Mirror Symmetry and the Dual Proof

Proof.

The Fano/LG correspondence was established through the work of Hori–Vafa [hori-vafa-2000] via gauged linear sigma models and Givental [givental-1996] via equivariant Gromov–Witten theory. The key argument proceeds in three steps.

Step 1: Gauged Linear Sigma Model (GLSM). For \(X = \mathbb{CP}^{n-1}\), consider the \(U(1)\) GLSM with \(n\) chiral fields of charge 1. In the geometric phase (large Fayet–Iliopoulos parameter \(r \gg 0\)), the low-energy theory is the non-linear sigma model on \(\mathbb{CP}^{n-1}\). In the Landau–Ginzburg phase (\(r \ll 0\)), the theory reduces to a Landau–Ginzburg model on \(\check{X}\) with superpotential \(W\).

Step 2: Superpotential from disk counting. The superpotential \(W\) counts holomorphic disks bounded by Lagrangian fibers. For \(X = \mathbb{CP}^{n-1}\), the toric structure provides the computation: \(W = \sum_{i=1}^{n} x_i\) subject to \(\prod x_i = q\), where \(q = e^{-t}\) is the exponentiated Kähler parameter.

Step 3: Specialization to \(\mathbb{CP}^1\). For \(n = 2\), the constraint \(x_1 x_2 = q\) and change of variables \(x = x_1/\sqrt{q}\) gives \(W = \sqrt{q}(x + 1/x)\). Setting \(q = 1\) (the natural normalization), we obtain \(W = x + 1/x\) on \(\check{X} = \mathbb{C}^*\).

(See: Hori–Vafa [hori-vafa-2000), Givental \cite{givental-1996], Auroux [auroux-2007]} □

Proof.

Step 1: Toric construction. \(\mathbb{CP}^1\) is a toric variety with moment polytope \(\Delta = [0, 1] \subset \mathbb{R}\). The vertices \(\{0, 1\}\) correspond to the two torus-fixed points (the poles of \(S^2\)). Each vertex contributes one monomial to the superpotential:

where \(v_1 = 1\) and \(v_2 = -1\) are the inward-pointing primitive normals to the facets.

Step 2: Uniqueness. The superpotential is determined by the fan of \(\mathbb{CP}^1\) (or equivalently, its moment polytope). The fan of \(\mathbb{CP}^1\) in \(\mathbb{R}\) consists of two rays: \(\mathbb{R}_{\geq 0}\) and \(\mathbb{R}_{\leq 0}\). The primitive generators are \(+1\) and \(-1\), giving exactly two monomials \(x\) and \(x^{-1}\). No other toric structure on \(\mathbb{CP}^1\) exists.

Step 3: Verification via GW theory. The genus-0 Gromov–Witten potential of \(\mathbb{CP}^1\) is:

where \(N_d = 1\) for all \(d\) (every degree-\(d\) map \(\mathbb{CP}^1 \to \mathbb{CP}^1\) contributes 1 after appropriate counting). The mirror statement matches this with the period integrals of \(W = x + 1/x\) (sec:170-thimbles).

(See: Givental [givental-1996), Hori–Vafa \cite{hori-vafa-2000], Auroux–Katzarkov–Orlov [auroux-katzarkov-orlov-2006]} □

Proof.

Step 1: Compute \(W'(x) = \frac{d}{dx}(x + x^{-1}) = 1 - x^{-2}\). Setting \(W'(x) = 0\) gives \(x^2 = 1\), so \(x = \pm 1\).

Step 2: Non-degeneracy: \(W''(x) = 2x^{-3}\), so \(W''(+1) = 2\) and \(W''(-1) = -2\). Both are non-zero, confirming Morse singularities.

Step 3: The Milnor number at an isolated critical point of a function on \(\mathbb{C}^*\) equals \(\dim_\mathbb{C} \mathcal{O}_{\mathbb{C}^*, x_0}/(W')\). At \(x = \pm 1\), the local ring has \(W' = 1 - x^{-2} = (x^2 - 1)/x^2\), which vanishes to first order, giving \(\mu = 1\) at each point.

