Skip to main content
Log in

The DT-Instanton Equation on Almost Hermitian 6-Manifolds

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This article investigates a set of partial differential equations, the DT-instanton equations, whose solutions can be regarded as a generalization of the notion of Hermitian–Yang–Mills connections. These equations owe their name to the hope that they may be useful in extending the DT-invariant to the case of symplectic 6-manifolds. In this article, we give the first examples of nonabelian and irreducible DT-instantons on non-Kähler manifolds. These are constructed for all homogeneous almost Hermitian structures on the manifold of full flags in \(\mathbb {C}^3\). Together with the existence result we derive a very explicit classification of homogeneous DT-instantons for such structures. Using this classification we are able to observe phenomena where, by varying the underlying almost Hermitian structure, an irreducible DT-instanton becomes reducible and then disappears. This is a non-Kähler analogue of passing a stability wall, which in string theory can be interpreted as supersymmetry breaking by internal gauge fields as in Anderson et al. (J High Energy Phys 09:026, 2009).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Any nearly Kähler manifold is Einstein with positive scalar curvature. Hence, if it is complete, must actually be compact.

  2. The two irreducible HYM connections existing for \(s<s_0\) are actually gauge equivalent, see Remark 6.2. However, the gauge transformation exchanging them fixed the reducible HYM connection existing at \(s=s_0\).

References

  1. Anderson, L.B., Gray, J., Lukas, A., Ovrut, B.: Stability walls in heterotic theories. J. High Energy Phys. 09, 026 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  2. Baulieu, L., Kanno, H., Singer, I.M.: Special quantum field theories in eight and other dimensions. Commun. Math. Phys. 194(1), 149–175 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  3. Baulieu, L., Losev, A., Nekrasov, N.: Chern–Simons and twisted supersymmetry in various dimensions. Nucl. Phys. B 522(1–2), 82–104 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bryant, R.L.: On the geometry of almost complex 6-manifolds. Asian J. Math. 10(3), 561–605 (2006)

    Article  MathSciNet  Google Scholar 

  5. Bryant, R.L.: Some remarks on \({\rm G}_2\)–structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, pp. 75–109 (2006)

  6. Charbonneau, B., Harland, D.: Deformations of nearly Kähler instantons. Commun. Math. Phys. 348(3), 959–990 (2016)

    Article  ADS  Google Scholar 

  7. Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 3(1), 1–26 (1985)

    Article  MathSciNet  Google Scholar 

  8. Donaldson, S.K., Segal, E.P.: Gauge theory in higher dimensions, II, Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics. Surv. Differ. Geom. 16, 1–41 (2011)

    Article  Google Scholar 

  9. Foscolo, L.: Deformation theory of nearly Kähler manifolds. J. Lond. Math. Soc. 95(2), 586–612 (2017)

    Article  MathSciNet  Google Scholar 

  10. Iqbal, A., Vafa, C., Nekrasov, N., Okounkov, A.: Quantum foam and topological strings. J. High Energy Phys. 04, 011 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  11. Lubke, M., Teleman, A.: The Kobayashi–Hitchin Correspondence. World Scientific, Singapore (1995)

    Book  Google Scholar 

  12. Oliveira, G.: Monopoles in Higher Dimensions, Ph.D. Thesis (2014)

  13. Oliveira, G.: Calabi–Yau monopoles for the Stenzel metric. Commun. Math. Phys. 341(2), 699–728 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  14. Tanaka, Y.: A weak compactness theorem of the Donaldson–Thomas instantons on compact Kähler threefolds. J. Math. Anal. Appl. 408(1), 27–34 (2013)

    Article  MathSciNet  Google Scholar 

  15. Tanaka, Y.: A removal singularity theorem of the Donaldson–Thomas instanton on compact Kähler threefolds. J. Math. Anal. Appl. 411(1), 422–428 (2014)

    Article  MathSciNet  Google Scholar 

  16. Tanaka, Y.: On the moduli space of Donaldson–Thomas instantons. Extracta mathematicae 31(1), 89–107 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Thomas, R.P.: Derived categories for the working mathematician, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999). AMS/IP Stud. Adv. Math. 23, 349–361 (2001)

    Article  Google Scholar 

  18. Thomas, R.P.: Gauge Theory on Calabi–Yau manifolds, Ph.D. Thesis (1997)

  19. Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39(S1), S257–S293 (1986)

    Article  MathSciNet  Google Scholar 

  20. Verbitsky, M.: Hodge theory on nearly Kähler manifolds. Geom. Topol. 15(4), 2111–2133 (2011)

    Article  MathSciNet  Google Scholar 

  21. Wang, H.C.: On invariant connections over a principal fibre bundle. Nagoya Math. J. 13, 1–19 (1958)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank Benoit Charbonneau, Gäel Cousin and Lorenzo Foscolo for helpful conversations regarding this article. We are particularly thankful for Benoit Charbonneau’s comments and carefully reading a previous version of this article.

Gonçalo Oliveira is supported by Fundação Serrapilheira 1812-27395, by CNPq grants 428959/2018-0 and 307475/2018-2, and FAPERJ through the program Jovem Cientista do Nosso Estado E-26/202.793/2019.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gavin Ball.

