# How to construct/characterize “Thermal” sections ?

There were errors in the way I framed the question last time. So doing a major revision this time.

Consider $SU(2)$ as a homogeneous space $SU(2)\times SU(2)/SU(2)$ and sections of this principle bundle $x \mapsto (g_l(x),g_r(x))$ which are compatible with the projection map $(g_l(x),g_r(x)) \mapsto g_lg_r^{-1} = x$. Hence diagonal action on any section by a map from $SU(2)$ to itself ("gauge function") is compatible with this projection.

Now consider two elements $A$ and $B$ of $SU(2)$ which are acting on the base $SU(2)$ as $x \mapsto AxB^{-1}$. With respect to this a section of the bundle will be called "thermal" (there are physics motivations) if,

$$\sigma(AxB^{-1}) = (A,B).\sigma(x)$$

So the condition of being a thermal section seems to be a guage invariant constraint if one restricts to gauges which have the symmetry that $h(x)=h(AxB^{-1})$.The gauge map acts as $\sigma(x) \mapsto \sigma(x).(h(x),h(x))$.

(All the $.$'s are the standard group multiplication in in $SU(2)\times SU(2)$)

And by the first criteria of what is a valid section all sections can be gauge transformed into one another another since any section giving $(g_l(x),g_r(x))$ is gauge equivalent to the "canonical section" $(I,x^{-1})$ by the gauge function $g_l(x):SU(2)\mapsto SU(2)$ since $g_l(x)g_r(x)^{-1}=x$)

Is there a known mathematically concept equivalent to this? Like any mechanism by which given an $A$ and $B$ and a homogeneous space $G/H$, one would be able to manufacture "thermal" sections for it?

For the specific homogeneous space given it also happens that the push forward of the standard basis in the tangent space at identity of $SU(2)$ ("Pauli matrices") by the thermal section gives a vielbein for the standard metric on $SU(2)$!.

How generic is this?

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What is the action of SU(2) on SU(2)xSU(2)? – Orbicular Mar 3 '10 at 18:26
I don't understand this question. I don't even understand the notation. The projection map seems odd to me. I would expect that $(g_l(x),g_r(x)$ would get mapped back to $x$. Also what does $(A,B).\sigma(x)$ actually mean? – José Figueroa-O'Farrill Mar 3 '10 at 19:13
$g_l(x)$ and $g_r(x)$ are both elements of $SU(2)$ and usually chosen (atleast in the literature I am familiar with) such that $g_l(x)g_r(x)^{-1} = x$ $(A,B)$ is an element of $SU(2)\times SU(2)$ and $\sigma(x)$ is also an element of $SU(2)\times SU(2)$. So the $.$ is just group multiplication in $SU(2)\times SU(2)$. More explicitly if $\sigma(x)=(g_l(x),g_r(x))$ then $(A,B).\sigma (x) = (A.g_l(x),B.g_r(x))$. – Anirbit Mar 4 '10 at 5:59
Titles in the form of a question are strongly encouraged. Someone reading the list of questions has no idea what you want to know about thermal sections. – Ben Webster Mar 6 '10 at 21:46
Thanks for the advice. Changed the title. – Anirbit Mar 7 '10 at 14:57

This is still poorly written. As I understand it:

Let $G$ be a group. Consider the principle $G$-bundle $G\times G\rightarrow G$ where the action of $G$ on $G\times G$ is $g\colon (g_1,g_2)\mapsto (gg_1,gg_2)$ and the projection map is $(g_1,g_2)\mapsto g_1g_2^{-1}$.

A section $\sigma\colon G\rightarrow G\times G$ is thermal if $\sigma(AxB^{−1})=(A,B)\sigma(x)$ for all $x\in G$ (the base) and all $A,B\in G$.

An example of a section is $x\mapsto (x,1)$. Another example is $x\mapsto (1,x^{-1})$.

Now, what is the question?

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I was asking whether there is a "method" by which one can construct thermal sections on a homogeneous space G/H for a given A and B? Thinking of $S^3$ as $SU(2)\times SU(2)/SU(2)$, I know of "a" thermal section on it for a particular A and B. But that is a pure guess. If I want to say do it on $S^5$ thinking of that as $SO(6)/SO(5)$ then I can't hope to have another lucky guess. :) – Anirbit Mar 15 '10 at 10:17
And for homogeneous spaces one thinks of the group $G$ to act on the space $G/H$ transitively. Why are you thinking of the action in the reverse direction for the case of $S^3$ where $G=SU(2)\times SU(2)$ and $H=SU(2)$? – Anirbit Mar 15 '10 at 10:20