Welcome to my homepage.

About me

I am a 4-th year Ph.D student at Tsinghua University, Yau Mathematical Sciences Center. Here is my google scholar and CV.

E-mail:
h-zhang21@mails.tsinghua.edu.cn
zhanghao1999math@gmail.com

Academic Experiences

Research Descriptions

I am interested in vertex operator algebras and their applications in conformal field theory. Recently, I am working on finite logarithmic chiral CFTs with Bin Gui.

In the previous literature, rational chiral CFTs have been investigated for more than a decade. Due to diverse backgrounds of experts in this area, many different approaches have been introduced to give a precise formulation of the sewing-factorization theorem. However, it seems that the literature doesn’t provide a clear and detailed account of how one approach is related to the other.

Unfortunately, there are even more discrepancies when it comes to finite logarithmic chiral CFTs. Due to Miyamoto’s observation that traces are not enough to consider self-sewing, the VOA community usually uses pseudo-traces and associative algebras in their formulation of sewing-factorization theorem. The most recent progress is Huang’s proof of modular invariance of pseudo-traces of intertwining operators, which is known as the geometric construction of (genus 1) conformal blocks. On the other hand, the TQFT community prefers coends to deal with this problem. This is (conjecturally) known as the categorical construction of (genus 1) conformal blocks. In particular, the categorical S-transformation is defined in terms of coends.

In Huang’s proof of rigidity of the tensor category of a C_2-cofinite and rational VOA, it is highly non-trivial to prove that the modular S-transform coincides with the categorial S-transform. For C_2-cofinite VOAs, the definition of categorical S-transformation is defined using Lyubashenko’s coend. Therefore, it is necessary to compare the SL_2(Z)-representation obtained from pseudo-traces with the one coming from the coends.

In my three-part series paper “Analytic Conformal Blocks of C_2-cofinite Vertex Operator Algebras” joint with Bin Gui, we focused on finite logarithmic chiral CFTs and gave a coend formulation of the sewing-factorization theorem. We proved that disjoint sewing is a coend and gives a categorical construction of conformal blocks of arbitrary genus.

In the second paper, we proved that (higher genus) pseudo-traces come from disjoint sewing. In a future paper, we will prove the inverse part, which will establish the relationship between pseudo-traces and coends we want.

See my talk at SUSTech for the introduction to the relationship between disjoint sewing and coends. If you have any questions, feel free to send me an e-mail!

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