Reading Supersymmetric Field Theories on Three-Manifolds

How can a quantum field theory keep supersymmetry when spacetime is curved? This paper makes the question concrete: specify a metric and additional external fields, solve an equation for a supersymmetry parameter, and use the resulting symmetry to construct actions and study observables. A metric alone is generally insufficient.

The paper is Supersymmetric Field Theories on Three-Manifolds, by Cyril Closset, Thomas T. Dumitrescu, Guido Festuccia and Zohar Komargodski. Use the freely available arXiv:1212.3388v3 PDF, dated 22 October 2013. Its journal record is JHEP 05 (2013) 017, published on 6 May 2013. The arXiv history identifies v1 on 14 December 2012, v2 on 12 February 2013, and describes v3 as having minor changes. The equation numbers here refer to v3; printed page 3 is PDF viewer page 4 because the title page is unnumbered.

Start with why this problem is useful. You can use the map below before knowing the geometric machinery. For the equation-reading assignment, first complete supergravity as background sources, the field dictionary and the Killing spinor equations.

Separate the three questions

The setting is three-dimensional $\mathcal N=2$ theory with a continuous R symmetry: four real Poincaré supercharges before Euclidean continuation. The paper uses two independent complex label families in Euclidean signature. The dimension of their solution spaces determines the background-preserved symmetry; it is unrelated to a dynamical gauge-group rank.

Read the introduction’s separation of the construction, printed pp. 1–3/PDF pp. 2–4, as three questions:

1. Which backgrounds preserve a symmetry? Freeze the bosonic supergravity fields and set the gravitini, the fermionic background sources, to zero. A surviving transformation must keep those values fixed. Its spinor parameter therefore solves a Killing spinor equation.
2. Which matter action has that symmetry? Couple the matter multiplets to those same sources with the required nonminimal terms. Finding a spinor has not yet checked a guessed matter action.
3. What observable can be extracted? The final application relates a specified squashed-sphere response to flat-space current and stress-tensor correlators. That requires the complete source family and a prescription for contact terms, not just a metric deformation.

The background auxiliary fields are prescribed data. They are not eliminated using gravitational equations of motion. The paper obtains the rigid equations and matter constructions without presenting a complete nonlinear three-dimensional supergravity action; the introduction states that boundary explicitly. These distinctions are the reason to read the source construction before applying a convenient sphere formula.

Introduction: identify every term

Use this first assignment with the three preparation lessons above.

Read in v3	What to identify	Detailed teaching calculation
(1.1), printed 3/PDF 4	Which quantities are potentials and which are dual field strengths?	The dual background fields
(1.2), printed 3/PDF 4	Why is the central charge different from an R charge?	Fields and their units
(1.3), printed 3/PDF 4	Why must a zero gravitino also have zero variation?	Preserving the frozen background
(1.4)–(1.5), printed 4/PDF 5	Which signs change between the two charge sectors?	Both Killing spinor equations

Here is the working dictionary. Length coordinates are used, so a derivative has mass dimension one.

Paper symbol	Meaning and convention
$g_{\mu\nu}$, $\nabla_\mu$	A real positive metric and its Levi-Civita spin derivative; the metric has dimension zero.
$A_\mu$	Background R connection, of dimension one; it is not a dynamical matter gauge field. It may be complex, while allowed R-gauge parameters are real.
$C_\mu$, $V^\mu$	Graviphoton potential of component dimension zero, and its dual field strength $V^\mu=-i\epsilon^{\mu\nu\rho}\partial_\nu C_\rho$, of dimension one.
$B_{\mu\nu}$, $H$	Two-form potential of component dimension zero, and $H=\tfrac i2\epsilon^{\mu\nu\rho}\partial_\mu B_{\nu\rho}$, of dimension one.
$\zeta,\widetilde\zeta$	Independent commuting two-component labels, of R charges $+1,-1$ and dimension $-1/2$. A tilde does not impose complex conjugation.
$\psi_\mu,\widetilde\psi_\mu$ in (1.3)	Background gravitini. The lessons often capitalize them as $\Psi_\mu,\widetilde\Psi_\mu$ to distinguish them from a matter fermion.
$Z$ in (1.2)	The flat supersymmetry algebra’s central-charge operator. Its eigenvalue has dimension one; the Killing labels themselves have zero central charge.
$\epsilon^{\mu\nu\rho}$, $\gamma_\mu$	The metric volume tensor and curved gamma matrices. In an oriented orthonormal frame, $\epsilon_{123}=1$ and $\gamma_a\gamma_b=\delta_{ab}+i\epsilon_{abc}\gamma_c$.

