Cobordism-valued intersection theory on $\overline{\mathcal{M}}_{0,n}$

This paper refines the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism to derive inductive formulas for cobordism-valued psi-class intersections and explicit classes up to $n=8$, including their images in $K$-theory.

The Gist

Imagine you are an architect trying to count and categorize every possible way a family of trees can grow, branch, and connect. In mathematics, specifically in a field called algebraic geometry, these "trees" are actually shapes called curves (specifically, genus-zero curves, which look like spheres or bubbles). When these curves have specific points marked on them (like distinct leaves or flowers), they form a complex landscape called a moduli space (denoted as $M_{0,n}$).

For a long time, mathematicians could only take a "black and white" photo of these landscapes. They could count how many ways the curves could intersect, but they only saw the numbers (like 1, 2, or 5). This is like looking at a map and just seeing the distance between cities.

Benjamin Ellis-Bloor's paper is about upgrading that map to 3D, full-color, high-definition holograms. He introduces a new way of measuring these shapes called cobordism.

The Core Idea: From "Counting" to "Classifying"

Think of Cobordism as a super-advanced sorting machine.

• The Old Way (Chow Ring/Cohomology): Imagine you have a pile of different fruits (apples, oranges, bananas). The old method just counts them: "There are 5 fruits." It tells you the total number but loses the identity of the fruit.
• The New Way (Cobordism): This method doesn't just count; it asks, "Is this fruit an apple? Is it a banana? Is it a hybrid?" It assigns a unique "genetic code" (a class in the Lazard ring) to every shape. This code tells you not just how many shapes there are, but what kind of shapes they are made of.

The paper focuses on the simplest case: genus zero (spherical shapes) with $n$ marked points. The goal is to find the "genetic code" for the entire landscape of these shapes.

The Problem: The "String Equation" is Broken

Mathematicians have a famous rule called the String Equation. It's like a recipe that tells you how to calculate the properties of a landscape with $n+1$ points based on a landscape with $n$ points.

• In the old world (Chow ring): If you add a new point to your shape, the recipe says, "The total count stays the same, just shift things around." It's a simple, clean rule.
• In the new world (Cobordism): When you try to use this same recipe, it breaks! Why? Because in this new, high-definition world, adding a point doesn't just shift things; it actually adds new material. The "pushforward" (moving the shape from a complex space to a simple one) isn't zero anymore; it's a complex, multi-layered object.

The Solution: A Recursive Recipe

Ellis-Bloor fixes the broken recipe. He derives a new, more complex String Equation that works for cobordism.

Here is the analogy:
Imagine you are building a tower of blocks.

1. The Old Rule: "To build a tower of height $N$, just take the tower of height $N-1$ and add one block on top." (Simple).
2. The New Reality: "To build a tower of height $N$, you take the tower of height $N-1$, but you also have to glue in a whole new wing of the building, and that wing has its own internal structure that depends on how the blocks are arranged."

The paper provides the exact formula for this "gluing." It involves:

• The Universal Formal Group Law: Think of this as the "physics engine" of the universe of shapes. It dictates how two shapes merge. The paper uses a specific, universal version of this engine that works for all possible shapes.
• Recursive Steps: The formula says, "To find the code for $n$ points, look at the code for $n-1$ points, subtract some specific interactions, and add a correction term that involves splitting the shape into two smaller pieces (like a tree splitting into two branches)."

The Results: The "Genetic Codes"

The paper doesn't just give the theory; it does the heavy lifting and calculates the actual codes for small numbers of points ($n$ up to 8).

• For $n=3$: The code is just 1 (the simplest shape).
• For $n=4$: The code is $u_1$ (which corresponds to a projective line, or a circle).
• For $n=5$: The code is $4u_1^2 - 3u_2$. This looks like a complex algebraic soup, but it's actually a precise description of the shape's "DNA."

The author also shows how to translate these complex codes back into simpler languages:

• Chow Ring (The "Black and White" photo): If you ignore the complex details, the formula simplifies to a famous combinatorial number (multinomial coefficients).
• K-Theory (A "Color" photo): If you look at it through a different lens (K-theory), the formula changes slightly but remains elegant, involving a variable $\beta$ that acts like a "twist" factor.

Why Does This Matter?

1. Universality: Because algebraic cobordism is the "universal" theory, solving the problem here solves it for every other theory (like K-theory or Chow rings) automatically. It's like finding the master key that opens every door.
2. Precision: It allows mathematicians to distinguish between shapes that look the same in a simple count but are fundamentally different in their structure.
3. Foundation: These calculations are the building blocks for Gromov-Witten invariants, which are used in string theory (physics) to understand how strings move through space-time. By refining these invariants, the paper helps physicists get a more accurate picture of the universe.

Summary in One Sentence

Benjamin Ellis-Bloor has upgraded the mathematical "ruler" used to measure the shapes of curved surfaces, creating a new, ultra-precise formula that accounts for the complex internal structure of these shapes, allowing us to calculate their properties with a level of detail previously impossible.

Technical Summary

Here is a detailed technical summary of the paper "Cobordism-valued intersection theory on $M_{0,n}$" by Benjamin Ellis-Bloor.

1. Problem Statement

The paper addresses the calculation of Gromov-Witten invariants for the moduli space of stable genus-zero curves with $n$ marked points ($M_{0,n}$), specifically when the target space is a point. While these invariants are well-understood in rational cohomology (via the Chow ring) and $K$-theory, this work aims to compute them in algebraic cobordism ($\Omega^*$).

