tt* Geometry and Ground-State Bundles

tt* geometry equips the supersymmetric ground states of a family of two-dimensional $(2,2)$ theories with a Hermitian metric, Berry connection, and chiral-ring action. Compatibility among these structures produces nonlinear differential equations—the tt* equations—and a flat connection depending on a spectral parameter. The construction adds norm and transport information that a chiral ring alone cannot see, but it requires a finite-rank, gapped ground-state bundle; vacuum collisions and continuum thresholds are genuine boundaries of that description.

Required background. We use the analysis of continuum states, boundaries, and ground-state bundles and massive Landau–Ginzburg vacua and soliton charges. Helpful background. Effective twisted-superpotential vacua provide gauge-theory examples of parameter-dependent isolated vacua.

The ground-state bundle

Place the theory on a spatial circle of circumference $L$ and let $t^i$ be complex supersymmetric couplings. Assume on an open parameter region $\mathcal M^\circ$ that:

1. the Hamiltonian $H(t,\bar t)$ is self-adjoint on a common dense domain;
2. there are $N<\infty$ normalizable zero-energy states;
3. a positive gap separates them from the rest of the spectrum;
4. the spectral projector onto the zero-energy subspace varies smoothly.

The ground states form a rank-$N$ vector bundle $\mathcal V\to\mathcal M^\circ$. In a local frame $|a(t,\bar t)\rangle$, its Hermitian metric is

$$g_{a\bar b}=\langle\bar b|a\rangle.$$

The orthogonal projector $P_0$ defines the Berry connection. In a general frame its coefficients are

$$(A_i)_a^{\ b}=g^{b\bar c}\langle\bar c|\partial_i a\rangle, \qquad (A_{\bar i})_a^{\ b}=g^{b\bar c}\langle\bar c|\partial_{\bar i}a\rangle, \qquad D_i=\partial_i+A_i, \qquad D_{\bar i}=\partial_{\bar i}+A_{\bar i}.$$

with the usual frame-transformation law. Only its holonomy and curvature are gauge invariant; individual connection coefficients are not.

The gap hypothesis is what makes $P_0$ smooth. Without it, the resolvent formula for $\partial_iP_0$ develops small denominators and Berry transport can mix the putative ground subspace with continuum states.

Chiral-ring action on vacua

Let $\phi_i$ be chiral operators associated with the couplings $t^i$. Multiplication in supercharge cohomology acts on ground states:

$$\phi_i|a\rangle=(C_i)_a^{\ b}|b\rangle+Q|\cdots\rangle.$$

After projection, $C_i=P_0\phi_iP_0$ is an endomorphism of $\mathcal V$. Associativity and commutativity of the chiral ring imply

$$[C_i,C_j]=0.$$

The conjugate antichiral operators define $\bar C_{\bar i}$, related to $C_i$ by the Hermitian metric and the chosen real structure. In a holomorphic frame, $C_i$ can be holomorphic even though $g$ is not.

Knowing only the matrices $C_i$ does not determine $g$. The ring is algebraic; the metric remembers how physical bra and ket vacua are paired. tt* equations couple the two.

Deriving the tt* equations

Varying a supersymmetric coupling inserts an integrated descendant of $\phi_i$. Supersymmetric Ward identities move supercharges through the cylinder amplitude. Contributions from paired excited states cancel, while contact terms project back to the ground states. The result is

$$[D_i,D_j]=0, \qquad [D_{\bar i},D_{\bar j}]=0,$$ $$D_iC_j=D_jC_i, \qquad D_{\bar i}C_j=0,$$

and the central equation

$$[D_i,D_{\bar j}]=-[C_i,\bar C_{\bar j}].$$

Together with their complex conjugates, these are the tt* equations of Cecotti and Vafa 1991. Overall signs can move between the curvature convention and the definition of $\bar C$; the flat-connection check below fixes internal consistency.

In a holomorphic frame with $D_{\bar i}=\partial_{\bar i}$ and $D_i=\partial_i+g^{-1}\partial_i g$, the central equation becomes

$$\partial_{\bar j}(g^{-1}\partial_i g) =-[C_i,g^{-1}C_j^\dagger g].$$

This is nonlinear because the physical adjoint of $C_j$ itself depends on $g$.

The spectral-parameter connection

Introduce $\zeta\in\mathbb C^*$ and define

$$\nabla_i(\zeta)=D_i+\zeta^{-1}C_i, \qquad \nabla_{\bar i}(\zeta)=D_{\bar i}+\zeta\bar C_{\bar i}.$$

Compute the mixed curvature:

$$[\nabla_i,\nabla_{\bar j}] =[D_i,D_{\bar j}] +\zeta[D_i,\bar C_{\bar j}] +\zeta^{-1}[C_i,D_{\bar j}] +[C_i,\bar C_{\bar j}].$$

The tt* equations make each coefficient vanish. Pure $(2,0)$ and $(0,2)$ curvatures vanish similarly, so

$$\nabla(\zeta)^2=0 \qquad\text{for every }\zeta\ne0.$$

Conversely, expanding flatness in powers of $\zeta$ recovers the tt* equations. This Lax form connects tt* geometry to integrable systems and makes Stokes phenomena visible as $\zeta\to0$ or $\infty$.

