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

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

**When:** Thursdays @ 3:30pm-4:30pm**Where: **Online at Zoom Meeting ID: 926 8339 9101. Contact Dr. Rosenberg for passcode.

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. Topics for 2017-2018 and 2018-2019 are listed below to give you an idea of what sort of things we tend to cover.

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.

Students (advanced undergraduates or graduate students) who want to participate can get 1-3 credits 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 and references for 2021-2022

The topic for 2021-2022 is the **Sachdev-Ye-Kitaev model**. Here are a few basic references:

- Juan Maldacena and Douglas Stanford, Comments on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818.
- The Sachdev-Ye-Kitaev model, a summary and bibliography.
- Subir Sachdev, the SYK model.
- Vladimir Rosenhaus, An Introduction to the SYK Model, J. Phys. A: Math. Theor. 52, 323001 (2019), arXiv:1807.03334.
- Douglas Stanford and Edward Witten, Fermionic localization of the Schwarzian theory, J. High Energy Physics 2017 10, 008.

### Topics and references for 2020-2021

Those interested in this RIT might also be interested in the Wales Mathematical Physics-Physical Mathematics Seminar and the Western Hemisphere Colloquium on Geometry and Physics.

The topic for Fall 2020 and the first half of Spring 2021 (up to Spring Break) is **tropical geometry** and its applications to physics. Here are a few basic references:

- Mark Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, 114,
*American Mathematical Society, Providence, RI,*2011, draft available here. - Brief introduction to tropical geometry, by Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, and Kristin Shaw, arXiv:1502.05950
- Geometry in the tropical limit by Ilia Itenberg and Grigory Mikhalkin, arXiv:1108.3111.

Also, here are notes by Tristan Hubsch from his talk on March 11.

Starting April 2021 we plan to switch to *N*=4 SYM (super-Yang-Mills) theory and connections to the geometric Langlands program, following the 2009 Bourbaki talk by Frenkel, entitled *Gauge Theory and Langlands Duality*. For current work on this topic, you might want to see the lecture by David Ben-Zvi at the Westerm Hemisphere Colloquium on Geometry and Physics.

### Topics and references for 2019-2020

The topic for 2019-2020 is **invertible TQFTs and symmetry-protected topological (SPT) phases of matter**. Here are a few basic references:

- M. Atiyah, An introduction to topological quantum field theories,
*Turkish J. Math.*21 (1997), no. 1, 1–7. available here. - M. Atiyah, Topological quantum field theories, Publ. Math. IHES no. 68 (1988), 175–186 (1989). available here.
- Daniel Freed and Greg Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023, arXiv:1208.5055.
- D. Freed, M. Hopkins, J. Lurie, and C. Teleman, Topological quantum field theories from compact Lie groups.
*A celebration of the mathematical legacy of Raoul Bott,*367–403, CRM Proc. Lecture Notes, 50,*Amer. Math. Soc., Providence, RI,*2010, arXiv:0905.0731. - V. and T. Ivancevic, Undergraduate Lecture Notes in Topological Quantum Field Theory, arXiv:0810.0344.
- D. Freed and M. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527.
- D. Freed and M. Hopkins, Invertible phases of matter with spatial symmetry, arXiv:1901.06419.
- D. Gaiotto and T. Johnson-Freyd, Symmetry Protected Topological phases and Generalized Cohomology, arXiv:1712.07950.
- Wikipedia article on SPT order, with lots of references from the physics literature.
- A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology, arXiv:1403.1467.
- A. Kapustin and A. Turzillo, Equivariant topological quantum field theory and symmetry protected topological phases,
*J. High Energy Phys**.*2017, no. 3, 006, arXiv:1504.01830. - E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys.
**88**, 035001 (2016), arXiv:1508.04715. - D. Freed,
*Lectures on Field theory and topology*. CBMS Regional Conference Series in Mathematics, 133.*American Mathematical Society, Providence, RI,*2019. - A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). doi:10.1063/1.3149495, arXiv:0901.2686
- Jonathan Campbell, Homotopy Theoretic Classification of Symmetry Protected Phases, arXiv:1708.04264.
- E. Witten and K. Yonekura, Anomaly Inflow and the
η -Invariant, arXiv:1909.08775 - Sri Tata, Notes on dimer models and Spin/Pin TQFTs
- En-Jui (Eric) Kuo, Notes on Kitaev's periodic table

### Topics and references for 2018-2019

The topic for **fall 2018** was *the AdS/CFT correspondence*. Here is a list of references to get started:

- Juan Maldacena, The gauge/gravity duality, arXiv:1106.6073.
- Horatiu Nastase, Introduction to AdS-CFT, arXiv:0712.0689.
- Makoto Natsuume, AdS/CFT Duality User Guide, arXiv:1409.3575.
- Sean A. Hartnoll, Andrew Lucas, Subir Sachdev, Holographic quantum matter, arXiv:1612.0732.
- Davide Gaiotto, Juan Maldacena, The gravity duals of N=2 superconformal field theories, arXiv:0904.4466.
- O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Large N Field Theories, String Theory and Gravity, arXiv:hep-th/9905111.
- Juan Maldacena, The Large N limit of superconformal field theories and supergravity, arXiv:hep-th/9711200.
- James Lindesay and Leonard Susskind, The Holographic Universe, World Scientific, 2004.
- Jonas Probst, Applications of the Gauge/Gravity Duality, Ph.D. thesis, Oxford, 2018.
- Raman Sundrum, From Fixed Points to the Fifth Dimension, arXiv:1106.4501.
- Vladimir Rosenhaus, An introduction to the SYK model, arXiv:1807.03334.
- Gábor Sárosi, AdS
_{2}holography and the SYK model, arXiv:1711.08482.

The topic for **spring 2019** was *Bridgeland stability*. We started with a little background on algebraic geometry and the physics motivation, and then gave a quick introduction to triangulated categories, before getting to the main topic. Here is a list of references:

- Emanuele Macrì and Benjamin Schmidt, Lectures on Bridgeland Stability, arXiv:1607.01262.
- Emanuele Macrì and Paolo Stellari, Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces, arXiv:1807.06169.
- François Charles, Conditions de stabilité et géométrie birationnelle [d'après Bridgeland, Bayer-Macrì, ...] (Bourbaki talk), arXiv:1901.02930.
- Arend Bayer, A tour to stability conditions on derived categories (lecture notes).
- Ciaran Meachan, Moduli of Bridgeland-Stable Objects, PhD thesis, Univ. of Edinburgh, 2012.
- Claudio Fontanari and Diletta Martinelli, Why should a birational geometer care about Bridgeland stability conditions?, arXiv:1605.04803.
- Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, arXiv:1401.4949.
- Daniel Huybrechts, Introduction to stability conditions, arXiv:1111.1745.
- Tom Bridgeland, Spaces of stability conditions, arXiv:math/0611510.
- Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317-345.
- Michael R. Douglas, D-branes, categories and
𝒩=1 supersymmetry,*J. Math. Phys.*42 (2001), no. 7, 2818–2843. - Tom Bridgeland and Ivan Smith, Quadratic differentials as stability conditions,
*Publ. Math. Inst. Hautes Études Sci.*121 (2015), 155–278.

### Topics and references for 2017-2018

The topic for **fall 2017** was *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** was *generalized geometry* and its applications to physics (especially supersymmetry and string theory). We began 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.

### 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

### Detailed schedule posted below.