Eric Sharpe - Examples of homological projective duality in physics

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We physicists speak of Kahler phases, which are the chambers in VGIT. One description of result is as LG models: V//C × = ( ) C 4 (1) ⊕ C2 (−2) //C × V// + C × : LG model on Tot( O(-2) 2 --> P 3 ) Kahler r V// − C × : LG model on Tot( O(-1) 4 --> P 1 [2,2] ) W = ∑ a both with p a Q a (φ) = ∑ ij φ i A ij (p)φ j
All phases of GLSM’s can be described as LG models, and ultimately this is why work such as BFK relates GLSM phases to hpd, via matrix factorizations. That said, we usually don’t stop at LG. We can often use the `renormalization group’ to give an alternative description of the low-energy dynamics of the theory (which for some purposes may be more complicated than a LG model).
Page 1 and 2: Examples of homological projective
Page 3 and 4: What did hpd teach us? Prior to ~ 2
Page 5 and 6: Outline: * physical realization of
Page 7 and 8: Prototype for the exotic GLSM’s I
Page 9: GLSM for P 3 [2,2] (=T 2 ): Looks l
Page 13 and 14: The other phase is more exciting: V
Page 15 and 16: A more important subtlety is the fa
Page 17 and 18: General decomposition conjecture Co
Page 19 and 20: Quick consistency check: A sheaf on
Page 21 and 22: GLSM’s Let’s now return to our
Page 23 and 24: V// − C × : Double cover P 1 Ber
Page 25 and 26: Next simplest example: // + : LG on
Page 27 and 28: The next example in the pattern is
Page 29 and 30: Check that we are seeing K’s nonc
Page 31 and 32: Summary so far: CP 7 [2,2,2,2] This
Page 33 and 34: More examples: CI of 2 quadrics in
Page 35 and 36: D-brane probes of nc resolutions Le
Page 37 and 38: Our D-brane probes of this Landau-G
Page 39 and 40: When is a matrix factorization `poi
Page 41 and 42: We’re interested in Landau-Ginzbu
Page 43 and 44: Example: Family [C 2 /Z 2 ] x,y ×
Page 45 and 46: So far: When the LG model flows in
Page 47 and 48: Case of an nc resolution, cont’d:
Page 49 and 50: So far: * examples of hpd realizing
Page 51 and 52: Let’s work through this in more d
Page 53 and 54: As a warm-up, let’s outline the G
Page 55 and 56: P 7 [2,2,2,2], cont’d In principl
Page 57 and 58: P 7 [2,2,2,2], cont’d After norma
Page 59 and 60: Result for P 7 [2,2,2,2]: n N 1 512
Page 61 and 62:
Applying the same method, Compare G
Page 63:
Summary: * physical realization of
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Eric Sharpe - Examples of homological projective duality in physics

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