# Core Research Area SM: String topology and mirror symmetry

### Leading Researchers

String topology and mirror symmetry are both directly inspired by developments in string theory, but are not typically mentioned together in one sentence. The topics in this core research area bring these subjects together in studies involving holomorphic curves and the algebraic structures arising from them.

String topology, introduced by Chas and Sullivan [CS1], [CS2]
is an interpretation of open-closed string field theory in the language of algebraic
topology (see also [OL] for background). The most
satisfactory treatments to date identify the (co)homology of the free loop space with
some version of Hochschild (co)homology, where the set of possible operations is now
essentially understood [Wa]. By far the most important
of these operations for symplectic applications are the loop product and the loop and
string brackets, including their homotopy versions. The basic link between string
topology and symplectic topology comes via the study of holomorphic curves with boundary
on a Lagrangian submanifold. Noticing that the compactification of such moduli spaces
can be described in terms of string topology of this submanifold, Fukaya [Fu2] has outlined a proof that any prime, orientable
closed Lagrangian 3-manifold in **C**^{3} must be diffeomorphic to a product
S^{1} x Σ for some closed surface Σ. To extend his approach to the
study of other Lagrangian embedding problems, it seems central to understand the string
topology for other classes of closed manifolds. A specific thesis topic in this
direction could aim to treat connected sums, particularly of K(π,1)'s, with the goal
of proving vanishing results for the operations which could serve as a starting point
for a variant of Fukaya's arguments.

The work on mirror symmetry considered here concerns the toric degeneration point of view developed in the joint program of one of the applicants with Mark Gross [GS1], [GS2], [GS3]. At the center of this program is a class of maximal degenerations of certain algebraic varieties whose central fibre is a union of toric varieties. An example is the Dwork family X_{0}X_{1}X_{2}X_{3}+t(X_{0}^{4}+X_{1}^{4}+X_{2}^{4}+X_{3}^{4})=0 of quartic surfaces in **P**^{3} with central fibre t=0 the union of four coordinate planes. Our program provides *canonical* such families out of purely discrete data [GS3], much as toric varieties are encoded by fans, but our families involve an inductively constructed wall structure and hence are highly non-linear. Our wall structure is a higher dimensional and tropical analogue of the gradient flow trees in two dimensions of [KS]. The discrete data consists of a topological manifold B along with a decomposition into integral polyhedra *P* and the structure of a fan at each vertex of *P*. Mirror symmetry then works as a perfect duality on the discrete data. The program can be partially extended to also include more singular degenerations that occur naturally from the classification theoretic point of view [GHK]. Note that the construction of the family is entirely complex-geometric without any explicit reference to mirror symmetry. The wall structure is nevertheless conjectured to encode certain Floer-theoretic invariants of the mirror side, as verified in some easy cases in [GPS], [GHK] and [CLT], [Lau]. This makes contact with the Floer-theoretic mirror construction [Fu1], [Ab2].

During the first funding period there have been two major new techniques introduced into the program. (1) The existence of a canonical basis of sections of the ample line bundle that our construction comes with [GS5]. A comprehensive treatment of these generalized theta functions is currently being finalized. There is a conjectured relation to symplectic cohomology of the mirror side [GHK],[Pa],[Fa]. (2) A theory of log Gromov-Witten invariants [GS4], [AC]. This theory is necessary for computing Gromov-Witten invariants from the central fibre. An ongoing project jointly with Dan Abramovich, Qile Chen and Mark Gross concerns a gluing formula for log Gromov-Witten invariants generalizing [Li], which treated the case with only two irreducible components meeting along a smooth divisor. In this context a generalization of log Gromov-Witten invariants occur naturally that admit negative virtual touching orders relative a divisor. These are called *punctured invariants*. The (straightforward) generalization of log Gromov-Witten theory to this case is currently being in the process of rewriting. In a degeneration situation an unobstructed punctured curve is expected to deform to a map from a Riemann surface with boundary circles mapping to a vanishing cycle, see [GS6] for some discussion.

Possible topics to be considered here include algebraic structures in symplectic cohomology via logarithmic Gromov-Witten theory, the symplectic interpretation of walls in our construction, the geometric quantization interpretation of our theta functions and the investigation of generalized local mirror symmetry suggested by Aganagic and Vafa in the context of knot contact homology [AV][AENV].

