denoted by primes, to a mainly negative one used in this article can be done through implying that , , and . Additionally, the fields are often redefined as , , , , , .\n"},"parts":[{"template":{"target":{"wt":"refn","href":"./Template:Refn"},"params":{"group":{"wt":"nb"},"1":{"wt":"The action and supersymmetry variations depend on the [[metric signature]] used. Transforming from a mainily positive signature, denoted by primes, to a mainly negative one used in this article can be done through implying that , , and . Additionally, the fields are often redefined as , , , , , .\n"}},"i":0}}]}"> [nb 7]
They are useful for constructing the Killing spinor equations and finding the supersymmetric ground states of the theory since these require that the fermionic variations vanish.
Since type IIA supergravity has p-form field strengths of even dimensions, it also admits a nine-form gauge field . But since is a scalar and the free field equation is given by , this scalar must be a constant. [12] : 115 Such a field therefore has no propagating degrees of freedom, but does have an energy density associated to it. Working only with the bosonic sector, the ten-form can be included in supergravity by modifying the original action to get massive type IIA supergravity [15] : 89–90
where is equivalent to the original type IIA supergravity up to the replacement of and . Here is known as the Romans mass and it acts as a Lagrange multiplier for . Often one integrates out this field strength tensor resulting in an action where acts as a mass term for the Kalb–Ramond field.
Unlike in the regular type IIA theory, which has a vanishing scalar potential , massive type IIA has a nonvanishing scalar potential. While the supersymmetry transformations appear to be realised, they are actually formally broken since the theory corresponds to a D8-brane background. [14] : 668 A closely related theory is Howe–Lambert–West supergravity [16] which is another massive deformation of type IIA supergravity, [nb 8] but one that can only be described at the level of the equations of motion. It is acquired by a compactification of eleven-dimensional MM theory on a circle.
Compactification of eleven-dimensional supergravity on a circle and keeping only the zero Fourier modes that are independent of the compact coordinates results in type IIA supergravity. For eleven-dimensional supergravity with the graviton, gravitino, and a 3-form gauge field denoted by , then the 11D metric decomposes into the 10D metric, the 1-form, and the dilaton as [13] : 308
Meanwhile, the 11D 3-form decomposes into the 10D 3-form and the 10D 2-form . The ten-dimensional modified field strength tensor directly arises in this compactification from .
Dimensional reduction of the fermions must generally be done in terms of the flat coordinates , where is the 11D vielbein. [nb 9] In that case the 11D Majorana graviton decomposes into the 10D Majorana gravitino and the Majorana fermion , [9] : 268 [nb 10] although the exact identification is given by [14] : 664
where this is chosen to make the supersymmetry transformations simpler. [nb 11] The ten-dimensional supersymmetry variations can also be directly acquired from the eleven-dimensional ones by setting . [nb 12]
The low-energy effective field theory of type IIA string theory is given by type IIA supergravity. [15] : 187 The fields correspond to the different massless excitations of the string, with the metric, 2-form , and dilaton being NSNS states that are found in all string theories, while the 3-form and 1-form fields correspond to the RR states of type IIA string theory. Corrections to the type IIA supergravity action come in two types, quantum corrections in powers of the string coupling , and curvature corrections in powers of . [15] : 321–324 Such corrections often play an important role in type IIA string phenomenology. The type IIA superstring coupling constant corresponds to the vacuum expectation value of , while the string length is related to the gravitational coupling constant through . [12] : 115
When string theory is compactified to acquire four-dimensional theories, this is often done at the level of the low-energy supergravity. Reduction of type IIA on a Calabi–Yau manifold yields an theory in four dimensions, while reduction on a Calabi–Yau orientifold further breaks the symmetry down to give the phenomenologically viable four-dimensional supergravity. [13] : 356–357 Type IIA supergravity is automatically anomaly free since it is a non-chiral theory.
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In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics. It is a special case of 4D N = 1 global supersymmetry.
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In supersymmetry, eleven-dimensional supergravity is the theory of supergravity in the highest number of dimensions allowed for a supersymmetric theory. It contains a graviton, a gravitino, and a 3-form gauge field, with their interactions uniquely fixed by supersymmetry. Discovered in 1978 by Eugène Cremmer, Bernard Julia, and Joël Scherk, it quickly became a popular candidate for a theory of everything during the 1980s. However, interest in it soon faded due to numerous difficulties that arise when trying to construct physically realistic models. It came back to prominence in the mid-1990s when it was found to be the low energy limit of M-theory, making it crucial for understanding various aspects of string theory.
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