Deformed Hermitian Yang–Mills equation

In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a D-brane in the B-model (commonly called a B-brane) of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger[1] in the case of Abelian gauge group (the unitary group ), and by Leung–YauZaslow[2] using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.

Definition

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In this section we present the dHYM equation as explained in the mathematical literature by Collins-Xie-Yau.[3] The deformed Hermitian–Yang–Mills equation is a fully non-linear partial differential equation for a Hermitian metric on a line bundle over a compact Kähler manifold, or more generally for a real  -form. Namely, suppose   is a Kähler manifold and   is a class. The case of a line bundle consists of setting   where   is the first Chern class of a holomorphic line bundle  . Suppose that   and consider the topological constant

 

Notice that   depends only on the class of   and  . Suppose that  . Then this is a complex number

 

for some real   and angle   which is uniquely determined.

Fix a smooth representative differential form   in the class  . For a smooth function   write  , and notice that  . The deformed Hermitian Yang–Mills equation for   with respect to   is

 

The second condition should be seen as a positivity condition on solutions to the first equation. That is, one looks for solutions to the equation   such that  . This is in analogy to the related problem of finding Kähler-Einstein metrics by looking for metrics   solving the Einstein equation, subject to the condition that   is a Kähler potential (which is a positivity condition on the form  ).

Discussion

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Relation to Hermitian Yang–Mills equation

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The dHYM equations can be transformed in several ways to illuminate several key properties of the equations. First, simple algebraic manipulation shows that the dHYM equation may be equivalently written

 

In this form, it is possible to see the relation between the dHYM equation and the regular Hermitian Yang–Mills equation. In particular, the dHYM equation should look like the regular HYM equation in the so-called large volume limit. Precisely, one replaces the Kähler form   by   for a positive integer  , and allows  . Notice that the phase   for   depends on  . In fact,  , and we can expand

 

Here we see that

 

and we see the dHYM equation for   takes the form

 

for some topological constant   determined by  . Thus we see the leading order term in the dHYM equation is

 

which is just the HYM equation (replacing   by   if necessary).

Local form

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The dHYM equation may also be written in local coordinates. Fix   and holomorphic coordinates   such that at the point  , we have

 

Here   for all   as we assumed   was a real form. Define the Lagrangian phase operator to be

 

Then simple computation shows that the dHYM equation in these local coordinates takes the form

 

where  . In this form one sees that the dHYM equation is fully non-linear and elliptic.

Solutions

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It is possible to use algebraic geometry to study the existence of solutions to the dHYM equation, as demonstrated by the work of Collins–Jacob–Yau and Collins–Yau.[4][5][6] Suppose that   is any analytic subvariety of dimension  . Define the central charge   by

 

When the dimension of   is 2, Collins–Jacob–Yau show that if  , then there exists a solution of the dHYM equation in the class   if and only if for every curve   we have

 [4]

In the specific example where  , the blow-up of complex projective space, Jacob-Sheu show that   admits a solution to the dHYM equation if and only if   and for any  , we similarly have

 [7]

It has been shown by Gao Chen that in the so-called supercritical phase, where  , algebraic conditions analogous to those above imply the existence of a solution to the dHYM equation.[8] This is achieved through comparisons between the dHYM and the so-called J-equation in Kähler geometry. The J-equation appears as the *small volume limit* of the dHYM equation, where   is replaced by   for a small real number   and one allows  .

In general it is conjectured that the existence of solutions to the dHYM equation for a class   should be equivalent to the Bridgeland stability of the line bundle  .[5][6] This is motivated both from comparisons with similar theorems in the non-deformed case, such as the famous Kobayashi–Hitchin correspondence which asserts that solutions exist to the HYM equations if and only if the underlying bundle is slope stable. It is also motivated by physical reasoning coming from string theory, which predicts that physically realistic B-branes (those admitting solutions to the dHYM equation for example) should correspond to Π-stability.[9]

Relation to string theory

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Superstring theory predicts that spacetime is 10-dimensional, consisting of a Lorentzian manifold of dimension 4 (usually assumed to be Minkowski space or De sitter or anti-De Sitter space) along with a Calabi–Yau manifold   of dimension 6 (which therefore has complex dimension 3). In this string theory open strings must satisfy Dirichlet boundary conditions on their endpoints. These conditions require that the end points of the string lie on so-called D-branes (D for Dirichlet), and there is much mathematical interest in describing these branes.

