Reflection Positivity, Unitarity, and Hermitian TQFTs

A Hermitian TQFT intertwines orientation reversal of bordisms with a dagger or complex-conjugation operation in its target. Reflection positivity is stronger: when a bordism is doubled across a reflection hypersurface, the induced sesquilinear form on the state space must be positive definite. For a semisimple two-dimensional Frobenius theory this reduces to positivity of the trace weights in a self-adjoint idempotent basis. Duals, adjoints, and cobordism-hypothesis data alone do not imply that inequality. The result is a positive functorial topological theory in the declared Euclidean setting, not by itself a reconstruction theorem for a general Lorentzian QFT.

Required background. Bordism categories and symmetric monoidal TQFTs supply cutting and duality; the cobordism hypothesis supplies full dualizability; Osterwalder–Schrader reflection positivity supplies the general Euclidean positivity mechanism; and Hilbert positivity and unitary evolution distinguish positive inner products from formal adjoints.

Helpful background. Topological order and invertible phases supplies the physical comparison without identifying mathematical TQFT positivity with microscopic unitarity.

Hermitian functoriality and reflected doubles

Let $\overline M$ denote the bordism obtained from $M$ by reversing its orientation and exchanging incoming and outgoing boundaries. A Hermitian target has a conjugate-linear involution and a dagger on morphisms. A Hermitian TQFT satisfies coherent equivalences

$$Z(\overline\Sigma)\simeq\overline{Z(\Sigma)}, \qquad Z(\overline M)=Z(M)^\dagger.$$

Now cut a closed reflected manifold $D_\Sigma M$ into $M$ and its mirror $\overline M$ along a closed hypersurface $\Sigma$. If $M$ prepares a state $v=Z(M)\in Z(\Sigma)$, functoriality gives

$$Z(D_\Sigma M)=\langle v,v\rangle_\Sigma.$$

Reflection positivity requires this number to be nonnegative for every state prepared on one side; after quotienting any null vectors, positivity is definite on the physical state space. For an extended theory, the reflection structure must be compatible at every lower codimension, not merely on closed partition functions. Freed and Hopkins distinguish reflection structures, Hermitian values, and extended positivity in Freed and Hopkins 2021, §§4.3 and 8.1, pp. 29–32 and 63–68.

This is the topological specialization of the Osterwalder–Schrader idea. A general OS theorem additionally needs a Euclidean field algebra, regularity, covariance, and analytic reconstruction data. A finite TQFT has no local propagating energy spectrum to reconstruct, so positivity here should not be advertised as a proof of unitarity for an arbitrary metric-dependent theory.

The semisimple Frobenius test

Let a two-dimensional oriented TQFT have

$$A=\bigoplus_{i=1}^{r}\mathbb C p_i, \qquad p_ip_j=\delta_{ij}p_i, \qquad \varepsilon(p_i)=\theta_i\ne0.$$

Choose the involution $p_i^*=p_i$ and complex conjugation on coefficients. Reflection of a cap supplies the Hermitian form

$$\langle a,b\rangle=\varepsilon(a^*b).$$

Writing $a=\sum_i a_i p_i$ gives

$$\langle a,a\rangle =\sum_{i=1}^{r}\theta_i\,|a_i|^2.$$

Therefore the theory is positive for this real structure precisely when every $\theta_i>0$. Nondegeneracy of the Frobenius pairing only requires $\theta_i\ne0$; it does not fix the signs. Positivity is thus additional to the algebraic sewing axioms.

The exact first application is to reflect a two-dimensional bordism across its boundary circle, derive this state-space form, and test it in a semisimple Frobenius model. The result is returned to topological order and invertible phases, where microscopic phase diagnostics remain the subject. Here the calculation is exact, finite-dimensional, and conditional on the chosen involution and trace.

As an independent sewing check, glue the reflected cylinder to states $a$ and $b$ in either order. Hermitian functoriality requires

$$\langle Z(M)a,b\rangle =\langle a,Z(\overline M)b\rangle.$$

In the idempotent basis this becomes ordinary adjointness of diagonal matrices with the positive weight matrix $\operatorname{diag}(\theta_i)$. The check simultaneously tests the reflection convention and the gluing pairing.

Dualizability without positivity

Take $A=\mathbb C p_+\oplus\mathbb C p_-$ with

$$\varepsilon(p_+)=1, \qquad \varepsilon(p_-)=-1.$$

The bilinear Frobenius form is nondegenerate, the algebra is semisimple, and the associated point object is fully dualizable in the Morita $2$-category. All oriented cutting and gluing relations still hold. Yet

$$\langle p_-,p_-\rangle=-1.$$

The negative-norm state is the adversarial fixture demanded by the classification boundary. The strongest surviving conclusion is a Hermitian or algebraic TQFT with duals and gluing, not a reflection-positive theory. Neither changing terminology to “unitary” nor invoking the cobordism hypothesis repairs the failed inequality; one must change the trace or restrict to a positive subtheory in a way compatible with all bordisms.

Reflection positivity also depends on the tangential structure. For pin or spin theories the reflection lift and fermion parity enter the conjugation law. A positive oriented model cannot simply be reused after changing that structure without checking the new involution.

Exercises

Show that positivity forces all semisimple trace weights to be real and positive.

Solution

Since $p_i=p_i^*$, its norm is $\langle p_i,p_i\rangle=\theta_i$. A Hermitian positive-definite form has real positive diagonal values on every nonzero vector, hence $\theta_i>0$ for all $i$. Conversely, positive weights make $\sum_i\theta_i|a_i|^2$ positive for every nonzero $a$.

Does a positive partition function on every closed surface prove reflection positivity?

Solution

No. Reflection positivity tests the full Gram matrix of states prepared by bordisms, including off-diagonal pairings and lower-codimension compatibility. Closed partition functions probe only selected contractions. A positive list of those scalars need not make every state-space form positive.

References

• Atiyah, Michael F. “Topological Quantum Field Theory.” Publications Mathématiques de l’IHÉS 68 (1988): 175–186. DOI; Open PDF.
• Freed, Daniel S., and Michael J. Hopkins. “Reflection Positivity and Invertible Topological Phases.” Geometry & Topology 25 (2021): 1165–1330. DOI; Open PDF.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.