Which geometric data does the partition function see?

Can a partition function remain unchanged when the space on which the fields live becomes curved? A useful research exercise is to construct the deformation, verify its supersymmetry, and ask exactly which conclusion follows. The answer involves more than the metric: the preserved foliation, transverse complex structure, background bundles, and contact prescription all matter.

Prepare with the reproduction method, global spinor patching, the linearized current calculation, and source derivatives and contact terms. The project below is an established-result reproduction followed by a training extension. It is not a proposal that compatible-metric independence is new or unresolved.

Specify the quantity and the geometry

Use a three-dimensional N=2 theory, with four real supercharges in its Lorentzian parent. For a concrete model take one free chiral multiplet, no dynamical gauge group or superpotential, scalar R-charge zero, and flavor charge $+1$. A constant background flavor scalar $m>0$ gives a real fermion mass and positive scalar mass squared $m^2$. This choice is not a superconformal R-assignment. We will construct a particular preserved odd charge $Q$, labeled by a commuting spinor; we do not assume that all parent charges survive.

The observable is the normalized change $\mathcal F(\varepsilon)-\mathcal F(0)$, where $\mathcal F=-\log Z$, with the same supersymmetric regulator and local source counterterms throughout. A continuous logarithm is used in a neighborhood with $Z\ne0$. A single chiral has a fermion contact-term prescription; we do not set those terms to zero by assumption. Keeping that prescription is part of defining the observable, even for free matter.

Take a three-torus with length coordinates

$$\tau\sim\tau+\beta,\qquad x\sim x+2\pi L,\qquad y\sim y+2\pi L, \qquad \operatorname{vol}_{\rm coord}=d\tau\wedge dx\wedge dy, \tag{D01.1}$$

where $\beta,L>0$. Fix the periodic spin structure in all three directions, a trivial R-bundle, and the product identifications. Periodic fermions in the $\tau$ direction give an index-type path integral, not a thermal partition function. The positive mass removes the scalar zero-mode problem at the undeformed point; it does not dispense with a regulator. CK19, section 3.1 explains the index and its spectral qualifications.

A transversely holomorphic foliation, or THF, consists here of the circles generated by $\xi=\partial_\tau$ and a complex coordinate on the two directions transverse to them. Use the site’s convention $z=x-iy$. On overlapping patches the transverse changes must be holomorphic, $z'=F(z)$, while the leaf coordinate can shift by a real function of $z,\bar z$. The torus identifications above are a particularly simple example. A compatible metric can change lengths without changing those leaves or transverse coordinate transitions.

Our first family is

$$ds^2=d\tau^2+e^{2f(x)}(dx^2+dy^2),\qquad f(x)=\varepsilon\cos(x/L),\qquad \eta=d\tau . \tag{D01.2}$$

The parameter $\varepsilon$ is dimensionless. The positive exponential gives a smooth metric for every finite real $\varepsilon$. Neither $\xi$ nor $z$ changes. This is a compatible-metric deformation, rather than a deformation of the THF. The distinction is the one formalized in G, section 5.4.

Reproduce the finite background

Use the oriented frame $e^1=d\tau$, $e^2=e^f dx$, $e^3=e^fdy$ and $\gamma=(\sigma_3,-\sigma_1,-\sigma_2)$, where $\sigma_i$ are the Pauli matrices. Cartan’s equation $de^a+\Omega^a{}_b\wedge e^b=0$ gives $\Omega_{23}=-f'(x)dy$; the other independent connection forms vanish. Thus the spin connection is $\Gamma=-if'(x)dy\,\gamma_1/2$.

The charge-one parameter equation in P (1.4) is

$$(\nabla_a-iA_a)\zeta =-\frac H2\gamma_a\zeta-iV_a\zeta -\frac12\epsilon_{abc}V_b\gamma_c\zeta. \tag{D01.3}$$

Here $A$ is the paper’s R-connection, $H$ its scalar auxiliary source, and $V=-i*dC$ the dual field strength of the central-charge source $C$. Their frame components have dimension inverse length. They are background fields, not fields to be eliminated by an equation of motion.