### References

[AC] D. Abramovich, Q. Chen:
*Logarithmic stable maps to Deligne-Faltings pairs I*,
Annals of Mathematics **180** (2014), 455--521.

[AV] M. Aganagic, C. Vafa:
*Large N Duality, Mirror Symmetry, and a Q-deformed
A-polynomial for Knots*,
preprint arXiv:1204.4709 [hep-th], 27pp.

[AENV] M. Aganagic, T. Ekholm, L. Ng, C. Vafa:
*Topological Strings, D-Model, and Knot Contact Homology*,
preprint arXiv:1304.5778 [hep-th], 154pp.

[CLT] K. Chan, S-C. Lau, H-H. Tseng:
*Enumerative meaning of mirror maps for toric Calabi-Yau
manifolds*, Adv. Math.\ **244** (2013), 605--625.

[CS1] M. Chas and D. Sullivan:
*String topology*,
arXiv:math.GT/9911159, 1--24.

[CS2] M. Chas and D. Sullivan:
*Closed String Operators in Topology Leading to Lie
Bialgebras and higher String Algebra*,
in: O.A. Laudal, R. Piene (eds.), The Legacy of Niels Henrik
Abel, Springer, 2004, 771--784.

[Fa] O. Fabert:
*Classical mirror symmetry for open Calabi-Yau manifolds*,
preprint arXiv:1310.6014 [math.DG], 20pp.

[Fu1] K. Fukaya:
*Multivalued Morse theory, asymptotic analysis and mirror symmetry*,
in ``Graphs and patterns in mathematics and theoretical physics,''
205--278, Proc. Sympos. Pure Math. 73,
Amer. Math. Soc. 2005.

[Fu2] K. Fukaya:
*Application of Floer homology of Langrangian submanifolds to
symplectic topology*,
in: Paul Biran et al. (eds.), Morse Theoretic Methods in
Nonlinear Analysis and in Symplectic Topology, Springer,
2006, 231--276.

[GHK] M. Gross, P. Hacking, S. Keel:
*Mirror symmetry for log Calabi-Yau surfaces I*,
preprint arXiv:1106.4977v1 [math.AG], 144pp.

[GPS] M. Gross, R. Pandharipande, B. Siebert:
*The tropical vertex*, Duke Math. J. **153** (2010), 297--362.

[GS1] M. Gross, B. Siebert:
*Mirror symmetry via logarithmic degeneration data I*,
J. Differential Geom. **72** (2006), 169--338.

[GS2] M. Gross, B. Siebert:
*Mirror symmetry via logarithmic degeneration data II*,
J. Algebraic Geom. **19** (2010), 679--780.

[GS3] M. Gross, B. Siebert:
*From real affine to complex geometry*,
Ann. of Math. **174** (2011), 1301--1428.

[GS4] M. Gross and B. Siebert: *Logarithmic
Gromov-Witten invariants*, J. of the AMS, **26** (2013),
451--510.

[GS5] M. Gross, B. Siebert:
*Theta functions and mirror symmetry*,
preprint arXiv:1204.1991 [math.AG], 43pp.

[GS6] M. Gross, B. Siebert:
*Local mirror symmetry in the tropics*,
preprint arXiv:1404.3585 [math.AG], 27pp.,
to appear in: Proceedings of the ICM 2014.

[KS] M. Kontsevich, Y. Soibelman,
*Affine structures and non-Archimedean analytic spaces.*,
in: The unity of mathematics (P. Etingof, V. Retakh,
I.M. Singer, eds.), 321--385, Progr. Math. 244,
Birkh\"auser 2006.

[Lau] S.-C. Lau:
*Gross-Siebert's slab functions and open GW invariants
for toric Calabi-Yau manifolds*,
preprint arXiv:1405.3863 [math.AG], 13pp.

[Li] J. Li:
*A degeneration formula of GW-invariants*,
J. Differential Geom. **60** (2002) 199--293.

[OL] J. Latschev and A. Oancea (eds.):
*Free loop spaces in geometry and topology*, IRMA Lectures in Mathematics and Theoretical Physics, EMS Publishing House 2015.

[Pa] J. Pascaleff:
*On the symplectic cohomology of log Calabi-Yau surfaces*,
preprint arXiv:1304.5298 [math.SG], 48pp.

[Wa] N. Wahl:
*Universal operations in Hochschild homology*,
J. reine angew. Math., published online 2014,doi: 10.1515/crelle-2014-0037, arXiv:1212.6498.