The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.
We develop a unified spectral and complex analytic theory of the Lame equation through its canonical Heun realization. Starting with ellipsoidal separation, we derive essential elements, including the Lame-preserving anharmonic parameter action and finite accessory spectral polynomials. We construct the global Heun mon...
In this article, we give a global construction of the Virasoro conformal blocks on Teichm\"uller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which enc...
Guillaume Baverez, C. Guillarmou, Antti Kupiainen et al.· 0 citations
We extend the algebro-geometric construction of bifiltered affine schemes that encode both the Higgs branch and the Macdonald index of 4d $\mathcal{N}=2$ superconformal field theories to theories with non-trivial Higgs branches. For the Argyres--Douglas theories $\mathcal{D}_p(SU(N))$, we find that a universal potentia...
We study the asymptotic and TQFT-type properties of the Kashaev--Luo--Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a hyperbolic cone struct...
Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. I...
We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have a...
Suresh Govindarajan, Aditya Jain, Akhila Sadanandan et al.· 0 citations
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