Step 4: The Euler characteristic relation \(\sum_i \mu_i = \chi(\mathbb{CP}^1) = 2\) confirms consistency. □

Proof.

The fiber \(W^{-1}(t)\) is defined by \(x + 1/x = t\), or equivalently \(x^2 - tx + 1 = 0\). The discriminant of this quadratic (in \(x\)) is \(\Delta = t^2 - 4\). The fiber degenerates precisely when \(\Delta = 0\), i.e., when \(t = \pm 2\). At these values, the two roots coalesce: \(x = t/2 = \pm 1\).

The Picard–Lefschetz formula gives the monodromy: for a loop \(\gamma\) around \(t = +2\) in the base, the monodromy acts on the vanishing cycle \(\delta\) by \(T_\gamma(c) = c + \langle c, \delta \rangle \cdot \delta\), where \(\langle \cdot, \cdot \rangle\) is the intersection form. □

Proof.

The identification follows from three independent routes:

Route 1 (Toric): The toric fan of \(\mathbb{CP}^1\) has two rays (primitive generators \(\pm 1\)), corresponding to two fixed points under the \(\mathbb{C}^*\)-action. The superpotential \(W = x + 1/x\) has one monomial per ray. Each critical point (\(W' = 0\)) occurs at a fixed point of the involution \(x \mapsto 1/x\), which mirrors the antipodal map exchanging the poles.

Route 2 (Topological): The total Milnor number \(\mu_{\mathrm{tot}} = 2\) equals \(\chi(\mathbb{CP}^1) = 2\). This is not a coincidence: for a Lefschetz fibration on a Fano manifold, the Milnor numbers of the mirror superpotential recover the Betti numbers of the original variety. For \(\mathbb{CP}^1\): \(b_0 = b_2 = 1\), \(b_1 = 0\), and \(\chi = 2\).

Route 3 (Physical): Under the dictionary (prop:P14-Ch170-mirror-dict), poles of \(S^2\) map to critical points of \(W\). The monopole at the north pole (positive magnetic charge) maps to \(x = +1\) (positive critical value), and the anti-monopole at the south pole maps to \(x = -1\) (negative critical value). The symmetry \(W \mapsto -W\) under \(x \mapsto -x\) mirrors the charge conjugation symmetry of the monopole.

(See: Ch 159 (\Stwo = \mathbb{CP)^1 identification), Ch 162 (motivic structure)} □

Proof.[Proof sketch (Seidel)]

The proof proceeds through several steps.

Step 1: Generating objects. On the B-side, \(D^b(\mathrm{Coh}(\mathbb{CP}^1))\) is generated by the exceptional collection \(\mathcal{O}, \mathcal{O}(1)\) (Beilinson's theorem, thm:P14-Ch170-beilinson). On the A-side, \(\mathrm{MF}(W)\) is generated by two objects corresponding to the two critical points of \(W\).

Step 2: Morphism matching. The Hom-spaces on both sides are computed and matched:

where \(E_\pm\) are the matrix factorizations corresponding to the critical points \(x = \pm 1\).

Step 3: \(A_\infty\) structures. Seidel verifies that the higher \(A_\infty\) products (\(m_k\) for \(k \geq 3\)) match on both sides, establishing the equivalence at the \(A_\infty\) level.

Step 4: Categorical equivalence. Since both sides have two generators with matching Hom-spaces and \(A_\infty\) structures, the functor between them is an equivalence of \(A_\infty\)-categories, which descends to an equivalence of triangulated categories.

(See: Seidel [seidel-2001), \cite{seidel-2008-book], Auroux–Katzarkov–Orlov [auroux-katzarkov-orlov-2006]} □

Proof.

