# Twisted Holography and Koszul Duality

During Fall 2021, Dylan Butson, Brian Williams, and I are running a learning seminar on the work of Kevin Costello and collaborators on twisted string theory, twisted holography, and relations to koszul duality. The seminar takes place online on **TUESDAYS from 10:00-12:00 EST** unless otherwise stated. Those interested in participating should feel free to contact one of the organizers for a zoom link.

## Schedule:

Date | Speaker | Title and Abstract | Recording |
---|---|---|---|

October 6 | Kevin Costello | ## From physical holography to Koszul dualityAbstract: I will review how the physics statement of holography can be "twisted" to yield a statement that can be formulated mathematically. I'll try to start at the beginning, and sketch the idea of twisting supersymmetric theories and supergravity, before moving on to trying to explain what holography looks like once you twist. | Video Slides |

October 11 | Brian Williams | ## A user's guide to holographyAbstract: In this talk I want to explain some very basic ideas that appear in twisted holography. In fact, I won’t say anything about supersymmetric twists or AdS/CFT. Rather, I will introduce the basic mathematical ideas involved with framing holographic dualities; the most important of which is the concept of Koszul duality. I’ll work through a simplest example of Koszul duality and will draw parallels with a story in topological field theory. | Video Slides |

October 19 | Surya Raghavendran | ## Topological strings and Twisted supergravityAbstract: To formulate a holographic correspondence at the level of twists, it is useful to have a mathematical way of discussing twists of superstrings. Costello-Li conjecture that certain twists of superstrings are equivalent to topological strings. The goal of this talk will be to learn to work with the mathematical outputs of this conjecture. I'll introduce topological string analogues of open string field theory and closed string field theory. The former will recover twists of supersymmetric gauge theories, and the latter will contain twists of supergravity. I'll then discuss how to codify some ways in which the open and closed sectors interact. | Video |

October 26 | Dylan Butson | ## Overview of the twisted holography programAbstract: I'll outline the general set-up for formulating twisted holography type conjectures following Kevin's program: I'll briefly recall the mathematical avatars of open and closed string field theories discussed in Surya's talk, explain how to use these to (somewhat) systematically extract classical field theory configurations from the string theory input data, and state Kevin's meta-conjecture about the expected holographic principle satisfied by quantizations of these setups, using the perspective on Koszul duality explained in Brian's talk (and his cool new paper with Natalie Paquette! https://arxiv.org/abs/2110.10257). In the remaining time, I'll discuss some expectations, methods, and challenges for carefully formulating such conjectures at the quantum level, and briefly survey some of the existing work in the field that we'll be hearing about in later talks. | Video Slides |

November 2 | Benjamin Gammage | ## Skew howe duality from holographyAbstract: After an introduction to computations in the A-model, we spend the talk bringing together all the previously discussed ingredients (except backreaction) from the mathematical approach to holography by working in detail through a simple A-model example (due to Surya) which will recover skew Howe duality. | Video Slides |

November 9 | Brian Williams | ## Brane couplings and BackreactionsAbstract: An important part of the physical holography story is the how an ambient bulk theory is modified in the presence of defects, or branes. We’ll introduce this so-called `backreaction’. We will then go through a few examples related to the topological string and twists of the superstring. | Video Slides |

November 16 | Nafiz Ishtiaque | ## Physical approach to topological holographyAbstract: I will review some basics of holographic duality like the decoupling limit and discuss how topological holography arises from the traditional physical approach of computing Feynman and Witten diagrams. I will discuss an elementary example, involving a topological analog of AdS_3/CFT_2 duality with defects. In algebraic terms, the duality will be presented as an isomorphism between Yangian algebras. Furthermore, I will present this as a supersymmetric subsector of the familiar AdS_5/CFT_4 duality. | Video Slides |

November 23 | Davide Gaiotto | ## Twisted HolographyAbstract: I will review the twisted holography setup relating the protected chiral algebra of N=4 SYM to the B-model on SL(2,C). | Video (Passcode: ?Nq!^BzE0p) |

