Variation of Hodge Structure and tt* Geometry,
University of Hamburg, Winter Term 2017/2018
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)
Sessions this term:
Date
|
Topic
|
Room, Time
|
19.10.2017
|
I-VHS for elliptic curves, notes
|
H3, 12:15
|
26.10.2017
|
I-VHS for elliptic curves, notes
|
H3, 12:15
|
26.10.2017
|
I-VHS for elliptic curves, notes
|
H2, 14:15!
|
06.11.2017
|
Exercise
session, discussion sheet 1 Please contact Martin Vogrin regarding
the dates
|
Geo 432, 12:15 |
09.11.2017
|
I-VHS for elliptic curves, notes
(page 24 corrected) |
H3, 12:15
|
16.11.2017
|
II-Variation of Hodge Structure, notes |
H3, 12:00
|
20.11.2017
|
Exercise
session, discussion sheet 2 |
Geo 432, 12:15
|
23.11.2017
|
II-Variation of Hodge Structure, notes |
H3, 12:05 |
30.11.2017
|
Will take place on another date!
|
other date tbd
|
04.12.2017
|
Exercise
session, discussion sheet 3, fill
poll |
Geo 432, 12:15
|
07.12.2017
|
II-Variation of Hodge Structure, notes |
H3, 12:05 |
14.12.2017
|
III- Special Geometry, notes |
H3, 12:05 |
21.12.2017
|
III- Special Geometry, notes |
H3, 12:05 |
11.01.2018
|
III- Special Geometry, by Florian
Beck on Freed's
paper |
H3, 12:05 |
18.01.2018
|
IV-tt* Geometry, notes
|
H3, 12:05 |
| 19.01.2018 |
Exercise
session, discussion sheet 4
|
Geo 233, 12:15
|
25.01.2018
|
IV-tt* Geometry, notes
, see also section 3 of this
paper.
|
H3, 12:05 |
01.02.2018
|
V-Advanced topics: Semi-Infinite
Hodge Structures, by Martin Vogrin based on Barannikov's
paper
|
H3, 12:05 |
| 02.02.2018 |
Exercise
session, discussion sheet 5
|
Geo 233, 12:15
|
Literature:
A. Ceresole, R.
D’Auria, S. Ferrara, W. Lerche, J. Louis, and T.
Regge. Picard-Fuchs equations, special geometry
and target space duality. In Mirror symmetry, II, volume 1 of AMS/IP Stud. Adv. Math., pages 281–353. Amer. Math. Soc.,
Providence, RI, 1997.
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.
Course
description
Exercise sheets
Contact Martin
Vogrin regarding the exact solution discussion
times and location
Sheet
1
Sheet
2
Sheet
3
Sheet
4
Sheet
5
Research
projects
|
