# One-Plaquette Action and SU(2)'s Irreducible Representations

I have a typical single-plaquette partition function for a gauge-field $$Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$ with $U$ as the product of the the $U$'s assigned to each link around a plaquette. Now the $U$'s are irreducible representations of my group elements, which in my case is SU(2), and lets take the 1/2 representation as an example, then define the character as $\Xi_{r}\equiv \text{Tr}[U]$ which for our case is $\Xi_{1/2}=\text{Tr}[U]$. Now I have to take the product of these representations, however (and here's my question), how do I know which group elements/that-element's-representation to assign to each link?

I'm not sure how to compute the Trace without knowing first how to do the product of the link's representations, but I don't even know how to assign the elements to the links.

Thanks,

-

Okay. Thanks for that ;) – kηives May 22 '12 at 3:09
There is no representation on a link, there is a 2 by 2 matrix, an element of SU(2), on each link. The trace of the product is the matrix trace of the matrix product. Do you want to know how to simulate this? You need a metropolis step, and there are tricks for making good updates over big regions. – Ron Maimon May 22 '12 at 6:24

## 1 Answer

I'm not completely sure what OP is asking(v1). However here is my interpretation.

How do I know which group elements [...] to assign to each link?

The group element $U_{\ell}\in SU(2)$ affiliated with a link $\ell\in L$ is not fixed. One is supposed to integrate over all possible group values of $U_{\ell}\in SU(2)$. Phrased differently, the link variables $(U_{\ell})_{\ell\in L}$ are the dynamical variables of the model.

The integration measure in the integral for $Z$ reads

$$[dU]~=~ \prod_{\ell\in L} dU_{\ell}$$

Here $dU_{\ell}$ typically denotes the Haar measure for $SU(2)$.

-

Oh, duh, just like stat-mech.  Thanks. – kηives May 22 '12 at 0:39
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9326463341712952, "perplexity_flag": "head"}