November 30 | Kasia Budzik | ## Giant gravitons in twisted holographyAbstract: I will talk about the correlation functions of determinant operators in the chiral algebra subsector of N = 4 SYM, which are dual to the Giant Graviton branes in the B-model on SL(2,C). For each large-N saddle of the correlation functions of determinants, we will define a spectral curve in SL(2,C), which we will identify with the worldsheet of the dual Giant Graviton brane. | Video Slides |

December 7 | Jihwan Oh | ## Review of twisted M-theoryAbstract: I will review some basics of twisted M-theory mostly based on 1610.04144, 1705.02500, 1907.06495. | Video Slides |

December 14 | Yehao Zhou | ## Algebras of observables of 4d and 5d topological-holomorphic Chern-Simons theoriesAbstract: This talk is an addendum to Nafiz and Jihwan’s talks. I will explain how to properly define the large-N limit of the algebras of observables in 4d and 5d Chern-Simons theories discussed in previous talks. And then discuss some features of these algebras, for example generators and relations, coproducts, etc. | Video Slides |

December 21 | Ji Hoon Lee | ## A microscopic expansion for superconformal indicesAbstract: I discuss a novel expansion of superconformal indices of U(N) gauge theories at finite N. When a holographic description is available, the formula expresses the index as a sum over stacks of "giant graviton" branes in the dual string theory. We derive these contributions in gauge theory by counting determinant operators and their modifications. Surprisingly, the expansion seems to be exact: the sum over strings and branes seems to capture the degeneracy of states expected from saddle geometries, while also reproducing the correct degeneracies at lower orders of charges. We thus conjecture that determinant operators and their modifications, dressed with usual operators of the multi-trace form, exhaust the Q-cohomology at finite N. Based on 2109.02545 and work in progress with D. Gaiotto. | Slides |

## References:

### Koszul Duality

- Koszul Duality in Quantum Field Theory - Natalie Paquette, Brian Williams

### Background on Twisted Supergravity, BCOV theory, and Topological Strings

- Twisted supergravity and its quantization - Kevin Costello, Si Li
- Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model - Kevin Costello, Si Li
- Quantization of open-closed BCOV theory, I - Kevin Costello, Si Li
- Anomaly cancellation in the topological string - Kevin Costello, Si Li
- Twisting pure spinor superfields, with applications to supergravity - Ingmar Saberi, Brian Williams
- Maximally twisted eleven-dimensional supergravity - Richard Eager, Fabian Hahner
- Twisted S-duality - Surya Raghavendran, Philsang Yoo

### Examples in Twisted String Theory

- Twisted Holography - Kevin Costello, Davide Gaiotto
- Topological Holography: The Example of The D2-D4 Brane System - Nafiz Ishtiaque, Faroogh Moosavian, Yehao Zhou
- Twisted Supergravity and Koszul Duality: A case study in $\mathrm{AdS_3}$ - Kevin Costello, Natalie Paquette
- Giant gravitons in twisted holography - Kasia Budzik, Davide Gaiotto

### Examples in Twisted $\Omega$-deformed M-theory

- M-theory in the $\Omega$-background and 5-dimensional non-commutative gauge theory - Kevin Costello
- Holography and Koszul duality: the example of the M2 brane - Kevin Costello
- Aspects of $\Omega$-deformed M-theory - Davide Gaiotto, Jihwan Oh
- Twisted M2 brane holography and sphere correlation functions - Jacob Abajian, Davide Gaiotto
- Miura operators, degenerate fields and the M2-M5 intersection - Davide Gaiotto, Miroslav Rapcak
- Feynman diagrams and $\Omega$-deformed M-theory - Jihwan Oh, Yehao Zhou
- Twisted holography of defect fusions - Jihwan Oh, Yehao Zhou
- A domain wall in twisted M-theory - Jihwan Oh, Yehao Zhou
- Evidence for an Algebra of $G_2$ Instantons - Jihwan Oh, Yehao Zhou