Consider the exact answer

$$H=V=C=0,\qquad A=-\frac{f'}2dy-\frac i2df,\qquad \zeta=e^{f/2}u_+,\qquad u_+=\begin{pmatrix}1\\0\end{pmatrix}. \tag{D01.4}$$

An overall constant supplies the parameter’s engineering dimension. To verify every component, use $\gamma_1u_+=u_+$. The $x$ equation contains $\partial_x\zeta=f'\zeta/2$ and $-iA_x\zeta=-f'\zeta/2$, which cancel. The $y$ equation contains $\Gamma_y\zeta=-if'\zeta/2$ and $-iA_y\zeta=+if'\zeta/2$, which also cancel. The $\tau$ equation is zero on both sides. The parameter is periodic and nowhere zero in a global frame, so this verifies a global solution, not just a local algebraic kernel.

The imaginary term $-i\,df/2$ is a globally defined Euclidean source. It has not been removed by an ordinary real $U(1)_R$ gauge transformation. Deleting it while keeping $\zeta$ fixed would leave a residual $f'\zeta/2$ in the $x$ equation. Complex source continuation and a real gauge redundancy are different operations.

For the flavor multiplet take $\sigma_F=m$, $a_F=D_F=0$ and zero background fermions. This obeys the sufficient condition $a_F=-mC$, $D_F=-mH$ in P (6.26). That condition preserves any charges already admitted by the supergravity background; it does not manufacture four charges on a generic geometry.

There is an independent geometric check. In the paper’s convention, a round sphere has negative scalar curvature. Direct coordinate Christoffels give

$$R_P=2e^{-2f}f'' =-\frac{2\varepsilon}{L^2}e^{-2\varepsilon\cos(x/L)}\cos(x/L). \tag{D01.5}$$

It is nonzero for $\varepsilon\ne0$, so the deformation is not merely a coordinate rewriting of the flat metric. Periodicity makes $\int\sqrt g\,R_P=0$, but not $\int\sqrt g\,R_P^2$. Define the modified Bessel function by $I_n(t)=(2\pi)^{-1}\int_0^{2\pi}e^{t\cos u}\cos(nu)\,du$ for integer $n$. Using $\cos^2u=(1+\cos2u)/2$ gives

$$\int_{T^3}\sqrt g\,R_P^2 =\frac{8\pi^2\beta\varepsilon^2}{L^2} \left[I_0(2\varepsilon)+I_2(2\varepsilon)\right]>0. \tag{D01.6}$$

This checks curvature, orientation-independent positivity, dimensions, and the flat limit at once.

Separate a tangent calculation from a finite conclusion

Write $f=\varepsilon f_1$. At zero, the source variation from (D01.4) is precisely the $\tau$-independent deformation of the earlier R-multiplet calculation. With its current normalization, define

$$\begin{aligned} \mathcal O_\perp={}&T_{22}+T_{33} -\frac i2(\partial_xj_{R2}+\partial_yj_{R3})\\ &-\frac12(\partial_xj_{R3}-\partial_yj_{R2})-\partial_\tau J_Z =\frac i2Q\mathcal W,\\ \mathcal W={}&\widetilde{\mathcal S}_{2,2}-i\widetilde{\mathcal S}_{3,2}. \end{aligned} \tag{D01.7}$$

Here $\widetilde{\mathcal S}$ is the opposite-charge supercurrent and $J_Z$ is the scalar central-current partner. The canonical derivation is in the linearized lesson; no superspace assumption is hidden in this identity. For our profile, periodic integration by parts yields

$$S'(0)=-\int d^3x\,f_1\mathcal O_\perp =Q\left[-\frac i2\int d^3x\,f_1\mathcal W\right]. \tag{D01.8}$$

For example, the term $f_1\partial_xj_{R3}/2$ becomes $-f_1'j_{R3}/2$, reproducing the real $A_y$ source. The divergence partner gives the imaginary $A_x$ term. Ignoring either source would spoil the match.

An invariant integration cycle and an anomaly-free regulated $Q$ Ward identity imply $\mathcal F'(0)=0$. Our exact solution of the geometric equation does not by itself turn (D01.8) into a proof at every $\varepsilon$. The action, charge, measure and composite operators all change. In a fixed source prescription,

$$\mathcal F''(0)=\langle S''(0)\rangle -\langle S'(0)S'(0)\rangle_c, \tag{D01.9}$$

including the local terms at coincident insertions. Computing only the separated-point product misses part of the response.