Additional information

Communicated by S. Gukov.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A. The topology of the bundles \(P_{\beta }\)

Appendix A. The topology of the bundles \(P_{\beta }\)

Recall that the bundles \(P_{\beta }\) are constructed via \(P_{\beta }= \mathrm{SU}(3) \times _{(T^2, \lambda _{\beta })} \mathrm{SO}(3)\), where

$$\begin{aligned} \lambda _{\beta } = \mathrm {diag}( e^{\frac{i}{2}\beta } , e^{-\frac{i}{2}\beta } ) \in \mathrm{SU}(2)/\mathbb {Z}_2 . \end{aligned}$$

Let \(V_{\beta }=P_{\beta } \times _{SO(3)} \mathbb {R}^3\) be the vector bundle associated with respect to the standard representation SO(3)-representation, and consider the \(\mathrm{U}(2)\) bundle

$$\begin{aligned} E_{\beta }=\mathrm{SU}(3) \times _{(T^2 , \tilde{\lambda }_{\beta } )} \mathbb {C}^2 , \end{aligned}$$

where

$$\begin{aligned} \tilde{\lambda }_{\beta } = \mathrm {diag}( e^{i\beta } , 0 ) \in \mathrm{U}(2), \end{aligned}$$

and the ismorphism. This has the property that the \(\mathrm{U}(2)\)-adjoint bundle of \(E_{\beta }\) splits as \(\mathfrak {u}_{E_{\beta }} \cong \underline{\mathbb {R}} \oplus V_{\beta }\) and

$$\begin{aligned} w_2(V_{\beta })= c_1(E_{\beta }) \mod 2, \ \ \ p_1(V_{\beta }) = c_1(E_{\beta })^2 - 4 c_2(E_{\beta }) . \end{aligned}$$

We shall now compute the Chern classes of the bundles \(E_{\beta }\) using Chern-Weyl theory. For this we must equip \(E_{\beta }\) with a connection which we choose to be the standard invariant connection given by

$$\begin{aligned} A_{\beta }= \beta \otimes \mathrm {diag}(i,0). \end{aligned}$$

This has curvature \(F_{\beta } = d\beta \otimes \mathrm {diag}(i,0)\) and so

$$\begin{aligned} c_1(E_{\beta })=-\frac{1}{2\pi }[d \beta ] , \ \ c_2(E_{\beta })= \frac{1}{4 \pi ^2} [d \beta ] \cup [d \beta ]. \end{aligned}$$

Furthermore, a computation using the Maurer–Cartan equations shows that

$$\begin{aligned} d\beta _1^2 + d\beta _2^2 + d \beta _1 \wedge d\beta _2 = d \mathop {\mathrm {Im}}((\eta _1+i\theta _1) \wedge (\eta _1+i\theta _1) \wedge (\eta _1+i\theta _1)), \end{aligned}$$

and so in \(H^4(\mathbb {F}_2, \mathbb {Z})\) we have

$$\begin{aligned}{}[d\beta _1] \cup [d \beta _2 ]= - [d\beta _1] \cup [d \beta _1 ] - [d\beta _2] \cup [d \beta _2 ]. \end{aligned}$$

So, writing \(\beta = k \beta _1 + l \beta _2\) we compute that

$$\begin{aligned} w_2(V_{\beta })=-\frac{1}{2\pi }\left( k[d \beta _1] + l [d \beta _2] \right) \mod 2 , \end{aligned}$$

while

$$\begin{aligned} p_1(V_{\beta })= & {} \frac{1}{4 \pi ^2} [d \beta ] \cup [d \beta ] \\= & {} \frac{1}{4 \pi ^2} \left( k^2 [d \beta _1] \cup [d \beta _1 ] + 2kl [d \beta _1] \cup [d \beta _2] + l^2 [d \beta _2] \cup [d \beta _2] \right) \\= & {} \frac{1}{4 \pi ^2} \left( k(k-2l) [d \beta _1] \cup [d \beta _1 ] + l(l-2k) [d \beta _2] \cup [d \beta _2] \right) . \end{aligned}$$

In particular, when \(\beta \) is one of the roots \(r_1\), \(r_2\), \(r_3\) we respectively obtain

$$\begin{aligned}&w_1(P_{r_1}) = -\frac{1}{2\pi } [d \beta _1] \mod 2 , \ \ \ p_1(V_{r_1})= -\frac{3}{4 \pi ^2} [d \beta _1] \cup [d \beta _1 ] , \\&w_1(P_{r_2}) = \frac{1}{2\pi }[d \beta _2] \mod 2 , \ \ \ p_1(V_{r_2})= -\frac{3}{4 \pi ^2} [d \beta _2] \cup [d \beta _2] , \\&w_1(P_{r_3}) = -\frac{1}{2\pi }\left( [d \beta _1] - [d \beta _2] \right) \mod 2 , \\&p_1(V_{r_3})= \frac{1}{4 \pi ^2} \left( 3 [d \beta _1] \cup [d \beta _1 ] + 3 [d \beta _2] \cup [d \beta _2] \right) , \end{aligned}$$

so these three bundles are all topologically different.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ball, G., Oliveira, G. The DT-Instanton Equation on Almost Hermitian 6-Manifolds. Commun. Math. Phys. 388, 819–844 (2021). https://doi.org/10.1007/s00220-021-04206-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-04206-8

Navigation