

The homology H(M) is defined using a more basic object C(M) called a "chain complex". There are many different ways to define a chain complex C(M), but they all compute the same homology H(M). One difficulty in String Topology is that, up to now, there is no nice way to define the String Topology multiplication on a chain complex C(LM) computing H(LM) -- it is only defined directly on homology H(LM).

String topologists believe that if one could define the operations on the level of chains C(LM), then deeper algebraic structures in Strong Topology would be revealed, leading to the definition of a Topological Conformal Field Theory (TCFT, a mathematical object coming out of quantum physics) from LM.

Our first project is to define a chain complex C(LM) which computes H(LM), upon which we can define a version of the String Topology multiplication on chains, and so construct (at least the genus 0 part of) the TCFT.

We will do this using a new homology theory called "Kuranishi homology" recently defined by the PI. Kuranishi homology has the property that problems to do with "transversality" usually disappear. Since the difficulty of defining string topology operations on chains is mostly about transversality, Kuranishi homology is a good choice.

Now let M be a symplectic manifold, and N a Lagrangian submanifold. A major tool in symplectic geometry is the study of "J-holomorphic curves", 2-dimensional manifolds C in M whose boundary dC is a 1-dimensional manifold (loop) in N. So there is a natural map from families of J-holomorphic curves to the loop space LN, mapping C to dC.

Our second project is to prove some conjectures by Fukaya. These say that using the families of J-holomorphic curves in (M,N), we can define chains in C(LN) satisfying equations involving the chain-level String Topology operations defined in the first project.

By combining this with facts about the topology of LN, we expect to prove important new restrictions on the topology of Lagrangian submanifolds in simple symplectic manifolds, such as flat spaces and projective spaces.

The immediate beneficiaries of this research will be Mathematicians working in String Topology, and Mathematicians working in Symplectic Geometry. There will be longer term benefits to Mathematicians in the area of Symplectic Geometry known as Symplectic Field Theory, and to String Theorists in Theoretical Physics interested in open strings with boundary in Lagrangians (A-branes).

String Topologists will benefit from the research by gaining a new chain model KC_*(LN) for the homology of the loop space LN over Q, on which string topology operations are defined at the chain level without transversality assumptions, satisfy the expected identities on the nose, and which is simple to use, and close to the intuitive notion of 'chains on loop space'. This should lead to a better understanding of string topology, and further advances. Mathematicians and Physicists interested in Quantum Field Theories will also gain new families of examples of TCFTs.

Symplectic Geometers will benefit from the research through the proof of exciting conjectures by Fukaya which place new restrictions on the topology of Lagrangians in fundamental examples of symplectic manifolds, such as flat space C^n and projective manifolds CP^n. It is likely that these methods can be developed further, leading to more restrictions and a better understanding of the allowed topologies for Lagrangian submanifolds.

Symplectic Field Theorists will benefit because our methods should lead to new perspectives on the relation between the String Topology of a manifold N and the Symplectic Field Theory of its cotangent bundle T^*N. String Theorists will benefit because our results -- both the construction of the TCFT in the String Topology part, and the material on chains on loop space of a Lagrangian in the Symplectic Geometry part -- should have meaning in String Theory, and lead to a more sophisticated understanding of open strings.

The beneficiaries will be made aware of the research by the posting of preprints on the arXiv and publication in journals, by talks by the PI and PDRA at seminars in UK universities and at national and international conferences, and through discussions between the PI, PDRA and their contacts in the String Topology and Symplectic Geometry communities.

The PI and PDRA will be undertaking the impact activities, which will primarily be writing papers, both research and expository, and giving seminar and conference talks. The PI has many years experience in these activities. He will mentor the PDRA in academic writing, through commenting on drafts, and in giving talks, and will where possible arrange invitations for them to give seminars, for instance in the Oxford Algebraic and Symplectic Geometry seminar, co-organized by the PI, and to attend relevant conferences. In this way, the PDRA will increase his abilities in technical writing and in giving presentations, both important transferrable skills.