Euclidean Superspace, Conjugation, and Field-Space Complexification

Four-dimensional Euclidean supersymmetry is not obtained by replacing $t$ with $-i\tau$ while retaining every Lorentzian dagger relation. The connected spin group changes from $SL(2,\mathbb C)$ to $SU(2)_L\times SU(2)_R$; the two Euclidean Weyl representations are independent, and pointwise complex conjugation does not exchange them. A Euclidean supersymmetric functional integral therefore requires three separate records: the complexified algebra, an antilinear reflection or real structure when one exists, and the integration cycle in complex field space.

Required background. Supercovariant Derivatives, Chirality, and Integrability supplies the Lorentzian chiral multiplet and its conjugation. Euclidean Correlators and Schwinger Functions supplies Euclidean source, contour, and reflection-positivity distinctions.

Helpful background. Branches, Sheets, Continuation, and Monodromy supplies the analytic-continuation data needed when singularities or multivalued parameters are present.

The Euclidean spinor representations

In Lorentzian signature,

$$\operatorname{Spin}^+(1,3)\simeq SL(2,\mathbb C),$$

and complex conjugation exchanges the left and right Weyl representations. This makes

$$(\psi_\alpha)^\dagger=\bar\psi_{\dot\alpha}$$

a Lorentz-covariant local statement.

In four-dimensional Euclidean signature,

$$\operatorname{Spin}(4) \simeq SU(2)_L\times SU(2)_R.$$

A left spinor transforms as $(\mathbf2,\mathbf1)$ and a right spinor as $(\mathbf1,\mathbf2)$. Complex conjugation of an $SU(2)$ doublet is equivalent, using its invariant antisymmetric tensor, to another doublet of the same $SU(2)$ factor. It does not turn $(\mathbf2,\mathbf1)$ into $(\mathbf1,\mathbf2)$. Hence a single Euclidean Weyl pair admits no pointwise Lorentzian-style Majorana relation.

Zumino’s original analysis states this obstruction as the absence of a Hermitian minimal supersymmetry algebra in four-dimensional Euclidean space; its simplest useful form is complex, with independent spinorial parameters Zumino 1977, pp. 369–371. Symplectic-Majorana conditions can exist after adding a suitable internal doublet, but that is a different real form with more structure, not the continuation of one Lorentzian Weyl spinor by notation alone.

One explicit Wick-rotation convention

Set

$$x^0=-ix^4, \qquad \partial_0=i\partial_4,$$

and choose Euclidean matrices

$$\sigma_E^m = (\sigma^1,\sigma^2,\sigma^3,i\mathbf1), \qquad \widetilde\sigma_E^m = (\sigma^1,\sigma^2,\sigma^3,-i\mathbf1).$$

They satisfy

$$\sigma_E^m\widetilde\sigma_E^n +\sigma_E^n\widetilde\sigma_E^m =2\delta^{mn}\mathbf1.$$

The Lorentzian differential contractions continue as

$$\sigma^\mu\partial_\mu \longrightarrow \sigma_E^m\partial_m, \qquad \bar\sigma^\mu\partial_\mu \longrightarrow -\widetilde\sigma_E^m\partial_m.$$

The second minus sign is often absorbed into a phase redefinition of the right-handed spinor. That redefinition must also be made in the supersymmetry parameter, kinetic term, and transformation rules. This illustrates why $t=-i\tau$ is only one line of a spinor continuation.

After continuation, a convenient complexified algebra is

$$\{Q_\alpha,\widetilde Q_{\dot\beta}\} =2(\sigma_E^m)_{\alpha\dot\beta}P_m, \qquad \{Q,Q\}=\{\widetilde Q,\widetilde Q\}=0,$$

with $P_m=i\partial_m$ in the coordinate representation. The tilde is a representation label. It is not a claim that $\widetilde Q=Q^\dagger$.

Van Nieuwenhuizen and Waldron give a continuous spinor Wick rotation and display the phase and doubling choices needed to relate Lorentzian and Euclidean actions van Nieuwenhuizen and Waldron 1996, pp. 29–36. Their construction is one consistent prescription; other Euclidean real forms are possible, but mixing prescriptions is not.

