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Resolution of Stringy Singularities by Non-commutative Algebras

David Berenstein, Robert G. Leigh

TL;DR

The paper develops a non-commutative geometric framework for resolving stringy singularities via algebras that are locally finite over their centers. It treats D-branes as coherent modules over a non-commutative algebra ${\cal A}$ and identifies the center ${\cal Z}{\cal A}$ as the commutative target space for closed strings, with singularities resolved when local algebras are regular (i.e., admit finite projective resolutions of length $d+1$). Through crossed-product constructions and detailed examples (orbifolds, orbifolds of orbifolds, and conifolds), it derives local quivers from Ext groups, computes brane intersections, and connects these to the derived category of coherent sheaves; closed-string data are captured by cyclic/Hochschild homology, enabling Betti-number calculations for global non-commutative Calabi–Yau spaces. The work provides a concrete, calculable path to extract topological and brane data in singular backgrounds, offering a non-commutative resolution paradigm that complements and extends geometric resolutions in string theory.

Abstract

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.

Resolution of Stringy Singularities by Non-commutative Algebras

TL;DR

The paper develops a non-commutative geometric framework for resolving stringy singularities via algebras that are locally finite over their centers. It treats D-branes as coherent modules over a non-commutative algebra and identifies the center as the commutative target space for closed strings, with singularities resolved when local algebras are regular (i.e., admit finite projective resolutions of length ). Through crossed-product constructions and detailed examples (orbifolds, orbifolds of orbifolds, and conifolds), it derives local quivers from Ext groups, computes brane intersections, and connects these to the derived category of coherent sheaves; closed-string data are captured by cyclic/Hochschild homology, enabling Betti-number calculations for global non-commutative Calabi–Yau spaces. The work provides a concrete, calculable path to extract topological and brane data in singular backgrounds, offering a non-commutative resolution paradigm that complements and extends geometric resolutions in string theory.

Abstract

In this paper we propose a unified approach to (topological) string theory on certain singular spaces in their large volume limit. The approach exploits the non-commutative structure of D-branes, so the space is described by an algebraic geometry of non-commutative rings. The paper is devoted to the study of examples of these algebras. In our study there is an auxiliary commutative algebraic geometry of the center of the (local) algebras which plays an important role as the target space geometry where closed strings propagate. The singularities that are resolved will be the singularities of this auxiliary geometry. The singularities are resolved by the non-commutative algebra if the local non-commutative rings are regular. This definition guarantees that D-branes have a well defined K-theory class. Homological functors also play an important role. They describe the intersection theory of D-branes and lead to a formal definition of local quivers at singularities, which can be computed explicitly for many types of singularities. These results can be interpreted in terms of the derived category of coherent sheaves over the non-commutative rings, giving a non-commutative version of recent work by M. Douglas. We also describe global features like the Betti numbers of compact singular Calabi-Yau threefolds via global holomorphic sections of cyclic homology classes.

Paper Structure

This paper contains 16 sections, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. Crossed products
  4. A first example: ${\mathbb{C}}^2/{\mathbb{Z}}_n$
  5. Unorbifolding the orbifold
  6. Orbifold of an Orbifold and discrete torsion
  7. Lifting issues
  8. ${\mathbb{C}}_2/{\mathbb{Z}}_4$
  9. ${\mathbb{C}}^2/\hat{E}_8$
  10. ${\mathbb{C}}^3/{\mathbb{Z}}_4$
  11. Non-orbifold singularities
  12. Regularity, resolutions and intersection theory
  13. Counting open strings and quiver building
  14. Closed string states
  15. Betti Numbers for global non-commutative Calabi Yau spaces
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Small coordinate patch around $\alpha_0,\beta_0$
  • Figure 2: Quiver diagram for the $A_{n-1}$ singularity
  • Figure 3: Quiver diagram of the $\hat{E}_8$ singularity drawn using the two step technique of BJL3
  • Figure 4: Noncommutative cover of singularity without discrete torsion: the special points correspond to the intersection of two curves of singularities
  • Figure 5: Noncommutative cover of singularity with discrete torsion: at the special points the fractional branes have monodromies BL