Abstract
We study n-point correlation functions for a vertex operator algebra V on a Riemann surface of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras V L and describe their holomorphic and modular properties. We also compute the genus two Heisenberg vector n-point function and the Virasoro vector one point function. Comparing with the companion paper, when a pair of tori are sewn together, we show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The normalized partition functions of a lattice theory V L are compatible, each being identified with the genus two Siegel theta function of L.
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Notes
- 1.
- 2.
Here we include an additional subscript of either ε or ρ to distinguish between the two formalisms.
- 3.
Note that d is denoted by β in [18].
- 4.
- 5.
Two graphs are isomorphic if they have the same labelled vertices and directed edges.
- 6.
Note: in [17] the functional dependence on β, here denoted by a superscript, was omitted.
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Acknowledgements
Geoffrey Mason was supported by the NSF and NSA.
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Appendix
Appendix
We list here some corrections to [17] and [18] that we needed above.
-
(a)
Display (27) of [17] should read
$$\displaystyle{ \epsilon (\alpha,-\alpha ) =\epsilon (\alpha,\alpha ) = (-1)^{(\alpha,\alpha )/2}. }$$(132) -
(b)
Display (45) of [17] should read
$$\displaystyle{ \gamma (\varXi ) = (a,\delta _{r,1}\beta + C(r,0,\tau )\alpha _{k} +\sum _{l\neq k}D(r,0,z_{kl},\tau )\alpha _{l}). }$$(133) -
(c)
As a result of (a), displays (79) and (80) of [17] are modified and now read
$$\displaystyle\begin{array}{rcl} F_{N}(\mathrm{e}^{\alpha },z_{1};\mathrm{e}^{-\alpha },z_{ 2};q)& =& \epsilon (\alpha,-\alpha )\frac{q^{(\beta,\beta )/2}} {\eta ^{l}(\tau )} \frac{\exp ((\beta,\alpha )z_{12})} {K(z_{12},\tau )^{(\alpha,\alpha )}}, {}\end{array}$$(134)$$\displaystyle\begin{array}{rcl} F_{V _{L}}(\mathrm{e}^{\alpha },z_{1};\mathrm{e}^{-\alpha },z_{ 2};q)& =& \epsilon (\alpha,-\alpha ) \frac{1} {\eta ^{l}(\tau )} \frac{\varTheta _{\alpha,L}(\tau,z_{12}/2\pi \mathrm{i})} {K(z_{12},\tau )^{(\alpha,\alpha )}}. {}\end{array}$$(135) -
(d)
The expression for ε(τ, w, χ) of display (172) of [18] should read
$$\displaystyle{ \epsilon (\tau,w,\chi ) = -w\sqrt{1 - 4\chi }\left (1 + \frac{1} {24}w^{2}E_{ 2}(\tau )(1 - 4\chi ) + O(w^{4})\right ) }$$(136)
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Mason, G., Tuite, M.P. (2014). Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II. In: Kohnen, W., Weissauer, R. (eds) Conformal Field Theory, Automorphic Forms and Related Topics. Contributions in Mathematical and Computational Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43831-2_7
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