A chiral multiplet becomes an independent pair

The Lorentzian fields are

$$\Phi=(A,\psi_\alpha,F), \qquad \Phi^\dagger=(A^*,\bar\psi_{\dot\alpha},F^*).$$

After complexification and continuation, write instead

$$\Phi_E=(A,\psi_\alpha,F), \qquad \widetilde\Phi_E =(\widetilde A,\widetilde\psi_{\dot\alpha},\widetilde F).$$

The Euclidean chiral constraints can be written

$$\widetilde D_{\dot\alpha}\Phi_E=0, \qquad D_\alpha\widetilde\Phi_E=0,$$

but

$$\widetilde A\ne A^*, \qquad \widetilde\psi\ne\psi^\dagger, \qquad \widetilde F\ne F^*$$

as algebraic identities on the complexified Euclidean field space. The two superfields transform independently under the complexified algebra.

The scalar part of a canonical kinetic term illustrates the continuation cleanly:

$$\int\mathrm d^4x_M\, \partial_\mu A^*\partial^\mu A \quad\longrightarrow\quad \int\mathrm d^4x_E\, \partial_m\widetilde A\,\partial_m A.$$

Only after the Euclidean action and contour have been chosen may one impose a cycle such as $\widetilde A=A^*$ for the bosonic variables. Fermionic Berezin variables are integrated independently even when an antilinear reflection relates their correlation functions. Auxiliary fields can require an additional phase or rotated contour so that the Gaussian integral represents the continued Lorentzian theory.

A compact translation record is therefore

Layer	Lorentzian statement	Euclidean statement
Representation	$\bar\psi$ is the conjugate right Weyl spinor	$\psi$ and $\widetilde\psi$ belong to independent $SU(2)$ factors
Algebra	$\bar Q=Q^\dagger$ in a unitary real form	$Q$ and $\widetilde Q$ first define a complex algebra
Fields	Barred components are pointwise conjugates	Tilded components are independent coordinates on complex field space
Reality	Hilbert-space adjoint at one spacetime point	An antilinear involution may include Euclidean-time reflection
Path integral	Real-time contour and $e^{iS}$	Chosen middle-dimensional cycle and $e^{-S_E}$

Reflection is not pointwise conjugation

For Lorentzian reconstruction, the relevant Euclidean antilinear operation is typically an Osterwalder–Schrader reflection

$$\Theta:\quad (\tau,\mathbf x)\longmapsto(-\tau,\mathbf x),$$

combined with complex conjugation and a spinor matrix that exchanges the two Euclidean chiral representations. Schematically,

$$\Theta A(\tau,\mathbf x) =\widetilde A(-\tau,\mathbf x),$$

while the spinor formula contains a convention-dependent matrix implementing the reflected frame. Because $\Theta$ changes the point, it is not the same assertion as $\widetilde A(\tau,\mathbf x)=A(\tau,\mathbf x)^*$.

Reflection positivity is a property of Euclidean correlation functions and their supported test functionals. It is neither guaranteed by a supersymmetric action nor equivalent to the positivity of the bosonic integrand. The foundational reconstruction conditions are stated by Osterwalder and Schrader 1973, pp. 83–112; the exact hypotheses and scalar reflection form used on this site are developed on Euclidean Correlators and Schwinger Functions.

Complexification and the integration cycle

A Euclidean supersymmetric path integral has the schematic form

$$Z_E = \int_{\mathcal C} \mathcal D A\,\mathcal D\widetilde A\, \mathcal D\psi\,\mathcal D\widetilde\psi\, \mathcal D F\,\mathcal D\widetilde F\; e^{-S_E}.$$

The holomorphic field space and the cycle $\mathcal C$ answer different questions:

• the complexified space is where the supersymmetry algebra and holomorphic constraints act;
• the cycle selects an actual integral and encodes convergence and reality;
• an antilinear involution may map the cycle to itself and support a reconstruction interpretation;
• a chosen supercharge $Q_{\rm loc}$ may preserve the cycle even when the full complex supersymmetry algebra does not.

