Deformed Hermitian Yang–Mills equation

In this section we present the dHYM equation as explained in the mathematical literature by Collins-Xie-Yau.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. 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 $(1,1)$-form. Namely, suppose $(X,\backslash omega)$ is a Kähler manifold and $[\backslash alpha]\; \backslash in\; H^\{1,1\}(X,\backslash mathbb\{R\})$ is a class. The case of a line bundle consists of setting $[\backslash alpha]=c\_1(L)$ where $c\_1(L)$ is the first Chern class of a holomorphic line bundle $L\backslash to\; X$. Suppose that $\backslash dim\; X\; =\; n$ and consider the topological constant

:$\backslash hat\; z([\backslash omega],\; [\backslash alpha])\; =\; \backslash int\_X\; (\backslash omega\; +\; i\; \backslash alpha)^n.$

Notice that $\backslash hat\; z$ depends only on the class of $\backslash omega$ and $\backslash alpha$. Suppose that $\backslash hat\; z\backslash ne\; 0$. Then this is a complex number

:$\backslash hat\; z([\backslash omega],\; [\backslash alpha])\; =\; r\; e^\{i\; \backslash theta\}$

for some real $r>0$ and angle $\backslash theta\backslash in\; [0,2\backslash pi)$ which is uniquely determined.

Fix a smooth representative differential form $\backslash alpha$ in the class $[\backslash alpha]$. For a smooth function $\backslash phi:\; X\; \backslash to\; \backslash mathbb\{R\}$ write $\backslash alpha\_\{\backslash phi\}\; =\; \backslash alpha\; +\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash phi$, and notice that $[\backslash alpha\_\{\backslash phi\}]\; =\; [\backslash alpha]$. The deformed Hermitian Yang–Mills equation for $(X,\backslash omega)$ with respect to $[\backslash alpha]$ is

:$\backslash begin\{cases\}\backslash operatorname\{Im\}(e^\{-i\backslash theta\}\; (\backslash omega\; +\; i\; \backslash alpha\_\{\backslash phi\})^n)\; =\; 0\backslash \backslash$

\operatorname{Re}(e^{-i\theta} (\omega + i \alpha_{\phi})^n) > 0.\end{cases}

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 $\backslash operatorname\{Im\}(e^\{-i\backslash theta\}\; (\backslash omega\; +\; i\; \backslash alpha\_\{\backslash phi\})^n)\; =\; 0$ such that $\backslash operatorname\{Re\}(e^\{-i\backslash theta\}\; (\backslash omega\; +\; i\; \backslash alpha\_\{\backslash phi\})^n)\; >\; 0$. This is in analogy to the related problem of finding Kähler-Einstein metrics by looking for metrics $\backslash omega\; +\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash phi$ solving the Einstein equation, subject to the condition that $\backslash phi$ is a Kähler potential (which is a positivity condition on the form $\backslash omega\; +\; i\; \backslash partial\; \backslash bar\; \backslash partial\; \backslash phi$).

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

:$\backslash operatorname\{Im\}((\backslash omega+i\backslash alpha)^n)=\backslash tan\; \backslash theta\; \backslash operatorname\{Re\}((\backslash omega+i\backslash alpha)^n).$

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 $\backslash omega$ by $k\backslash omega$ for a positive integer $k$, and allows $k\backslash to\; \backslash infty$. Notice that the phase $\backslash theta\_k$ for $(X,k\backslash omega,[\backslash alpha])$ depends on $k$. In fact, $\backslash tan\; \backslash theta\_k\; =\; O(k^\{-1\})$, and we can expand

:$(k\backslash omega\; +\; i\; \backslash alpha)^n\; =\; k^n\; \backslash omega^n\; +\; i\; n\; k^\{n-1\}\; \backslash omega^\{n-1\}\backslash wedge\; \backslash alpha\; +\; O(k^\{n-2\}).$

Here we see that

:$\backslash operatorname\{Re\}((k\backslash omega\; +\; i\; \backslash alpha)^n)\; =\; k^n\; \backslash omega^n\; +\; O(k^\{n-2\}),\backslash quad\; \backslash operatorname\{Im\}((k\backslash omega\; +\; i\; \backslash alpha)^n)\; =\; nk^\{n-1\}\; \backslash omega^\{n-1\}\backslash wedge\; \backslash alpha\; +\; O(k^\{n-3\}),$

and we see the dHYM equation for $k\backslash omega$ takes the form

:$C\; k^\{n-1\}\; \backslash omega^n\; +\; O(k^\{n-3\})\; =\; n\; k^\{n-1\}\; \backslash omega^\{n-1\}\; \backslash wedge\; \backslash alpha\; +\; O(k^\{n-3\})$

for some topological constant $C$ determined by $\backslash tan\; \backslash theta$. Thus we see the leading order term in the dHYM equation is

:$n\backslash omega^\{n-1\}\backslash wedge\; \backslash alpha\; =\; C\; \backslash omega^n$

which is just the HYM equation (replacing $\backslash alpha$ by $F(h)$ if necessary).

