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String Theory  Winding Strings 
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#1
Dec2610, 06:13 AM

P: 14

The answers will probably be "it ain't as simple as that" but here's trying, anyway.
1. Is current thinking that strings have to wind "around" something? If so, must the compactified space have at least one hole? 2. With just a single hole, presumably there can be just one topologically distinct string configuration (ignoring winding number and sense)? 3. With two holes, three string configurations  or maybe more with "twists"? 4. Three holes give seven ways of winding  plus twisted ways? 5. How many holes are favoured, and are there enough winding configurations to accommodate SuSy? Thanks in advance. Paul 


#2
Dec2610, 01:16 PM

P: 407

considers all other nontrivial cycles as linear combinations of the basic wrappings. For your question of two holes, the answer is essentially yes, but to be precise, you must specify more data (like if the space is compact or not). For example, a compact Rieman surface with two holes has four basis cycles  two "around" the holes and one "around" each neck. Lets denote this basis by (b1,b2,a1,a2). Then a general string wraps around a cycle of the form n1 b1+ n2 b2+m1 a1 + m2 a2. One says that the string has wrapping numbers (n1,n2,m1,m2) with respect to that basis; these numbers can also be viewed as certain charges. 


#3
Dec2610, 02:18 PM

P: 14

@surprised, that was really helpful  thanks a lot! It's moved me on a fraction and even nudged me gently to go back to my 30yearold linear algebra, group theory and diff. geometry. They may just make a bit more sense this time around...



#4
Dec2610, 04:39 PM

P: 337

String Theory  Winding Strings
This question may be just a sign of my extreme ignorance, but is there some reason that one couldn't have strings connected to each other at vertices in some sort of giant branching tree in order to keep the individual strings from shrinking to nothing? 


#5
Dec2610, 06:38 PM

P: 14

@surprised, you've made me think even more about "twists". Are your wrapping numbers allowed to be negative? Even though it may be impossible to specify winding sense around an isolated hole, there must be configurations of several holes where the net result of a cycle does depend on the (relative) individual senses.
Sorry i can't give you a picture, but think of an eastwest double bagel  two holes and a neck. Wind the string over the northwest, under the neck and onwards under and into the eastern hole, over the top of the southeastern limb, down again to go under the southwestern limb on the western bagel and back to the start. Now i'm fairly convinced this cycle has a different handedness from the one where all the overs and unders are reversed. So the quantum numbers need to be distinguishable (even if, as might be the case here, the two configurations have the same energy). My next problem might be dismissed on the grounds of compactness or some other "nice" topological requirement? Find a bit of the space where two bagellike pieces come close but don't meet. (#1) Wrap the string over the left hole, over the right, wind back down and under both to complete the cycle. Give this wrapping numbers (1,1). (#2) Wrap over the left, dip under the right, complete a figureeight path. Give this wrapping numbers (1,1) and think of it as #1 with a 180 twist. (#3) Make as with #1 but give it a 360 twist so the wrapping numbers are again (1,1). Now #1 and #3 are obviously not equivalent in handedness or energetically, so i guess it's anathema to string theory. If i pumped up the shape enough, the bagels would separate and the twisted bits of string would ping back and wind around *something"? Conclusion: strings aren't allowed to twist around themselves (to the extent of touching, anyway)? 


#6
Dec2710, 01:27 AM

P: 407

There is a number of papers on such "string junctions", one is for example: http://arXiv.org/pdf/hepth/9804210 


#7
Dec2710, 01:50 AM

P: 407

Note that for your example with two holes, there are 4 basic cycles and thus 4 wrapping numbers to be specified, so in order to determine whether two configurations are the equivalent, you must compare all four wrapping numbers. 


#8
Dec2710, 06:35 AM

P: 14

Lovely, ta very much!



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