Organized by Mina Aganagic, Ivan Danilenko, Peter Koroteev, and Miroslav Rapcak

Weekly on **Mondays 2:10 PM** (Pacific Time)

**Some talks will be given in Zoom** using the following link: http://berkeley.zoom.us/j/93328405860?pwd=Um1GbHBCSUJMdUlWWnd0ZVMxQmwwdz09

For in-person meetings: at 402 Physics South

We have a lunch on the 4th floor of Physics South before the seminar

Jan 24 | Monica Kang | in-person | 4d SCFTs with a=c | |

Jan 31 | Sasha Voronov | zoom | Mysterious Triality | Slides |

Feb 7 | Gregory Moore | zoom | Summing Over Bordisms In 2d TQFT | |

Feb 14 | No Seminar |
|||

Feb 21 | No Seminar |
|||

Feb 28 | Marcos Marino | zoom | Resurgence and Quantum Topology | |

Mar 7 | Benjamin Gammage | in-person | 2-categorical 3d mirror symmetry | |

Mar 14 | Max Zimet | in-person | Orbifolds, gauge theory, and hyperkahler geometry | |

Mar 21 | No Seminar |
|||

Mar 28 | Sunghyuk Park | in-person | R-matrix and a q-series invariant of 3-manifolds | |

Apr 4 | Raphael Bousso | in-person | Singularities From Entropy | |

Apr 11 | Brian Williams | in-person | A holomorphic approach to fivebranes | |

Apr 18 | Justin Hilburn | in-person | 3d Mirror Symmetry and 2-categories O | |

Apr 25 | Andrey Smirnov | in-person | Bethe ansatz problem for Nakajima varieties | |

May 2 | Tudor Dimofte | in-person | Line operators, flat connections, and VOA’s | |

May 9 | Denis Auroux | zoom | Lagrangian Floer theory and analytic functions on Riemann surfaces | Video |

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

Fall 2021, Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

I will present a particular set of 4d \(\mathcal{N}=2\) SCFTs that can be labeled with a pair of Lie groups of type ADE. For specific choices, we get infinitely many theories arising from this construction that have their two central charges to be identical: a=c (without taking any large N limit). Interestingly, the Schur indices of these theories are identical to that of \(\mathcal{N}=4\) super Yang-Mills upto rescaling fugacities. I will further utilize this construction to present various 4d \(\mathcal{N}=1\) SCFTs with a=c.

Mysterious duality was discovered by Iqbal, Neitzke, and Vafa in 2002 as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series \(E_k\). It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics which gives rise to the same \(E_k\) symmetry pattern. I will present a sequence of topological spaces, starting with the four-sphere \(S^4\), and then forming its iterated cyclic loop spaces \(L_c^k S^4\), within which we will see the \(E_k\) symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its \(E_k\) symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space \(L_c^k S^4\) is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of (11-k)-dimensional supergravity as the defining equations of the Sullivan minimal model of \(L_c^k S^4\). This gives an explicit duality between rational homotopy theory and physics. Thereby, Iqbal, Neitzke, and Vafa’s mysterious duality between algebraic geometry and physics is extended to a triality involving algebraic topology, with the duality between topology and physics made explicit, i.e., demystified. The mystery is now transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. This is a report on the recent workarXiv:2111.14810 [hep-th] with Hisham Sati.

Some recent work in the quantum gravity literature has considered what happens when the amplitudes of a TQFT are summed over the bordisms between fixed in-going and out-going boundaries. We will comment on these constructions. The total amplitude, that takes into account all in-going and out-going boundaries can be presented in a curious factorized form. This talk reports on work done with Anindya Banerjee and is based on the paper on the e-print arXiv 2201.00903.

Quantum field theories and string theories often lead to perturbative series which encode geometric information. In this lecture I will argue that, in the case of complex Chern-Simons theory, perturbative series secretly encode integer invariants, related in some cases to BPS counting. The framework which makes this relation possible is the theory of resurgence, where perturbative series lead to additional non-perturbative sectors, and the integer invariants arise as Stokes constants. I will illustrate these claims with explicit examples related to quantum invariants of hyperbolic knots. If time permits, I will mention similar results in topological string theory.

3d mirror symmetry predicts equivalences between topological twists of dual 3d \(\mathcal{N}=4\) theories, which we would like to understand as equivalences between their 2-categories of boundary conditions. Unfortunately, it is not known how to describe these 2-categories mathematically, although the B-side has been partially understood from work of Kapustin-Rozansky-Saulina and Arinkin. For abelian gauge theories, we propose that perverse schobers provide a good model for the 2-category of A-twisted boundary conditions, and we show that this can be used to prove “homological 3d mirror symmetry” for such theories. Mathematically speaking, we give a spectral description of the 2-category of spherical functors. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

I will describe a physically-motivated conjectural approach (developed with Shamit Kachru and Arnav Tripathy) to the construction of hyperkahler deformations of \( (\mathbb{R}^{4-r} \times T^r)/\Gamma \) with \( 0 \leq r \leq 4 \), where \(\Gamma\) is a finite group, as well as recent work with Arnav Tripathy which proves some of these conjectures. This generalizes Kronheimer’s construction of ALE manifolds in the \(r=0\) case, but for \(r\neq 0\) it involves an infinite-dimensional hyperkahler quotient. I will explain how to reinterpret this as a gauge theory problem on the dual torus, where one is now interested in a moduli space of singular equivariant instantons. Many results from Kronheimer’s work generalize to this setting, and conversely we obtain some novel results which apply to the ALE setting. I will explain how the \(r\geq 2\) case interacts with open string mirror symmetry, enumerative geometry, and another construction of hyperkahler manifolds. Finally, I will specialize to the \(r=1\) case, the rigorous study of which we have recently completed, and which serves as a model for many of the phenomena we expect to find at larger values of \(r\).