The dHYM equation may also be written in local coordinates. Fix $p\backslash in\; X$ and holomorphic coordinates $(z^1,\backslash dots,z^n)$ such that at the point $p$, we have

:$\backslash omega\; =\; \backslash sum\_\{j=1\}^n\; i\; dz^j\; \backslash wedge\; d\backslash bar\; z^j,\backslash quad\; \backslash alpha\; =\; \backslash sum\_\{j=1\}^n\; \backslash lambda\_j\; i\; dz^j\; \backslash wedge\; d\backslash bar\; z^j.$

Here $\backslash lambda\_j\; \backslash in\; \backslash mathbb\{R\}$ for all $j$ as we assumed $\backslash alpha$ was a real form. Define the Lagrangian phase operator to be

:$\backslash Theta\_\{\backslash omega\}(\backslash alpha)\; =\; \backslash sum\_\{j=1\}^n\; \backslash arctan(\backslash lambda\_j).$

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

:$\backslash Theta\_\{\backslash omega\}(\backslash alpha)\; =\; \backslash phi$

where $\backslash phi\; =\; \backslash theta\backslash mod\; 2\backslash pi$. In this form one sees that the dHYM equation is fully non-linear and elliptic.

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.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.Collins, T.C. and Yau, S.T., Moment maps, nonlinear PDE, and stability in mirror symmetry. arXiv preprint 2018, {{arXiv|1811.04824}}.Collins, T.C. and Shi, Y., Stability and the deformed Hermitian–Yang–Mills equation. arXiv preprint 2020, {{arXiv|2004.04831}}. Suppose that $V\backslash subset\; X$ is any analytic subvariety of dimension $p$. Define the central charge $Z\_V([\backslash alpha])$ by

:$Z\_V([\backslash alpha])\; =\; -\backslash int\_V\; e^\{-i\backslash omega\; +\; \backslash alpha\}.$

When the dimension of $X$ is 2, Collins–Jacob–Yau show that if $\backslash operatorname\{Im\}(Z\_X([\backslash alpha]))>0$, then there exists a solution of the dHYM equation in the class $[\backslash alpha]\backslash in\; H^\{1,1\}(X,\backslash mathbb\{R\})$ if and only if for every curve $C\backslash subset\; X$ we have

:$\backslash operatorname\{Im\}\backslash left(\backslash frac\{Z\_C([\backslash alpha])\}\{Z\_X([\backslash alpha])\}\backslash right)>0.$

In the specific example where $X=\backslash operatorname\{Bl\}\_p\; \backslash mathbb\{CP\}^n$, the blow-up of complex projective space, Jacob-Sheu show that $[\backslash alpha]$ admits a solution to the dHYM equation if and only if $Z\_X([\backslash alpha])\backslash ne\; 0$ and for any $V\backslash subset\; X$, we similarly have

:$\backslash operatorname\{Im\}\backslash left(\backslash frac\{Z\_V([\backslash alpha])\}\{Z\_X([\backslash alpha])\}\backslash right)>0.$A. Jacob, and N. Sheu, The deformed Hermitian–Yang–Mills equation

on the blow-up of P^n, arXiv preprint 2020, {{arXiv|2009.00651}}

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.Chen, G., The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. math. (2021) 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 $\backslash omega$ is replaced by $\backslash varepsilon\; \backslash omega$ for a small real number $\backslash varepsilon>0$ and one allows $\backslash epsilon\backslash to\; 0$.

In general it is conjectured that the existence of solutions to the dHYM equation for a class $[\backslash alpha]\; =\; c\_1(L)$ should be equivalent to the Bridgeland stability of the line bundle $L$. 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.Douglas, M.R., Fiol, B. and Römelsberger, C., Stability and BPS branes. Journal of High Energy Physics, 2005(09), p.006.

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 $X$ 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.

File:D3-brane et D2-brane.PNG

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 $X$.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). 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 $Y\backslash subset\; X$ and a holomorphic vector bundle $E\backslash to\; Y$ over it (here $Y$ would be viewed as the support of the coherent sheaf $E$ over $X$), possibly with a compatible Chern connection on the bundle.

This Chern connection arises from a choice of Hermitian metric $h$ on $E$, with corresponding connection $\backslash nabla$ and curvature form $F(h)$. Ambient on the spacetime there is also a B-field or Kalb–Ramond field $B$ (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 $B$, which commonly denotes the magnetic field strength).Freed, D.S. and Witten, E., Anomalies in string theory with $ D $-branes. Asian Journal of Mathematics, 3(4), pp. 819–852. Mathematically the B-field is a gerbe or bundle gerbe over spacetime, which means $B$ consists of a collection of two-forms $B\_i\; \backslash in\; \backslash Omega^2(U\_i)$ for an open cover $U\_i$ 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).Laine, K., Geometric and topological aspects of Type IIB D-branes. Master's thesis (advisor Jouko Mickelsson), University of Helsinki This B-field has the property that when pulled back along the inclusion map $\backslash iota:\; Y\; \backslash to\; X$ the gerbe is trivial, which means the B-field may be identified with a globally defined two-form on $Y$, written $\backslash beta$. The differential form $\backslash alpha$ discussed above in this context is given by $\backslash alpha\; =\; F(h)\; +\; \backslash beta$, and studying the dHYM equations in the special case where $\backslash alpha\; =\; F(h)$ or equivalently $[\backslash alpha]\; =\; c\_1(L)$ should be seen as turning the B-field off or setting $\backslash beta\; =\; 0$, 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 $(Y,E)$ in spacetime equipped with a B-field $B$, and is derived from the corresponding equations of motion for A-branes through mirror symmetry. Mathematically the A-model describes D-branes as elements of the Fukaya category of $X$, special Lagrangian submanifolds of $X$ 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 $Y=X$.

• Hermitian Yang–Mills connection
• Yang–Mills connection
• Thomas–Yau conjecture

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Category:Geometry

Category:String theory

Category:Differential geometry

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