**Organizers:** Richard Wentworth (Math), Tristan Hubsch (Physics, Howard Univ.), Jonathan Rosenberg (Math), Amin Gholampour (Math)

**Other Faculty Participants:** Joel Cohen (Math), Paul Green (Math, emeritus)

**When:** Thursdays @ 3:30pm-4:30pm**Where:** PHY 1117 (This is the Center for Particle & String Theory, located in what used to be the physics chair's office in the Toll Physics Building adjacent to the Mathematics department.)

This interdisciplinary RIT will aim to foster interactions between mathematicians and physicists on topics of mutual interest. It will roughly follow the example of a similar RIT from 2010-2011 and from the last three years. The topic for 2016-17 was mirror symmetry.

It is not assumed that participants already be knowledgeable in *both* math and physics, just in some aspect of one or the other. Relevant math topics are differential geometry, representation theory, algebraic topology, and algebraic geometry. Relevant physics topics are classical and quantum field theories, and supersymmetry.

The organization meeting for Fall 2017 is scheduled for September 7th. Students (advanced undergraduates or graduate students) who want to participate can get credit as MATH 489 (undergrad) or 689 (graduate) if they wish, by contacting the organizers.

We have a wiki where participants can exchange comments and revise notes. Contact the math/physics computer helpdesk if you need a login id and password.

Topics from previous years:

- Supermanifolds, topology, and integration --- a few references:

- S. J. Gates
*, Ectoplasm has no topology, hep-th/9709104 and hep-th/9809056.* - E. Witten,
*Notes On Supermanifolds and Integration*, 1209.2199. - The Berezin integral.
- S. J. Gates and G. Tartaglino-Mazzucchelli,
*Ectoplasm and superspace integration measure for 2D supergravity with four spinorial supercurrents*, 0907.5264. - S. J. Gates and A. Morrison,
*A Derivation of an Off-Shell N = (2,2) Supergravity Chiral Projection Operator*, 0901.4165.

- S. J. Gates
- Dimensonal reduction in supersymmetry
- For example, S. J. Gates and T. Hubsch,
*On dimensional extension of supersymmetry: From worldlines to worldsheets*, 1104.0722, deals with reduction from 1+1 to 0+1 dimensions.

- For example, S. J. Gates and T. Hubsch,
- Adinkras and combinatorics--- a few references:

- Yan Zhang,
*The combinatorics of adinkras.* - Yan Zhang,
*Adinkras for mathematicians*, 1111.6055. - Greg Landweber, Bibliography on adinkras.
- C. Doran, K. Iga, G. Landweber, and S. Mendez-Diez,
*Geometrization of N-extended 1-dimensional supersymmetry algebras*, 1311.3736. - T. Hübsch and G.A. Katona,
*On the Construction and the Structure of Off-Shell Supermultiplet Quotients*, Int. J. Mod. Phys. A27 (2012) 1250173, 1202.4342. - C.F. Doran, T. Hübsch, K.M. Iga and G.D. Landweber,
*On General Off-Shell Representations of Worldline (1D) Supersymmetry*, Symmetry 6 no. 1, (2014) 67–88, 1310.3258.

- Yan Zhang,
- Super-Riemann surfaces and physical applications
- The Haag-Łopuszański-Sohnius Theorem and its variants. This is the supersymmetric analogue of the better-known Coleman-Mandula Theorem.
- Mirror symmetry (2016-2017). In the fall, we followed a somewhat ad hoc approach based on looking at a lot of examples (e.g., elliptic curves and the quintic Calabi-Yau). In the spring, we followed the multi-author book published by AMS, Dirichlet Branes and Mirror Symmetry. The preface and Chapter 1 can be downloaded from the AMS website; Chapter 2 is at arXiv:hep-th/0609042. An electronic version of the whole book is available at http://www.claymath.org/publications/online-books

### Topics and references for 2017-2018

The topic for **fall 2017** is *topological states of matter*. In no particular order, here is a list of references:

- Emil Prodan and Hermann Schulz-Baldes, "Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics", arXiv:1510.08724.
- Daniel Freed and Greg Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023, arXiv:1208.5055.
- A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). doi:10.1063/1.3149495, arXiv:0901.2686
- F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly". Phys. Rev. Lett. 61, 2015–2018 (1988).
- C.L. Kane and E.J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005), arXiv:cond-mat/0411737
- C.L. Kane and E.J. Mele,
**Z**_{2}Topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005), arXiv:cond-mat/0506581. - C.L. Kane and E.J. Mele, Topological Mirror Superconductivity, Phys. Rev. Lett. 111, 056403 (2013), arXiv:1303.4144.
- Jean Bellissard, Noncommutative Geometry and the Quantum Hall Effect, Proceedings of the International Conference of Mathematicians (Zürich 94), Birkhäuser (1995).
- J. Bellissard, A. van Elst, H. Schulz-Baldes, The Non Commutative Geometry of the Quantum Hall Effect (longer version of #8 above), arXiv:cond-mat/9411052.
- David Tong: Lectures on the Quantum Hall Effect, Univ. of Cambridge, arXiv:1606.06687.
- Edward Witten, Three Lectures On Topological Phases Of Matter, arXiv:1510.07698.
- Ralph M. Kaufmann, Dan Li, and Birgit Wehefritz-Kaufmann, Notes on topological insulators, Rev. Math. Phys., 28(10), 1630003, 2016, arXiv:1501.02874.
- Anton Akhmerov, Jay Sau, et al., Topological condensed matter, an online course.

The topic for **spring 2018** is *generalized geometry* and its applications to physics (especially supersymmetry and string theory). We will begin with the first reference listed, Hitchin's notes. Here is a short list of basic references:

- N. Hitchin, Lectures on Generalized Geometry, arXiv:1008.0973.
- M. Zabzine, Lectures on Generalized Complex Geometry and Supersymmetry, arXiv:hep-th/0605148.
- Dimitrios Tsimpis, Generalized geometry lectures on type II backgrounds, arXiv:1606.08674.
- N. Hitchin, Generalized Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308, arXiv:math/0209099.
- M. Gualtieri, Generalized Kahler geometry, arXiv:1007.3485.
- G. Cavalcantri and M. Gualtieri, Generalized complex geometry and T-duality, arXiv:1106.1747.