My research so far revolves around the conjectural topological invariants, $\hat{Z}$ and $F_K$, which are integer coefficient $q$-series with interesting modular properties and are believed to have categorifications. I'm working to find a mathematical definition for these invariants, as well as studying their properties in various contexts.

- With Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski and Piotr Sułkowski

**$\hat{Z}$ at large $N$: from curve counts to quantum modularity**

preprint, submitted to CMP [arXiv:2005.13349] - With Sergei Gukov, Po-Shen Hsin, Hiraku Nakajima, Du pei and Nikita Sopenko

**Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants**

preprint [arXiv:2005.05347] **Large color $R$-matrix for knot complements and strange identities**

preprint [arXiv:2004.02087]- With Sungbong Chun, Sergei Gukov and Nikita Sopenko

**3d-3d correspondence for mapping tori**

preprint, submitted to JHEP [arXiv:1911.08456] **Higher rank $\hat{Z}$ and $F_K$**

SIGMA 16 (2020), 044, 17 pages [journal | arXiv:1909.13002]

Inspired by earlier works by many people (e.g. [LZ,H] among many others), the existence of a $q$-series valued invariant $\hat{Z}$ (pronounced "Z-hat" or "Zed-hat") for 3-manifolds was conjectured in [GPV,GPPV]. More recently, $\hat{Z}$ for 3-manifolds with a toral boundary (e.g. knot complements) was studied in [GM], and it was named $F_K$ for the knot complement $Y = S^3\setminus K$.

One reason we are interested in this new conjectural invariant is its integrality (i.e. its coefficients are integers) which hints that it may have a categorification. Indeed, from physics point of view [GPV,GPPV], these invariants should be the graded Euler characteristics of certain Hilbert spaces of BPS states. A hope is that categorification of these conjectural invariants (i.e. a 3-manifold analogue of Khovanov homology) will deepen our understanding of 3- and (smooth) 4-manifolds.

Another reason is that these $q$-series are expected to have interesting modular properties. A remarkable example given in [GM] is the (orientation reversed) Brieskorn homology sphere $Y=-\Sigma(2,3,7)$, in which case we have

$\displaystyle\quad\hat{Z}(Y;q) = \sum_{n=0}^{\infty}\frac{q^{n^2}}{(q^{n+1})_n} = 1+q+q^3+q^4+q^5+2q^7+q^8+2q^9+q^{10}+2q^{11}+q^{12}+3q^{13}+\cdots,$

which is exactly $\mathcal{F}_0(q)$, one of Ramanujan's mock theta functions of order 7. It would be very interesting to understand these (quantum) modular objects through the lens of logarithmic vertex algebras and non-semisimple modular tensor categories, as conjectured in [CCFGH].

[LZ] R. Lawrence, D. Zagier - Modular forms and quantum invariants of 3-manifolds (1999)

[H] K. Hikami - On the Quantum Invariant for the Brieskorn Homology Spheres (2004)

[GPV] S. Gukov, P. Putrov, C. Vafa - Fivebranes and 3-manifold homology (2016)

[GPPV] S. Gukov, D. Pei, P. Putrov, C. Vafa - BPS spectra and 3-manifold invariants (2017)

[CCFGH] M. C. N. Cheng, S. Chun, F. Ferrari, S. Gukov, S. Harrison - 3d Modularity (2018)

[GM] S. Gukov, C. Manolescu - A two-variable series for knot complements (2019)