The unbroken-phase program of "Space-Time and Topological Orbifolds": combining chiral topological sigma-model sectors, the mass-shell condition, and complex saddles with independent left/right moduli

Witten's 1988 paper "Space-Time and Topological Orbifolds" studies string propagation on a complexified spacetime quotiented by imaginary translations, interprets topological sigma-model correlators as scattering of strings with self-dual momentum and winding, and generalizes the Gross-Mende high-energy saddle to strings carrying both momentum and winding, finding that the saddle-point action becomes complex and consistency forces the world-sheet insertion points z_r and their conjugates to be varied as independent complex variables. The paper explicitly leaves open how these observations fit together: the physical meaning of the imaginary parts of the complexified target coordinates; the absence of a mass-shell condition and of any rationale for integrating over vertex insertion points; the status of complex saddle points with non-real action and whether a Wick rotation makes them physical; the rule combining the purely left-moving topological sigma model and its parity reversal into a single theory reproducing the conventional broken phase; and the consequences of independent left- and right-moving Riemann-surface moduli.

Background

In 1988 Edward Witten proposed, in "Space-Time and Topological Orbifolds" (Phys. Rev. Lett. 61, 670), to test the idea that topological quantum field theories are related not to the usual phases of physical theories but to a hypothetical phase of string theory in which general covariance and the higher symmetries of string theory are unbroken. The ingredients are as follows. A two-dimensional topological sigma model is a sigma model (a field theory of maps from a two-dimensional world sheet with coordinates $z,\bar z$ into a target manifold) whose target $M$ is complex and which possesses a fermionic Becchi-Rouet-Stora-Tyutin (BRST) symmetry generated by a nilpotent operator $Q$; its Hamiltonian is a $Q$-commutator, so only BRST ground states are physical, and correlation functions of BRST-invariant operators are computed exactly by classical instanton solutions, which here are holomorphic maps $\partial_{\bar z}X^{i}=0$ (the zero-action instantons of Floer and Gromov theory). Witten takes the target coordinates to be $d$ complex variables $X^{i}$, so spacetime is complexified, and quotients $M$ by the group $G$ of imaginary shifts $X^{i}\to X^{i}+2\pi i k^{i}$ with real $k^{i}$. String propagation on the orbifold $M'=M/G$ requires twisted sectors with boundary condition $X^{i}(\sigma+\pi)=X^{i}(\sigma)+2\pi i k^{i}$, each containing a single physical ground state with vertex operator $V_{k^{i}}$; the relevant instanton for $n$ insertions is $X^{i}_{\mathrm{class}}(z)=\sum_{r}k^{i}_{(r)}\ln(z-z_{r})$ and its action vanishes. In parallel, Gross and Mende had shown that very-high-energy, fixed-angle string scattering is governed by a classical world-sheet solution, $X^{i}_{\mathrm{GM}}=\frac{i}{2}\sum_{r}p^{i}_{(r)}[\ln(z-z_{r})+\ln(\bar z-\bar z_{r})]$, with a nonzero action and with the insertion points $z_{r}$ fixed by a saddle-point condition; macroscopic 'cosmic' strings, described instead by large windings $a^{i}_{(r)}$ with solution $X^{i}_{\mathrm{cosmic}}=\frac{i}{2}\sum_{r}a^{i}_{(r)}[\ln(z-z_{r})-\ln(\bar z-\bar z_{r})]$, are the winding-dominated counterpart: macroscopic fundamental strings, introduced as astronomical objects by Witten in 1985 ("Cosmic superstrings") and studied in collision by Polchinski in 1988. The vertex operator of an ordinary closed string carrying momentum $p$ and winding $a$ is $W_{p,a}=\exp\{\frac{1}{2}[(p+a)\cdot X_{L}+(p-a)\cdot X_{R}]\}$, where $X_{L},X_{R}$ are the left- and right-moving pieces of the target coordinate field; the holomorphic instanton $X_{\mathrm{class}}$ corresponds precisely to the self-dual case $p^{i}=a^{i}$, so the topological sigma model on $M'=M/G$ describes strings with a self-dual pattern of momenta and windings, and its parity reversal describes the anti-self-dual one. The scattering of strings that are energetic both through momentum and through winding is governed by the combined classical solution $X^{i}_{\mathrm{tot}}=\frac{i}{2}\sum_{r}[(p^{i}_{(r)}+a^{i}_{(r)})\ln(z-z_{r})+(p^{i}_{(r)}-a^{i}_{(r)})\ln(\bar z-\bar z_{r})]$, with action $I(z_{r},\bar z_{r})=-\frac{1}{2}\sum_{r<s}(p_{(r)}\cdot p_{(s)}+a_{(r)}\cdot a_{(s)})[\ln(z_{r}-z_{s})+\ln(\bar z_{r}-\bar z_{s})]-\frac{1}{2}\sum_{r<s}(p_{(r)}\cdot a_{(s)}+a_{(r)}\cdot p_{(s)})[\ln(z_{r}-z_{s})-\ln(\bar z_{r}-\bar z_{s})]$; this action is generically complex when both momenta and windings are nonzero. The saddle-point equations $\partial I/\partial z_{r}=0$ and $\partial I/\partial\bar z_{r}=0$ are then an overdetermined system if one keeps the usual reality condition $\bar z_{r}=(z_{r})^{*}$, so consistency requires treating $z_{r}$ and $\bar z_{r}$ as independent complex variables, meaning that the Riemann-surface moduli seen by left-moving modes are independent of those seen by right-moving modes. Two further pieces of standard terminology appear in the sequel: topological gravity denotes the topological field theory of two-dimensional gravity (the world-sheet metric degrees of freedom), whose coupling to a topological sigma model defines a topological string, with amplitudes given by integrals over the moduli space of Riemann surfaces; and Lefschetz thimbles are the cycles of steepest descent attached to complex saddle points, the modern tool for defining integrals with complex-valued actions. Witten remarks that coupling to topological gravity might supply the rationale for integrating over the vertex insertion points, though he does not pursue this. Witten's paper closes by acknowledging that the construction is 'guesswork': the correct assumptions for these problems were, in his words, unclear, and the combination of the chiral theory with its parity reversal 'remains to be discovered'.