The complex-source and real-gauge-parameter distinction is stated before P (2.10), printed p. 8/PDF p. 9. For spinor index order and the paper’s negative round-sphere curvature sign, keep the spinor and curvature reference alongside Appendix A.

Move all terms of (1.4)–(1.5) to the left. If $q_R=+1$ labels $\zeta$ and $q_R=-1$ labels $\widetilde\zeta$, both equations read

$$\left[ \nabla_\mu-iq_RA_\mu+\frac H2\gamma_\mu +iq_RV_\mu+\frac{q_R}{2}\epsilon_{\mu\nu\rho}V^\nu\gamma^\rho \right]\varepsilon_{q_R}=0. \tag{PC00.1}$$

Every term has dimension $1/2$ after acting on the label. The $H$ term has the same sign in both equations; the $A$ and $V$ terms change sign. There is no extra $-izC_\mu$ term for these centrally neutral parameters. The connection $A-V/2$ used in a matter derivative is not a replacement for $A$ in (PC00.1) with the other terms left untouched.

The fixed-label odd operations have R charges opposite to their labels: $\delta_\zeta$ lowers a component’s R charge, while $\delta_{\widetilde\zeta}$ raises it. This does not change the charges used in the differential equation.

A map through the rest of the paper

The introduction is the entrance, not a requirement to read every appendix before solving one example. Choose a row by the question you want to answer. The links lead to completed teaching calculations; the dedicated Section 5 reading guide adds a more detailed equation map for additional supercharges.

Paper passage and checked landmark	Reading goal	Teaching route
Section 2: (2.15), printed 10/PDF 11	Recover the rigid gravitino variation from current/source couplings.	The R-multiplet and the supplied variation
Section 3: (3.1)–(3.8), printed 10–12/PDF 11–13	Separate differential uniqueness, pointwise bilinears and global spin/R lifts.	The Section 3 guide maps the full argument and reconstructs a spinor from its charged bilinear.
Section 4: (4.1)–(4.21), printed 12–16/PDF 13–17	Follow transverse integrability into a local spinor and then global patching.	The Section 4 guide maps the construction and checks a cylinder-to-annulus overlap with its spin and R seams.
Section 5: (5.1)–(5.14), printed 17–20/PDF 18–21	Distinguish opposite-charge pairs, maximal local enhancement and global transport.	The Section 5 guide
Section 6: (6.1), printed 21/PDF 22; (6.32), printed 29/PDF 30	Read the source-dependent algebra and then check an action.	Section 6 guide, with the full component/action map and a compensated R/flavor-source calculation
Section 7: (7.1), printed 30/PDF 31	Compare the first-order curved coupling with flat current operators.	Section 7 guide reconstructs a current preimage and tests its first-order scope, building on the canonical flat-space check.
Section 8: (8.2)–(8.3), printed 34/PDF 35	Relate the real sphere response to a separated-point correlator, retaining the source and contact prescription.	The Section 8 guide connects squashing response with the free-chiral check.
Appendices A and B	Look up conventions, then the definitions used by the transverse geometry.	The Appendices A/B guide maps the conventions and reconstructs the spin-curvature identity; its separate geometry entrance introduces the compatible connection.
Appendices C and D: (C.1), printed 43/PDF 44; (D.1)–(D.19), printed 45–48/PDF 46–49	Compare the four-dimensional construction with its twisted reduction.	The Appendices C/D guide maps the complete reduction. The four-dimensional rigid background derives (C.1); twisted reduction derives the 3d source and multiplet dictionary. Neither branch is a prerequisite for the introduction.

The map distinguishes an original paper argument from a teaching reconstruction. In particular, the site’s explicit examples do not expand the paper’s local statements into a classification of all compact quotients. Section 7 concerns the first-order deformation around flat space; it does not by itself prove unrestricted finite independence of a partition function.