Step 1 (Exceptional pair): \(\mathrm{Hom}(\mathcal{O}(1), \mathcal{O}) = H^0(\mathcal{O}(-1)) = 0\) since \(\mathcal{O}(-1)\) has no global sections on \(\mathbb{CP}^1\). The Ext groups \(\mathrm{Ext}^i(\mathcal{O}(1), \mathcal{O}) = H^i(\mathcal{O}(-1)) = 0\) vanish for all \(i\) by Serre duality: \(H^1(\mathcal{O}(-1)) \cong H^0(\mathcal{O}(-1))^* = 0\). This confirms semiorthogonality.

Step 2 (Generation): Every coherent sheaf on \(\mathbb{CP}^1\) decomposes as a direct sum of a torsion sheaf and a vector bundle. By Grothendieck's theorem, every vector bundle splits as \(\bigoplus_i \mathcal{O}(n_i)\). The Euler exact sequence \(0 \to \mathcal{O}(-1) \to \mathcal{O}^{\oplus 2} \to \mathcal{O}(1) \to 0\) shows \(\mathcal{O}(-1) \in \langle \mathcal{O}, \mathcal{O}(1) \rangle\). By twisting, all \(\mathcal{O}(n)\) lie in \(\langle \mathcal{O}, \mathcal{O}(1) \rangle\). Torsion sheaves (structure sheaves of points) are cokernels of maps between line bundles, hence also generated.

(See: Beilinson [beilinson-1978), Ch 162 (motivic decomposition), Ch 171 (rigidity)] □

Proof.

Step 1 (Classification): Matrix factorizations of \(W\) over \(R = \mathbb{C}[x, x^{-1}]\) are classified by the Knörrer periodicity theorem and direct computation. Since \(W\) has exactly two non-degenerate critical points, each contributes one indecomposable matrix factorization (by the Koszul duality between critical points and MF generators).

Step 2 (Hom computation): The Hom-spaces in \(\mathrm{MF}(W)\) are:

These match the Hom-spaces of \((\mathcal{O}, \mathcal{O}(1))\) in thm:P14-Ch170-beilinson, confirming the HMS correspondence.

Step 3 (Uniqueness): Since \(D^b(\mathrm{Coh}(\mathbb{CP}^1))\) is generated by two exceptional objects and \(\mathrm{MF}(W)\) is also generated by two objects with identical Hom-data, the equivalence \(E_+ \leftrightarrow \mathcal{O}\) and \(E_- \leftrightarrow \mathcal{O}(1)\) is the unique categorical match (up to autoequivalence).

(See: Orlov [orlov-2004), Dyckerhoff \cite{dyckerhoff-2011], Ch 162 (exceptional collection), Ch 171 (categorical rigidity)} □

Proof.

Step 1 (Thimble construction): Write \(x = re^{i\phi}\). Then \(\mathrm{Re}(W) = (r + 1/r)\cos\phi\). At the critical point \(x = +1\) (\(r = 1, \phi = 0\)), the steepest-descent direction satisfies \(\nabla \mathrm{Re}(W) \cdot \dot{\gamma} < 0\). Since \(\mathrm{Re}(W)\) achieves its maximum value \(2\) at \(x = +1\) (on the unit circle), the thimble \(\gamma_+\) descends from this maximum along both directions on the real axis: toward \(x \to 0^+\) and \(x \to +\infty\).

Step 2 (Intersection): The thimble \(\gamma_-\) from \(x = -1\) descends along the negative real axis: toward \(x \to 0^-\) and \(x \to -\infty\). The two thimbles intersect transversally at one point in the universal cover (the monodromy action permutes sheets), giving \(\langle \gamma_+, \gamma_- \rangle = \pm 1\).

Step 3 (Picard–Lefschetz): The monodromy of the fiber around \(t = +2\) acts by the Picard–Lefschetz transformation: \(T_{+2}(\gamma) = \gamma + \langle \gamma, \delta_+ \rangle \cdot \delta_+\), where \(\delta_+\) is the vanishing cycle at \(x = +1\). Similarly for \(t = -2\).

(See: Seidel [seidel-2008-book), Arnold–Gusein-Zade–Varchenko \cite{agv-singularities]} □

Proof.

