The elliptic genus and hidden symmetry (Q5940420)

The partition function (elliptic genus) of a coupled complex bosonic and fermionic quantum field on \(S^1\times \mathbb{R}\) is evaluated. The equations, Wess-Zumino equations or Landau-Ginzburg equations, are determined by a superpotential \(V\), an \(\widetilde n\)-th degree polynomial, \(\widetilde n\geq 2\). The complex scalar fields \(\varphi= \{\varphi_i\}\) and the Dirac fields \(\psi= \{\psi_{\alpha,i}\}\) have \(n\) and \(2n\) components respectively. They are twist fields, that is a translation about the circle results in each component of the field being multiplied by a phase proportional to a real parameter \(\varphi\in (0,2\pi]\). Twisting partially breaks supersymmetry [\textit{A. Jaffe}, Proc. Natl. Acad. Sci. USA 97, No. 4, 1418-1422 (2000; Zbl 0980.58004)]. Half the supercharges are translation and twist invariant.The elliptic genus can be written as a function of the invariant charges denoted by \(Q\). The partition function is \[ {\mathcal Z}^V= \text{Tr}_{\mathcal H} (\Gamma e^{-i\theta J-i\sigma P-\beta H}). \] Here \({\mathcal H}={\mathcal H}^b \otimes{\mathcal H}^f\) is the Hilbert space of the theory, where the bosonic and fermionic Hilbert spaces \({\mathcal H}^b\) and \({\mathcal H}^f\) are the symmetric and respectively the skew-symmetric tensor algebras over the one particle space \({\mathcal K}\), the direct sum of \(2n\)-copies of \(L^2(S^1)\). \(\Gamma\) is \((-1)^{N^f} \), \(N^f\) the fermion number operator, \(J=J(V)\) the generator of the twist symmetries of \(H=H(V)\), the Hamiltonian of the theory which is determined by \(V\), and \(P\) is the momentum operator. \(Q\) is related to \(H\) an \(P\) by the equation \(Q^2= H+P\). In [\textit{E. Witten}, Int. J. Mod. Phys. A 9, 4783-4800 (1994)], it is suggested to compute \({\mathcal Z}^V\), consider \({\mathcal Z}^{\lambda V}\), then it should be independent of \(\lambda\), so the evaluation of \({\mathcal Z}\) reduces to the case \(V=0\). But to show continuity of \({\mathcal Z}^{\lambda V}\) with respect to \(\lambda\) and to evaluate \({\mathcal Z}^V\) when \(V=0\) (this case is ill-defined) are both difficult problems. In this paper, the author overcomes these difficulties mainly by applying his results in; [`Twist fields and constructive quantum field theory', in preparation, hereafter referred to as [1], and \textit{O. Grandjean} and \textit{A. Jaffe}, J. Math. Phys. 41, No. 6, 3698-3763 (2000; Zbl 0974.58022), hereafter, referred to as [2]], for a class of \(V\), \(J\) and \(P\). The answer is expressed in terms of Jacobi's theta function of the second kind \(\vartheta_1 (\tau, \theta)\), where \(\tau= (\sigma+ i\beta)/l\), \(l\) is the radius of \(S^1\). Hence there exists a hidden SL\((2,\mathbb{Z})\)-symmetry in the theory. The outline of the paper is as follows: In Section 1, assumptions on \(V\), \(J\) and \(P\), quasi-homogeneous (QH), elliptic property (EL), twist relations (TR) and normalization condition (NC), are stated. QH means existence of \(n\) constants \(\Omega_i\) called quasi-homogeneous weights, such that \(0<\Omega_i \leq 1/2\) and \(V(z)=\sum^n_{i=1} \Omega_iz_i \partial V(z)/ \partial z_i\). EL means existence of a constant \(M\) such that \(|\partial^\alpha V|\leq \varepsilon|\partial V|^2+M\) and \(|z|^2+ |V|\leq M(|\partial V |^2+1)\), for a given \(\varepsilon>0\). TR determines twist angles \(\chi\) to be \(\chi^b_i= \Omega_i\varphi\), \(\chi^f_{1,i}= \Omega_i\varphi\) and \(\chi^f_{2,i}= (1-\Omega_i) \varphi\), where \(\varphi\in (0,\pi]\) is a single twisting parameter. NC determines \(J\) and \(P\) to be \(P\Omega_{\text{vac}}=0\) and \(J \Omega_{ \text{vac}} =-\frac 12\widehat c\Omega_{\text{vac}}\), where \(\widehat c= \sum^n_{i=1} (1-2\Omega_i)\). Then it is shown \[ {\mathcal Z}^V(\tau, \theta, \varphi) =z^{\widehat c/2} \prod^n_{i=1}{f{\vartheta_1} \bigl(\tau, (1-\Omega_i) (\theta- \varphi\tau) \bigr) \over\vartheta_1 \bigl(\tau, \Omega_i (\theta-\varphi \tau)\bigr)}.\tag{1} \] By this formula, it follows \(\lim_{\varphi\to 0}{\mathcal Z}^V= \prod^n_{i=1} \vartheta_1 (\tau,(1-\Omega_i) \theta)/ \varphi_1 (\tau,\Omega_i \theta)\), which was obtained in [\textit{T. Kawai}, \textit{Y. Yamada}, and \textit{S.-K. Yang}, Nucl. Phys B 414, No. 1-2, 191-212 (1994; Zbl 0980.58500)]. The integer-valued index \(\text{Index}_\Gamma(Q)\) of the supercharge is obtained as \[ \text{Index}_\Gamma(Q) =\lim_{\theta\to 0}\Bigl( \lim_{\varphi \to 0}{\mathcal Z}^V \Bigr)= \lim_{\varphi \to 0}\Bigl( \lim_{\theta \to 0} {\mathcal Z}^V\Bigr)= \prod^n_{i=1} \left({1\over \Omega_1}- 1\right). \] These results are summarized in Section 2. The supercharge \(Q=Q(V)\) with a real parameter \(\lambda\in [0,1]\) is constructed in Section 3. \(Q(\lambda)\) is expressed as \(Q_0+ \lambda Q_I\). Mollifiers \(Q_\Lambda (\lambda)\), \(\Lambda\in [0,\infty]\) of \(Q(\lambda)\) such that \(Q_0(\lambda) =Q_0\) and \(Q_\infty (\lambda)=Q(\lambda)\) are constructed in Section 4. The Hamiltonian \(H(\lambda V)\) and \(Q(\lambda V)\) are related by \(Q(\lambda V)^2 =H(\lambda V)+P\). Approximating Hamiltonians \(H_\Lambda\) are defined similarly by using \(Q_\Lambda\). It is shown in [1] that the self-adjoint quantum field twist Hamiltonian \(H(V)\) is the norm-resolvent limit of \(H_\Lambda(V)\) and \(e^{-\beta H(V)}\) is trace class for \(\beta>0\). Moreover, \(H\) and \(H_\Lambda\) both commute with \(e^{i\beta J+i \sigma P}\) and \(\Gamma=(-2)^{N^f}\) (stated as Prop. 1.1). In Section 5, several estimates on the Hamiltonian and supercharge are given. The author says that precise proofs of estimates are given in [1] and detailed definitions of several operators appearing in this Section are given in [2]. By using results in Section 5, differentiability of the map \(\lambda\to {\mathcal Z}_\Lambda^{\lambda V}\) and \(\partial {\mathcal Z}_\Lambda^{\lambda V}/\partial \lambda =0\) (Th.6.1), and estimate \(|{\mathcal Z}_\Lambda^{\lambda V}-{\mathcal Z}^0 |\leq M\lambda^\alpha\), \(0<\lambda\leq 1\) and \(0\leq\alpha <2/(\widehat n-1)\) (Th.6.2) are proved in Section 6. By Theorems 6.1 and 6.2, we have \({\mathcal Z}^V (\tau,\theta, \varphi)= {\mathcal Z}^V_\Lambda (\tau,\theta, \varphi)= {\mathcal Z}^0 (\tau,\theta, \varphi)\). This function is shown to be holomorphic for all \(\tau\in \mathbb{H}\), the upper halfplane, for all \(\theta\in \mathbb{C}\), and extends to a holomorphic function of \(\varphi\). These are shown in Section 7. Then apply the results in [2], \({\mathcal Z}^0\) is evaluated in Section 7 and (1) is obtained.