algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The -Yang-Mills equation (or -YM equation) arises from a generalization of the Yang-Mills action functional with a function, which scales the norm of the curvature form. Notable special cases include exponential Yang-Mills connections, -Yang-Mills connections and Yang-Mills-Born-Infeld connections with positive and negative sign.
Let be a strictly increasing -function with . Let:
(Wei 22)
Since is a function, one can also consider the following constant:
(Baba & Shintani 23, Definition 4.8, Baba 23)
Let be a compact Lie group with Lie algebra and be a principal -bundle with a orientable Riemannian manifold . Let be its adjoint bundle. The -Yang-Mills action functional (or -YM action functional) is given by:
(Baba & Shintani 23, Definition 3.1, Baba 23)
For a flat connection (with ), one has . Hence is required to avert divergence for a non-compact manifold , although this condition can also be left out as only the derivative is of further importance.
A connection is called -Yang-Mills connection if it is a critical point of the -Yang–Mills action functional, hence if:
for all smooth families with . This is the case iff the -Yang–Mills equation (or -YM equation) is fulfilled:
(Baba & Shintani 23, Corollary 3.4, Baba 23)
For a -Yang-Mills connection , its curvature is called -Yang-Mills field (or -YM field).
A -Yang–Mills connection/field with:
(or for normalization) is called (normed) exponential Yang–Mills connection/field. In this case, one has . The exponential and normed exponential Yang-Mills action functional are denoted with and respectively. (Matsura & Urakawa 95)
is called -Yang–Mills connection/field. Usual Yang–Mills connections/fields are exactly the -Yang–Mills connections/fields. In this case, one has . The -Yang-Mills action functional is denoted with .
or is called Yang–Mills–Born–Infeld connection/field (or YMBI connection/field) with negative or positive sign respectively. In these cases, one has and respectively. The Yang-Mills-Born-Infeld action functionals with negative and positive sign are denoted with and respectively.
(Baba & Shintani 23, Example 3.2, Wei 22, Baba 23)
Analogous to (weakly) stable Yang-Mills connections and (weakly) stable Yang-Mills-Higgs connections, one can also consider the positivity of the second derivative in -Yang-Mills theory. is called a stable -Yang-Mills connection (or stable -YM connection), if:
for all smooth families with . It is called weakly stable if only holds. A -Yang–Mills connection, which is not weakly stable, is called unstable. If is a (weakly) stable or unstable -Yang-Mills connection, its curvature is also called (weakly) stable or unstable -Yang-Mills field.
(Baba & Shintani 23, Definition 3.6, Baba 23)
For a Yang-Mills connection with constant curvature, its stability as Yang-Mills connection implies its stability as exponential Yang-Mills connection.
(Matsura & Urakawa 95, Corollary 6.2)
Every non-flat exponential Yang-Mills connection over with and:
is flat.
(Baba & Shintani 23, Proposition 4.14, Baba 23)
Every non-flat Yang-Mills-Born-Infeld connection over with and:
is flat.
(Baba & Shintani 23, Proposition 4.13)
For , every non-flat -Yang-Mills connection over is unstable.
(Baba & Shintani 23, Theorem 1.2 and Corollary 4.12, Baba 23, Baba 23)
For , every non-flat -Yang-Mills connection over is unstable.
For , every non-flat Yang-Mills-Born-Infeld connection with positive sign over is unstable.
For , every non-flat -Yang-Mills connection over the Cayley plane is instable.
(Baba 23)
Fumiaki Matsura, Hajime Urakawa, On exponential Yang-Mills connections (1995), DOI:10.1016/0393-0440(94)00041-200041-2)
Shihshu Walter Wei, On exponential Yang-Mills fields and p-Yang-Mills fields (2022), arXiv:2205.03016
Kurando Baba, Kazuto Shintani, A Simons type condition for instability of F-Yang-Mills connections (2023), arXiv:2301.04291
Kurando Baba, On instability of F-Yang-Mills connections (2023), Slides
See also:
Last revised on November 5, 2024 at 02:10:52. See the history of this page for a list of all contributions to it.