Step 1: Parametrize the unit circle by \(x = e^{i\theta}\), so \(dx/x = i\, d\theta\) and \(W = e^{i\theta} + e^{-i\theta} = 2\cos\theta\). The integral becomes:

(absorbing the factor of \(i\) by standard conventions for period integrals).

Step 2: This is the standard integral representation of the modified Bessel function:

With \(z = -2/\hbar\), we obtain \(\Pi_0 = 2\pi I_0(2/\hbar)\) (using \(I_0(-z) = I_0(z)\)).

Step 3 (Classical limit): As \(\hbar \to \infty\), \(2/\hbar \to 0\), and \(I_0(0) = 1\), so \(\Pi_0 \to 2\pi\). This equals \(\mathrm{Area}(S^2)/2 = 4\pi/2\), connecting the mirror period to the fundamental area scale of TMT.

Step 4 (Semiclassical): As \(\hbar \to 0^+\), the saddle-point approximation at \(\theta = 0\) (the maximum of \(-2\cos\theta/\hbar\)) gives \(\Pi_0 \sim \sqrt{2\pi / (2/\hbar)} \cdot e^{2/\hbar} = \sqrt{\pi\hbar}\, e^{2/\hbar}\).

(See: Watson [watson-bessel), Ch 166 (period duality)] □

Proof.

Step 1 (\(HH^2\) computation): The Hochschild–Kostant–Rosenberg (HKR) theorem gives an isomorphism:

for a smooth projective variety \(X\). For \(X = \mathbb{CP}^1\): The first term vanishes because \(\dim \mathbb{CP}^1 = 1\) so \(\bigwedge^2 T = 0\). The second uses \(T_{\mathbb{CP}^1} \cong \mathcal{O}(2)\) and \(H^1(\mathbb{CP}^1, \mathcal{O}(2)) = 0\) (Kodaira vanishing, since \(\deg \mathcal{O}(2) = 2 > 2g - 2 = -2\)). The third uses \(H^2(\mathbb{CP}^1, \mathcal{O}) = 0\) (dimension).

Step 2 (Deformation theory): In the deformation theory of \(A_\infty\)-categories, \(HH^2\) classifies infinitesimal deformations and \(HH^3\) classifies obstructions. Since \(HH^2 = 0\), no infinitesimal deformations exist — the category is rigid.

Step 3 (Transfer to mirror): By HMS (thm:P14-Ch170-hms-seidel), \(D^b(\mathrm{Coh}(\mathbb{CP}^1)) \cong \mathrm{MF}(W)\) as \(A_\infty\)-categories. An equivalence of \(A_\infty\)-categories preserves Hochschild cohomology:

Therefore \(\mathrm{MF}(W)\) is also rigid: the LG model admits no deformations.

Step 4 (Physical consequence): If TMT could be deformed to an alternative theory TMT', then either:

• The deformation appears on the B-side (algebraic): impossible, since \(HH^2(\mathbb{CP}^1) = 0\).
• The deformation appears on the A-side (symplectic): impossible, since \(HH^2(\mathrm{MF}(W)) = 0\).

No deformation channel exists on either side. The theory is rigid.

(See: Kontsevich [kontsevich-1994), Keller \cite{keller-2006], Ch 171 (seven-link rigidity chain)} □

Proof.

Step 1 (Period ring structure): From Ch 166, the TMT period ring is \(R_{\mathrm{TMT}} = \mathbb{Q}[\pi, 1/\pi]\), and the involution \(\iota\) acts by \(\pi \mapsto 1/\pi\), fixing the rational coefficients.

Step 2 (Mirror action on periods): Under mirror symmetry, the B-model period \(\Omega_B = \oint \omega\) (depending on complex structure, involving \(\pi\)) maps to the A-model period \(\Omega_A = \int_\gamma e^{-W/\hbar}\) (depending on symplectic structure). For the fundamental period (thm:P14-Ch170-periods), \(\Pi_0 \to 2\pi\) in the classical limit, confirming that the A-model period involves \(\pi\) in the same ring.