Localization often deforms $\mathcal C$ into a union of steepest-descent cycles or treats fields and their tilded partners as independent. Such a deformation is valid only if the integrand is holomorphic in the required region, the endpoints and asymptotics are controlled, and no singularity is crossed. Supersymmetry of the formal integrand does not prove contour equivalence.

Gauge theories add another layer: the Euclidean connection may be integrated over a real compact-gauge contour, complexified to solve saddle equations, or decomposed into independent holomorphic variables. The global gauge form, bundle sector, and gauge-fixing contour must accompany that choice.

Reverse continuation has hypotheses

Starting from a Euclidean formula, one cannot infer a Lorentzian unitary theory merely by setting $x^4=ix^0$. Reverse continuation requires:

1. Euclidean covariance and suitable regularity of the correlation functions;
2. an antilinear reflection and reflection positivity when a Hilbert-space reconstruction is claimed;
3. analytic domains reaching the desired Lorentzian boundary values;
4. compatible spinor, gauge, and source conventions;
5. a contour whose deformation back to the Lorentzian prescription is justified.

A complex supersymmetric saddle or a $Q$-closed observable may be mathematically useful without satisfying all of these requirements. The resulting claim should then remain Euclidean or cohomological rather than being promoted to a Lorentzian unitary statement.

Common pitfalls

Bar does not always mean complex conjugation. In Euclidean supersymmetry, using a bar as an index label and as an adjoint operation in the same formula hides the independent-field structure. Tildes make the distinction visible.

A real bosonic contour need not preserve all supersymmetries. The complex algebra may move the contour. State which supercharge or real form preserves the chosen cycle.

Reflection positivity is not manifest action reality. An action can be invariant under an antilinear reflection while its local density is not pointwise Hermitian; conversely, a real-looking density need not yield the required reflected quadratic form.

Exercises

1. Check the Euclidean sigma algebra

Verify the relation for $m=n=4$ and for one mixed pair $(m,n)=(i,4)$.

Solution

For $m=n=4$,

$$2(i\mathbf1)(-i\mathbf1)=2\mathbf1.$$

For $(i,4)$,

$$\sigma^i(-i\mathbf1)+(i\mathbf1)\sigma^i=0.$$

These are respectively $2\delta^{44}\mathbf1$ and $2\delta^{i4}\mathbf1$.

2. Separate field space from contour

Does imposing $\widetilde A=A^*$ on one integration cycle restore the algebraic identity $\widetilde\Phi_E=\Phi_E^\dagger$ everywhere in complexified superspace?

Solution

No. It selects a middle-dimensional subset of the complex bosonic field space. The complexified supersymmetry transformations, saddle equations, and allowed contour deformations are still defined using independent coordinates. Moreover, the spinors and auxiliaries require their own phase and reflection prescriptions.

3. Diagnose a localization claim

A $Q_{\rm loc}$-exact deformation leaves the formal derivative of the integrand equal to a $Q_{\rm loc}$ variation. What additional check is needed for the integral to be deformation independent?

Solution

One must show that the chosen integration cycle is $Q_{\rm loc}$ invariant or that boundary contributions vanish, and that no pole, Stokes wall, or singular locus invalidates the contour deformation. The algebraic $Q_{\rm loc}$ identity alone does not control integration boundaries.

Continue

Off-Shell Closure and Auxiliary Fields returns to Lorentzian component closure. Rigid Supersymmetry on Curved Backgrounds uses the Euclidean real-form and contour distinctions on nontrivial geometries.

References

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI. Open PDF.

• van Nieuwenhuizen, Peter, and Andrew Waldron. “On Euclidean Spinors and Wick Rotations.” Physics Letters B 389 (1996): 29–36. DOI. arXiv:hep-th/9608174.

• Zumino, Bruno. “Euclidean Supersymmetry and the Many-Instanton Problem.” Physics Letters B 69 (1977): 369–371. DOI. CERN record.