Topological field theories in two dimensions, Math 595, Fall 2025

Basic information

Official designation: MATH 595 Section TFT.
Lectures: MWF 1:00pm–1:50pm in 148 Armory Hall.
Instructor: James Pascaleff
- Email: jpascale@illinois.edu; Office: 805 W. Pennsylvania, Room 306; Phone: (217) 244-7277.
- Office hours: Mondays 2pm–2:50pm in 136 Armory, Wednesdays 12pm–12:50pm in 137 Armory. (Note room numbers differ by one.)
Course website: jpascale.web.illinois.edu/courses/595fa25
Prerequisites: Manifolds. Abstract linear algebra. Homology.

Course description

Two-dimensional topological field theories (2D TFT) are algebraic structures whose operations are determined by surfaces and the ways that surfaces can be decomposed and put back together. The main examples of these structures have a topological, geometric, or algebraic origin. They arise in many different contexts, including physics, symplectic geometry, algebraic geometry, algebraic topology and homological algebra.

This course is intended for students with a wide variety of backgrounds, but some familiarity with topology and geometry (surfaces, manifolds, homology) will be assumed.

There are several variants of the concept of a 2D TFT. This course will aim to survey these notions and explain where they come up in mathematics. Topics may include:

2D cobordism category and commutative Frobenius algebras. Following the Atiyah-Segal definition.
Moduli spaces of surfaces and cohomological field theories. Frobenius manifolds and CohFTs, with a view toward the reconstruction results of Dubrovin, Kontsevich-Manin, Givental, and Teleman. Examples from algebraic geometry (e.g. Gromov-Witten theory).
Operads of surfaces and their algebras. Gerstenhaber algebras, Batalin-Vilkovisky algebras, hypercommutative algebras. Examples from homological algebra and symplectic geometry.
Open-closed theories. Allowing surfaces with boundary conditions. Moore-Segal classification theorem, \(A_{\infty}\) categories, and Costello’s theory.
Theories with domain walls. Allowing surfaces with “seams.” Connections to the theory of 2-categories.

Policies

Homework: There will be homework problems in connection with the lectures, not necessarily on a regular schedule. Since this is a graduate topics course, the homework is optional, unless you are a student whose grade in this course really matters. If you are in the latter category, and in particular if you are un undergraduate, please contact the instructor.
Homework submission: Homework may be submitted at any time during the semester, either on paper or by email. Homework submitted after the last day of the finals period (Thursday, December 18, 2025) will not be accepted. The instructor will strive to provide timely feedback.
Student presentations: In the final section of the course, students will study and present a topic related to the course material, under the instructors supervision.
Disability accommodations: Students who require special accommodations should contact the instructor as soon as possible. Any accommodations must be requested at least one week in advance and will require a letter from DRES.

Literature

Reference on cobordisms:

J. W. Milnor, Lectures on the h-cobordism theorem, Princeton Univ. Press, Princeton, NJ, 1965.

The original paper that introduced TQFT:

M. F. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. No. 68 (1988), 175–186 (1989).

Reference for many aspects of TQFT:

F. S. Quinn III, Lectures on axiomatic topological quantum field theory, in Geometry and quantum field theory (Park City, UT, 1991), 323–453, IAS/Park City Math. Ser., 1, Amer. Math. Soc., Providence, RI.

2D TQFT:

R. Dijkgraaf, A geometrical approach to two-dimensional Conformal Field Theory. Dissertation, Utrecht University (1989). See particularly Chapter 3.
B. Dubrovin, Geometry of 2D topological field theories, in Integrable systems and quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes in Math. Fond. CIME/CIME Found. Subser., 1620 , Springer, Berlin.
S. F. Sawin, Direct sum decompositions and indecomposable TQFTs, J. Math. Phys. 36 (1995), no. 12, 6673–6680.
L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), no. 5, 569–587.
J. Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, 59, Cambridge Univ. Press, Cambridge, 2004.

Finite group gauge theory:

D. S. Freed, Lectures on Topological Quantum Field Theory. See pages 22–25.
R. H. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no.~2, 393–429.
D. S. Freed and F. S. Quinn III, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no.~3, 435–472.