Massive vacua and the canonical basis

In a massive Landau–Ginzburg theory with isolated critical points $x_a$, one can choose a semiclassical vacuum basis localized near each $x_a$. Chiral multiplication is then approximately diagonal:

$$(C_i)_a^{\ b}\simeq \delta_a^{\ b}\,\phi_i(x_a).$$

Off-diagonal entries of the metric arise from tunneling solitons. At large circle size they are exponentially suppressed by

$$\exp(-L M_{ab}),$$

where $M_{ab}$ is a BPS soliton mass when such a soliton exists. Their phases and jumps encode soliton multiplicities. When central-charge rays align, the preferred asymptotic basis changes by a Stokes matrix; the smooth physical metric remains the invariant object.

The Stokes matrix is not obtained from vacuum critical values alone. It depends on which gradient-flow trajectories exist and on their signed degeneracies.

Worked family: $W=X^3/3-uX$

For $u\ne0$,

$$W'(X)=X^2-u=0, \qquad X_\pm=\pm\sqrt u.$$

The critical values are

$$W_+=-\frac23u^{3/2}, \qquad W_-=+\frac23u^{3/2}.$$

Thus

$$\Delta W=W_--W_+=\frac43u^{3/2},$$

and in the normalization $Z=2\Delta W$ the BPS mass is

$$M_{+-}=\frac83|u|^{3/2}.$$

A loop $u\mapsto e^{2\pi i}u$ sends $\sqrt u\mapsto-\sqrt u$ and exchanges the two vacua. The vacuum bundle therefore has nontrivial permutation monodromy even before one computes its Hermitian metric.

At $u=0$, the critical points collide, $W''=2X$ vanishes, and the soliton mass goes to zero. The gap hypothesis fails. The correct object can be a singular extension, a conformal tt* system, or a larger bundle including the new light states; it is not the smooth rank-two massive bundle continued without qualification.

Massive and conformal regimes

In a massive theory, tt* describes finitely many vacua and soliton tunneling. At a conformal point, operator-state correspondence relates the ground-state metric to two-point functions of chiral primaries, and scaling dimensions constrain asymptotics; the relation between massive deformations and conformal classification is developed in Cecotti and Vafa 1993. The limit is subtle:

• the circumference $L$ introduces the dimensionless combinations $L^{1-\Delta_i}t^i$;
• relevant couplings can drive exponential massive asymptotics;
• colliding vacua can create irregular singularities in the $\zeta$ connection;
• marginal directions may have monodromy and operator mixing;
• noncompact SCFTs can have a continuum and no finite-rank normalizable ground bundle.

Boundary conditions at the conformal point and in the massive asymptotic region are part of a tt* solution. The differential equations alone admit unphysical solutions with the wrong positivity or singularity behavior.

Relation to Berry phases and indices

An index counts ground states with signs and may remain constant while the Berry holonomy changes continuously. The chiral ring tells how protected operators act. tt* combines both and adds the metric:

Datum	Rank/count	Ring action	Norms	Parallel transport
Witten index	yes, signed	no	no	no
Jacobi or quantum ring	indirectly	yes	only a topological pairing	no
Berry connection	fixed rank assumed	no	compatible metric needed	yes
tt* geometry	yes	yes	yes	yes

This explains why two theories with isomorphic rings can still have different tt* data, and why a mirror claim can be tested more sharply by matching the full flat family $\nabla(\zeta)$.

Failure modes and repairs

Vacuum collision. Enlarge the low-energy description and impose singular boundary conditions derived from the light theory; do not use a nondegenerate Hessian formula.

Continuum threshold. Specify an infrared regulator or a scattering-state completion. A finite matrix Berry connection may be replaced by an operator-valued connection.

Changing Hilbert-space domain. If boundary conditions vary with parameters, include their contribution to the connection; differentiating vectors in inequivalent domains is not defined.

Non-Hermitian continuation. Complexifying couplings is useful for holomorphy, but the physical tt* metric is defined on a real slice with a positive inner product. Stokes data away from that slice do not by themselves prove unitarity.

Exercises

1. Verify that flatness of $\nabla(\zeta)$ implies the central tt* equation.

Solution

The coefficient of $\zeta^0$ in $[\nabla_i,\nabla_{\bar j}]$ is $[D_i,D_{\bar j}]+[C_i,\bar C_{\bar j}]$. Setting the curvature to zero yields $[D_i,D_{\bar j}]=-[C_i,\bar C_{\bar j}]$.

2. In the cubic family, explain the three effects of taking $u$ once around zero.

Solution

$\sqrt u$ changes sign, so the two critical points are exchanged. Since $u^{3/2}$ also changes sign, the two critical values are exchanged and the oriented soliton central charge reverses. A local vacuum frame therefore returns only after a permutation, giving nontrivial bundle monodromy.

3. Why is a constant Witten index insufficient to guarantee a smooth tt* bundle?

Solution

An index can stay fixed when zero-energy states meet a continuum in boson–fermion pairs or when vacua collide without changing the signed count. Smooth Berry projection requires an actual spectral gap and normalizable states, conditions the index does not test.

References

• Cecotti, S., and Vafa, C. “Topological–Anti-Topological Fusion.” Nuclear Physics B 367 (1991): 359–461. doi:10.1016/0550-3213(91)90021-O.
• Cecotti, S., and Vafa, C. “On Classification of $N=2$ Supersymmetric Theories.” Communications in Mathematical Physics 158 (1993): 569–644. doi:10.1007/BF02096804; arXiv:hep-th/9211097.