The core challenge is that algebraic cobordism, constructed by Levine and Morel, is the universal oriented cohomology theory. Unlike the Chow ring, where the pushforward of the structure sheaf of the universal curve $\pi_* (1)$ vanishes, in cobordism, this pushforward is non-zero and carries significant geometric information. The author seeks to:

1. Derive a "string equation" in algebraic cobordism that relates the cobordism class $[M_{0,n+1}]$ to $[M_{0,n}]$.
2. Provide inductive formulas for psi-class intersections (integrals of products of first Chern classes of tautological line bundles) on $M_{0,n}$.
3. Explicitly calculate these classes for $n \leq 8$ in terms of the generators of the Lazard ring $L_*$.

2. Methodology

The author employs a combination of geometric constructions and formal group law theory:

• Algebraic Cobordism Framework: The work utilizes the geometric presentation of $\Omega^*$ by Levine and Pandharipande, where cobordism classes are represented by projective morphisms modulo "double point relations." This allows for the use of explicit blow-up formulas.
• Keel's Iterated Blow-up Construction: The paper relies on Keel's description of $M_{0,n}$ as an iterated blow-up of $M_{0,n-1} \times \mathbb{P}^1$ along smooth centers. The author analyzes the normal bundles of these blow-up centers to apply Levine-Pandharipande's blow-up formula.
• Key Technical Lemma (Lemma 1.5): A crucial step is the explicit calculation of the normal bundle $N_{D_T/B_k}$ for the blow-up centers. The author proves that this normal bundle decomposes as $(L_a^\vee \otimes L_b^\vee) \oplus L_b^\vee$, where $L_a$ and $L_b$ are tautological line bundles on the boundary divisors.
• Universal Formal Group Law: Since $\Omega^*$ is governed by the universal formal group law (FGL) $x +_\Omega y$, the calculations involve manipulating power series in the Lazard ring $L_* \cong \mathbb{Z}[u_1, u_2, \dots]$. The author derives specific power series expansions (e.g., for the inverse of the FGL and specific rational functions involving the FGL) to handle the pushforwards of projective bundles appearing in the blow-up formula.
• Inductive Recursion: By combining the blow-up formula with Quillen's formula for the pushforward of projective bundles, the author derives a recursive relation (Theorem 1.1) that expresses integrals on $M_{0,n+1}$ in terms of integrals on $M_{0,n}$ and boundary divisors.

3. Key Contributions

A. The Cobordism String Equation (Theorem 1.1 & 4.1)

The paper generalizes the classical string equation (which holds in the Chow ring) to algebraic cobordism.

• In the Chow ring, $\int_{M_{0,n+1}} 1 \cdot \psi^d = \sum \int_{M_{0,n}} \psi^{d/\psi_i}$.
• In cobordism, the formula is significantly more complex due to the non-vanishing of $\pi_*(1)$. It includes:
  1. A term involving the cobordism class of $\mathbb{P}^1$ (denoted $[P^1]$).
  2. Corrections involving the inverse of the formal group law ($\bar{\psi}$).
  3. A sum over boundary divisors $D_T$ involving a specific power series $b_{ij}$ derived from the universal FGL.

B. Explicit Generators and Calculations

The author fixes a specific set of generators for the Lazard ring $L_*$ (related to projective spaces and blow-ups) and uses the derived recursion to compute the cobordism classes $[M_{0,n}]$ for $n \leq 8$.

• Example Results:
  • $[M_{0,3}] = 1$
  • $[M_{0,4}] = u_1$
  • $[M_{0,5}] = 4u_1^2 - 3u_2$
  • $[M_{0,6}] = 31u_1^3 - 30u_1 u_2 + 17u_3$
  • (Higher $n$ expressions are provided in the text and Appendix).

C. Connection to Chow and K-Theory

The paper demonstrates how the general cobordism formula specializes to known results:

• Chow Ring: Mapping to the additive formal group law ($x +_A y = x+y$) recovers the standard multinomial coefficient formula for psi-class intersections.
• K-Theory: Mapping to the multiplicative formal group law ($x +_K y = x + y - \beta xy$) recovers formulas related to Quantum K-theory (Lee's work), providing a closed formula for twisted psi-class intersections.

4. Results

• Inductive Formulas: The paper provides a complete algorithm to calculate any psi-class intersection $\int_{M_{0,n}} \psi^d$ in $\Omega^*$.
• Explicit Tables: Appendix A contains tables of all cobordism-valued psi-class intersections for $n \leq 8$, expressed in terms of the generators $u_i$.
• Verification: The results for $n \leq 6$ agree with previous rational cobordism calculations by Coates and Givental, but the current work provides integral formulas (over $\mathbb{Z}$) rather than just rational ones.

5. Significance

• Universality: By working in algebraic cobordism, the results automatically imply formulas for any oriented cohomology theory (via the orientation map $\Omega^* \to A^*$). This unifies calculations in cohomology, K-theory, and other theories.
• Resolution of a Conjecture/Gap: Givental had noted the lack of a simple string equation in quantum cobordism. This paper derives such an equation specifically for the genus-zero, point-target case, filling a gap in the literature.
• Computational Power: The explicit formulas for $n \leq 8$ provide concrete data for testing conjectures in enumerative geometry and string theory that rely on cobordism invariants.
• Methodological Advancement: The detailed analysis of normal bundles in Keel's construction and the application of the blow-up formula in the context of the universal FGL offers a robust toolkit for future calculations in algebraic cobordism on moduli spaces.

In summary, this paper successfully lifts the intersection theory of $M_{0,n}$ from the Chow ring to the universal setting of algebraic cobordism, providing a recursive framework and explicit integral formulas that generalize and unify previous results in cohomology and K-theory.