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Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. operators I'm interested in would replace boundary conditions not with Dirichlet conditions but with conditions that jump at some discrete points in the domain. For example, if the boundary conditions were $$\phi(x)= \begin{cases}0, &\quad x< x_0 \\ 1 &\quad x_0 < x < y_0 \\ 0 &\quad x > y_0\\ \end{cases},$$ then that would correspond to b.c.c. operators inserted at $x_0,y_0$.

Motivated by considerations from Schramm-Loewner-Evolution (SLE), it should be expected that b.c.c. operators for the (say, self-dual) free boson CFT are related to the degenerate Virasoro field $\phi_{2,1}$ at $c=1$. $SLE(\kappa)$ is a theory of random conformally-invariant curves between boundary points which is expected to be described by a theory of $c = \frac{(8-3\kappa)(\kappa-6)}{2\kappa}$. In the case of b.c.c. operators above, the random curve would be the it is expected (c.f. lecture notes of Peltola) that correlations of the form above for general are related to chiral conformal blocks of full-plane correlators $\langle \phi_{2,1}(x_0) \phi_{2,1}(y_0) \rangle$.

In particular, at $\kappa=4$, SLE is expected to be related to a level line of the Gaussian Free Field (i.e. free boson CFT) that connects the two jump points at $x_0,y_0$. As such, I'm wondering if there is a known description of b.c.c. operators in this theory, or related ones like free fermion BCFTs with various boundary conditions.

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  • $\begingroup$ One can often learn about bcc operators for fermions by using bosonization. But I'm worried that $\phi = 0$ Dirichlet and $\phi = 1$ Dirichlet are inidistinguishable in the usual operator algebra for the free boson which does not include $\phi$ itself. $\endgroup$
    – Connor Behan
    Commented Jun 9 at 0:23
  • $\begingroup$ Right, locally the boundary conditions are not different. But somehow the jump $\phi =0 \to \phi = 1$ on the boundary is represented by the derivatives $\partial \phi$ which are in the algebra. So the point is that the jump itself should be expressible in the operator algebra. $\endgroup$
    – Joe
    Commented Jun 10 at 17:59
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    $\begingroup$ Crossposted to mathoverflow.net/q/473430/13917 $\endgroup$
    – Qmechanic ♦
    Commented Jun 18 at 10:57

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