G states compatible-metric independence at fixed THF, with locally holomorphic dependence on THF and flavor-bundle moduli. Its introduction specifies counterterm and anomaly assumptions and says that the analysis displayed there is linearized. G, introduction and (6.10)–(6.17)

The finite implication has a precise logical form: if the completed Ward argument applies at each point of a connected, nonsingular family with the same holomorphic bundle data, auxiliary-source class and contact prescription, then $d\mathcal F/d\varepsilon=0$ throughout that family and its integral vanishes. This is stronger input than the one tangent we reproduced. It also requires control of the integration contour and of any zero modes or singularities encountered along the path. A statement that allowed moduli can affect $Z$ does not say that every theory detects every modulus.

Watch an unprotected part of the spectrum change

For our R-neutral free scalar, (D01.4) and P (6.32) give the actual quadratic operator $-\Delta_g+m^2$. Restrict to modes independent of $\tau$ and write $\phi=e^{iny/L}v(u)$, $u=x/L$, with nonzero integer $n$. A Laplacian eigenvalue $\lambda$ obeys

$$\left[-\frac{d^2}{du^2}+n^2\right]v(u) =\Lambda e^{2\varepsilon\cos u}v(u),\qquad \Lambda=L^2\lambda,\qquad v(u+2\pi)=v(u). \tag{D01.10}$$

The scalar action eigenvalue is $m^2+\lambda$. This is a generalized eigenproblem because the metric changes the inner-product weight. Its lowest eigenvalue in this fixed $n$ sector minimizes

$$\Lambda[v]=\frac{\int_0^{2\pi}(|v'|^2+n^2|v|^2)\,du} {\int_0^{2\pi}e^{2\varepsilon\cos u}|v|^2\,du}. \tag{D01.11}$$

Bounding the denominator and then trying $v=1$ gives $n^2e^{-2|\varepsilon|}\le\Lambda_0\le n^2/I_0(2\varepsilon)$. The upper bound is not an exact eigenvalue: a constant $v$ does not solve (D01.10) at nonzero warp.

For $n=1$, set $v=1+\varepsilon v_1+\cdots$ with zero mean for each correction. At first order $-v_1''=2\cos u$, so $v_1=2\cos u$. Averaging the second-order equation gives $0=\Lambda_2+\langle2\cos u\,v_1+2\cos^2u\rangle=\Lambda_2+3$. Translation $u\mapsto u+\pi$ reverses $\varepsilon$ without changing the spectrum. Therefore $\Lambda_0=1-3\varepsilon^2+O(\varepsilon^4)$.

As a finite-input check, a Fourier variational calculation gives $\Lambda_0=0.89828363$ at $\varepsilon=0.2$ and $0.61072878$ at $\varepsilon=0.5$. Independently integrating the even periodic differential equation reproduces these eight displayed digits; halving the integration step reduces its error by approximately $16$, as expected for fourth-order integration. These are converged numerical diagnostics, not rigorous spectral error certificates.

Thus ordinary scalar levels move. Metric independence concerns the complete supersymmetric path integral with its fermions, auxiliary fields and measure. A cutoff on scalar eigenvalues alone is not a supersymmetric regulator, and its changing determinant is not a counterexample.

Change one more input while keeping the relevant sources fixed

For a bounded extension, keep $f=\varepsilon\cos(x/L)$ and change only the metric’s leaf connection:

$$\eta=d\tau+q(x)dy,\qquad q(x)=\alpha\cos(x/L),\qquad ds^2=\eta^2+e^{2f}(dx^2+dy^2). \tag{D01.12}$$

The real parameter $\alpha$ is dimensionless. The vector $\xi=\partial_\tau$, leaves and transverse coordinate transitions remain fixed. The metric-dual one-form $\eta$ changes; it is not by itself the fixed THF data.

Set $t=e^{-2f}q'$, a quantity of dimension inverse length. Cartan’s connection in the frame $(\eta,e^fdx,e^fdy)$ is

$$\Omega_{12}=\frac t2e^3,\qquad \Omega_{13}=-\frac t2e^2,\qquad \Omega_{23}=-\frac t2e^1-f'dy.$$

Solving (D01.3), or applying P (4.5), (4.14), (4.19), gives

$$\begin{gathered} V=t\eta=*d\eta,\qquad H=\frac{it}{2},\qquad A=t\eta-\frac{f'}2dy-\frac i2df,\qquad \zeta=e^{f/2}u_+,\\ C=iq\,dy,\qquad B=-\frac q2d\tau\wedge dy. \end{gathered} \tag{D01.13}$$

Here $B$ is the background two-form with $H=i*dB$. All potentials are global and periodic; $V=-i*dC$ and $d*V=d^2\eta=0$. For example the first frame equation has $\Gamma_1\zeta=-it\zeta/4$, so its left side is $-5it\zeta/4$; the right side is $-H\zeta/2-it\zeta$, the same result. The transverse components are checked in Exercise 2.