Problem Statement

Give a precise, predictive world-sheet formulation of the 'unbroken phase' program proposed in Witten's "Space-Time and Topological Orbifolds" (1988): starting from the topological sigma-model Lagrangian $\mathcal{L}=\int dz\,d\bar z\,(g_{\bar i i}\partial_{z}\bar X^{\bar i}\partial_{\bar z}X^{i}+ig_{\bar i i}\chi^{\bar i}_{+}\partial_{\bar z}\psi^{i}+ig_{i\bar i}\chi^{i}_{-}\partial_{z}\psi^{\bar i})$ on the orbifold $M'=M/G$ of the complexified flat target by imaginary shifts, together with its parity reversal, the work should resolve, within one coherent framework, the five issues the paper explicitly leaves open: (1) the physical interpretation of the imaginary parts of the complexified target coordinates $X^{i}$, and whether their suppression by the imaginary-shift orbifold is legitimate or must be replaced by an unbroken-phase description that retains them; (2) the origin of a mass-shell condition on the twisted-sector labels $k_{(r)}$ and of the integration over the vertex insertion points $z_{r}$ in the topological-orbifold correlators; (3) the mathematical and physical status of saddle points whose action $I(z_{r},\bar z_{r})$ is not real, including whether and how a Wick rotation of world-sheet time renders such saddles physically meaningful; (4) an explicit rule for combining the purely left-moving and purely right-moving (anti-self-dual) topological sectors into a single theory that reproduces the conventional broken phase of string theory, with its correct spectrum and interactions, background dependence arising only through the combination rule; and (5) a concrete derivation and justification of the generalized saddle-point procedure in which $z_{r}$ and $\bar z_{r}$ are varied as independent complex variables, and the consequences of independent left- and right-moving Riemann-surface moduli for amplitude computations. The formulation must state the correct assumptions (which, in the paper's own words, were unclear) and its consequences must reproduce the known broken-phase results in the limiting cases where all windings vanish (the Gross-Mende high-energy saddle), where all momenta vanish (the macroscopic-string, winding-dominated saddle), and where momenta and windings coincide (the self-dual topological correlators). Each of the five parts is an independently required component of the resolution, and the resolutions must be mutually consistent within the single framework.

References

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