This is an advanced graduate course for master and phd students as well as researchers who are interested in complex algebraic-geometric structures which originate from the physics of supersymmetric theories. The course will discuss the variation of Hodge structure and the associated special Kähler geometry, it will then proceed to a generalization of special Kähler geometry known as the topological anti-topological fusion geometry or tt* geometry.

Prerequisites:
A good knowledge of complex geometry and familiarity with algebraic geometry are required. A background in physics of supersymmetric theories is beneficial for the course but not necessary.

Topics of the course:

(1) Intuition for variation of Hodge structure from elliptic curves
(2) Variation of Hodge structure
(3) Special Geometry
(4) tt∗ geometry
(5) Advanced topics, e. g. quantum cohomology, twistorial structure of tt∗equations, algebraic structure of tt∗ equations, relation to Hitchin systems.

Logistics:

The course takes place on:

Thursdays, 12:15 - 13:45 in H3 (Geomatikum)

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A. Cox and S. Katz. Mirror symmetry and algebraic geometry. 2000.

James Carlson, Stefan Müller-Stach, and Chris Peters. Period mappings and period domains, volume 85 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2003.

Sergio Cecotti and Cumrun Vafa. Topological–anti-topological fusion. Nuclear Phys. B, 367(2):359–461, 1991.

B. Dubrovin. Geometry and integrability of topological-antitopological fusion. Comm. Math. Phys., 152(3):539–564, 1993.

Daniel S. Freed. Special Kahler manifolds. Commun. Math. Phys., 203:31–52, 1999.

Martin A. Guest. From quantum cohomology to integrable systems, volume 15 of Oxford Grad- uate Texts in Mathematics. Oxford University Press, Oxford, 2008.

Claire Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cambridge Stud- ies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, 2007. Translated from the French by Leila Schneps.

Variation of Hodge Structure and tt* Geometry, University of Hamburg, Winter Term 2017/2018