To keep the flavor data fixed, continue its background as

$$\sigma_F=m,\qquad a_F=-mC,\qquad D_F=-mH, \qquad \mathcal A_F\equiv a_F+i\sigma_F\eta=im\,d\tau. \tag{D01.14}$$

The complex connection $\mathcal A_F$ packages the flavor holomorphic line-bundle data: its relevant components define the holomorphic transition problem. Keeping the whole connection fixed is a sufficient, stronger condition than keeping only its equivalence class. This application of G, section 6.3 avoids accidentally changing a flavor source while comparing geometries. The central potential’s flat part is fixed as $C=i\eta-i\,d\tau$, and we make no extra shift by the auxiliary freedoms denoted $U,\kappa$ in the sources.

For $\alpha\ne0$, $a_F,D_F,H$ are complex. Start from the convergent massive free theory at $\alpha=0$ and use a specified analytic continuation near that point, deforming the field contour only when its integral remains justified. Do not call this a positive action on an unchanged real contour, or infer a globally unrestricted continuation. For a first project, fix $\varepsilon=0.2$, $mL=1$, $\beta/L=1$ and vary $\alpha$ near zero. Reconstruct the sources, verify (D01.14), and test the Ward response with a regulator that preserves the chosen charge. The geometric identities supply exact expected outputs; a regulator-level partition-function calculation is the next task, not a result asserted by the scalar check above.

Read the literature with the claim boundaries intact

The literature review for this dossier was made on 20 October 2026. Read P, sections 4 and 7 for reconstruction and the flat current test; then G, sections 5–6 for the geometric interpretation. The later twisted-holomorphic formulation develops a nonlinear argument in four dimensions and sketches its three-dimensional extension in section 6.1. It retains measure, counterterm and anomaly qualifications. CK19, section 3.3 places these structures in the later framework of three-dimensional partition functions. This sequence supports an established-result training problem; the dated search is not evidence of novelty or an exhaustive classification of subsequent work.

Two translation details prevent false disagreements. G uses $A^{(R)}=A_P-3V/2$ and calls $x+iy$ its complex coordinate, the conjugate of our $z$. CK19 additionally warns in footnote 10 that its reconstruction uses the opposite $\eta$ relative to P. Equations (D01.3)–(D01.14) use P’s conventions directly. Changing the name of the complex coordinate in a source comparison is different from varying its torus periods in a physical family.

Exercises

1. Change the harmonic and distinguish a zero integral from flatness

Replace $f$ by $\varepsilon\cos(px/L)$ with a positive integer $p$. Find the supersymmetry sources, scalar curvature, and integrated curvature square. Does the vanishing of the integrated scalar curvature show that the metric is flat, or that its supersymmetric partition function is constant?

Hint 1

Differentiate $f$ before rescaling the integration variable. There are $p$ full periods in the $x$ circle.

Hint 2

The average of a periodic function is unchanged by replacing its argument by $pu$, but each derivative brings a factor $p$.

Solution

The global spinor is $e^{f/2}u_+$ and $H=V=C=0$. The R-source is $A_y=\varepsilon p\sin(px/L)/(2L)$, $A_x=i\varepsilon p\sin(px/L)/(2L)$, $A_\tau=0$. The same two component cancellations verify supersymmetry. Curvature is $R_P=-2\varepsilon p^2e^{-2f}\cos(px/L)/L^2$. Multiplication by $\sqrt g=e^{2f}$ removes the exponential, so its integral vanishes by periodicity. Squaring instead gives

$$\int\sqrt g\,R_P^2 =\frac{8\pi^2\beta\varepsilon^2p^4}{L^2} \left[I_0(2\varepsilon)+I_2(2\varepsilon)\right].$$

For nonzero $\varepsilon$ this is positive. The metric is curved despite its zero integrated signed curvature. Neither curvature integral evaluates $Z$; the Ward identity, global source data and quantum prescription remain necessary. The $p^4$ growth also warns that a small amplitude does not imply small curvature when the wavelength is short.