Restore a constraint in the first equation

In (1.1), the statement $\nabla_\mu V^\mu=0$ appears next to the definition of $V$. Show that it follows from that definition on a curved coordinate patch, rather than being an auxiliary equation of motion. Does the same argument imply that $H$ is constant?

Hint 1

Use $\nabla_\mu V^\mu=(\sqrt g)^{-1}\partial_\mu(\sqrt g\,V^\mu)$. The alternating symbol with entries $0,\pm1$ is not the coordinate volume tensor.

Hint 2

Write $\epsilon^{\mu\nu\rho}=\varepsilon^{\mu\nu\rho}/\sqrt g$. Compare the symmetry of two partial derivatives with that of the symbol.

Solution

For a smooth local potential,

$$\nabla_\mu V^\mu =-\frac{i}{\sqrt g}\varepsilon^{\mu\nu\rho} \partial_\mu\partial_\nu C_\rho=0. \tag{PC00.2}$$

The two derivatives are symmetric in $\mu,\nu$, so their contraction with the antisymmetric symbol vanishes. The cancellation includes the metric dependence through $\sqrt g$; treating $\epsilon^{\mu\nu\rho}$ as a constant tensor would miss that step. It is the local field-strength identity behind the background dictionary, with no matter or gravitational equation of motion used.

There is no corresponding condition $dH=0$ from the definition alone: $dB$ is already a three-form, and its exterior derivative is a four-form, identically zero in three dimensions. Other supersymmetry requirements may constrain $H$, but that is another calculation. Local potentials may also require patches globally.

Exit calculation: change a phase into an amplitude

Work on a flat rectangular three-torus with length coordinates $x^j\sim x^j+L_j$, periodic spin structure in all directions and a trivial R line bundle. Set $V=H=0$, and choose the smooth, dimensionless periodic function $f(x)=\lambda\sin(2\pi x^1/L_1)$, where $\lambda$ is real. For $A=df$, the two families are $\zeta=e^{if}c$ and $\widetilde\zeta=e^{-if}\widetilde c$, with arbitrary constant columns.

Now change the fixed source to $A=-i\,df$. Find both solution spaces, check their global boundary conditions, and decide whether an allowed R-gauge transformation relates this background to $A=0$. Keep the distinction between a complex source and a real gauge parameter.

Hint 1

With $V=H=0$, use $(\partial-iA)\zeta=0$ and $(\partial+iA)\widetilde\zeta=0$. Check the factor $(-i)(-i)$ before choosing an exponential.

Hint 2

The new connection is exact and flat, but its imaginary part is invariant under $A\mapsto A+d\lambda_R$ for real $\lambda_R$. A positive amplitude is not a phase.

Solution

For the changed source the two operators are $\partial-df$ and $\partial+df$. Therefore

$$\zeta=e^f c,\qquad \widetilde\zeta=e^{-f}\widetilde c. \tag{PC00.3}$$

Differentiating cancels the respective connection term in every direction. Since $f$ is periodic and finite, these columns are smooth, nowhere zero when their constant column is nonzero, and satisfy the chosen spin boundary conditions. Multiplying any solution by $e^{-f}$ or $e^f$ respectively makes it constant, so these are the complete spaces: two complex directions in each sector, the maximal complexified parent count.

For $\lambda\ne0$ this is not related to $A=0$ by a real R-gauge transformation. Its imaginary part is $-df$, which such transformations leave unchanged. One explicit invariant diagnostic, using the fixed real metric, is

$$\int_{T^3}|\operatorname{Im}A|^2\,d^3x =\frac{\lambda^2}{2} \left(\frac{2\pi}{L_1}\right)^2L_1L_2L_3>0. \tag{PC00.4}$$

It is zero for $A=0$. A formal complex change of variables relates the two differential operators, but it is not an allowed real R-gauge transformation in the paper’s stated convention. This calculation establishes supersymmetric background solutions, not equality of arbitrary matter generating functionals or their integration contours.

You can now locate the main equations, distinguish a source identity from a supersymmetry condition, and check what a change of source actually does. Continue with the Section 2 guide to understand the origin of the gravitino variation, or with the Section 3 guide to discover what one nonzero solution implies.