Donaldson–Thomas theory

<h2 id="definition-and-examples">Definition and examples</h2>
The basic idea of <a href="/facts/Gromov%E2%80%93Witten_invariant/EJwrClYB">Gromov–Witten invariants</a> is to probe the geometry of a space by studying pseudoholomorphic maps from <a href="/facts/Riemann_surface/qoj8ogv8">Riemann surfaces</a> to a smooth target. The moduli stack of all such maps admits a virtual fundamental class, and intersection theory on this stack yields numerical invariants that can often contain enumerative information. In similar spirit, the approach of Donaldson–Thomas theory is to study curves in an algebraic three-fold by their equations. More accurately, by studying ideal sheaves on a space. This moduli space also admits a virtual fundamental class and yields certain numerical invariants that are enumerative.
Whereas in Gromov–Witten theory, maps are allowed to be multiple covers and collapsed components of the domain curve, Donaldson–Thomas theory allows for nilpotent information contained in the sheaves, however, these are integer valued invariants. There are deep conjectures due to Davesh Maulik, <a href="/facts/Andrei_Okounkov/ME33ML8t">Andrei Okounkov</a>, <a href="/facts/Nikita_Nekrasov/c77b5q7e">Nikita Nekrasov</a> and <a href="/facts/Rahul_Pandharipande/9FhB30g5">Rahul Pandharipande</a>, proved in increasing generality, that Gromov–Witten and Donaldson–Thomas theories of algebraic three-folds are actually equivalent.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> More concretely, their generating functions are equal after an appropriate change of variables. For Calabi–Yau threefolds, the Donaldson–Thomas invariants can be formulated as weighted Euler characteristic on the moduli space. There have also been recent connections between these invariants, the motivic Hall algebra, and the ring of functions on the quantum torus.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>The moduli space of lines on the <a href="/facts/Quintic_threefold/SHvYr27J">quintic threefold</a> is a discrete set of 2875 points.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The virtual number of points is the actual number of points, and hence the Donaldson–Thomas invariant of this moduli space is the integer 2875.</li>
<li>Similarly, the Donaldson–Thomas invariant of the moduli space of <a href="/facts/Conic/teqgLptX">conics</a> on the quintic is 609250.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<h3>Definition</h3>
For a Calabi–Yau threefold 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> and a fixed cohomology class 
 
 
 
 α
 ∈
 
 H
 
 even
 
 
 (
 Y
 ,
 
 Q
 
 )
 
 
 {\displaystyle \alpha \in H^{\text{even}}(Y,\mathbb {Q} )}
 
 there is an associated moduli stack 
 
 
 
 
 
 M
 
 
 (
 Y
 ,
 α
 )
 
 
 {\displaystyle {\mathcal {M}}(Y,\alpha )}
 
 of coherent sheaves with Chern character 
 
 
 
 c
 (
 
 
 E
 
 
 )
 =
 α
 
 
 {\displaystyle c({\mathcal {E}})=\alpha }
 
. In general, this is a non-separated Artin stack of infinite type which is difficult to define numerical invariants upon it. Instead, there are open substacks 
 
 
 
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 
 
 {\displaystyle {\mathcal {M}}^{\sigma }(Y,\alpha )}
 
 parametrizing such coherent sheaves 
 
 
 
 
 
 E
 
 
 
 
 {\displaystyle {\mathcal {E}}}
 
 which have a stability condition 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 imposed upon them, i.e. 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
-stable sheaves. These moduli stacks have much nicer properties, such as being separated of finite type. The only technical difficulty is they can have bad singularities due to the existence of obstructions of deformations of a fixed sheaf. In particular
 
 
 
 
 
 
 
 
 T
 
 [
 
 
 E
 
 
 ]
 
 
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 
 
 
 ≅
 
 
 Ext
 
 
 1
 
 
 (
 
 
 E
 
 
 ,
 
 
 E
 
 
 )
 
 
 
 
 
 
 Ob
 
 
 [
 
 
 E
 
 
 ]
 
 
 (
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 )
 
 
 
 ≅
 
 
 Ext
 
 
 2
 
 
 (
 
 
 E
 
 
 ,
 
 
 E
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}T_{[{\mathcal {E}}]}{\mathcal {M}}^{\sigma }(Y,\alpha )&\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})\\{\text{Ob}}_{[{\mathcal {E}}]}({\mathcal {M}}^{\sigma }(Y,\alpha ))&\cong {\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\end{aligned}}}
 
Now because 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is Calabi–Yau, Serre duality implies
 
 
 
 
 
 Ext
 
 
 2
 
 
 (
 
 
 E
 
 
 ,
 
 
 E
 
 
 )
 ≅
 
 
 Ext
 
 
 1
 
 
 (
 
 
 E
 
 
 ,
 
 
 E
 
 
 ⊗
 
 ω
 
 Y
 
 
 
 )
 
 ∨
 
 
 ≅
 
 
 Ext
 
 
 1
 
 
 (
 
 
 E
 
 
 ,
 
 
 E
 
 
 
 )
 
 ∨
 
 
 
 
 {\displaystyle {\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}}\otimes \omega _{Y})^{\vee }\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})^{\vee }}
 
which gives a perfect obstruction theory of dimension 0. In particular, this implies the associated virtual fundamental class
 
 
 
 [
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 
 ]
 
 v
 i
 r
 
 
 ∈
 
 H
 
 0
 
 
 (
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 ,
 
 Z
 
 )
 