2. Complete the shear check and compensate the flavor source

For (D01.12), let $u_f=e^{-f}f'$. Verify the second and third frame components of (D01.3). Then derive $V=-i*dC$, $H=i*dB$ and the fixed complex flavor connection, including the signs. What remains to be checked before extending a partition-function equality to arbitrary $\alpha$?

Hint 1

Use $\gamma_3u_+=i\gamma_2u_+$, $E_2=e^{-f}\partial_x$, and $E_3=e^{-f}(\partial_y-q\partial_\tau)$.

Hint 2

In the frame, $V=(t,0,0)$ and $A=(t,-iu_f/2,-u_f/2)$. Also $dB=q'd\tau\wedge dx\wedge dy/2$.

Solution

The connections are $\Gamma_2=it\gamma_2/4$ and $\Gamma_3=-iu_f\gamma_1/2+it\gamma_3/4$. For $a=2$, the derivative $E_2\zeta=u_f\zeta/2$ cancels $-iA_2\zeta=-u_f\zeta/2$, leaving $it\gamma_2\zeta/4$. The right side is $-it\gamma_2\zeta/4+t\gamma_3\zeta/2$, equal to it by the stated gamma identity. For $a=3$, $E_3\zeta=0$ and the $u_f$ terms cancel. Both sides reduce to $it\gamma_3\zeta/4$, since the right side is $-it\gamma_3\zeta/4-t\gamma_2\zeta/2$.

Next $dC=iq'dx\wedge dy$, and $*(dx\wedge dy)=e^{-2f}\eta$, so $-i*dC=t\eta$. The two-form derivative in Hint 2 gives $i*dB=it/2$. The periodic potentials have no nonzero flux through a closed coordinate two-cycle. Finally $a_F=-imqdy$, $D_F=-imt/2$ give $a_F+im\eta=im d\tau$, with no change of flavor holonomies or transition functions of this complex connection.

These are exact finite-$\alpha$ background identities. They do not establish convergence of the continued matter integral for all $\alpha$, the absence of zero-mode crossings, or a quantum Ward identity for a chosen regulator. Those issues, together with fixed counterterms, decide whether a finite partition-function comparison is justified.

3. Change the transverse complex structure instead

On the same coordinate torus consider the flat metric $ds^2=d\tau^2+e^{2s}dx^2+e^{-2s}dy^2$ with constant real $s$. Find a compatible complex coordinate and its periods. Express it in terms of $z=x-iy$ and $\bar z$. Can an infinitesimal periodic coordinate change remove its constant antiholomorphic mixing? Does the geometric dependence result require a nonzero response to $s$ in our free theory?

Hint 1

Take $w=e^sx-ie^{-s}y$. Compute its two periods before attempting to rescale it.

Hint 2

The integral of a derivative of a smooth periodic function over the transverse torus vanishes. The mixing coefficient at $s=0$ is a nonzero constant.

Solution

The periods of $w$ are $2\pi Le^s$ and $-2\pi iLe^{-s}$. With the ordered $x,y$ cycles their ratio is $-ie^{-2s}$; its negative imaginary part reflects our $x-iy$ convention. The usual opposite ordering or conjugate convention reverses that sign. The ratio varies, unlike the conformal factor in (D01.2).

Moreover $w=\cosh s\,z+\sinh s\,\bar z$. Dividing by the constant $\cosh s$ leaves an antiholomorphic coefficient $\tanh s=s+O(s^3)$. An infinitesimal periodic reparametrization $z\mapsto z+v(z,\bar z)$ would have to satisfy $\partial_{\bar z}v=1$ to remove it, up to a sign. Integrating this equation over the torus gives zero on the left and nonzero area on the right, a contradiction. Large changes of cycle basis can give discrete modular identifications; they do not erase this local continuous deformation.

Thus the fixed-THF metric-independence argument does not apply to the $s$ family. It allows a dependence on this modulus but does not demand one. An index may fail to detect it in a particular theory. A zero response here would therefore not prove that the two THFs are equivalent.

The next bounded calculation is to reproduce the shear identities at the stated parameters and then specify a regulator for the full free multiplet. If a proposed determinant varies, first check its source family, zero modes, contact terms and charge-preserving cutoff. A verified mismatch under all those conditions would be informative; a changing scalar eigenvalue alone is the expected starting point.