Path Integral Derivations Of K-Theoretic Donaldson Invariants
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908562182111232 |
|---|---|
| author | Kim, Heeyeon Manschot, Jan Moore, Gregory W. Tao, Runkai Zhang, Xinyu |
| author_facet | Kim, Heeyeon Manschot, Jan Moore, Gregory W. Tao, Runkai Zhang, Xinyu |
| contents | We consider 5d $\mathcal{N}=1$ SU(2) super Yang-Mills theory on $X\times S^1$, with $X$ a closed smooth four-manifold. A partial topological twisting along $X$ renders the theory formally independent of the metric on $X$. The theory depends on the spin structure and the circumference $R$ of $S^1$. The coefficients of the $R$-expansion of the partition function are Witten indices, which are identified with $L^2$-indices of Dirac operators on moduli spaces of instantons. The partition function encodes BPS indices for instanton particles on a spatial manifold $X$, and these indices are special cases of K-theoretic Donaldson invariants. When the 't Hooft flux of the gauge theory is nonzero and $X$ is not spin, the 5d theory can be anomalous, but this anomaly can be canceled by coupling to a line bundle with connection for the global $U(1)$ ``instanton number symmetry''. For $b_2^+(X)>0$ we can derive the partition function from integration over the Coulomb branch of the effective 4d low-energy theory. When $X$ is toric we can also use equivariant localization with respect to the $\mathbb{C}^* \times \mathbb{C}^*$ symmetry. The two methods lead to the same results for the wall-crossing formula. We also determine path integrals for four-manifolds with $b_2^+(X)>1$. Our results agree with those for algebraic surfaces by Göttsche, Kool, Nakajima, Yoshioka, and Williams, but apply to a larger class of manifolds. When the circumference of the circle is tuned to special values, the path integral is associated with the 5d superconformal $E_1$ theory. Topological invariants in this case involve generalizations of Seiberg-Witten invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path Integral Derivations Of K-Theoretic Donaldson Invariants Kim, Heeyeon Manschot, Jan Moore, Gregory W. Tao, Runkai Zhang, Xinyu High Energy Physics - Theory Mathematical Physics Algebraic Geometry Differential Geometry We consider 5d $\mathcal{N}=1$ SU(2) super Yang-Mills theory on $X\times S^1$, with $X$ a closed smooth four-manifold. A partial topological twisting along $X$ renders the theory formally independent of the metric on $X$. The theory depends on the spin structure and the circumference $R$ of $S^1$. The coefficients of the $R$-expansion of the partition function are Witten indices, which are identified with $L^2$-indices of Dirac operators on moduli spaces of instantons. The partition function encodes BPS indices for instanton particles on a spatial manifold $X$, and these indices are special cases of K-theoretic Donaldson invariants. When the 't Hooft flux of the gauge theory is nonzero and $X$ is not spin, the 5d theory can be anomalous, but this anomaly can be canceled by coupling to a line bundle with connection for the global $U(1)$ ``instanton number symmetry''. For $b_2^+(X)>0$ we can derive the partition function from integration over the Coulomb branch of the effective 4d low-energy theory. When $X$ is toric we can also use equivariant localization with respect to the $\mathbb{C}^* \times \mathbb{C}^*$ symmetry. The two methods lead to the same results for the wall-crossing formula. We also determine path integrals for four-manifolds with $b_2^+(X)>1$. Our results agree with those for algebraic surfaces by Göttsche, Kool, Nakajima, Yoshioka, and Williams, but apply to a larger class of manifolds. When the circumference of the circle is tuned to special values, the path integral is associated with the 5d superconformal $E_1$ theory. Topological invariants in this case involve generalizations of Seiberg-Witten invariants. |
| title | Path Integral Derivations Of K-Theoretic Donaldson Invariants |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2509.23042 |