 
 {\displaystyle [{\mathcal {M}}^{\sigma }(Y,\alpha )]^{vir}\in H_{0}({\mathcal {M}}^{\sigma }(Y,\alpha ),\mathbb {Z} )}
 
is in homological degree 
 
 
 
 0
 
 
 {\displaystyle 0}
 
. We can then define the DT invariant as
 
 
 
 
 ∫
 
 [
 
 
 
 M
 
 
 
 σ
 
 
 (
 Y
 ,
 α
 )
 
 ]
 
 v
 i
 r
 
 
 
 
 1
 
 
 {\displaystyle \int _{[{\mathcal {M}}^{\sigma }(Y,\alpha )]^{vir}}1}
 
which depends upon the stability condition 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 and the cohomology class 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
. It was proved by Thomas that for a smooth family 
 
 
 
 
 Y
 
 t
 
 
 
 
 {\displaystyle Y_{t}}
 
 the invariant defined above does not change. At the outset researchers chose the Gieseker stability condition, but other DT-invariants in recent years have been studied based on other stability conditions, leading to wall-crossing formulas.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
<h2 id="facts">Facts</h2>
<ul><li>The Donaldson–Thomas invariant of the moduli space M is equal to the weighted <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of M. The weight function associates to every point in M an analogue of the <a href="/facts/Milnor_number/9gADdHo2">Milnor number</a> of a hyperplane singularity.</li></ul>
<h2 id="generalizations">Generalizations</h2>
<ul><li>Instead of moduli spaces of sheaves, one considers moduli spaces of <a href="/facts/Derived_category/pBQHnFYo">derived category</a> objects. That gives the Pandharipande–Thomas invariants that count stable pairs of a Calabi–Yau 3-fold.</li>
<li>Instead of integer valued invariants, one considers <a href="/facts/Motive_(algebraic_geometry)/o0rNsaPo">motivic</a> invariants.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Enumerative_geometry/NSq0g3bY">Enumerative geometry</a></li>
<li><a href="/facts/Gromov%E2%80%93Witten_invariant/EJwrClYB">Gromov–Witten invariant</a></li>
<li><a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a></li>
<li><a href="/facts/Quantum_cohomology/0QaSnto0">Quantum cohomology</a></li></ul>

<ul><li><a href="/facts/Simon_Donaldson/zkQ5xq5u">Donaldson, Simon K.</a>; <a href="/facts/Richard_Thomas_(mathematician)/aFhgN0s7">Thomas, Richard P.</a> (1998), "Gauge theory in higher dimensions", in Huggett, S. A.; Mason, L. J.; Tod, K. P.; Tsou, S. T.; Woodhouse, N. M. J. (eds.), The geometric universe (Oxford, 1996), <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, pp. 31–47, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-850059-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1634503">1634503</a></li>
<li><a href="/facts/Maxim_Kontsevich/7No1RaTQ">Kontsevich, Maxim</a> (2007), <a href="https://www.ihes.fr/~maxim/TEXTS/DTinv-AT2007.pdf">Donaldson–Thomas invariants</a> (PDF), Mathematische Arbeitstagung, Bonn{{citation}}: CS1 maint: location missing publisher (link)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R. (2006). "Gromov–Witten theory and Donaldson–Thomas theory, I". Compositio Mathematica. 142 (5): 1263–1285. arXiv:math/0312059. doi:10.1112/S0010437X06002302 (inactive 15 December 2024). S2CID 5760317.{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link) <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bridgeland, Tom (1 June 2018). "Hall algebras and Donaldson-Thomas invariants". In De Fernex, Tommaso (ed.). Proceedings of Symposia in Pure Mathematics. Vol. 97.1. Providence, Rhode Island: American Mathematical Society. pp. 75–100. doi:10.1090/pspum/097.1/01670. ISBN 978-1-4704-3577-6. <a href="978-1-4704-3577-6" target="_blank">978-1-4704-3577-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Harris, Joe (1 December 1979). "Galois groups of enumerative problems". Duke Mathematical Journal. 46 (4): 685–724. doi:10.1215/S0012-7094-79-04635-0. ISSN 0012-7094. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Katz, Sheldon (11 August 2010). "On the finiteness of rational curves on quintic threefolds". Compositio Mathematica. 60 (2). Martinus Nijhoff Publishers: 151–162. ISSN 0010-437X. Retrieved 20 November 2024. <a href="https://eudml.org/doc/89802" target="_blank">https://eudml.org/doc/89802</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Szendrői, Balázs (9 June 2016). "Cohomological Donaldson–Thomas theory". In Bouchard, Vincent; Doran, Charles; Méndez-Diez, Stefan; Quigley, Callum (eds.). String-Math 2014 - Conference Proceedings. Proceedings of Symposia in Pure Mathematics. Vol. 93. Providence, Rhode Island: American Mathematical Society. pp. 363–396. arXiv:1503.07349. doi:10.1090/pspum/093. ISBN 978-1-4704-1992-9. MR 3526001. <a href="978-1-4704-1992-9" target="_blank">978-1-4704-1992-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Thomas, R. P. (2000). "A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations". Journal of Differential Geometry. 54 (2): 367–438. arXiv:math/9806111. doi:10.4310/jdg/1214341649. MR 1818182. <a href="https://projecteuclid.org/euclid.jdg/1214341649" target="_blank">https://projecteuclid.org/euclid.jdg/1214341649</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kontsevich, Maxim; Soibelman, Yan (2008-11-16). "Stability structures, motivic Donaldson-Thomas invariants and cluster transformations". arXiv:0811.2435 [math.AG]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Donaldson–Thomas theory open-in-new

Donaldson–Thomas theory