Moduli spaces of Riemann surfaces:

Many references are available, my presentation is similar in spirit to:

C. J. Earle Jr. and J. Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43.
J. W. Robbin and D. A. Salamon, A construction of the Deligne-Mumford orbifold, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611–699.

Conformal Field Theories

G. B. Segal, The definition of conformal field theory, in Topology, geometry and quantum field theory, 421–577, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge.
A. Henriques, The functorial approach to chiral 2D CFT, Lecture notes (2025). Particularly see the section on “conformal welding.”
E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285.
K. J. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165–214.
T. Kimura, J. D. Stasheff and A. A. Voronov, On operad structures of moduli spaces and string theory, Comm. Math. Phys. 171 (1995), no. 1, 1–25.

Cohomological Field Theories

M. Kontsevich and Y. I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562.
Y. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, 47, Amer. Math. Soc., Providence, RI, 1999.
R. V. Pandharipande, Cohomological field theory calculations, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, 869–898, World Sci. Publ., Hackensack, NJ.

Homework

Lecture notes

August 25, Introduction to the course, Quantum Mechanics.
August 27, TQFT Axioms.
August 29, TQFT Axioms, continued.
September 3, Consequences of the axioms.
September 5, One-dimensional TQFTs.
September 8, finish previous lecture notes.
September 10, Begin 2D TQFTs.
September 12, Example: Group algebra of a finite abelian group. Notes by Aline Leite Vilela D’Oliveira.
September 15, Algebraic structures from a 2D TQFT.
September 17, Frobenius algebras: relating equivalent definitions.
September 19, Construction of the coproduct on a Frobenius algebra.
September 22, From Frobenius algebras to 2D TQFTs.
September 24, Semisimple algebras, group algebras.
September 29, Semisimple Frobenius algebras and their TQFTs.
October 1, 2D Gauge theory with finite gauge group.
October 3, continue previous topic.
October 6, continue previous topic.
October 8, Finishing off the calculation.
October 10, Conformal structures on surfaces.
October 13, Moduli spaces of conformal structures.
October 15, Examples of moduli spaces.
October 17, More examples of moduli spaces.
October 20, continue previous topic.
October 22, Definition of Conformal Field Theory.
October 24, Topological Conformal Field Theories.
October 27, Cohomology-level structure of a TCFT.
October 29, Algebraic structures up to homotopy.
October 31, More algebraic structures.
November 3, Batalin-Vilkovisky algebras. (Notes originally written for the “What is…?” seminar. Homological grading convention is used here.)
November 5, Deligne-Mumford compactification of moduli spaces.
November 7, Stratification of the Deligne-Mumford space.
November 10, Cohomological Field Theories.
November 12, Curve counting and Gromov-Witten theory.
November 14, Topological part of a CohFT and Quantum Cohomology rings.
November 17.
November 19.
November 21.
December 1, (tentative) Dheeran Wiggins, Symmetric monoidal categories and the functorial formulation of TQFT.
December 3, (tentative) Xianhao An, Decompositions of TQFTs.
December 5, (tentative) Zhengbo Qu, Contractiblity of \(\mathcal{J}(S)\).
December 8, (tentative) Ea Thompson, Operads.
December 10.

Possible topics for presentations

More aspects of the classification of 2D TQFT
- Durhuus-Jonsson studies unitary field theories and shows that they are always semisimple.
- Sawin studies direct sum decompositions of TQFT.
- Abrams has a number of interesting examples.
Hurwitz numbers from TQFT
- The main thing will be to explain what Hurwitz numbers count, and how this connects to the 2D Dijkgraaf-Witten TQFT.
Formulation of TQFT/CFT/TCFT as a symmetric monoidal functor
Operads and moduli spaces of Riemann surfaces
Connectedness/contractibility of the space of almost complex structures on a surface
Cohomology of \(\bar{M}_{0,n}\)
String topology BV algebra
Gerstenhaber/BV algebra structure on Hochschild Cohomology
Gromov-Witten invariants (e.g. Kontsevich-Manin axioms)
Kontsevich formula counting rational plane curves
Open/closed theories (Lazaroiu, Moore-Segal)
Suggest your own topic!