\(\hat{Z}\) is a 3d TQFT whose existence was predicted by S. Gukov, D. Pei, P. Putrov, and C. Vafa in 2017 using 3d/3d correspondence. To each 3-manifold equipped with a \(\mathrm{spin}^c\) structure, \(\hat{Z}\) is supposed to assign a q-series with integer coefficients that is categorifiable and provides an analytic continuation of Witten-Reshetikhin-Turaev invariants. In 2019, S. Gukov and C. Manolescu initiated a program to mathematically construct \(\hat{Z}\) via Dehn surgery, and as part of that they conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series \(F_K(x,q)\). In this talk, I will explain how to prove their conjecture for a big class of links by “inverting” the R-matrix state sum.

[Joint work with Arvin Shahbazi-Moghaddam.] We prove a singularity theorem based on the covariant entropy bound. The theorem connects singularities to quantum information, and it eliminates the assumption of noncompactness from Penrose’s 1965 theorem. A quantum extension of our theorem further eliminates the null energy condition. It can be applied to an evaporating black hole, where it leads to an interesting puzzle.

In almost all situations, the twist of a supersymmetric QFT has the structure of a holomorphic QFT. I’ll review general aspects of holomorphic QFT while drawing parallels to the familiar situation of chiral CFT. I will then define a holomorphic model which we propose describes the minimal twist of the six-dimensional superconformal theory associated to the Lie algebra sl(2). A primary source of evidence for this proposal is via an approach to holography in the twisted setting using Koszul duality.

n this talk I will explain how 3d mirror symmetry predicts an equivalence between 2-categories associated to dual pairs of hyperkahler quotients. The first 2-category is of an algebro-geometric flavor and was described by Kapustin/Rozansky/Saulina. The second category depends on symplectic topology and has a conjectural description in terms of the 3d generalized Seiberg-Witten equations. Both of these 2-categories are expected to categorify category O for symplectic resolutions in the sense of Braden/Licata/Proudfoot/Webster. I will also explain joint work with Ben Gammage and Aaron Mazel-Gee on proving 3d mirror symmetry for hypertoric varieties. This is a sequel to the work Ben Gammage described in his talk earlier this semester.

It was shown by M.Aganagic and A.Okounkov that the relative insertions in the quantum K-theory of Nakajima varieties are equivalent to non-singular descendent insertions. Among other things, this result leads to integral representation for solutions of qKZ equations and explicit description of the Bethe vectors.

From representation theoretic viewpoint, the result of Aganagic-Okoukov deals with the "slope" zero qKZ. In my talk I discuss generalizations of qKZ equations to an arbitrary slope. I also discuss the result of H. Dinkins describing the integral solutions of qKZ equations and the corresponding Bethe vectors for general slopes. These results point to existence of an enumerative theory generalizing the relative quasimap count for Nakajima varieties.

3d N=4 gauge theories admit two topological twists, “A” and “B”, which are expected to give rise to extended TQFT’s. Aspects of these TQFT’s are starting to be defined mathematically. My goal in this talk is to discuss recent developments in understanding line operators (the value of the TQFT’s on a circle), which are expected to form a dg braided tensor category in the A and B twists — analogous to familiar structure of Wilson lines in 3d Chern-Simons theory. I will motivate two definitions of the categories of line operators, one intrinsically 3-dimensional (following work of Hiilburn-Yoo and Hilburn-Raskin, and work with Garner-Geracie-Hilburn) and one via boundary VOA’s (following work of Costello-Creutzig-Gaiotto and Ballin-Niu, and work in progress with Ballin-Creutzig-Niu). I will then explain how continuous global symmetries, which are prevalent in 3d N=4 theories, lead to deformations of the categories of line operators, enhancing them to certain sheaves of categories. Finally, I will touch on a recent application to non-semisimple TQFT’s related to quantum groups at roots of unity, based on work with Creutzig-Garner-Geer.

The goal of this talk will be to describe the relation between generators of Lagrangian Floer cohomology on a surface and functions on its mirror -- both locally on building blocks such as cylinders, pairs of pants, and mirrors of pairs of pants, and globally on elliptic curves, higher genus surfaces, and their mirrors. The common theme throughout will be that Floer theory on cylindrical portions of a surface gives a geometric interpretation of Laurent series expansions of analytic functions on the mirror. The less routine parts of the story build on Heather Lee's thesis and on joint work with Alexander Efimov and Ludmil Katzarkov.