 
Open strings with endpoints fixed on D-branes

In the B-model of topological string theory, homological mirror symmetry suggests D-branes should be viewed as elements of the derived category of coherent sheaves on the Calabi–Yau 3-fold  .[10] This characterisation is abstract, and the case of primary importance, at least for the purpose of phrasing the dHYM equation, is when a B-brane consists of a holomorphic submanifold   and a holomorphic vector bundle   over it (here   would be viewed as the support of the coherent sheaf   over  ), possibly with a compatible Chern connection on the bundle.

This Chern connection arises from a choice of Hermitian metric   on  , with corresponding connection   and curvature form  . Ambient on the spacetime there is also a B-field or Kalb–Ramond field   (not to be confused with the B in B-model), which is the string theoretic equivalent of the classical background electromagnetic field (hence the use of  , which commonly denotes the magnetic field strength).[11] Mathematically the B-field is a gerbe or bundle gerbe over spacetime, which means   consists of a collection of two-forms   for an open cover   of spacetime, but these forms may not agree on overlaps, where they must satisfy cocycle conditions in analogy with the transition functions of line bundles (0-gerbes).[12] This B-field has the property that when pulled back along the inclusion map   the gerbe is trivial, which means the B-field may be identified with a globally defined two-form on  , written  . The differential form   discussed above in this context is given by  , and studying the dHYM equations in the special case where   or equivalently   should be seen as turning the B-field off or setting  , which in string theory corresponds to a spacetime with no background higher electromagnetic field.

The dHYM equation describes the equations of motion for this D-brane   in spacetime equipped with a B-field  , and is derived from the corresponding equations of motion for A-branes through mirror symmetry.[1][2] Mathematically the A-model describes D-branes as elements of the Fukaya category of  , special Lagrangian submanifolds of   equipped with a flat unitary line bundle over them, and the equations of motion for these A-branes is understood. In the above section the dHYM equation has been phrased for the D6-brane  .

See also

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References

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  1. ^ a b Marino, M., Minasian, R., Moore, G. and Strominger, A., Nonlinear instantons from supersymmetric p-branes. Journal of High Energy Physics, 2000(01), p.005.
  2. ^ a b Leung, N.C., Yau, S.T. and Zaslow, E., From special lagrangian to hermitian–Yang–Mills via Fourier–Mukai transform. Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319–1341.
  3. ^ Collins, T.C., XIIE, D. and YAU, S.T.G., The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics. Geometry and Physics: Volume 1: A Festschrift in Honour of Nigel Hitchin, 1, p. 69.
  4. ^ a b Collins, T.C., Jacob, A. and Yau, S.T., (1, 1) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions. Camb. J. Math. 8 (2020), no. 2, 407–452.
  5. ^ a b Collins, T.C. and Yau, S.T., Moment maps, nonlinear PDE, and stability in mirror symmetry. arXiv preprint 2018, arXiv:1811.04824.
  6. ^ a b Collins, T.C. and Shi, Y., Stability and the deformed Hermitian–Yang–Mills equation. arXiv preprint 2020, arXiv:2004.04831.
  7. ^ A. Jacob, and N. Sheu, The deformed Hermitian–Yang–Mills equation on the blow-up of P^n, arXiv preprint 2020, arXiv:2009.00651
  8. ^ Chen, G., The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. math. (2021)
  9. ^ Douglas, M.R., Fiol, B. and Römelsberger, C., Stability and BPS branes. Journal of High Energy Physics, 2005(09), p.006.
  10. ^ Aspinwall, P.S., D-Branes on Calabi–Yau Manifolds. In Progress in String Theory: TASI 2003 Lecture Notes. Edited by MALDACENA JUAN M. Published by World Scientific Publishing Co. Pte. Ltd., 2005. ISBN 9789812775108, pp. 1–152 (pp. 1–152).
  11. ^ Freed, D.S. and Witten, E., Anomalies in string theory with $ D $-branes. Asian Journal of Mathematics, 3(4), pp. 819–852.
  12. ^ Laine, K., Geometric and topological aspects of Type IIB D-branes. Master's thesis (advisor Jouko Mickelsson), University of Helsinki