Instantons and wormholes in $N=2$ supergravity

In this paper, we construct Euclidean instanton and wormhole solutions in $d=4$, $N=2$ supergravity theories with hypermultiplets. The analytic continuation of the hypermultiplet action, involving pseudoscalar axions, is discussed using the approach originally developed by Coleman, which determines the appearance of boundary terms. In particular, we investigate the conditions obtained by requiring the action to be positive definite once the boundary terms are taken into account. The case of two hypermultiplets parameterizing the coset $G_{2,2}/SU(2)\times SU(2)$ is studied in detail. Orientifold projections that reduce the supersymmetry to $N=1$ are also discussed.

Figure 1

Root diagram of $SU(2,1)$ with identification of the symmetry generators.

Figure 2

The $G_{2,2}$ root diagram. $Y_0$ and $H$ are the Cartan generators.

Figure 3

Action of regular RR wormhole solutions as a function of ${\gamma }_1$ and ${\gamma }_2$ for fixed values of $\varphi$ and $y$ at $\tau =0$. The plot is obtained setting $e^{-{\varphi }_0}=0.001$, $y_0=10$ and $q_{E_{q1}}^{\prime }=q_{E_{p0}}^{\prime }=1$. Extremal instantons correspond to an absolute minimum of the action for ${\gamma }_1={\gamma }_2=0$.

Figure 4

Action of regular NS-NS wormhole solutions as a function of ${\gamma }_1$ and ${\gamma }_2$ for fixed values of $\varphi$ and $y$ at $\tau =0$. The plot is obtained setting $e^{-{\varphi }_0}=0.001$, $y_0=10$, and $q_E^{\prime }={\tilde{q}}_{Y_{-}}^{\prime }=1$. Extremal instantons correspond to an absolute minimum of the action.