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. It requires a finite-rank normalizable ground-state sector that is spectrally isolated in the finite-circle problem. A continuum threshold can destroy that sector; a collision of semiclassical vacua instead destroys the canonical eigenline description and need not destroy the full tt* bundle.

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. The circle is important: an infinite-volume mass gap may close at a conformal point even though the finite-circle Ramond Hamiltonian still has a discrete, isolated ground-state sector. 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 isolation 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 other states. Degeneracy inside the rank-$N$ ground space is allowed and is handled by the non-Abelian connection; loss of separation from states outside that space is the obstruction.

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. The topological theory also supplies a holomorphic bilinear pairing $\eta_{ab}$, whereas

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

is the physical positive Hermitian pairing on the unitary slice. The real structure must make $g$ and $\eta$ compatible; neither pairing can be reconstructed from the multiplication table alone. In a holomorphic frame, $C_i$ and $\eta$ 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

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

and the central equation

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

Together with $[C_i,C_j]=[\bar C_{\bar i},\bar C_{\bar j}]=0$, these are the tt* equations of Cecotti and Vafa 1991, §§3–4. Overall signs can move between the order chosen for curvature commutators and the definition of $\bar C$; the next formula fixes the convention used here.

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

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

Indeed, with $D_i=\partial_i+A_i$, $D_{\bar j}=\partial_{\bar j}$, and $A_i=g^{-1}\partial_i g$, one has $[D_i,D_{\bar j}]=-\partial_{\bar j}A_i$. This makes the displayed metric equation equivalent to $[D_i,D_{\bar j}]=-[C_i,\bar C_{\bar j}]$. It is nonlinear because the physical adjoint of $C_j$ itself depends on $g$.

The spectral-parameter connection

Introduce $\zeta\in\mathbb C^*$. Absorb the circumference and the dimensions of the couplings into the normalization of $C_i$, 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

For a rank-two example, the chapter’s phase–mirror–tt* map follows the $\mathbb{CP}^1$ ring through its mirror critical values and square-root monodromy. At $q=0$ it marks the boundary of that gapped canonical chart, not an automatic loss of the finite-circle bundle rank.

In a massive Landau–Ginzburg theory with distinct nondegenerate critical points $x_a$, the Jacobi ring is semisimple. Its primitive idempotents define a canonical eigenline decomposition in which chiral multiplication is algebraically diagonal,

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

The physical normalized ground states become localized near these idempotents at large $L$; that asymptotic identification, unlike the algebraic diagonalization, receives corrections. Off-diagonal entries of the metric arise from tunneling solitons and have leading exponential scale

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

where $M_{ab}=|Z_{ab}|$ when a BPS soliton exists; power-law prefactors are not shown. For fixed couplings, the asymptotic flat sections of $\nabla(\zeta)$ jump when $\arg\zeta$ crosses a BPS ray determined by $Z_{ab}$. The corresponding Stokes factor has an off-diagonal signed integer $\mu_{ab}$, the protected soliton index, once an ordering and orientation convention are fixed.

As couplings vary, rays can align and reorder. Individual $\mu_{ab}$ can then jump while the appropriately ordered product of Stokes factors remains invariant. The critical values determine the rays and BPS bounds, not the integers $\mu_{ab}$: those require the existence and signed count of gradient-flow trajectories Cecotti and Vafa 1993, §§2–4 (PDF).

Worked family: W = X³/3 − 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

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

Critical values alone would give only this BPS bound. For $u>0$ and canonical Kähler metric, the real trajectory

$$X(x^1)=\sqrt u\,\tanh\!\left(\sqrt u\,(x^1-x_0)\right)$$

solves $\partial_1X=u-X^2$ and connects $X_-$ to $X_+$. It therefore supplies one elementary soliton trajectory up to translation; the sign of its protected index depends on the orientation convention. Rotating $u$ rotates the corresponding central-charge ray without changing this local count away from a wall.

A loop $u\mapsto e^{2\pi i}u$ sends $\sqrt u\mapsto-\sqrt u$ and exchanges the two critical points. Thus the canonical eigenline decomposition has transposition monodromy. This is not yet the Berry holonomy of the full rank-two ground-state bundle: that gauge-invariant statement requires the connection and metric.

At $u=0$, the critical points collide, $W''=2X$ vanishes, and the soliton mass goes to zero. The infinite-volume massive description and the two canonical idempotents fail, but the Jacobi ring

$$\mathbb C[X]/(X^2)$$

still has dimension two. On a finite circle, the conformal theory can retain two normalizable Ramond ground states separated from higher finite-size levels, so a rank-two tt* bundle may extend across $u=0$ in a noncanonical frame. What must be reanalyzed is the regularity of $P_0$, the metric’s conformal boundary condition, and the now nonsemisimple ring—not automatically the rank.

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, §§3–5 (PDF). 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 canonical eigenvalues make the semisimple frame singular even when the full connection extends;
• 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. Abandon the primitive-idempotent basis and the nondegenerate-Hessian approximation. Determine whether the full finite-circle projector and rank extend in a regular frame; if they do, impose the conformal or singular boundary condition selected by the light theory.

Continuum threshold. Test normalizability and separation from scattering states. If either fails, specify an infrared regulator or a scattering-state completion; the ordinary finite Hermitian bundle no longer follows from $P_0$.

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 what taking $u$ once around zero proves—and what it does not prove.

Solution

$\sqrt u$ changes sign, so the two critical points and their critical values are exchanged; the oriented soliton central charge reverses. This proves transposition monodromy of the canonical eigenline labels. It does not by itself compute Berry holonomy of the full rank-two bundle, because that requires parallel transport with $D$.

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 canonical vacua collide without changing the signed count. Smooth Berry projection requires normalizable states and spectral isolation from the rest of the finite-circle spectrum, 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. Open PDF.