The Geometry of Supersymmetric Partition Functions (1309.5876v2)

Abstract: We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z_M. Our primary focus is the dependence of Z_M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N=1 theories with a U(1)_R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z_M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z_M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N=2 theories with a U(1)_R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z_M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S^3 x S^1 or S^2 x S^1, which are related to supersymmetric indices, and manifolds diffeomorphic to S^3 (squashed spheres). In examples where Z_M has been calculated explicitly, our results explain many of its observed properties.

• The paper demonstrates that the partition function for 4D N=1 and 3D N=2 theories depends holomorphically on moduli while remaining independent of the metric.
• It employs rigorous analysis on complex manifolds and transversely holomorphic foliations to relate partition functions to the moduli of holomorphic vector bundles.
• This result provides a robust framework for exploring supersymmetric indices and advanced computational techniques in varied geometric settings.

Overview of the Geometry of Supersymmetric Partition Functions

The paper under discussion explores the geometry of supersymmetric field theories residing on compact manifolds, specifically focusing on the parameter dependence of their partition functions. The prominent emphasis of this paper is the partition function $Z_M$'s dependence on the manifold $M$'s geometry and the background gauge fields linked with continuous flavor symmetries. The exploration extends over four-dimensional $N = 1$ theories with a $U(1)$ R-symmetry where $M$ serves as a complex manifold with a Hermitian metric, and three-dimensional $N = 2$ theories associated with a $U(1)_R$ symmetry, necessitating a transversely holomorphic foliation (THF) with a compatible metric.

The authors present a fundamental result showing that in both four- and three-dimensional theories, the partition function is independent of the metric. It, however, displays a holomorphic dependency on the respective moduli, such as complex structure moduli in four dimensions and THF moduli in three dimensions.

This analytical undertaking elucidates how $Z_M$ corresponds holomorphically to vector bundles over $M$ defined by background gauge fields, demonstrating its dependency on bundle moduli. The paper further ventures to calculate instances of $Z_M$ in explicit settings and relates these evaluations to observed characteristics, furnishing theoretical underpinnings for certain documented empirical properties.

Key Findings and Results

• Four-dimensional Analysis: For $N = 1$ supersymmetric theories on complex manifolds, the partition function $Z_M$ is verified to depend holomorphically on the complex structure's moduli. The metric independence establishes $Z_M$ as a potential topological invariant, though this delineation does not fully hold across all deformations beyond flat space.
• Three-dimensional Examination: Analogous results are extended to $N = 2$ theories on three-dimensional manifolds punctuated by THFs. Here, too, $Z_M$ shows a holomorphic relation to the THF's moduli. Moreover, THF deformations elucidate the partition function's limited parameter space, specifically holomorphic deformation parameters affecting $Z_M$.
• Background Gauge Fields: The work advocates that $Z_M$ retains holomorphic dependency on the anti-holomorphic parts of background gauge fields, linking it intricately to the moduli of holomorphic vector bundles on $M$.
• Implications for Supersymmetric Theories: The findings carry pivotal implications for the theoretical construction of partition functions tied to supersymmetric theories on specified geometric backgrounds. It posits potential pathways for understanding phenomena such as the supersymmetric index within the four-dimensional context and its parallel in three dimensions.

Theoretical and Practical Implications

The research contained within leverages complex manifold theories to gain insights into supersymmetric field theory, forecasting the potential for further explorations into the nature of partition functions linked to broader classes of supersymmetric spaces. It lays out a framework for evaluating the functionality and implications of supersymmetry in distinct geometrical contexts, carving a path for future analytical and computational advancements in supersymmetric field theories.

The discourse presents robust results that bolster the theoretical fabric of deformations in supersymmetric backgrounds and their partition functions—a theme central to the modern field of theoretical physics. Practically, these results pave the ground for pursuing advanced computational techniques to explore new classes of supersymmetric theories and potential geometric configurations.

In conclusion, this examination of the geometry of supersymmetric partition functions delivers substantial contributions to the theoretical understanding of supersymmetric fields configured on compact manifolds. Its analysis provides an essential basis for ongoing research and imparts potential frameworks for future investigations in the theoretical physics community. The insights on dependencies and independence in partition functions tied to manifold geometry and gauge fields remain particularly illuminating for subsequent endeavors in this area of paper.