Why is the interpolation between two connections related via a gauge transformation still a connection?

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I am studying the theory of anomalies in gauge field.



Let $A$ be a gauge field (or a connection for mathematicians). Let $A_U$ be an equivalent gauge related via a local gauge transformation
$$A_U=U^-1dU+U^-1AU$$
Then, there is a interpolating between the two
$$A(s)=sA+(1-s)A_U$$
where $sin[0,1]$.



Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?




differential-geometry fiber-bundles connections gauge-theory




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    up vote
    1
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    1












    I am studying the theory of anomalies in gauge field.



    Let $A$ be a gauge field (or a connection for mathematicians). Let $A_U$ be an equivalent gauge related via a local gauge transformation
    $$A_U=U^-1dU+U^-1AU$$
    Then, there is a interpolating between the two
    $$A(s)=sA+(1-s)A_U$$
    where $sin[0,1]$.



    Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?




    differential-geometry fiber-bundles connections gauge-theory




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite
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      up vote
      1
      down vote

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      I am studying the theory of anomalies in gauge field.



      Let $A$ be a gauge field (or a connection for mathematicians). Let $A_U$ be an equivalent gauge related via a local gauge transformation
      $$A_U=U^-1dU+U^-1AU$$
      Then, there is a interpolating between the two
      $$A(s)=sA+(1-s)A_U$$
      where $sin[0,1]$.



      Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?




      differential-geometry fiber-bundles connections gauge-theory




      share|cite|improve this question











      I am studying the theory of anomalies in gauge field.



      Let $A$ be a gauge field (or a connection for mathematicians). Let $A_U$ be an equivalent gauge related via a local gauge transformation
      $$A_U=U^-1dU+U^-1AU$$
      Then, there is a interpolating between the two
      $$A(s)=sA+(1-s)A_U$$
      where $sin[0,1]$.



      Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?





      differential-geometry fiber-bundles connections gauge-theory





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      share|cite|improve this question




      share|cite|improve this question









      asked Jul 15 at 20:08









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