Step 3 (Identification): The involution \(\iota\) on \(R_{\mathrm{TMT}}\) exchanges \(g^2 = 4/(3\pi)\) with \(4\pi/3\), sending small coupling to large coupling. Mirror symmetry similarly exchanges the “volume” (\(\sim \pi\)) with its reciprocal (\(\sim 1/\pi\)). The rational content (\(4/3\) in the coupling) is preserved by both operations. Since both \(\iota\) and mirror symmetry are involutions on \(R_{\mathrm{TMT}}\) that exchange \(\pi\) and \(1/\pi\) while fixing \(\mathbb{Q}\), they must coincide (as the automorphism group of \(R_{\mathrm{TMT}}\) fixing \(\mathbb{Q}\) is \(\mathbb{Z}/2\mathbb{Z}\), generated by \(\pi \mapsto 1/\pi\)).

(See: Ch 166 (period-inverse duality), Ch 162 (motivic periods)) □

Proof.

Step 1 (Toric degeneration): The toric structure of \(\mathbb{CP}^1\) (the \(\mathbb{C}^*\)-action by \([z_0 : z_1] \mapsto [\lambda z_0 : z_1]\)) provides a toric degeneration: the family \(\{X_t\}_{t \to 0}\) where \(X_t \cong \mathbb{CP}^1\) degenerates to the union of two components (the closures of the two torus orbits) meeting at a point. In the tropical limit, this becomes the interval \([0, 1]\).

Step 2 (Log structure): The degeneration provides a log Calabi–Yau structure on the central fiber. The Gross–Siebert machinery uses this log structure to define a scattering diagram and count tropical disks.

Step 3 (Disk counting): A tropical disk is a balanced weighted graph with one unbounded edge (the “output”) and boundary on a reference fiber of the SYZ-type fibration. For \(\mathbb{CP}^1\) with its standard fibration to \([0, 1]\):

• From vertex 0: one tropical disk with weight \(+1\), contributing monomial \(x^{+1} = x\).
• From vertex 1: one tropical disk with weight \(-1\), contributing monomial \(x^{-1} = 1/x\).

No higher-order corrections (“wall-crossing”) occur because \(\mathbb{CP}^1\) has no interior lattice points in its moment polytope (equivalently, the scattering diagram is trivial).

Step 4 (Uniqueness): The tropical computation is combinatorial and depends only on the fan (or equivalently, the moment polytope \([0, 1]\)). Since the fan of \(\mathbb{CP}^1\) is unique, the tropical mirror is unique: \(W = x + 1/x\).

(See: Gross–Siebert [gross-siebert-2006), \cite{gross-siebert-2010], Auroux [auroux-2007]} □

Proof.

Step 1: The B-model formulation is the content of the entire TMT book: the interface \(S^2 \cong \mathbb{CP}^1\) carries line bundles, whose sections are monopole harmonics, whose overlap integrals produce physical constants.

Step 2: The A-model formulation translates every B-model object via the mirror dictionary (prop:P14-Ch170-mirror-dict). Monopole harmonics become sections of matrix factorizations. Overlap integrals become oscillatory integrals with kernel \(e^{-W/\hbar}\). The coupling \(g^2 = 4/(3\pi)\) appears as a period of \(W\).

Step 3: The categorical equivalence \(D^b(\mathrm{Coh}(\mathbb{CP}^1)) \cong \mathrm{MF}(W)\) is an exact isomorphism of \(A_\infty\)-categories, not an approximation. Every computation in the B-model has an exact A-model counterpart, and vice versa.

Step 4: The double rigidity (\(HH^2 = 0\) on both sides) means any deformation attempt is blocked from both directions simultaneously. This is strictly stronger than B-side rigidity alone: even if one could imagine an exotic deformation invisible to the B-model, the A-model (an independent mathematical framework) also forbids it.

(See: All results of this chapter; Ch 171 (seven-link chain, total rigidity)) □