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UID:724634898eec0765cf033d43ab063f0e
CATEGORIES:Gauge Theory Learning Seminar
CREATED:20250425T115952
SUMMARY:Seiberg-Witten equations in all dimensions
LOCATION:Hill 705 and on Zoom
DESCRIPTION:Abstract: I will describe a generalisation of the Seiberg-Witten equations 
 to a Spin-c manifold of any dimension. The equations are for a U(1) connect
 ion A and spinor phi and also an odd-degree differential form b (of inhomog
 eneous degree). Clifford action of the form is used to perturb the Dirac op
 erator D_A. The first equation says that (D_A+b)(phi)=0. The second equatio
 n involves the Weitzenböck remainder for D_A+b, setting it equal to q(phi),
  where q(phi) is the same quadratic term which appears in the usual Seiberg
 -Witten equations. This system is elliptic modulo gauge in dimensions congr
 uent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to tak
 e two copies of the system, coupled via b. If time permits, I will also des
 cribe a variant of these equations which make sense on manifolds with a Spi
 n(7) structure. The most important difference with the familiar 3 and 4 dim
 ensional stories is that compactness of the space of solutions is, for now 
 at least, unclear. This is joint work with Partha Ghosh and, in the Spin(7)
  setting, Ragini Singhal.\n
X-ALT-DESC;FMTTYPE=text/html:<p><strong data-olk-copy-source="MessageBody">Abstract:</strong>&nbsp;I wil
 l describe a generalisation of the Seiberg-Witten equations to a Spin-c man
 ifold of any dimension. The equations are for a U(1) connection A and spino
 r phi and also an odd-degree differential form b (of inhomogeneous degree).
  Clifford action of the form is used to perturb the Dirac operator D_A. The
  first equation says that (D_A+b)(phi)=0. The second equation involves the 
 Weitzenböck remainder for D_A+b, setting it equal to q(phi), where q(phi) i
 s the same quadratic term which appears in the usual Seiberg-Witten equatio
 ns. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 
 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of
  the system, coupled via b. If time permits, I will also describe a variant
  of these equations which make sense on manifolds with a Spin(7) structure.
  The most important difference with the familiar 3 and 4 dimensional storie
 s is that compactness of the space of solutions is, for now at least, uncle
 ar. This is joint work with Partha Ghosh and, in the Spin(7) setting, Ragin
 i Singhal.</p>
CONTACT:Joel FIne (Université Libre de Bruxelles)
DTSTAMP:20260830T193002
DTSTART;TZID=America/New_York:20250429T110000
DTEND;TZID=America/New_York:20250429T120000
SEQUENCE:0
TRANSP:OPAQUE
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